Logic's lost genius - The life of Gerhard Gentzen 9780821835500, 2007060550, 9781470428129

Table of contents :
Cover
Contents
Preface to the English edition
Introduction
Early youth and abitur
1928-1938—Weimar Republic and National Socialism in peace. From the beginning of studies to the extension of the unscheduled assistantship for another year with effect from 1 October 1938
1939-1942—From the beginning of the war to dismissal from the Wehrmacht and the wartime habilitation under Helmut Hasse
The fight over “German logic” from 1940 to 1945: A battle between amateurs
Recovery and docent position 1942 to 1944
Arrest, imprisonment, death and Nachlass
Conclusion
Tables of the life of Gerhard Gentzen
Appendix A: Gentzen and geometry, by C. Smoryński
Appendix B: Hilbert’s programme, by C. Smoryński
Appendix C: Three lectures, by Gerhard Gentzen
Appendix D: From Hilbert’s programme to Gentzen’s programme, by Jan von Plato
Bibliography
Index
Back Cover

Citation preview

HISTORY OF MATHEMATICS • VOLUME 33

Logic’s Lost Genius The Life of Gerhard Gentzen

Eckart Menzler-Trott

American Mathematical Society • London Mathematical Society

Logic’s Lost Genius The Life of Gerhard Gentzen

Editorial Board American Mathematical Society London Mathematical Society Joseph W. Dauben Alex D. D. Craik Peter Duren Jeremy J. Gray Karen Parshall, Chair Robin Wilson, Chair Michael I. Rosen This work was originally published in German by Birkh¨ auser Verlag under the title c 2001. The present translation was created under license for the Gentzens Problem  American Mathematical Society and is published by permission. Photo of Gentzen in hat courtesy of the Archives of the Mathematisches Forschungsinstitut Oberwolfach. Reprinted with permission. 2010 Mathematics Subject Classification. Primary 01A60.

For additional information and updates on this book, visit www.ams.org/bookpages/hmath-33 Library of Congress Cataloging-in-Publication Data Menzler-Trott, Eckart. [Gentzens Problem. English] Logic’s lost genius : the life of Gerhard Gentzen / Eckart Menzler-Trott ; translated by Craig Smory´ nski and Edward Griffor. p. cm. (History of mathematics ; v. 33) ISBN 978-0-8218-3550-0 (alk. paper) 1. Gentzen, Gerhard. 2. Mathematicians—Germany—Biography. 3. Logic, Symbolic and mathematical. I. Title. QA29 .G467M46 510.92—dc22

2007 2007060550

AMS softcover ISBN: 978-1-4704-2812-9 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2007 by the American Mathematical Society. All rights reserved.  Reprinted by the American Mathematical Society, 2016. Printed in the United States of America. The American Mathematical Society retains all rights except those granted to the United States Government. ∞ The paper used in this book is acid-free and falls within the guidelines  established to ensure permanence and durability. The London Mathematical Society is incorporated under Royal Charter and is registered with the Charity Commissioners. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

21 20 19 18 17 16

As a 24th problem in my Paris talk I wanted to pose the problem: criteria for the simplicity of proof, or, to show that certain proofs are simpler than any others. In general, to develop a theory of proof methods in mathematics. Under a given set of conditions there can be but one simplest proof. Quite generally, if there are two proofs for a theorem, you must keep going until you have derived each from the other, or until it becomes quite evident what variant conditions (and aids) have been used in the two proofs. Given two routes, it is not right to take either of these two or to look for a third; it is necessary to investigate the area lying between the two routes. . . —David Hilbert (Hilbert’s notebook, Cod. Ms. D. Hil=600.3)

It is really astonishing how many highly gifted logicians died young (Nicod, Herbrand, Gentzen, Spector, Ramsey, etc.). —Kurt G¨ odel to Paul Bernays

Next I would like to call your attention to a frequently neglected point, namely the fact that Hilbert’s scheme for the foundations of mathematics remains highly interesting and important in spite of my negative results. What has been proved is only the specific epistemological objective which Hilbert had in mind cannot be obtained. This objective was to prove the consistency of the axioms of classical mathematics on the basis of evidence just as concrete and immediately convincing as elementary arithmetic. However, viewing the situation from a purely mathematical point of view, consistency proofs on the basis of suitably chosen stronger metamathematical presuppositions (as have been given by Gentzen and others) are just as interesting, and they lead to highly important insights into the proof theoretic structure of mathematics. —Kurt G¨ odel to Constance Reid

Contents Preface to the English Edition

xiii

Introduction 1. Gentzen’s Accomplishments 2. Aims of My Life Story of Gerhard Gentzen 3. Mathematical Logic and National Socialism: The Political Field

xvii xvii xviii xix

Chapter 1. Early Youth and Abitur 1. Gerhard Gentzen’s Birth 2. Gentzen’s Mother: Melanie Gentzen (1873-1968) 3. Gentzen’s Paternal Grandparents and His Father: The Attorney and Court Official Dr. jur. Hans Gentzen (1870-1919) 4. Youth and Student Days of Hans Gentzen 5. Gentzen’s Maternal Grandparents: The Physician Alfons Bilharz and Adele Bilharz 6. The Shining Example in Gentzen’s Family Tree: The Great, Successful Physician and Natural Scientist Maximilian Theodor Bilharz (1823-1862) 7. Gentzen’s Sister: Waltraut Sophie Margaret Gentzen (*1911) 8. The Schoolchild Gerhard Gentzen 9. The Death of the Father Means a Move and a New School 10. The Beginning of Gentzen’s Intellectual Activity 11. Gentzen’s Success at School 12. The Abitur on 29 February 1928

2 3 4

9 10 11 12 12 15 16

Chapter 2.

1. 2. 3. 4. 5. 6. 7.

1928-1938—Weimar Republic and National Socialism in Peace. From the Beginning of Studies to the Extension of the Unscheduled Assistantship for Another Year in Effect from 1 October 1938 Beginning of Studies in Greifswald Continuation of Study in G¨ ottingen Is G¨ ottingen the Centre of Mathematics for Gentzen? Continuing Studies in Munich A Semester in Berlin: Winter Semester 1930/31 Back in G¨ ottingen: Saunders Mac Lane and Gentzen as the “Type of a Scientifically Oriented Man” (Richard Courant) ¨ The Decision: Gentzen’s First Publication, “Uber die Existenz unabh¨ angiger Axiomensysteme zu unendlichen Satzsystemen” and His Programme for 1932

1 1 1

vii

21 21 22 23 23 26 27

30

viii

CONTENTS

8. 9. 10. 11. 12.

13. 14.

15.

16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

32. 33.

34. 35.

What Do Gentzen’s Intellectual Interests and Attitude in 1931 and 1932 Appear to Be? Political Language in Mathematics in 1918 Political Language by Hermann Weyl and David Hilbert Whom Did Gentzen Know in G¨ ottingen and What Did He Read? Gentzen’s Life in the Early Nazi Period. The Withdrawn Manuscript ¨ “Uber das Verh¨ altnis zwischen intuitionistischer und klassischer Arithmetik” of 15 March 1933 Gerhard Gentzen’s Dissertation “Untersuchungen u ¨ber das logische Schließen” of 12 July 1933 The Penetration of the Nazis into Mathematical Research at the University in G¨ ottingen 1933 and 1934—Or, Vahlen and Bieberbach vs. Weber and Wegner The State Examination with “Elektronenbahnen in axialsymmetrischen Feldern unter Anwendung auf kosmische Probleme” on 16 November 1933 Why Did Gentzen Join the SA? Gentzen’s Political Position: A Conjecture Gentzen in Financial Difficulties Financial Straits and Job Hunting Consistency Proof for Number Theory in Discussion with Paul Bernays “Widerspruchsfreiheit der reinen Zahlentheorie” Mirrored in the Correspondence of Bernays and Weyl Difficulties with the “Widerspruchsfreiheit der reinen Zahlentheorie” of 11 August 1935 The Unscheduled Assistantship under Hilbert from 1 November 1935: A Productive Period for Foundational Research Begins Consistency of Type Theory Revising the Proof of the “Widerspruchsfreiheit der reinen Zahlentheorie” “Die Widerspruchsfreiheit der reinen Zahlentheorie” Gentzen Was an Intellectual Independent Gentzen Expresses His Thanks to Turing Correspondence between Bernays and Ackermann 1936 to 1940 The Correspondence between Bernays and Gentzen Merrily Continues Invitation to the Parisian Descartes Congress in August 1937. The Invitation to Lecture to the DMV Conference in Bad Kreuznach on 21 September 1937: “Die gegenw¨artige Lage in der mathematischen Grundlagenforschung”. The Extension of the Tenure of the Unscheduled Assistantship on 1 October 1937 for a Year Jean Cavaill`es and Gerhard Gentzen Gentzen Becomes an “Associate” of the Publication of Scholz’s Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften “Die gegenw¨artige Lage in der mathematischen Grundlagenforschung” “Neue Fassung des Widerspruchsfreiheitsbeweises der reinen Zahlentheorie” 1938

33 34 34 37

38 41

46

51 52 53 54 55 57 58 59 62 63 64 65 69 71 72 75

77 82

84 86 88

CONTENTS

36. 37. 38. 39. 40. 41.

Bernays’ View of Gentzen’s Programme (Second Consistency Proof) Correspondence with Paul Bernays Extension of the Unscheduled Position for Another Year Taking Effect 1 October 1938 Bernays’ Views of Hilbert’s Programme and Gentzen’s Place in It Closing Thoughts on Paul Bernays Longer Notes

89 93 97 100 101 102

Chapter 3.

1. 2. 3. 4.

5. 6.

1939-1942—From the Beginning of the War to Dismissal from the Wehrmacht and the Wartime Habilitation under Helmut Hasse 1939: At the Highpoint of Reputation The Second Volume of Grundlagen der Mathematik of Hilbert and Bernays Appears Active Military Service at the Homefront as Radio Operator by the Flugwachkommando 1939/40: Preparation for Habilitation. “Beweisbarkeit und Unbeweisbarkeit von Anfangsf¨ allen der transfiniten Induktion in der reinen Zahlentheorie” 1941: Encouragement from Hellmuth Kneser 1942: Discharge from Military Service

ix

Chapter 4. The Fight over “German Logic” from 1940 to 1945: A Battle between Amateurs A Short Preface 1. Ludwig Bieberbach as Supporter of the Idea That the Validity of Mathematics Be Decided through World Views 2. An Advocate of Racial Purity Sets the Standards in German Mathematical Logic, Where Attempts Are Made to Confuse Scientific Results with Matters of Race 3. Gentzen as a “Witness” for a National-Racial Interpretation of the Mathematical Foundational Research through Steck and Requard 4. The Somewhat Sharper Point of View of Friedrich Requard 5. Ludwig Bieberbach and his Deutsche Mathematik 6. NS Ideology in Mathematics through Bieberbach Receives Negative Resonance Even within His Own Camp 7. Ludwig Bieberbach: Representative of “German Mathematics” 8. A Contemporary of Bieberbach’s in Exile: Johann L. Schmidt 9. Bieberbach and Intuitionism 10. Formalism and Proof Theory 11. Applied Mathematics as Folkish Mathematics 12. Nazis Criticised the Lack of Support from Mathematics 13. Mathematical Foundational Research Remains Unmolested by the Nazis 14. Attacks on Mathematical Logic from Without: Dingler, Steck, and May 15. The Attacked: Heinrich Scholz (1884-1956) 16. Bieberbach, Max Steck and Jænsch 17. Bieberbach and Erich Jænsch

117 117 119 125

128 135 139

141 141 142

143 144 147 150 152 153 159 160 160 162 163 167 169 176 186 188

x

CONTENTS

18.

19. 20. 21. 22. 23. 24. 25.

Steck’s Attack on Hilbert Leads to Bieberbach’s Commissioning a Defence of Mathematical Logic by H. Scholz and Publishing It in Deutsche Mathematik Interlude: May and Dingler Provide Arguments for Steck Steck and Scholz in Dispute Max Steck as Denouncing “Expert Witness” and Publicist The Exception: The Dedicated National Socialist, Logician and Historian of Mathematics, Oskar Becker, Remains Neutral Resistance as a Mathematician Was Possible under National Socialism Kurt Reidemeister’s Additional Contemplations on Politico-Scientific Power Play in “German Mathematics” Longer Notes

Chapter 5. Recovery and Docent Position 1942 to 1944 1. Final Discharge from the German Army 2. Hans Rohrbach Commandeers Gerhard Gentzen to Prague through the Osenberg Initiative 3. Gentzen’s Teaching Position in Prague: “Kepler’s Laws of Planetary Motion” 4. The First Courses in November 1943 5. The Last Known Scientific Letter of Gerhard Gentzen 6. Gerhard Gentzen in 1944: Teaching Functions, Computing Office, and War 7. Hans Rohrbach’s Report on the Conditions in the Mathematical Institute in Prague 8. Why Did Gentzen Banish Any Thought of Flight?

190 197 202 208 216 218 219 221 233 233 234 236 238 243 244 246 247

Chapter 6. Arrest, Imprisonment, Death and Nachlass 1. The Last Days of Freedom in the Private Sector 2. The Arrest of Gerhard Gentzen and the Awful Imprisonment 3. Gentzen’s Physical Death 4. Is Gentzen’s Death Understandable? 5. Rumours 6. Attempts to Rescue the Nachlass 7. The Deciphering of the Stenographic Notes

253 253 255 257 260 261 263 266

Conclusion 1. Misapprehensions about the Life of Gerhard Gentzen 2. Logic and Politics 3. Upshot

267 267 267 269

Tables of the Life of Gerhard Gentzen Chronology Contemporary Assessments of Gentzen Publications of Gentzen

273 273 278 281

Appendix A. Gentzen and Geometry ´ ski C. Smoryn

283

CONTENTS

Appendix B. Hilbert’s Programme ´ ski C. Smoryn 1. Constructive Prologue 2. Problems in Paris 3. Hilbert and Geometry 4. First Steps 5. Enter Brouwer 6. Back to Hilbert 7. Weyl Stirs Things Up 8. Hilbert Responds 9. More on Brouwer 10. Outbreak of Hostilities 11. The Formula Game 12. On the Infinite 13. A Fragile Truce 14. “Hilbert’s Programme” Is Born 15. Brouwer Takes Up Arms 16. Hilbert Finishes Off Brouwer 17. The Programme Expands 18. G¨ odel’s Theorem 19. Concluding Remarks

xi

291 291 293 296 300 301 308 310 312 322 323 324 325 327 329 331 332 334 335 339

Appendix C. Three Lectures Gerhard Gentzen 1. The Concept of Infinity in Mathematics 2. The Concept of Infinity and the Consistency of Mathematics 3. The Current Situation in Research in the Foundations of Mathematics

353

Appendix D. From Hilbert’s Programme to Gentzen’s Programme Jan von Plato 1. Mathematical Proof 2. Hilbert’s Programme 3. Gentzen’s Programme 4. Later Developments in Structural Proof Theory 5. References

367 367 373 383 396 401

Bibliography

405

Index

433

343 343 350

Preface to the English Edition In revising this book for translation I have cut down on the polemics, and I have incorporated new facts and material and what I have learned in these years, notably from some reviews of the German edition—especially the one of Dirk van Dalen— as well as from continual discussion with Jan von Plato and Craig Smory´ nski. I have taken a stand which is less critical of tradition, but more thorough acquaintance with what Gentzen and proof theory have done has pushed the concept of Gentzen as a “follower of the Hilbert Programme” further into the background. A clear exposition, description and interpretation of Hilbert’s Programme by Craig Smory´ nski in appropriate style, language and richness of ideas can be found in this book. Gentzen’s work should decisively not just be seen as contributing only to the post-G¨odelian development of Hilbert’s Programme. It this were true, Gentzen would be of almost no interest at all today to creative working mathematicians, but only a concern for antiquarians. Gerhard Gentzen (1909-1945) is the founder of modern structural proof theory. His permanent and sustained methods, rules and structures have resulted not only in a technical mathematical discipline called “proof theory” but also in applications in computer science (e.g. program verification) and all kinds of effective mathematics. Someday the performance of the pioneer Gerhard Gentzen will be seen like the creations of Heisenberg, Schr¨odinger or Dirac. Gentzen’s natural deduction, sequent calculi and ordinal proof theory certainly impress with their appearance, clarity and elegance. And, the techniques he developed in the years from 1931 to 1939 are now the minimum standard in proof theory. Gentzen wrote Bernays in a letter dated 3 March 1936: I also don’t know if I can claim any “priority” in all particulars; I wish only this once to survey further work on the programme from the now established points, the carrying out of which admittedly can require some years or decades.

And this programme was decisive for the post-G¨odelian period of logic. Following Jan von Plato, we recognise three stages in Gentzen’s ideas: the conviction that the consistency of arithmetic can be proven constructively arose from his ¨ manuscript “Uber das Verh¨ altnis zwischen intuitionistischer und klassischer Arithmetik” [On the relation between intuitionistic and classical arithmetic] withdrawn in March 1933. He proved this consistency in 1935 in his “Die Widerspruchsfreiheit der reinen Zahlentheorie” [The consistency of pure number theory]. And in his habilitation thesis submitted in 1939, “Beweisbarkeit und Unbeweisbarkeit von Anfangsf¨ allen der transfiniten Induktion in der reinen Zahlentheorie” [Provability and unprovability of initial cases of transfinite induction in pure number theory] he proved directly the unprovability of ε0 -induction in Peano Arithmetic. And he achieved all this on the basis of a calculus he constructed for such tasks in xiii

xiv

PREFACE TO THE ENGLISH EDITION

1932/1933. Even Paul Bernays agreed in 1938 that Gentzen had a place in foundational research in his own right, one that did not fall under the banner of “Hilbert’s Programme”. Jan von Plato describes the actual development of Gentzen and proof theory in detail in this book. But though a good number of passages have been revised, and though there are some small rearrangements in the order of treatment, the book has substantially remained the same. So those “innocent” of logic or proof theory should find it understandable. I cannot claim to have provided a definitive history of proof theory in the thirties and forties of the last century in Nazi Germany or a complete account of Gentzen, the man and the genius. Still I trust that this book gives a full and perspicuous presentation of the evidence and thus will be useful even to those who are not inclined to draw the same conclusions from it. My special thanks are to Craig Smory´ nski, who not only completed the laborious task of translation in a spirit of most pleasant and full collaboration in Westmont and different places in the world, but to whom is due the initiative and diligent working out of an improvement of ideas, content and style for the English edition. Everybody who knows his ideas will agree that they shed light on my dark thoughts. The responsibility for the content and for all that may be wrong with it, however, remains mine. The Department of Philosophy of the University of Helsinki has aided Craig and me with a visit where Craig, Sara Negri, Jan von Plato and I again could meet and discuss various aspects of Gentzen and his work. Some parts of this book were written in the beautiful atmosphere of the Mathematisches Forschungsinstitut of Oberwolfach. I acknowledge all this favour as a factor in the excellence of this book. The preparation of this edition was done under ideal working conditions. I hope that this book will be judged as a contribution to the philosophy and the history of logic and proof theory and not merely to the art of biography of mathematicians. This book is dedicated to Heidrun Trott and my children, Jacob Menzler, Laura Trott and Gwendolyn Trott. Eckart Menzler-Trott Munich, 1 June 2006

Acknowledgements. I wish first to express my thanks to Frau Waltraut Student, the sister of Gerhard Gentzen, who gave me access to the Bilharz family chronicles and during a visit lasting several days related much to me about her brother and the Bilharz family. Dipl.-Ing. G. Gentzen permitted me an insight into the life of Hans Gentzen through neatly marked extracts and turned over to me copies of letters concerning the death of Gerhard Gentzen. The late Saunders Mac Lane of Chicago told me of his study time in G¨ottingen and his fellow student Gentzen. Dr. Franz Krammer, in whose arms Gentzen died, instructed me on the circumstances of Gentzen’s death. Frau Erna Scholz sent me a photo of her husband, Heinrich Scholz, with Gentzen and willingly answered all questions. Martin Kneser placed at my disposal copies of the correspondence of his father, Hellmuth Kneser, with Gerhard Gentzen. Dr. Lothar Collatz (†), Dr. Ernst Mohr (†), Dr. Kurt Sch¨ utte (†), and Dr. Paul Lorenzen (†) have spoken as completely about their friend, colleague, and contemporary as was possible for them.

PREFACE TO THE ENGLISH EDITION

xv

Dr. Christian Thiel has let me examine the material of Gentzen’s Nachlass, so far as it has been deciphered, and constructed a painful list of my literal and typographical errors from the German and the present edition of the book. Contentual and formal criticism amounting to several pages were sent by Dr. Reinhard Siegmund-Schultze, and almost all of his suggestions have been taken into account in this edition. The reviews of the German edition by Dirk van Dalen, Petr H´ ajek, Ivor Grattan-Guinness, Albert C. Lewis, Rudolf Taschner, Volker Peckhaus, Peter Schreiber, and many others were very helpful, and I have been able to put some of their stimulation to good use here. Some material—items of information or suggestions—which I acknowledge at the appropriate places in the book, have been provided by: Dr. Norbert Schappacher, Dr. Justus Diller, Dr. Herbert Mehrtens, Jens Erik Fenstad, Dr. Volker R. Remmert, Dr. Harold Schellinx, Dr. Milan Vlach, Georg Kreisel, Gerd Robbel, Dr. Hans Hermes, Jan von Plato, Dirk van Dalen, Dr. Enno Folkerts, Dr. Gisbert Hasenjaeger, Dr. habil. Renate Tobies, Dr. F. L. Bauer, Ulrich Bieberbach, Dr. G¨ unther Engler, Dr. Gerhard H. R. Reisig, Dr. Moritz Epple, Dr. Hans Rohrurgen Kas¨ uske, and bach, Dr. Premysl Vihan, Dr. Olga Kraus, Dr. Ina Kersten, J¨ some others. I learned a lot about Gentzen’s logic from the writings of Wolfram Pohlers, Wilfried Buchholz, Helmut Schwichtenberg, Grisha Mints, Michael Rathjen, Sol Feferman, Wilfried Sieg and many others and thank them for their personal communications. odorn-Meyer (HandschriftMy thanks go to the libraries and archives: Petra Bl¨ enabteilung der Staats- und Universit¨ atsbibliothek Carl v. Ossietzky Hamburg); Dr. Laetitia Boehm (Archiv der Ludwig-Maximilian-Universit¨ at M¨ unchen); W. Schultze (Archiv der Humboldt Universit¨ at Berlin); Dr. Beat Glaus and Ms. Flavia Lanini (Archiv der ETH Z¨ urich); Dr. Ulrich Hunger (Universit¨ atsarchiv G¨ ottingen); Dieter Speck (Universit¨ atsarchiv Freiburg); Miroslav Kunstat (Archiv Univerzity Karlovy Prag); Frau H. Bertram (Mathematisches Institut Giessen); Berlin Document Center, Bundesarchiv Koblenz; Institut f¨ ur Zeitgeschichte; Matheararchiv Freiburg; Instimatisches Institut G¨ottingen; Deutsche Dienststelle; Milit¨ tut f¨ ur mathematische Logik und Grundlagenforschung M¨ unster; Bayerische Staatsbibliothek; Dombibliothek Freising; Hessische Staatsbibliothek Wiesbaden. For their constant support for decades I particularly and very warmly thank Antiquariat Kitzinger in Munich’s Schellingstraße and Antiquariat Renner, who have always gone out of their way to procure all books which on account of the consequences of war or chronic underfunding are not available in German libraries. Quick Note from the Main Translator. An old set of guidelines from the American Mathematical Society declared that the requirements for translating mathematics into English were, in order, a knowledge of the mathematics in question, a knowledge of English, and a knowledge of the language of the original. I have read mathematics and biography in German without any major difficulties in the past, but the present book has incredible breadth. The philosophical quotations of the National Socialists in Chapter 4, in particular, proved too much for me and I enlisted the aid of my friend Ed Griffor, whose knowledge of philosophy and the German language far exceeded my own. I owe him a debt of gratitude. As I was learning LATEX simultaneously with translating this book, the typography may not be up to par. For this I apologise to Eckart and the reader.

xvi

PREFACE TO THE ENGLISH EDITION

With respect to my appendix on Hilbert’s Programme, I should like to take the opportunity here to express my indebtedness to Dirk van Dalen, both for general stimulation and particular information on Brouwer and Hilbert. Craig Smory´ nski Westmont, 17 April 2007

Introduction 1. Gentzen’s Accomplishments Structural proof theory studies the general structure and properties of mathematical proofs. It was discovered by Gerhard Gentzen (1909-1945) in the first years of the 1930s and presented in his doctoral thesis “Untersuchungen u ¨ ber das logische Schließen” [Investigations on logical inference] in 1933.1 The setting of this task is neutral; it does not commit Gentzen to any specific view on the foundations of mathematics, be it formalism, intuitionism, or Cantorism. Gentzen presented first a general theory of the structure of mathematical proofs.2 The object of logic, in Gentzen’s view, is to study the general structure of proofs. It is a complete break with the logicist tradition of Frege, Peano, and Russell that Hilbert and his school had been pursuing and in which the notion of logical truth is basic. Proofs that follow a precise set of rules are called derivations, to distinguish them from most of the informal proofs found in mathematics.3

A more traditional view: Gerhard Gentzen’s work in the 1930s (see Gentzen 1969) has been the most influential for the development of modern reductive proof theory in practice. . . The aim of the Gentzen-Sch¨ utte-Takeuti line of development is what I call the constructive consistency proof rationale for reductive proof theory. . . ”4

But Gentzen went further than this, leaving behind the constraints of the Hilbert programme. The “true beginnings” of structural proof theory may be dated from the publication of the landmark paper Gentzen (1935).5 Interest in proofs as combinatorial structures in their own right was awakened. . . Nowadays there are more reasons. . . for studying structural proof theory. For example, automated theorem proving implies an interest in proofs as combinatorial structures; and in logic programming, formal deductions are used in computing.6

Indeed Gentzen-style systems and his ideas of natural deduction and sequent calculi dominate proof theory and very strongly influence computer science today. As Dirk van Dalen puts it, “Before Gentzen, proof theory was ‘hacker’s paradise’ ”; 1 Sara

Negri and Jan von Plato, Structural Proof Theory, Cambridge University Press, 2001,

p. xi.

2 Sara

Negri and Jan von Plato, “Hilbert’s last problem”, Arkhimedes 2002, no. 5, p. 4. von Plato, “Proof theory of classical and intuitionistic logic”, L. Haaparanta, ed., History of Modern Logic, Oxford University Press, Oxford, in press, pp. 2ff. 4 Solomon Feferman, “Does reductive proof theory have a viable rationale?”, p. 4 of the Internet version. Erkenntnis 53 (2000), 317–332. 5 A.S. Troelstra and H. Schwichtenberg, Basic Proof Theory, Second Edition, Cambridge University Press, Cambridge, 2000, p. ix. 6 Ibid. 3 Jan

xvii

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INTRODUCTION

after Gentzen, who was stimulated by G¨ odel’s work, there was “a method full of beauty and elegance.”7 In Germany especially it was difficult for logicians to appreciate Gentzen’s originality because the reception of his work was too often influenced by ignorance and the familiar perspective of Hilbert’s programme and Paul Bernays’ convenient assessments. In §2.2.1 of Gentzen (1936), Gentzen writes, “The theory of arbitrary mathematical proofs as objects is called ‘proof theory’ or ‘metamathematics’ ”, and he shows what is actually provable after G¨ odel’s results. Dirk van Dalen describes it with the following words: Gentzen was an exponent of the new generation in every regard. A generation which had begun to understand that G¨ odel wasn’t the end of logic but meant the beginning of a new and rich life of the subject. Gentzen found himself at the summit of the discipline as he brilliantly created methods to put into practice in logic, namely those of natural deduction and sequent calculi. With the help odel. of these methods he saw chances to do what was still to be done after G¨ He thoroughly investigated the problem of excluding contradictions from arithmetic. The adjective “brilliant” is correct here. . . Gentzen, who without fuss had relegated Hilbert’s age to the museum, was immature in his socio-political development.8

2. Aims of My Life Story of Gerhard Gentzen This is the story of the outer life of the mathematician and logician Dr.habil. Gerhard Gentzen; it is not a book about proof theory or its development.9 Gentzen belongs to the pioneers and founders of proof theory and is one of the “classics”10 of the field. Although this biography is a contribution to the history of mathematical logic under National Socialism, it is only partly a history of mathematics during these times.11 This book does two things: My biography explains Gentzen’s ideas and theorems to the layman and gives a full account of the life and tragic death of Gerhard Gentzen. The biography also clarifies Gentzen’s position as a mathematician under National Socialism. However, even for those selectively interested in the history 7 Dirk van Dalen, “Ein Logiker unter den Nazis”, pp. 30ff., in Mitteilungen der Deutschen Mathematiker-Vereinigung, 1/2003. 8 Ibid. 9 My work concentrates on the life of Gerhard Gentzen and is not a concise analysis of his work, although it does give indications. A first approach to Gentzen’s mathematical theory can be found in Jan von Plato’s contribution to this volume or in Gentzen’s works themselves. Whoever wants to incorporate the technical work into his reading of this volume is referred to Gentzen’s “Untersuchungen u ¨ ber das logische Schließen” (1935) and “Die Widerspruchsfreiheit der reinen Zahlentheorie” (1936). A very good introduction to structural proof theory with all necessary methods and results is given by Sara Negri and Jan von Plato, Structural Proof Theory, Cambridge University Press, Cambridge, 2001. 10 “Classics” are authors about whom, according to Robert Darnton, at least two biographies must be written: a glorification and an unmasking, for a true classic must first be someone who has something to hide. In diametric opposition to readings on the life and work of Gottlob Frege, I could have completely entered without a philology of suspicion, because Gentzen’s life, so far as it is known to us, lies open, straight and coherent before us. Who wants to see something else must squint, restrict himself methodically, write from emotion, resentment or rancour—or simply be prepared to be evil. 11 For a history of mathematics and mathematical logic in this period, see Sanford L. Segal, Mathematicians under the Nazis, Princeton University Press, Princeton, New Jersey, 2003. For German language literature on the period, cf. Chapter 4 of the present book.

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of mathematical logic under the Nazis, the mathematical and logical position of Gerhard Gentzen within a totalitarian system is also worth reading.12 Mathematical logic had political implications under National Socialism, whether it was deemed too formal and lacking of substance by some or recognised as a serious contribution to mathematics by others. The Nazis, in the form of certain functionaries and organisations, had individual representatives who supported, tolerated and promoted mathematical logic and its representatives in individual cases. Gentzen was one of these latter. He enjoyed the protection of Ludwig Bieberbach and the “Deutsche Forschungs-Gemeinschaft” (DFG) [German Research Council], as did the M¨ unster school surrounding Heinrich Scholz (Hermes, Schr¨oter, Behmann, Bachmann, Schweitzer, Ackermann, Kratzer). Hilbert’s programme was the unifying theme among German mathematicians despite the destruction of the G¨ottingen school by the Nazis. Almost all mathematicians shared Hilbert’s vision of a unified mathematics, whether they were Nazi supporters or opponents of Nazism. I don’t know any single mathematician who wanted to introduce, for example, intuitionism as an official fundamental philosophy of mathematics of the Nazis, as Weyl, Blumenthal or Menger had in the Weimar Republic. Furthermore the motivation of the Nazis for the protection of mathematical logic by Ludwig Bieberbach and others is still unexplained. Perhaps it was the desire of the Nazis for international recognition and because this field was one with international participation. Modern German mathematics stems from Klein, Hilbert and many other mathematicians of the 19th century, many of whom had in the meantime been driven away to exile or even been killed. The iron resolve of the Nazis to present the highest mathematical rationality as “German mathematics” to the rest of the world failed primarily because of the indignation of the mathematicians who survived exile and execution. There was no room for unwanted academic disputes that would disturb the efficiency of war-related work like cryptography, which was performed by mathematicians as politically diverse as Teichm¨ uller, Witt, and Hasenjæger.13

3. Mathematical Logic and National Socialism: The Political Field The founder of the journal Deutsche Mathematik, Ludwig Bieberbach, supported mathematical logic and even used his own funding by the DFG to promote the corresponding logical plans of Heinrich Scholz. Why? Ludwig Bieberbach, the author of the notorious essays “Pers¨onlichkeit und mathematisches Schaffen” [Personality and mathematical creation] and “Stilarten mathematischen Schaffens” [Styles of mathematical creation] defended the academic research in mathematical logic against attacks in the name of “Volk” and race. This requires clarification. 12 On

the history of mathematical logic under National Socialism (and the period preceding) there has till now been only the concise observations of Christian Thiel, “Folgen der Emigration deutscher und ¨ osterreichischer Wissenschaftstheoretiker und Logiker zwischen 1933 und 1945”, Berichte zur Wissenschaftsgeschichte 7 (1984), pp. 227-256 (here: pp. 248-252). My book only goes as deep technically as is still possible for a mathematically interested layman to follow. The specialists are possibly familiar with the deeper underlying ideas; these are expanded upon here in reviews and contemporary reports of discussions, so that one can form a more solid picture. 13 Cf. F.L. Bauer, Entzifferte Geheimnisse. Methoden und Maximen der Kryptologie, 2nd expanded edition, Springer Verlag, 1997.

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Natural scientists like to live with the illusion that their enterprises could really prosper only in free and democratic societies. The history of science in totalitarian societies should be merely a story of the oppression of “good” science or of one barely surviving in more-or-less unmolested niches. This idealised and ahistorical picture of science falls apart if one considers the crimes, transgressions and “achievements” that came about in the name of national socialist science. (Thomas Weber)

In mathematics it is not absolutely so. Also in the name of mobilisation or self-mobilisation for a political ideology, technically new mathematics is possible. Mathematical logic was promoted as a part of university mathematics to some extent, and internationally recognised mathematicians researched in the area, though uller. The with disgusting interference by the Nazis Dingler, Steck, Th¨ uring and M¨ mathematical logicians delivered results despite the personnel, financial and organisational “thinning out” of science by the Nazi Party. A serious science was also possible under National Socialism. However, this only succeeded with a kind of job sharing. Some representatives (H. Scholz, H. Hasse, W. S¨ uss and others) professionalised themselves as science organisers and undertook to negotiate with the Nazi Party and its ministers about the conditions for this science. Some representatives (Bieberbach, Vahlen, and others) identified themselves directly with the party, while some overshot the aim entirely (W. S¨ uss) or tried to keep themselves in a kind of balance (H. Hasse). It isn’t always and only a fault of the individuals and their individual choices; it might be a fault of the discipline and its organisation as well. I show the life of the mathematician Gentzen in important details. To be able to judge the mathematics of this time still requires many detailed studies. I have shown a life from the “files” and some testimonies. By the choice of the topics, the fortuity of the documents which came into my hands, I have described the outer life as reliably as I could. Gentzen’s life has, I am certain, nonetheless been lived quite differently. When I cite from Gentzen’s works and the reviews of his time, I try to show the field of arguments which were around this time to get to know the arguments for and against certain topics. The history of mathematics isn’t even recognised as a disciplinary task, let alone tackled as an interdisciplinary research project. The examination of the mathematics using means and methods of other sciences or humanities is still disgusting for many. The history of mathematics—as the late Nikolai Stuloff put it—states propositions which occasionally generate different opinions. It is possible that statements are made which cannot be categorised into right or wrong as in mathematics itself. An essential component of the historical work in the history of mathematics is the interpretation and explanation of texts, namely sources and secondary literature. A mathematical statement, once proved, seems timeless. The knowledge of a “historical prelude” which has led to this sentence must not necessarily be known for the understanding of the sentence. In the history of mathematics you will not find a theory, once formulated coherently and proved well, which ever had to be seen as wrong afterwards. But in physics one who treats the laws of falling bodies formulated by Galileo does not need to know at all the laws of falling bodies written down by Aristotle. One who teaches the conception of the heavens of Copernicus and Kepler doesn’t have to know the wrong conception of Claudius Ptolemy. However, in mathematics there are no such wrong theories. Take for example the theories of

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Archimedes and Apollonius. If the books are no longer learned, it is not because they are wrong but because their methods are simplified today. Their teachings are still valid today. One could still learn mathematics today in substance in the writings of Euclid, or Pappus or Diophantus. This is why one procedure in the history of mathematics is to put the value on the abstract, universal and history-independent validity of mathematics and to the inner development of the concepts and the discipline as a whole. Biographies in this concept of the history of mathematics are still like the hagiographies of elder churchmen or classical artists. The other group of historians emphasises that mathematical production or effective construction takes place within the cultural context, society and the communication codes of the mathematical community and therefore is determined by complex structures, conditional access and certain transmissions. It should be clear that there isn’t any conflict between the two groups. “Conceptual Historiography” is expanded by “Contextual Historiography” and their mutual interaction. One who investigates the development of mathematical concepts will have to make himself familiar with the results and methods of disciplines like anthropology, ethnology, philosophy, history, religion or psychology. Is there an interaction between mathematics and the history of ideas? Mathematical sentences are “timeless”, but the concepts, even some of their words, in which the theory is formulated are “children of their time”. Mathematics is a realm of liberty. So simplifications can be made many years after the justification of a lemma, and still undreamt coherences might be seen then, be formulated and taken more abstractly than the inventor might have ever imagined. Although the eternal truth of proved sentences is valid, some theories could often be formulated afterwards in a better, more elegant way and within applied sciences. I do hope I am making a small and fair contribution to the historiography of modern logic through the biography of Gerhard Gentzen. A bright and conceivable history of modern logic isn’t understandable without one’s biography using conceptual and contextual ideas. Moritz Epple (1999) writes: “It is only neither all about a story of mathematical concepts and ideas nor all about a collection of socio-biographical footnotes. . . for the row of the mathematical results. Rather it should be made clear at least at some important episodes, how mathematical knowledge produces and was used in concrete historical courses of action.” (p viii.) Moritz Epple doesn’t always succeed in it either, which is why he also includes socio-biographical inserts in his texts and theses. My biography shall nevertheless support the aim envisioned by Epple in the field of proof theory because, without clarification of historical facts, the different forms of evolving mathematical treatments, methodical and resulting knowledge, and epistemic configurations or its reflection are not once meaningfully describable.

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Therefore there is—standing completely unconnected beside each other—next to the problem of history of mathematics14 now a “contextual history of mathematics”.15 I myself feel somewhat obligated to the German tradition,16 which is embedded in the world history of mathematics.17

14 Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, New York, 1972. 15 Ronald Calinger, A Contextual History of Mathematics to Euler, Prentice-Hall, Upper Saddel River, 1999. 16 New exemplary literature in the German language: Herbert Mehrtens, Moderne. Sprache. Mathematik, Suhrkamp, Frankfurt am Main, 1990; Moritz Epple, Die Entstehung der Knotentheorie. Kontexte und Konstruktionen einer modernen mathematischen Theorie, Vieweg, Wiesbaden, 1999.— Craig Smory´ nski (1988, p. 16) once wrote: “Philosophy of mathematics was briefly concerned with foundations. Nowadays, philosophers are mostly interested in the philosophy of mathematics as a testing ground for their epistemological theories, and most mathematicians would find it pretty boring stuff.” In the meantime the history of mathematics has also become a sort of testing ground. Perhaps this has an end if the “descriptive phase” (C. Smory´ nski) is completed at least in the field of the newer mathematical logic of the 20th century. As long as this isn’t the case, my ideal lies between Jan of Plato, Creating Modern Probability. Its Mathematics, Physics and Philosophy in Historical Perspective, Cambridge University Press, Cambridge, 1994; and Dirk Van Dalen, Mystic, Surveyor, and Intuitionist. The Life of L. J. Brouwer. Vol 1: The Dawning Revolution and Vol. 2: Hope and Disillusion, Clarendon Press, Oxford, 1999 and 2005. Here it shows how splendid it is when two trained scientists write professional science history. The excellent use of the history of mathematics in a textbook for working mathematicians is shown in exemplary fashion by Smory´ nski in Logical Number Theory, vol. 1 (Springer, New York, 1991) and Vol. 2 (to appear). For all three a dictum of Hermann Weyl holds: “A scientist who writes on philosophy faces conflicts of conscience from which he will seldom extricate himself whole and unscathed; the open horizon and depth of philosophical thoughts are not easily reconciled with that objective clarity and determinacy for which he has been trained in the school of science” (p. v, Philosophy of Mathematics and Natural Science, Princeton University Press, Princeton, 1949). 17 Joseph W. Dauben and Christoph J. Scriba (eds.), Writing the History of Mathematics: Its Historical Development, Birkh¨ auser, Basel, 2002.

https://doi.org/10.1090/hmath/033/01

CHAPTER 1

Early Youth and Abitur 1. Gerhard Gentzen’s Birth Gerhard Carl Erich Gentzen was born on 24 November 1909 in the obstetric unit of the university hospital in Greifswald. He was the only son of Hans Gentzen, an attorney, notary, and court official; and Melanie Gentzen, n´ee Bilharz, a teacher of commerce. Baptism was on 28 March 1910.1 2. Gentzen’s Mother: Melanie Gentzen (1873-1968) Gentzen’s mother was 36 years old when he was born. In the Bilharz family chronicles she reminisced: I was born on 1 June 1873, a Whitsunday, in St. Louis (USA), the third child of my parents. Of my first years I have little to report. When I was 5 years old my parents made the journey across the ocean to Germany with their four children. We landed by Grandma Bilharz in “Eckh¨ ausle” [Little corner house] in Sigmaringen, which from then on would be our home. Here I attended the “Higher Daughters’ School”. At 17 I arrived in Reutlingen to attend the then very famous Womens’ Labour School, where I learned cutting and sewing. Following that I visited a nursery school in Karlsruhe for half a year in training to be the family nursery school teacher. After that I spent a year and a half as a domestic with the family of Superintendent Hermes in Halberstadt, which family had befriended us. In accordance with Uncle Viktor’s wishes I then took over the running of his household in the lower storey of “Eckh¨ ausle”, in which office I was replaced 6 years later by my younger sister Sophie. I myself took a position in Paris, which however proved to be a mistake. After returning to Germany, I came to M¨ uhlhausen in Thuringia to the family of a court official named Behring to assist his wife and look after their four children. There I met my later husband, the attorney Hans Gentzen, who while passing by stopped for a visit. After Behring’s departure I entered the Language and Commerce Institute of Frau Brewitz in Berlin, and after a year I took an examination as business teacher, in which profession I practised for 2 years with Frau Brewitz. After my engagement to Hans Gentzen, we were married in November 1908 [3 November 1908— EMT] and I moved into my new home in Bergen auf R¨ ugen.2

Johannes (familiarly: Hans) Anton Waldemar Gentzen and his new wife, Melanie, had a new house with a large garden built for them at Billrothstraße 16.3 The 1 Extract

2 Chronik

from the registry of the St. Nikolai evangelical parish, 1925, number 23, p. 293. Bilharz, entry on Gentzen Gentzen, p. 66. Melanie Gentzen died on 8 July 1968

in Rottweil. 3 The house would have been built in 1909 or 1910/1911. The entry by the real estate tax authority lists it in 1910 as “house and stables with laundry room” and in 1912 as “house with annex, courtyard, and house garden.” Hans Gentzen was the owner from 1910 to 1942 (Land Registry entry 2408 for field 14, plot 31 with 1095 square meters). Billrothstraße 16 was earlier called Joachimsberg 11; the exact time of the renaming is unknown. I thank the present (2004) 1

2

1. EARLY YOUTH AND ABITUR

street was named after a former occupant of one of the neighbouring houses, the surgeon Theodor Billroth (1829-1894), famous for having performed the first throat surgery.4

3. Gentzen’s Paternal Grandparents and His Father: The Attorney and Court Official Dr. jur. Hans Gentzen (1870-1919) Hans Gentzen’s father, born on 7 April 1839 in Pasewalk, was the senior primary school teacher Professor Dr. phil. Wilhelm Johann Carl Gentzen. He spent almost his entire life teaching mathematics at the Gymnasium in Stralsund. He pub¨ lished, among other things, “Uber die Bewegung eines Punktes auf einer gemeinen Kettenlinie” [On the motion of a point on a common catenary],5 presumably his dissertation.6 Dr. Wilhelm Gentzen died on 2 December 1919 in Stralsund. His wife, Agnes Alexandrine Alwine (n´ee Eisert), was born on 24 December 1846 in Stettin and had a French reform religious affiliation. She died a housewife on 2 December 1927 in Stralsund, having raised three sons. Hans Gentzen, who was born on 24 May 1870 in Stralsund, had two younger brothers: the future mining official Max Wilhelm Julius Gentzen (*8 May 1872 in Stralsund, †21 March 1940 in Stralsund) and the secondary school teacher and professor Erich Karl Hermann Gentzen (*25 March 1875 in Stralsund, †21 January 1957 in Berlin). Erich Gentzen had studied mathematics in Greifswald and most of his life taught mathematics at the Arndt-Gymnasium in Berlin-Dahlem. Of him it is reported: He had a villa in Dahlem and not only a pile of his own children, but also school boarders with whose money he could better his income. When I visited him in Dahlem in the winter of 1924/25, I hoped to inherit quite a few scientific books from him. But he said to me: I’ve thrown away all the mathematics books, but you can have as many novels as you want, and indicated at that an immense wall of bookshelves. He thought, what he used for instruction he knew by heart and all the rest no longer interested him. He was a rough diamond and could endure no Christmas celebration. On Christmas Eve he’d get drunk, set himself at the piano, and play “O, alte Burschenherrlichkeit” [Oh, good old student days] and other student and drinking songs.7

owner, J¨ urgen Kas¨ uske, for this information. He has become the de facto caretaker of legal records, a lot of the furnishings, and the old pear trees in the beautiful garden. From these records it appears that Hans Gentzen was a successful business lawyer. (Cf. further below.) 4 It would be purchased and renovated by the German Surgical Society. 5 With an enormous use of elliptic functions, Theta functions and such, this work calculates the time it takes for a point mass to fall from an arbitrary point on the hyperbolic cosine curve to the vertex. 6 Published by the Royal Governmental Press, Stralsund, 1872. 7 Biographical report on Dipl.-Ing. Hans Karl Gentzen (*8 August 1903 in Lehnitz, †18 October 1988 in Winsen/Aller) in possession of G. Gentzen. In the school chronicle, “The Arndt School in Dahlem, 1909/1910”, it says, “The school year started on 20 April 1909. . . During the Christmas holidays, Mr. Oberlehrer Gentzen fell ill and had to take leave for a quarter year. . . For our students, their parents, and other friends of our institution, the following lectures were organised: On 23 September, Mr. Oberlehrer Gentzen: ‘From the area of thermodynamics’, with experiments. . . On 24 november, Mr. Oberlehrer Gentzen: ‘On liquid air’, with experiments.” So one could already have legitimate hopes for scientific books.

4. HANS GENTZEN’S YOUTH

3

4. Youth and Student Days of Hans Gentzen Hans Gentzen graduated from the Stralsund Gymnasium with the Abitur in 1888. He had earlier been awarded the Reichenbach Medal for Diligence and Good Morals. This was given by a Swedish national foundation dating from the old Swedish days of the Free City of Stralsund. In prior times it had been given only once every 10 years, to the currently best student of the Gymnasium. Later the medal was awarded more frequently. Hans wanted to be an army officer, but the examining committee found him physically too weak and rejected him. So he decided to study law. Supposedly through the influence of fellow students he became a vegetarian and health freak. His rejection of all meat, his appearance (he wore long hair and sandals, walked year-round without stockings, and wore a cape), and a fanatical advocacy of the ascetic lifestyle brought much debate and rejection even within his own family: This Hans Gentzen visited my grandparents (the merchant Carl Friedrich Wilhelm Gentzen, third and second youngest brother of Wilhelm Gentzen [*6 December 1845 in Pasewalk, †15 February 1931 in Hannover— EMT]) and parents [the jurist Dr. jur. Hans August Karl Gentzen, *1 January 1876 in Breslau, †6 January 1957 in Hannover— EMT] in Berlin very frequently. If he rang the doorbell of my grandparents, he would spring in the moment the door opened, like a tree frog from a squatting position. Once inside he’d remove shoes and stockings “because it’s healthier!” He was a fanatic vegetarian and advocate of raw foods, his “lunch” he always took in the form of dried plums, beans, etc. from his jacket pocket. In the First World War as a soldier he ate from the field mess only potatoes and peas, not without first scraping a little fat off the potatoes.8 At Easter 1888 my cousin Hans Gentzen, oldest son of my father’s oldest brother, Uncle Wilhelm, came to Breslau to study law. He had, at not yet 18 years of age, taken the Abitur exam at the Gymnasium of his hometown of Stralsund. He was a highly gifted man and would have taken the exam at age 16 had his teachers not held him back. He possessed, alongside his astuteness and knowledge, something childish in his emotional life that even in his later years he never completely shed, and was an exceedingly good-natured, kind man with a small streak of stubbornness and an inclination toward eccentricity which remained with him his entire life. It is to be understood that his father had doubts about sending him to a college where he found no close connection to reliable persons and that he therefore selected Breslau for him where my parents lived. Uncle Wilhelm went so far as to house his son with my parents with the strict instruction to my father to keep my cousin away from contact with other students and to allow him no more freedom than one is accustomed to allow older schoolboys. Naturally that quickly became unbearable and remained so until he finally separated himself from my parents and found his own accommodations. After two semesters he left Breslau completely, to study further in Leipzig. His relation with my parents remained on the whole unspoilt, and he especially loved my mother with great affection and gratitude. In Leipzig my cousin evolved more and more into an eccentric, became an extreme vegetarian (as such for a long time he only took uncooked food), and hooked up with a painter Diefenbach who was known as an apostle of a natural lifestyle and in whose home in Vienna he even once spent his university break.9 8 Ibid. 9 Ibid.

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1. EARLY YOUTH AND ABITUR

Hans Gentzen was a follower of the ill-fated “Kohlrabi Apostle” Karl Wilhelm Diefenbach (1851-1913). Diefenbach lived in a quarry house in H¨ ollriegelskreuth on the Isar near Munich and founded Humanitas, Workshop for Religion, Art and Science. He attained artistic recognition through his “Per Aspera ad Astra”, a two-meter-high 68-meter-long silhouetted frieze consisting of 34 individual panels. This work, which hangs today in his birth city Hadamar, was reproduced in a famous “Leporello”, or fold-out book, by his considerably better-known student “Fidus” (Hugo H¨ oppener, 1868-1948). Diefenbach was regarded as an unbearable, uncompromising idealist, above all an extreme vegetarian and an extreme pacifist. Well-known as the defendant in the first German nudist trial, he attracted bailiffs and custody hearings like a beautiful flower draws bees. Moreover, he was barred from the Pinakothek (the famous art museum in Munich) on account of his monkish attire and disorderly hair. In search of peace of mind, he crossed the Alps on foot and eventually settled on the Italian isle of Capri, where he made some fine paintings which can still be seen there today. 5. Gentzen’s Maternal Grandparents: The Physician Alfons Bilharz and Adele Bilharz Richard Alexander Alfons Bilharz (*2 May 1836 in Sigmaringen, †23 May 1925 in Sigmaringen) was the younger brother of Theodor Bilharz, the widely travelled discoverer of the tropical disease bilharzia and one of the few Germans of the 19th century honoured today in Egypt. His wife Adele Bilharz, n´ee Fasnacht (*28 July 1851 in Murten, Switzerland, †14 October 1914 in Sigmaringen), an evangelist, was 18 when she married the Catholic Alfons. The mother of the Bilharz brothers, Elisa Bilharz, n´ee Fehr, was related to the patrician Zollikofer family of the Swiss canton of Thurgau, where she lived in Castle Altenklingen. Elisa was the oldest daughter of Salomon Fehr, an attorney and businessman who, for political reasons, had to flee to Canada— unaccompanied by wife and family. He had protested the method of establishing the Thurgau Restoration Constitution of 28 July 1814 and had thereby become a wanted man. Elisa was adopted by Sabine von Zollikofer, her mother’s also divorced sister. Thus, she grew up in an upper-class house and there got to know Joseph Anton Bilharz. In 1817 he became private secretary to Erbprinz [hereditary prince] Karl von Hohenzollern-Sigmaringen. Later he administered the princely archives and the court bank and advanced finally to chief accountant.10

Alfons was the seventh child among the siblings. Because of his inclination for reading and scientific study, his father deemed him a worthy successor to his gifted brother and called him “Theodor Secundus”.11 The aptness of this designation is borne out by his choice of medicine as a career. Theodor’s rˆole in Alfons’s training was manifold. From Egypt Theodor sent prepared slides, sectioned testicles and dissected penises. Alfons studied the testicles of castrati and found therefrom that testicles were independent glands, probably having an endocrine function, produced the main body of sperm, and were part of the regulatory cycle. From 10 Cf. the dissertation of Peter Maria Rob, “Alfons Bilharz (1836-1925). Ein Arzt zwischen ¨ Natur- und Geisteswissenschaften”, Kiel, 1987; published in Schleswig-Holsteinisches Arzteblatt, vol. 2 (1988), pp. 97-100. 11 Peter Maria Rob (1987), p. 17.

5. ALFONS BILHARZ

5

the atrophy of the prostate he elucidated the central regulatory problem resulting from castration. Under the supervision of the Viennese physiologist Josef Hyrtl (1810-1894)12 Alfons submitted a dissertation, “Descriptio anatomica organorum genitalium eunuchii ætiopis” [Anatomical description of the genital organ of black eunuchs— with accompanying remarks on clitoral and labial castration], and was promoted on 16 July 1859. Not too long after his promotion, Alfons interrupted his formal studies to spend seven months in Cairo with his brother. At Theodor’s side he studied schistosomes, concentrating his attention on the urogenital complications of schistosomiasis: There one day I found a small body, half the size of a lentil, under the mucous membrane that I must have released through an incision, thus probably a glandular secretion. Now it seemed to me the puzzle of the stone-forming “catarrh” was solved. . . Under the microscope the substance of the particle revealed itself to be organic, a colloid. It gave, so I thought, if freed accidentally the first cause and the kernel for an incrustation, thus for the formation of urinary calculus.13

In April 1860 Alfons returned to Berlin, where in 1861 he took the state exam and was licensed as a physician. Following the German student tradition of geographically widespread study, Alfons had studied in Freiburg, Heidelberg, W¨ urzburg, Berlin and Vienna. After his exams he further studied physiology under Emil DuBois-Reymond (1818-1896) in Berlin and mathematics under the physicist Gustav Robert Kirchhoff (1824-1887) in Heidelberg. On 2 May 1861 Alfons Bilharz began unpaid work in the electrophysically oriented laboratory of DuBois-Reymond, and after 14 days he wrote: I doubt not that I have stumbled upon a law of the greatest generality. . . it deals with nothing less than an entirely new view of electricity and heat, and in consequence of which a definition of matter in terms of force; finally I have therewith in my hands the key to the further development of chemistry and probably to the comprehension of crystallography. Yes, I know not which branch of the natural sciences will not be influenced in undreamt of ways by my law, because it simply is so general that it reaches to the outermost boundaries of knowledge which are at all accessible to the human mind. . . in any case I am unable to go back now; it must destroy me.14

From the beginning Alfons had placed the soul before the body and occupied himself with the connection between body and mind. During his formal studies he had reduced his medical obligations in favour of private humanistic studies. He was torn between physiology and metaphysics. And now it appears natural philosophical construction had replaced the toil of scientific induction. But brother Theodor breathed down his neck, and so Alfons continued with more exact studies. Together with O. Nasse, Alfons published the results of a study of electrotonic variation in nerves: They examined mechanical tetanus, to gain further electrophysiological knowledge, and discovered that the local excitability of a nerve could be modified 12 Josef Hyrtl (1810-1894) was one of the great anatomists of the rank of Virchow. His book Handbuch der praktischen Zergliederungskunst als Anleitung zu den Sections¨ ubungen und zur Ausarbeitung anatomischer Pr¨ aparate (Vienna: W. Braum¨ uller, 1860) is a classic. 13 Peter Maria Rob (1987), p. 64. 14 Ibid., p. 73.

6

1. EARLY YOUTH AND ABITUR

through physical and chemical influences. Superthreshold stimuli lead independent of the increase in stimulus strength to maximal propagating excitation; subthreshold stimuli however elicit only a local response. This phenomeon would be termed the “All-or-Nothing Law” by Bowditch in 1871. They observed further that after a response a momentary suppression of the excitability of the nerve remained, which however gradually increased until a normal electrotonus was attained. These observations coincide with what is today known by the concepts resting or membrane potential and refractory time.15

Nevertheless, with the death of his brother on 9 May 1862, Alfons Bilharz gave up his plan to define matter in terms of force or weight, heat, and electricity. “The plan, to further medical research by natural scientific methods, had failed.”16 His compulsory military service was spent in Rastatt. He decided after that to emigrate to America. The W¨ urzburger C.F. Castelhun who practised medicine in Saint Louis had encouraged him to do so already in 1861. In the meantime Adelheid (Adele) Fasnacht was already residing in North America. Her father had been a customs official and had associated himself with Garibaldi’s gang. After his death her mother opened a small shop. When she died, the children came under the care of their uncle, the contractor Jakob Fasnacht. Their stepsister, Rosalie, married a businessman, Emil Bandelier. In 1869 the children moved with the Bandelier family to the Swiss colony of Highland in Breese County, Illinois.17 There Alfons Bilharz married the 18-year-old Adele in East Saint Louis on 2 November 1869. They had five children, and he practised medicine in the state of Illinois for 12 years. On 1 September 1878, back in Germany for a new life, to span the gap between the humanities and the natural sciences he developed an epistemological interpretation of the Law of Conservation of Energy. He gave existence priority over thought. He attempted to ground philosophy, strictly separated from psychology, as a deductively constructed, systematic science as opposed to the merely empiricallypositivistically oriented natural sciences. He wanted to combine romantic idealism with natural scientific methods and a strict scientific approach, although these two cultures were already going their separate ways at the end of the 19th century. His so little noticed first work “Der heliozentrische Standpunkt der Weltbetrachtung. Grundlegung einer wirklichen Naturphilosophie” [The heliocentric vantage point of the world contemplation; outline of a true natural philosophy] possibly had considerable influence on Nietzsche’s concepts of eternal recurrence and the will to power. A case can probably be made that through the mediation of Bilharz Nietzsche took up the Law of Conservation of Energy and used it for his own conception.18 15 Ibid.,

p. 74. p. 99. 17 There is no indication that this concerned members of the religious community of Hutterers. However, the term “colony” and the date of resettlement do suggest a connection. 18 Peter Maria Rob (1988), p. 99. The book bore the title Alfons Bilharz. Der heliocentrische Standpunct der Weltbetrachtung. Grundlegung einer Naturphilosophie, Stuttgart: J. Cotta, 1879. For an indication of the possible correctness of this conjecture I refer to the book (p. 365) by Alwin Mittasch, Friedrich Nietzsche als Naturphilosoph, Kr¨ oner, Stuttgart, 1952. Mind you, Nietzsche also carefully studied Eugen D¨ uhring, Helmholtz, Robert Mayer, Emil DuBois-Reymond, Boscovich, Ludwig B¨ uchner, Fechner, Mach, Moleschott, African Spir, Virchow, Lasswitz, Vogt and Z¨ ollner along with many others, so that a sole attribution to Alfons Bilharz possesses no persuasiveness whatsoever. The traces of reading, underlinings and marginal notes of Nietzsche on the concepts of conservation of energy, power, causality, release, self-regulation, and freedom 16 Ibid.,

5. ALFONS BILHARZ

7

Eugen D¨ uhring, Arthur Schopenhauer, and Immanuel Kant were his idols before he developed his own thoughts. In 1882 in Sigmaringen he became acquainted with his friend and co-worker, the mathematician Portus Dannegger, who saw to it that Bilharz’s philosophical works were gradually published. From 1882 until 1907 Bilharz was director of the regional hospital in Sigmaringen. He permanently modernised the institution and initiated a number of new buildings. On 5 July 1894 he received the honorary title of “Sanit¨ atsrat” (Medical councillor), and on 22 June 1907 the more exalted honorary title “Geheimer Sanit¨ atsrat” (Privy medical councillor) was bestowed upon him. He gave up the direction of the institution in 1907 because of an eye complaint. As a general practitioner Alfons Bilharz was much loved; he was especially popular among women in childbed. He treated his mental patients very well and without force, so that the gratitude of the relatives of such patients was assured.19 He was often heard to say, “I must return to my fools; I belong with them.” In Sigmaringen, Adele gave birth to five more children, of whom one died early. She suffered a stroke on 11 October 1914, died three days later, and was buried on the 16th at the age of 63. At graveside Superintendent Theobald praised her for her “Swiss order and punctuality and German contentment.” She was characterised as a “hard worker” and “sympathetic life’s companion.” Her maternal earnestness was emphasised like the “infinite abundance of work,” her “self-denying devotion” to husband, house, and family: Reverence for a woman who would above all be a proper mother! Oh, how many German women have forgotten that in these past decades! How many have been infected by the French spirit and French poison!20

In old age, Alfons Bilharz was often depressed and longed to die. His writings had been completely ignored, disappointing him and robbing him of ambition. He died on 23 April 1925 at the age of 89. Presumably, Bilharz supplied financial support for the education of Gerhard Gentzen. An intellectual connection between the two cannot be verified. A disconnection is more likely: “Bilharz liked to work with analogies from mathematics; he compared, for example, knowledge with a mathematical equation and planned a ‘philosopical geometry’ on the basis of the ‘rectangularity of opposites’.”21 Just as with geometry, thought Bilharz, philosophy must be deductively constructed. But philosophical thought required a metalogic where the Law of the Excluded Middle is not valid. But since mathematics is based on this ‘Orthologic’,

await an analysis in “Nietzsche Studies” (in particular, Mittasch’s work appears to be superceded). Cf. also Alwin Mittasch, Von der Chemie zur Philosophie. Ausgew¨ ahlte Schriften und Vortr¨ age, Ulm, 1948. 19 The entire city was thankful. On 2 August 1936 the city was able to celebrate the 100th birthday of Alfons Bilharz. To this end the city unveiled a memorial plaque for Theodor and Alfons Bilharz. In the evening a small celebration took place. The Nazis hoped some of the glory would rub off on them. Even Gerhard Gentzen was present. Since 1975 a new plaque adorns the “Bilharz Apothecary Shop” in Eckh¨ ausle. 20 Chronik Bilharz, entry on Bertha Bilharz, p. 42. 21 Bruno Baron von Freytag L¨ oringhoff, “Alfons Bilharz”, p. 237, in: Neue Deutsche Biographie, volume 2, Duncker and Humblot, Berlin.

8

1. EARLY YOUTH AND ABITUR

the logic stands ‘entirely clueless’22 before the problem of identity. Via the concepts of equation, opposites and identity he attempted a reconciliation of existence and thought. Metaphysics was to be replaced by a ‘Protophysics’, out of which developed the Protologic (Metalogic). Its unique objects are content and form: in appearance a transformed form-being, seen from outside, is the formal exterior of our content-being or I, our body, which after the three thought categories size, space and time after the three directions of spatially extended body. . . This body, viewed from within, is called soul.

It has unfortunately not been handed down what Gerhard Gentzen might have said about the “Metaphysical elements of the mathematical science”,23 which Alfons Bilharz and P. Dannegger had written, if he even read the work.24 Bruno Baron von Freytag L¨oringhoff described it as worthless and damaging to Bilharz’s reputation. Alfons Bilharz felt overtaxed by the exact sciences. Early on he knew from Theodor the difference between knowledge (not knowing, half-knowing, knowing, better knowing) and science

and had to admit But for the secret of higher mathematics, namely the differential calculus, my poor skull proves itself fully refractory.25

He fared the same in physics. A concrete impotence blocked an immediate intuitive view of the whole of physics because I get lost in the details [more literally: I don’t see clearly the course of the threads in the particulars]. In one word, I barely understand physics and know above all not to labour lightly with its laws.26

The logician and historian of science Gerd Robbel27 conjectured a philosophical influence of Alfons Bilharz on the views of Gerhard Gentzen. I question this and see no immediate cause nor possibility of proof for a conjecture of this sort. Gentzen had, to be sure, written to his grandfather two or three times a year, but an exertion of influence on the part of the grandfather is neither documented nor handed down in family tradition. Also, Bilharz never sent his grandchild his philosophical works. Gentzen’s sister cannot remember any reading of his works by Gentzen. Certainly two works of Alfons Bilharz are to be found in the possession of Waltraut Student as an inheritance of her mother, Melanie Gentzen. They looked unread in 1988. I believe on the other hand, on the basis of the family legend of Alfons Bilharz as a 22 Alfons

Bilharz, Philosophie als Universalwissenschaft. Deductorisch dargestellt [Philosophy as universal science, deductively presented], Wiesbaden, 1912. 23 Verlag Tappen: Sigmaringen, 1880. 24 I suspect that statements about opposites standing perpendicular to each other could not have pleased Gentzen the mathematician. The needless metaphoric use of mathematical concepts in Philosophy is to this day incomprehensible to some mathematicians. Quite the opposite: the anger is great because the user arouses suspicion that he wants by devious means to obtain a sort of exactness for his unclear ideas, which exactness his metaphoric word usage simply does not yield. Neo-Kantianism tended to play with the mathematical usage of words in Philosophy. On his part, Alfons Bilharz used mathematical, formula-, and graphic-rich language for a clearer presentation of his thoughts. The private mathematical docent Dr. Hans Reichenbach even had to ward off a Bilharzian “solution” to Fermat’s Theorem. 25 Peter Maria Rob (1987), p. 76. 26 Ibid., p. 75. 27 Gerd Robbel, 1986, a and b.

6. THEODOR BILHARZ

9

clever philosopher who had nowhere achieved recognition, that Gentzen abhorred such metaphysical philosophising. Alfons Bilharz remained to his death disparagingly opposed to other philosophers and their systems, and remained intolerant and stubborn on these questions. 6. The Shining Example in Gentzen’s Family Tree: The Great, Successful Physician and Natural Scientist Maximilian Theodor Bilharz (1823-1862) The shadow of Maximilian Theodor Bilharz (*24 March 1823, †9 May 1862 in Cairo) loomed over Alfons Bilharz. Part of his inner turmoil must have lain therein, that his early deceased brother who was possibly more intelligent was in any case a successful and famous natural scientist. The firstborn was certainly an example for Gentzen. Theodor was first in Latin, Greek, and Hebrew. At 14 he had already written a drama and had translated the Frog-Mouse War himself. His poems approached the quality of those of Ludwig Uhland. During his Gymnasium time he trained himself to be a scientific researcher with the library inherited from his Great Uncle Kaspar Tobias von Zollikofer of St. Gallen. He had acquired permission from the heiress, his aunt Sabine von Zollikofer, to take some works from the collection prior to auction, in the choice of which he was assisted by the T¨ ubinger Professor Hugo von Mohl. He was also allowed to keep the butterfly and mineral collections as well as the collection of dried plants. Both were kept in a white, drawered cabinet. He studied medicine in T¨ ubingen and became simultaneously an excellent butterfly researcher. Alfons used Theodor’s library and was allowed use of his study room. His entire life he felt a constantly performed, time and again openly repeated, “inner thankfulness” towards his—in every respect—greater brother. In 1847 in T¨ ubingen, Theodor won the medical faculty’s prize competition on the blood of invertebrates and received a large gold medal for this dissertation. With that he later purchased for himself the “Oberhaus Microscope”,28 which he later used in Egypt in his extraordinarily fine researches on the electrical organ of the electric catfish. To be sure, Alfons Bilharz performed the bibliographic research for his mentor Theodor and therefrom formed the conception that the current of the gymnotus (a family of electrical fishes including the electric eel) flowed in the direction opposite to that of the malapterus (electric catfish), but Theodor ignored this suggestion of his brother. Subsequently, the correctness of Alfons’s idea came to light. For Theodor things progressed smoothly: in 1848 he took the state exam, was praised for his work on opium treatment for appendicitis, went to Freiburg, and became Prosektor [Chief of Pathology] in Kobelt. His former teacher Professor Wilhelm Grossinger received a call to become personal physician at the court of the viceroy Abba Pascha in Egypt and invited Theodor along. They left for Africa in 1850, and Theodor Bilharz became professor of anatomy at the medical school of Kasr-el-A¨ın, simultaneously holding the position of senior consultant of a department for internal medicine. He acquired the military rank of lieutenant colonel (Kaimakam). Theodor lived modestly in Old Cairo, where indeed he had an old servant available, but except for a clay water filter the furnishing and housekeeping 28 Born in Hesse, Georg Oberh¨ auser (1798-1868) is known for his fine microscope completed in Paris. From 1848 he supplied it with the well-known horseshoe shaped base, the so-called “continental model”.

10

1. EARLY YOUTH AND ABITUR

were of spartan simplicity. He wore a simple blue frock coat with a standup collar. An iron bedstead without a mattress— in its place a woolen blanket—and a folding table for writing next to an office cabinet characterised his lifestyle. In performing autopsies on the corpses of soldiers, he found parasitic flatworms, among which the Distomum hæmatobium, which lived in the blood of portal veins and caused the then widespread disease that Heinrich Meckel von Hemsbach later named bilharziosis or bilharzia. Von Hemsbach, who had visited Egypt as a patient, also renamed the parasite Bilharzia hæmatobium in 1856.29 Theodor’s skill in microscopy assured him a lasting reputation as a comparative anatomist. In 1858 he visited Europe and reported on his research to the great authorities of the period: Alexander von Humboldt, Johann M¨ uller, Emil DuBois-Reymond, Rudolf Virchow, Theodor Brugsch, Heinrich Gustav Magnus, and many others. In September he attended a meeting of natural scientists in Karlsruhe. On returning to Egypt, he devoted himself to the study of Egyptian history, in particular to hieroglyphics. In 1862, Duke Ernst von Coburg-Gotha invited Theodor to Ethiopia on a hunting trip. There, on 9 May 1862, Theodor Bilharz died of typhus. Theodor was a “cold” rationalist devoid of dogmatism who found success in the world, whereas his brother Alfons, who tried to unite the natural sciences, medicine, and philosophy, must be viewed by others in his efforts as research scientist and natural explorer as a failure. This could not have escaped the notice of the growing Gerhard Gentzen, who in the newly published second volume of Meyer’s Lexikon 30 would see only Theodor’s name listed. Moreover, there were many biographies and obituaries of Theodor which Gentzen may well have been acquainted with.

7. Gentzen’s Sister: Waltraut Sophie Margaret Gentzen (*1911) On 16 February 1911 Waltraut Gentzen31 was born into an intact, loving and harmonious family in Bergen. Gerhard and Waltraut got along unusually well. They almost never quarrelled. Gerhard was Waltraut’s intellectual rˆole model. He invented puzzles and jokes and entertained her with self-thought-up tales and narratives. He played with puppets, wrote poems for them, and performed short scenes with cardboard puppets. For his sister he built by hand a two-room dollhouse, which still exists today, of precisely mathematically calculated proportions. The furniture, carefully drawn on paper and then coloured, was manufactured with cardboard and paper stuck together. Nothing is glued. The clothes cabinet, stools, chests of drawers, beds and tables were assembled in such a sophisticated manner that on first view one doesn’t realise this. 29 Dr.

Johann Heinrich Meckel von Hemsbach (1821-1856) directed the Charit´e-Prosektur in Berlin from 1849 to 1856 along with Benno Reinhardt (1819-1852), while Rudolf von Virchow taught for a while in W¨ urzburg before resuming the position in 1856. On bilharzia, see John Farley, Bilharzia. A History of Tropical Medicine, Cambridge University Press, 1991. The battle against bilharzia has modern significance in that bilharzia causes ulceration of the genitals, thereby facilitating the spread of AIDS. 30 Bibliographisches Institut: Leipzig, 1925. 31 Waltraut long survived her brother and before she died a few years ago was most helpful to the author of this biography. Waltraut’s daughter, Barbara Student-Zwirn, has handed over 86 pieces of some of my research material as “Gentzen’s Teil-Nachlass” to Dr. Brigitte Parakenings, the director of the Philosophical Archive of the University of Konstanz.

8. THE SCHOOLCHILD GERHARD GENTZEN

11

Gerhard would wade in the river Sand with his sister and regale her with his own tales. On one occasion, in the old market in Stralsund, the pair saw a stargazer with an astronomical chart.

8. The Schoolchild Gerhard Gentzen In 1913 attorney Hans Gentzen received the title Justizrat [literal translation: Justice Councillor] and served in the voluntary fire department. In 1914 he was called up to serve in the home guard with the rank of Vizefeldwebel (Offizierstellvertreter) [“Vice-sergeant” and “officer-deputy”, respectively]. He was first assigned to the home defense of the island of R¨ ugen. In 1915 Gerhard Gentzen was ready for school but did not attend. Instead, his mother preferred to instruct him at home on the usual subjects.32 Following this he entered the Volksschule [elementary school] around Easter of 1916. In 1916 he wrote a small poem for his mother’s birthday dealing with the forest, the forest ranger, and flowers. In December, at the age of seven, he wrote a poem, perhaps less impressive than it should be in an unrhymed, arhythmic translation: Night It is night, No one is awake. In the wide distance shine the stars. A sentry stands before many a house, No one peeps out the door; The children are sleeping sound In their little nest.

His mother was so pleased with this poem that she copied it out for herself. On her birthday in 1917 he wrote another: Come, Mama, the weather is so nice We want to go a little while for a walk. And as they went, there came they soon Into a beautiful Forest. The child, he picked flowers there And went constantly from spot to spot. Then he made a flower-bunch And brought it with into the house. He stuck it in a pretty glass, That his mother enjoyed very much. And on mother’s birthday The child looked for flowers, as many as he liked. He placed them on the birthday table. The flowers smell ever and ever fresh.

Gentzen’s father returned to Bergen seriously ill in November 1918. He had had to stay in an unheated barracks in Stettin, probably doing service in a typing pool. He suffered from consumption.

32 Chronik

Bilharz.

12

1. EARLY YOUTH AND ABITUR

At Eastertime in 1918 Gerhard entered the Septima of the Bergen Realschule.33 He wrote essays like “The history of a rain drop” or “A visit to the smithy”. 9. The Death of the Father Means a Move and a New School Hans Gentzen was a successful business lawyer. He represented the financial interest of Hamburg mussel importers, Rostock paint factories, Berlin hemp and flax shops, Greifswald bookstores, R¨ ugen building firms and sawmills, as well as insolvency matters and the collection of monies from defaulting customers of Binz textile and fashion dealers. He also handled general mortgages, land registry matters, as well as questions of right-of-way, and had connections as far as Karlsbad. On 11 March 1919 Hans Gentzen succumbed to his galloping consumption— tuberculosis. His business successor was Justizrat H¨ansel. Gerhard’s grandfather Dr. Wilhelm Gentzen died nine months later on 2 December 1919. At the wish of her mother-in-law, Melanie Gentzen and her two children moved to Stralsund in the following year, 1920. The “years of good fortune”34 were over. Gerhard transferred to a humanistic Gymnasium. Grandmother Agnes, who stemmed from an old Stettin Huguenot family, was not easy in company, and the household in Schill-Straße 35 had to be kept in order. Melanie Gentzen received a small war victim’s and widow’s pension. This modest fortune was lost through war loans. The father had made no other provisions. Gerhard’s sister, Waltraut, recalls: Of grandmother Gentzen I have the memory that she was strict. I got from her my first piano lessons. Otherwise we were only together at mealtimes; she lived in her part of the flat, read English and French books, and to my knowledge concerned herself not at all with the grandchildren. It was different with Grandpa Bilharz, who happily played with us during our visits. We loved him very much. Unfortunately, because of the distance from North to South, our visits in Sigmaringen were rare. I remember in 1922 when we last visited Grandpa; he died in 1925. The year before we were perhaps once or twice in Sigmaringen.35

10. The Beginning of Gentzen’s Intellectual Activity Gerhard Gentzen wrote short dramas for a cardboard-and-paper theatre,36 like those manufactured around 1900 by the firm Schreiber in Esslingen. These finished sheets, which would be glued to cardboard, were a popular present for children. They came with scenery, figures, other accessories and printed stage plays. Gerhard devised his own dramas. Surviving are his “Der Beutel mit Gold” [The bag of gold], a stage play in three acts; “Weihnachten” [Christmas] in two acts; and “Ostern” [Easter] in two acts. These pieces testify to an inventive talent that knew how to express itself. They are effectively arranged for dramatic effect and, for his age, worked out in masterly fashion. 33 Secondary

school: The Bergen Realschule was almost a Gymnasium, differing from this latter class of college prep secondary schools primarily in not offering Latin. The Septima was not part of the proper Realschule curriculum, but a preparatory class. Gerhard had had only three years of schooling so far (including his mother’s instruction) and not the four customary in elementary school—in some of the German lands. 34 Cited from Gerd Robbel, “Aus der Gedankenwelt des jungen Gerhard Gentzen”, in: Berichte der Humboldt Universit¨ at Berlin, volume 12 (1986), number 6, pp. 51-57. 35 Ibid. 36 G. Gentzen’s theatre has been transferred to the Museum for Figure Theatre in Bochum.

10. THE BEGINNING OF GENTZEN’S INTELLECTUAL ACTIVITY

13

With Otto Michælis— the brother of Hertha Michælis— he founded the Dionysus Theatre. They wrote “Rosa von Tannenberg” and “Die Prinzessin im Zauberwald” [The princess in the magic forest]. They also manufactured marionettes, and Otto Michælis was responsible for lighting effects. A play would be reworked up to 5 times and seen up to 11 performances. The fairy tale plays came from Christmas or Easter plays and would be repeated for several years.37 One of Gentzen’s skills was that he could switch between a Berlin and a Bohemian dialect in performing different characters. In December 1922 the thirteen-year-old wrote to his grandfather, Alfons Bilharz: Dear Grandpapa! Because I know you are interested in my mathematical progress, I’m sending you the following self-discovered and self-proven theorems: I draw over the sides of an arbitrary triangle the equilateral triangle and join their vertices with the opposing corners of the given triangle. For the sake of brevity I call these lines pereunts (pereuntes, the through-goers).

Theorem 2 stated: The pereunts of a triangle are equal, intersect in a single point and form around it nothing but 60 degree angles. Proof:

For the Pythagorean Theorem he found his own proof. He ended the letter with wishes for a merry Christmas.38 At the same time he dedicated to his dear Grandpapa and his dear aunt Bertha, the oldest daughter of Alfons Bilharz, “Auf Manurga. Eine Indianergeschichte von Gerhard Gentzen” [On Manurga. An Indian story by Gerhard Gentzen]. This Indian story of 23 pages was told in the manner of a detective story and regurgitated a mixture of all kinds of things then common in childrens’ books. Cunning Chinese, upright German sea officers like Bernhard Maier, Indians, mountain lairs inaccessible to white men—everything was recalled that could be found at this time in exotic colonial novels, German war adventures at sea, in Friedrich Gerst¨ acker’s “Leatherstocking”, or even heard from the speech of adults about dirty coolies and colourfully clad foreigners, if one listened attentively. What is exceptional about Gentzen’s narration is not only its length, but also the interweaving of disparate subjects into a crime story, where even the strangest element still makes sense if one is able to accept the harmonious ending of the story, the Happy End as a goal in itself. It is basically a trivial adventure novel, but— how many youngsters of his age could write such a thing with such staying power and inventiveness?39 In 1923 he wrote again to his dear Grandpapa and his dear aunt Bertha that he had gotten both malt extract and tissue paper. He had himself received a motor and a book as well as his father’s chess pieces. He had performed a stage play for the Kasperl Theatre with rare puppets, played with his sister and her girlfriend Poch, and then built a crane which he operated with the given motor. And he was looking forward to returning to school. Between 1923 and 1924 grandfather Bilharz sent Gerhard a planetary chart, which his mother copied onto canvas. 37 Cf.

Max Eickemeyer, Das Kindertheater. Sein Bau und seine Einrichtung, Verlag J.F. Schreiber, Esslingen and Munich, 1900. 38 The letter is reproduced in its entirety in G. Robbel 1986b. Appendix A discusses the result and Gentzen’s proof in detail. 39 Information on curriculum and schools in the Weimar Republic can be found, for example, in Reinhard Dithmar (ed.), Schule und Unterricht in der Endphase der Weimarer Republik, Luchterhand, Neuwied, 1993. How this applies to Gentzen can only be speculated.

14

1. EARLY YOUTH AND ABITUR

New Year’s Eve was always celebrated with the Gentzen family in my parents’ house with lead drops [a fortune telling activity involving dripping molten lead into cold water— EMT] and before that parlour games. In playing cards Gerhard became very lively; he could in breathtaking fashion calculate the other players’ cards and make use of it in his own playing.40

In 1924 Grandpapa and Aunt Bertha again received the obligatory thank you letter. He thanked them for pear bread and a notebook. He had also received playing cards. However, his calculations of the planetary orbits were unfinished. It appears he liked to give evidence of the progress of his education. In 1925 the fifteen-year-old wrote a poem on the starry heavens:41 Bloody red it glows in the west And the sun sinks into the sea. The light of day dies, It grows darker around us. In the heaven’s twilight It sparkles: the first star! Venus it is, the Earth’s neighbour, And yet so far, so far. It grows darker on the firmament, Radiance flees the planet. Its hour is past When the twilight dies. Dark night! The stars glow High in heaven clear and light, And in distant arcs fly Meteors, nimbly and quick. Quietly stroll the planets In their established paths. Shine on us with their light, Calm gentle shimmer. And heaven’s largest arc Coils light a ribbon of stars Like a huge misty strip. The Milky Way it is called. Deep in the south on the horizon Moves a comet through the starry realm, And its tail in a long arc Hurries in advance, faint and pale. A cloud moves in front, Behind it it shines bright and clear; And the Moon’s light shimmer Outshines the host of stars.

40 Letter 41 Gerd

from Hertha Michælis (D¨ usseldorf, 7 February 1988) to the author. Robbel 1986b, p. 50, footnote 43.

11. GENTZEN’S SUCCESS AT SCHOOL

15

And in the East it grows lighter Starshine dies to nothingness; And a red morning shimmer Announces to us the coming of day.

Here I marvel at the poetic and at the same time exact description of the cosmos given by a youth42 who had carefully read43 Max Valier’s Der Sterne Bahn und Wesen. Gemeinverst¨ andliche Einf¨ uhrung in die Himmelskunde 44 [The stars orbit and nature: a generally comprehensible introduction to astronomy]. In February 1925 he wrote more chapters in 22 pages under the heading “Gedankenordnung I: Die Stellung der Mars im Sonnensystem und seine Monde” [Ordering of thoughts I: the position of Mars in the solar system and its moons]. In this he covered appearances like the polar-caps, canals, and temperature, and he discussed certain theses of Svante Arrhenius, Percival Lowell, and Giovanni Schiaparelli. He held it improbable—“everything speaks against it”— that men could live on Mars.45 He played chess and developed a number game. Also, he read novels by Karl May. Grandfather Bilharz died on 23 May 1925 at the age of 89, and Gerhard now concentrated his attention on school. 11. Gentzen’s Success at School Gentzen was encouraged by his teachers, who immediately recognised his talent. He had had no Latin the entire year at the Bergen school. His classmate and Conabiturient Burkhardt Schwerin, later senior teacher in Schwerin and Hamburg, reported: He acquired the requisite knowledge during summer vacation and placed himself from the beginning, as in almost all other subjects, at the top of the class. He became Primus [star pupil] and remained so until the Abitur. Dr. May, this highly musical, warm-hearted, empathetic and capable pædagogue integrated Gerhard Gentzen in his sensitive manner into the class and became a fatherly friend to him. In the 4th year Dr. Herbst, an excellent mathematician, took over the mathematics instruction. He recognised early on the great mathematical talent of Gerhard Gentzen and encouraged him. In the 8th year Dr. Herbst encouraged46 Gerhard Gentzen to write a term paper on mathematics. This work would be marked “with distinction”. The third person who moulded Gerhard 42 Cf. Fritz Kubli, “Kosmosvorstellungen von Kindern und die Astronomie in Unterricht”, in Uwe Hameyer and Thorsten Kapune (eds.), Weltall und Weldbild, Institut f¨ ur P¨ adagogik der Naturwissenschaften, Univerity of Kiel, 1984. 43 The book was found among his childhood books currently in the possession of his sister, Waltraut. On pages 164-174 one finds a chapter on Johannes Kepler and his Laws of Planetary Motion, intimate knowledge of which would yet play an important rˆ ole for him. 44 Leipzig: R. Voigtl¨ ander Verlag, 1924. 45 Gerhard largely had to gather this information himself from various sources, because the style is that of a child. My cursory reading of works and dictionaries of this period from 1911— beginning for example with Jean Henri Fabre’s Der Sternenhimmel [The starry heavens] (Stuttgart Franckische Verlagshandlung, 1911), over Ferdinand Meisel, Wandlungen des Weltbildes und des Wissens von der Erde. Das Weltbild der Gegenwart. Band I [Changes in the world view and of knowledge of the earth. The world view of the present. volume I] (Stuttgart and Berlin: Deutsche Verlagsanstalt)—up to 1922 (Max Valier) has yielded no indication that he could possibly have copied something out or neglected to cite a reference. 46 Dr. Herbst of Stralsund was later a good friend of Gentzen’s. As late as 1944 Gentzen visited him in Binz.

16

1. EARLY YOUTH AND ABITUR

Gentzen was Dr. Stengel, our principal. Dr. Stengel was a superb teacher, deeply filled and imbued with the ideals of humanism, a shining example for us Primaners [7th formers]. Gerhard Gentzen was a kind of outsider in a positive sense. Over his position as Primus he was never envied; it was anyway sacrosanct. He belonged to the class and the class liked him, who was always friendly and ready to help. He had no school friends in the true sense. His unspeakable love of mathematics filled his life completely. Because of his out of the ordinary talent and his high achievement, Gerhard Gentzen was admitted to the Studienstiftung des Deutschen Reiches.47 48

In 1926 he built his first radio using coils he wound himself. Presumably, Gerhard received— as his father had done before— the Reichenbach Medal awarded on 7 March 1927.49 There remains, however, no record to verify this. My last two school years I was at the Stralsund Realgymnasium [a gymnasium stressing modern languages, math, and science]. A year of this time I lived with the Gentzens. After my weekend visits with my parents in Putbus, Gerhard regularly picked me up at the harbour train station in Stralsund. We often took long walks in the Stralsund municipal forest. He also helped me often with the schoolwork, to the annoyance of his mother. In mathematics I was very poor. . . Once when I couldn’t cope with some geographical drawing, Gerhard took pity on me and completed the project in ten minutes. When I said nobody would take it to be my work, he had to introduce errors into it; the “correction” took 3 times as long. So we slowly grew to be friends. After school I attended a housekeeping school in Gernrode am Harz, so it resulted that Gerhard and I met each other in the Harz. I particularly remember one Sunday in Thule am Harz, where we were so cheerful that we rolled down the grass.50

12. The Abitur on 29 February 1928 In the “Report on the Abitur of Oberprimaner [8th Former] Gerhard Gentzen” we read the following: Of the subjects of instruction mathematics fascinated me the most, and I have occupied myself with it greatly. In 1922 I was enthusiastic about astronomy and shortly attempted to be able to predict the positions of the planets in the heavens. In 1924 I worked on the problem mathematically and solved it after many attempts. In the same year I applied myself to analytic geometry, over which I completed a term paper in my Unterprima [8th year] (1926/27). I would accordingly like to study mathematics.51

On 29 February 1928 Gerhard passed his Abitur “with distinction” and, at the request of the director of the school at which Gerhard had been the best student in his class, received a stipend from the Studienstiftung des Deutschen Volkes to allow 47 The name appears to be incorrect. The Studienstiftung des Deutschen Volkes was founded in 1925 as a registered charity under the name “Wirtschaftshilfe der Deutschen Studentenschaft” (roughly: Financial Aid for German Students). It was dissolved in 1934 by the Nazis and replaced by their own scholarship programme. The Studienstiftung was founded anew in 1948. Today, its various websites offer two English versions of its name: German National Merit Foundation and German National Academic Foundation.—Transl. 48 Undated letter from Dipl.-Ing. G. Gentzen. 49 The conjecture originates with G. Gentzen. 50 Letter from Hertha Michælis (D¨ usseldorf, 7 February 1988) to the author. 51 Gerd Robbel, 1986.

12. THE ABITUR ON 29 FEBRUARY 1928

17

him to carry out a course of studies at university. The assessment of the graduating Gentzen reads: He is the most gifted student the institution has had in a long time. His inclination and talent direct themselves in the first place towards mathematics and physics, especially astronomy, but throughout he always achieved “good” and “very good”. The teaching staff dealt in the Fall of 1926 with the question of whether or not to graduate Gentzen early. There was no doubt that he had achieved the goal of the Reifepr¨ ufung [“Certificate of Readiness”— diploma], but the teaching staff had the reservation that by his inclination to isolate himself and work intellectually by himself continually, he woud endanger his health if he should come to studies at so early an age. A discussion on the matter with his mother yielded independent of the staff’s reservations her pressing wish not to graduate her son early. Twice, in 1926 and 1927, the minister had distinguished Gentzen through the award of scholarships of 1000 marks. The works submitted thereby went far beyond what pupils usually achieve. Gentzen is by the way a quiet, serious youth, and in times past maintained a distance from his classmates; more recently it would appear that he has livened up a bit in this regard. He took part in historical and Greek-philosophical study groups. His pleasure in solving mathematical exercises allowed him to apply for a waiver of the mathematical dissertation. This last, the student work52 submitted to the PSK [abbreviation for “Prussian School Commission”— EMT] was marked as praiseworthy (G 5782/27 v. 25. Aug 27). It is to be assumed that G. in coming years will occupy an outstanding position in the area of science, if his health continues to improve.53

His diploma (“was 8 years at the gymnasium and in fact 2 in Prima”) records the following achievements: (1) (2) (3) (4) (5) (6) (7) (8) (9)

(10) (11) (12) (13) (14) (15) (16)

Religion: good German: good Latin: good Greek: very good; he took part in a philosophical study group. French: good English: good History (Civics): good; he took part in a study group. Geography: very good Mathematics: very good. He delivered a very good thesis. Remark: His outstanding natural ability for abstract thought leads him to an eager preoccupation with mathematical questions. His talent and diligence have repeatedly enabled him to carry out independent mathematical investigations. Physics: good Chemistry: — Biology: — Drawing and Art Instruction: very good Music: satisfactory Physical Education: good Handwriting: good

He graduates “with distinction”. The undersigned Board of Examiners has awarded him the diploma. Gentzen wishes to study mathematics. Stralsund, 29 February 1928 State Board of Examiners 52 This

53 Gerd

student work can no longer be found. Robbel 1986a, p. 57.

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1. EARLY YOUTH AND ABITUR

Among the papers at the Stralsund municipal archives there is also a protocol of an oral graduation exam in the subject of mathematics. The examiner was Gentzen’s teacher Dr. Herbst, with Prof. Dr. Kr¨ uger serving as an official witness from the State Ministry of Education: Dr. Herbst let it be known that he would put to the two following examinees questions which go beyond the scope of what would be expected of Gentzen in the school. . . Gentzen: I will derive the integrals for the conic sections a) for the parabola. Presentation of the area of the parabola. b) for the circle, for which the examinee derived the differential quotient of arcsin x/a c) for the hyperbola d) for the ellipse very good.54

Gentzen was not as solitary as he was portrayed. It is just that his friends, comrades, and acquaintances were not in the classroom. He was a lifelong friend with the neighbour’s daughter, Hertha Michælis. He played chess with Harald Frese and became friends again with Traugott Bartels, who had once broken with him. On 5 April (Easter) Gentzen was confirmed. He had been instructed by Superintendent Schmidt. His confirmation text was Romans 12,12: Be happy in hope, patient in sorrow, constant at prayer.55 Melanie Gentzen remembered the past and entered in the Chronicle: There followed years of great joy! Two children were granted us, Gerhard and Waltraut. In their own house with a beautiful, large garden they grew up happily. The First World War, which my husband took part in from beginning to end, returned him a very sick man in November 1918. He died at the end of March 1919. There now followed difficult years. In 1920, at the wish of my motherin-law I moved with the children to be with her in Stralsund. Gerhard was soon considered to be the best student at the local Gymnasium; Traute was successful at the Lyzeum [girls’ grammar school] and the Frauenschule [Literally the “Womens’ School”, the Frauenschule would have been an adjunct to the Lyzeum where girls learned housekeeping.]. Then came the disastrous inflation, which robbed us of our wealth and necessitated my taking a position as cutter in a large department store. After 5 years I gave up the position, as my mother-inlaw and children had pressing need of me. After the death of the former followed peaceful years. My highly gifted boy received a stipend.

Hertha Michælis recalled that she was a manager in the fashion house Zeck.56 But we want to look into the future with Melanie Gentzen: Upon completing school, my Traute trained in the Pestalozzi-Froebel House in Berlin to be a Kindergarten teacher and practised this profession until she became engaged with the survey-assessor Heinz Student, whom she followed to Liegnitz in 1937. I followed the children there in 1938. The war led Hans into the field (Italian captivity). After the collapse we fled, leaving all our possessions behind, taking 6-year old B¨ arbel Student and 3-year old Hans Lothar Student to “Eckh¨ ausle” in Sigmaringen, where we were received with the greatest affection by my sisters.57 54 Cited

from G. Robbel 1986a, p. 57. Footnote 2. 56 Letter from Hertha Michælis (D¨ usseldorf, 7 February 1988) to the author. Prof. Dr. Peter Schneider remarks that in all likelihood it was the department store Zeck. 57 Chronik Bilharz, entry Melanie Gentzen, p. 66. 55 Cf.

12. THE ABITUR ON 29 FEBRUARY 1928

19

Gerhard was allocated a stipend. Attending a higher school cost money in those days. Who couldn’t raise the monthly 60 marks had to distinguish himself with exceptional performance. For good performances there were various rewards. Good scholarly achievement thus often had little to do with ambition, but rather with material poverty and talent. I’d like to judge Gentzen’s youthful adventures and adolescent experiences: What might Gentzen have left behind if not for the loss of fatherly authority and love, the inevitable subordination of the house rule to the mother-in-law, and the absence of the mother through the obligation to earn money? And Gerhard would lose even more people who were important to him. But on the other hand, he had the solid support of his mother and sister, and his ideas, it seems, he could realise in school.

https://doi.org/10.1090/hmath/033/02

CHAPTER 2

1928-1938—Weimar Republic and National Socialism in Peace. From the Beginning of Studies to the Extension of the Unscheduled Assistantship for Another Year in Effect from 1 October 1938 1. Beginning of Studies in Greifswald Gerhard registered for mathematics and physics at the nearby University of Greifswald. Until the conclusion of his studies he received financial support through the Studienstiftung des Deutschen Volkes. He lived at Marienstraße 17 and wrote on 24 August 1928 to “Uncle Willi”, a school friend of his father who was active in the school in Stralsund but lived in Putbus, that he was content with his room with a view of the embankment. Before departing he had had to explain the operation of the radio to his mother and sister. He had also made contact with Helmut Michælis, who likewise studied in Greifswald. This was a brother of Hertha Michælis, his friend since youth. In Greifswald he also came to know Lothar Collatz (1910-1990),1 who came from Stettin and also studied mathematics. Lothar shared Gentzen’s love for sport, mental exercises, and astronomy. Why did Gentzen and Collatz leave the Mathematical Institute after only two semesters and go to G¨ ottingen? Lothar Collatz reported: We listened to Professor Hellmuth Kneser’s2 “Analytic Geometry” together and from the beginning of 1929 until the end of the winter semester Mr. and Mrs. Kneser invited Gentzen and me to evening meal in their apartment, which at that time was somewhat unusual; after all we were very young students in the second semester. Kneser showed us a variant of chess (blindfolded chess). Then he said to us: I would love to have you again as my students, but you must go to G¨ ottingen, where they have more to offer you than we have in Greifswald. He then gave a letter of recommendation for us to Professor Courant. So we both came to G¨ ottingen. In G¨ ottingen we were somewhat disturbed by the “mass production”. So we both went to Munich and listened to the superb lectures of Perron on Algebra II and of Carath´eodory on Function Theory. We knew that to complete a course of studies we would have to decide for a university. So we 1 *6 July 1910 in Arnsberg, †26 September 1990 in Varna, Bulgaria. Cf. the biographical remarks in “Lothar Collatz 1910-1990”, Hamburger Beitr¨ age zur Angewandten Mathematik, second revised edition. Series B, Report 23, December 1992; and Lothar Collatz, “Numerik”, pp. 269322, in: Gerd Fischer et al. (eds.), Ein Jahrhundert Mathematik 1890-1990, Vieweg, Wiesbaden, 1990. 2 Hellmuth Kneser (1898-1973), who was promoted by Hilbert in 1921, was assistant in G¨ ottingen from 1 October 1921 until 31 March 1925, where he habilitated on 1 December 1922. He worked in Greifswald from 1925 until 1937, thereafter in T¨ ubingen. (Cf. the entry on Kneser in Siegfried Gottwald, Hans-Joachim Ilgauds, and Karl-Heinz Schlote (eds.), Lexikon bedeutender Mathematiker, Bibliographisches Institut, Leipzig, 1990.)

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chose to go for a semester to Berlin where outstanding mathematical lectures were given. After our 6th semester, in spring of 1931, we had to choose a university for the completion. Gerhard decided for G¨ ottingen and I chose to take the exams in Berlin.3

This letter might correct somewhat the grandiose historical image many have of the much lauded attractiveness and uniqueness of mathematics in G¨ottingen. Good mathematics was also carried out in the “provinces”. Lothar Collatz valued Gentzen highly: In personal regards he was for me a true friend; he was reliable, honest, of strong character, conscientious, helpful and always had a fair and well-thoughtout judgement.4

2. Continuation of Study in G¨ ottingen So we find Gentzen and Collatz from 12 May 1929 in the Reinh¨ auser Landstraße; they had matriculated in G¨ ottingen on 22 April. He wrote again to “Uncle Willi” and “Aunt Iga” in Putbus and described the course of the day. The rooms of Collatz and Genzten were across from each other. In the morning breakfast was taken at Gerhard’s, lunch was eaten in the Mensa, and evenings they ate at Lothar’s. They visited the university daily for 5 to 6 hours. The lectures were hardly more difficult to understand than in Greifswald, but here much knowledge would be presupposed. He was pleased that a new mathematical institute would be built. Gentzen had also taken two philosophical lectures as well as a physics practicum on measurement and had made an arrangement to enable him to improve his stenography. With two fellow students he took long walks and planned a bigger tour to Kassel and the Weser River. On 2 December 1929, in a letter that was not copied out in one go, he thanked Uncle Willi and Aunt Iga for a necktie. On his birthday he went with fellow scholars to Schatzfeld on the southern edge of the Harz, where they broke up into several groups. They had agreed to meet again in G¨ ottingen and this in fact now took place. They read Max Weber’s Science as a Profession.5 He busied himself with fitting out his radio, playing chess, going to the movies and to theatre. The play “The Other Side” by Sheriff had not pleased him.6 On Monday the new Mathematical Institute would be opened. He would afterwards have participated in an internal celebration and attended two outstanding lectures.7 While Gentzen concentrated on Hilbert’s difficulty with mathematical foundations, Collatz would be inspired by Courant to a preoccupation with differential equations. From June to July 1929 both attended Hilbert’s lectures on Set Theory, and Collatz wrote up, as always, thorough lecture notes.8 From this course Gentzen 3 Letter

from Prof. Dr. L. Collatz to EMT of 31 March 1988. from Prof. Dr. L. Collatz to EMT of 28 January 1988. 5 Cf. the longer note at the end of this chapter. 6 Robert Cedric Sheriff’s “The Other Side” was a successful war drama of 1929 that took a critical look at war and was presented with an unemotional and sober objectivity. 7 The main speakers at the opening of the Mathematical Institute on 3 December 1929 were Hermann Weyl (cf. the note at the end of the chapter) and Theodor von K´ arm´ an. Their speeches were printed in Die Naturwissenschaften, 18th Year, 1930, volume 1, Weyl on pp. 4-11 and von K´ arm´ an following on pp. 12-16. Otto Neugebauer had conceived the new institute, and it was built in 1929 with money from the Rockefeller Foundation. Pages 1-4 of the volume contain an account of the G¨ ottingen Institute of the day by Neugebauer. 8 Lothar Collatz was an almost manic note-taker and documentalist and left behind many other papers, including two transcripts of Hilbert’s lectures: 1) Theory of Algebraic Invariants 4 Letter

4. CONTINUING STUDIES IN MUNICH

23

learned everything necessary about algebraic numbers, sets, logical entanglements, the paradoxes of Russell and Zermelo, the paradox of the class of all ordinals, Dedekind’s construction of number theory, permissible and impermissible inferences, the logic of tertium non datur, and the basic concepts of mathematical logic. On 17 July 1929 they both learned about “the problem of consistency”, the proof of consistency of numerical calculation, and consistency by admission of reduction. The accompanying proof methods with the resolution in individual proof threads and the epsilon axiom9 followed on 22, 25, and 29 July. Both would later assess these lectures as having been good. On 17 March 1903, the two left G¨ ottingen to study further in Munich. 3. Is G¨ ottingen the Centre of Mathematics for Gentzen? The lecture schedule for winter semester 1929 shows the Mathematical Institute together with the Mathematical-Physical Seminar still in Weender Landstraße 2 (currently the auditorium building) and Prinzenstraße 21. Both institutes were only first housed in Bunsenstraße 3 to 5 in the 1930 summer semester. Richard Courant was managing director with Hilbert, Landau, and Herglotz as professors. Lectures were also held by Emmy Noether and Felix Bernstein. In the 1930/31 winter semester, P. Bernays, K. Friedrichs, and H. Weyl joined the staff: Weyl as Hilbert’s successor, Hilbert becoming emeritus in 1930. Weyl was, above all, a mediator between mathematics and physics. The Institute for Mathematical Statistics led its own life nearby the Mathematical Institute. Founded in 1918, it resided under the direction of Felix Bernstein in the Paulinerstraße 19, and after the 1930/31 winter semester at Kurze Geismar 40. Important for Gentzen, however, would be Paul Hertz,10 who taught “Methods of the Exact Sciences” as assistant professor from 1921 to 1933. The students and visitors in G¨ ottingen are legion, among them for example Haskell Curry and Saunders Mac Lane. But Gentzen was an independent thinker, and independent thinkers don’t think much of descriptions they find but haven’t thought through themselves nor of impressions they might receive on the basis of what they heard or read of the thoughts of others. Gentzen remained deeply protestant. His motto could have been: Everything I believe and know comes from within and if it must be, I will write it myself. 4. Continuing Studies in Munich With the mathematicians Oskar Perron (1880-1975), known for his standard works on continued fractions, algebra and irrational numbers; Constantin Carath´eodory (1873-1950), who stood out in real and complex analysis and the calculus of variations; the topologist Heinrich Tietze (1880-1964); and the function theorist and (Winter semester 1929/30) (approximately 100 pages), and 2) Set Theory (Summer semester 1929) (approximately 88 pages). These transcriptions should be published. The Nachlass of Collatz is sorted till now only roughly by document type in 170(!) archive boxes. A first look-through revealed no documents pertaining to Gentzen. (Letter of 24 January 1997 from Frau Dr. Petra Bl¨ odorn-Meyer of the manuscript division of the State and Carl von Ossietzky University Library in Hamburg.) 9 The epsilon axiom would be a form of the axiom of choice. In place of quantifiers, Hilbert formulated his logical system by means of a sort of choice function denoted by the letter epsilon. This is discussed further in Appendix B. 10 A biographical sketch of Hertz can be found in the longer notes at the end of this chapter.

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founding father of multivariable complex analysis Friedrich Hartogs (1874-1943), the Mathematical Institute in Munich was still in its prime in 1930. Additional faculty included Liebman, Voss, Pringsheim (the father-in-law of the writer Thomas Mann), and Salomon Bochner (1899-1982), before the Nazis imposed severe restrictions on teaching and research activity. Gentzen lived during the summer semester of 1930 in upper Munich in Hohenzollernstraße 14, on the second floor. He registered as a student in the philosophical faculty and gave as his address Herzogstraße 82/2.11 He wrote to Hellmuth Kneser on 2 July 1930: Very respected Herr Professor! My fifth semester of studies has led me to Munich. It is particularly characterised by museum tours and walks, to get to know the beauties of the city and its environs. So I was, together mostly with my friend Lothar Collatz and his sister, on the Starnberger Lake, the Ammer Lake, twice in the Alps, in the Isar valley, etc. The Whitsun holidays I could spend with my uncle in Meran, where I saw much that was new. The two trips to the Alps were wholly successful. Beside these sorts of undertakings, however, science did not fare badly. There I could finally attend lectures on function theory, by Professor Carath´eodory. Somewhat better than this, with which I am however entirely pleased, I liked the Algebra lectures (part II) of Professor Perron. Further, I have in addition taken chemistry, astronomy, and an introduction to topology. In this latter the foundation of the field was dealt with from the theory of point sets in n-dimensional space. (Prof. Tietze) For a while I occupied myself privately with things that especially interested me. Still in summer break I thought about a presentation of the Peano curve and its non-differentiable limit function using triadic fractions with which I could prove the essential properties of these functions rather easily and entirely differently from the treatment in Bieberbach’s Differentialrechnung [Differential equations]. I don’t know if this presentation is already known because I could find no literature on Peano curves here. Then I have studied the book Theoretische Logik 12 [Theoretical Logic] by Hilbert and Ackermann, as I would like to come to a greater clarity on the foundations of mathematics. Now I will attempt to acquire the recent treatments of Hilbert on these matters. With devoted greetings and respects to your honoured wife I remain your Gerhard Gentzen13

In Hilbert’s writings he would encounter a political component of the foundational discussion: The great classics and creators of foundational research were Cantor, Frege and Dedekind; they found a sympathetic interpreter in Zermelo. . . An unfortunate conception of Poincar´e concerning the inference from n to n + 1, which had already been refuted two decades earlier through a precise proof by Dedekind, barred the path to progress. A new ban, one on impredicative statements would 11 Information

from a letter of 23 February 1988 from Frau Prof. Dr. Lætitia Boehm of the archives of the Ludwig Maximilian University (under Tgb A 29/B - 88): “Because the sheets for this time are missing, we can give you no further information on his studies.” 12 Cf. footnote 193, below. 13 The correspondence of Gentzen and Kneser is in the possession of Prof. Dr. Martin Kneser in G¨ ottingen—he is the son of Hellmuth Kneser—and I thank him for the possibility of citing it here.

4. CONTINUING STUDIES IN MUNICH

25

be enacted and upheld by Poincar´e, although Zermelo immediately gave a striking example against their ban and, besides, the ban offended against Dedekind’s result. Alas, too, the usually so excellent logic of Russell encouraged the heresy in its application to mathematics. So it happened that our beloved science— as concerned its innermost arithmetical nature and foundations—would be as though afflicted by an evil dream through two decades.14

Who had thus harmed mathematical science? A Frenchman and an Englishman, while the German international remained unheard. Why didn’t the Dutchman Brouwer come forward? Because Brouwer had indeed been invited to G¨ ottingen in 1919, and he had even been offered a professorship by the faculty in 1919—he stood in first place on the appointment list on 30 October 1919—but Hilbert could hardly have read his work, particularly the founding of intuitionism and, if so, then only in summaries. Brouwer later wrote: The contradictions which one repeatedly comes upon have called the formalistic critique to life, a critique which comes down to this, that the language accompanying mathematical thought-action is to be subjected to a mathematical examination. In such an examination the laws of theoretical logic present themselves as operators acting on basic formulas or axioms, and one sets oneself the goal to rearrange these axioms so that the linguistic working of the mentioned operators (which themselves are retained unchanged) can no longer be disturbed by the appearance of the linguistic form of a contradiction. On the achievement of this goal one need in no way despair, but nothing of mathematical value will have been gained: an incorrect theory unhampered by contradiction is thereby no less incorrect, just as a criminal policy unhampered by a reprimanding court is no less criminal.15

The familiarity and knowledge of writings in Europe were quite variable, that too because after losing the First World War the Germans were internationally isolated. For example, Laurent Schwarz wrote on p. 67 of his autobiography16 that in France until 1928 the books and ideas of Hilbert, van der Wærden, and others were hardly well-known and only a handful were known at all, in part because of chauvinism on the French side—the French rejected German participation in the International Congress of Mathematicians as late as 1924—a chauvinism which was foreign to Hilbert, who, even in the middle of a war, had written a lovely obituary of the French mathematician Gaston Darboux for the Academy of Sciences. Furthermore, until 1918 Hilbert knew little of Brouwer’s ideas on foundational research. Hilbert’s confrontation with Brouwerian intuitionism stemmed from Weyl’s writing Das Kontinuum (The continuum)17 and especially from the “Neuen Grundlagenkrise der Mathematik” [New crises in the foundations of mathematics].18 19 14 David Hilbert, “Probleme der Grundlagen der Mathematik”, Math. Ann. 17, Vol. 102 (1920), p. 2. 15 Luitzen E.J. Brouwer, “Uber ¨ die Bedeutung des Satzes vom ausgeschlossenem Dritten in der Mathematik, insbesondere in der Funktionentheorie”, Journal f. reine u. angewandte Math. 154 (1925), p. 1. 16 A Mathematician Grappling with His Century, Birkh¨ auser, Basel, 2001. 17 Das Kontinuum. Kritische Untersuchungen u ¨ber die Grundlagen der Analysis, Veit, Leipzig, 1918. 18 Math. Zeitschr. 10 (1921), pp. 39ff. 19 Cf. the biography by Dirk van Dalen, Mystic, Geometer, and Intuitionist. The Life of L.E.J. Brouwer, Vol. 1: The Dawning Revolution, Clarendon Press, Oxford, 1999. This is an excellent example of an intelligible and at the same time thoughtful biography.

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But Hilbert, like Gentzen, saw clearly the challenge beyond the polemics that grew with intuitionism and adapted Brouwer’s thought. Hilbert believed in a preestablished harmony of mathematics with the world, true to the Leibnizian model. (Cf. the end of Chapter 3.) Gentzen evidently found this persuasive. They would preserve the unity of mathematics by characterising the discipline with a unique method valid only in it. Hilbert fought the disintegration of mathematics into distinct independent disciplines and wrongly saw Brouwer as a representative of Chaos. Hilbert wanted to “rescue” the whole of mathematics and would thus allow no Verbotsdiktaturen.20 He desired the greatest possible freedom—` a la Kant with his Critique and System—and erected a metamathematics as a control authority to guarantee this unity. Because for him [i.e. Hilbert—EMT] mathematics had the significance of a worldview, and he approached every fundamental problem with the cast of mind of a conqueror, who is attempting to secure for mathematical thought as extensive a territory as possible. (Paul Bernays, “Die Bedeutung Hilberts f¨ ur die Philosophie der Mathematik” [Hilbert’s significance for the philosophy of mathematics], Die Naturwissenschaften 10 (1922), pp. 93-99, here p. 94.)

5. A Semester in Berlin: Winter Semester 1930/31 Gentzen matriculated in Berlin with the Philosophical Faculty of the Friedrich Wilhelms University on 29 October 1930 and studied mathematics. He ex-matriculated on 11 March 1931.21 I have no details on how Gentzen spent the winter semester of 1930/31 in Berlin, where his sister worked as a Kindergarten teacher. Waltraut Student gives as a reason for his Berlin interlude only that he wanted to be near her and offer encouragement for her exams to be a Kindergarten teacher, exams she passed successfully. His friend Collatz attended lectures by Richard von Mises (1883-1953), who emigrated to Istanbul in 1934. From von Mises he got the idea for his doctoral work, and Collatz counts as the “last student” Richard von Mises produced in Germany. That Gentzen attended lectures by Erhard Schmidt (1876-1959)—and also possibly lectures on physics by Erwin Schr¨ odinger—I can only conjecture, based on the fact that his friend Lothar Collatz attended these.22

20 Roughly: ban-dictatorships. Brouwer and Weyl would disallow certain familiar modes of inference, and Hilbert saw this as too restrictive. 21 Letter from Dr. W. Schultze of the Archives of the Humboldt University in Berlin of 15 January 1997. 22 Lothar Collatz, “Numerik”, pp. 269-322, here pp. 282 and 284, in: Gerd Fischer, et al. (eds.), Ein Jahrhundert Mathematik 1890-1990, Vieweg Verlag, Wiesbaden, 1990. A note about Erhard Schmidt: After the mathematician Issai Schur had warned him on account of the Kristallnacht not to leave the house, Prof. Dr. Erhard Schmidt came a few days later to visit. As Schur bitterly complained about the Nazis and Hitler, Schmidt defended the latter. “He said, ‘Suppose we had to fight a war to rearm Germany, unite with Austria, liberate the Saar and the German part of Czechoslovakia. Such a war would cost us half a million young men. But everybody would have admired our victorious leader. Now, Hitler has sacrificed half a million Jews and has achieved great things for Germany. I hope some day you will be recompensed, but I am still grateful to Hitler’.” (Menachem Max Schiffer, “Issai Schur: Some Personal Reminiscences”, pp. 177-181, here, p. 180, in: Heinrich Begehr (ed.), Mathematik in Berlin. Geschichte und Dokumentation. Zweiter Halbband, Shaker Verlag, Aachen, 1998.)

¨ 6. BACK IN GOTTINGEN

27

6. Back in G¨ ottingen: Saunders Mac Lane and Gentzen as the “Type of a Scientifically Oriented Man” (Richard Courant) On 24 April 1931 Gentzen again matriculated in G¨ ottingen, until 24 November 1933, and lived in the first floor of Schillerstraße 7. In 1931 the American student Saunders Mac Lane (1909-2005) met Gentzen: Gentzen and I were students of mathematical logic in G¨ ottingen at the same time (1931-1933). I knew Gentzen reasonably well, and had several conversations with him about logic (though I did not in any way have any influence on his important urzte Beweise in Logikkalk¨ ul” [Shortwork). After my own Ph.D. thesis (“Abgek¨ ened proofs in logical calculus], 61 pages, Diss. 1934, now published in a volume of my selected papers) had been written in English (June 1933) the authorities told me it must be written in German. I then asked Gentzen to translate it for me; he began his task, but so slowly that I finished the translation myself. I last saw him in July 1933. Gentzen subsequently sent me (to the USA) reprints of his important papers 1934-1937; I still have these copies, with his signature. . . Much later, I had occasion to use some Gentzen ideas about cut “eliminations” in some of my research.23

After his promotion in 1930 with research in proof theory, Jacques Herbrand (1908-1931)24 stayed on in Germany for another year and visited Berlin, Hamburg, and G¨ ottingen. In April 1931 Paul Bernays, who had already read Herbrand’s dissertation, had spoken with Herbrand, who had given him a copy of a letter G¨ odel had sent Herbrand.25 Perhaps Gentzen, who was only a year older than Herbrand, met him at Emmy Noether’s, perhaps together with Jean Cavaill`es. Jean Cavaill`es26 stayed in G¨ ottingen in 1931, when he certainly met Herbrand and Gentzen, who had returned from Munich in April and now studied there. Cavaill`es visited Emmy Noether in the course of their editing the correspondence between Cantor and Dedekind. On the 14th of July Herbrand was in G¨ ottingen. From there he sent his article “Recherches sur la th´eorie de la demonstration”27 to the Journal f¨ ur die reine und angewandte Mathematik, in which he generalised and in 23 Letter of 18 February 1988 to EMT. The use of Gentzen’s ideas can be found on page 392 of: Saunders Mac Lane, “Why commutative diagrams coincide with equivalent proofs, presented in honour of Nathan Jacobson”, Contemporary Mathematics 13 (1982), pp. 387-401. On the contents of Mac Lane’s dissertation as part of a projected structure theory for mathematics based on the principle of leading ideas to bring intuitive proof closer to formal logic and to shorten formal proofs by abbreviating routine sequences of steps, cf. section 5 (Logic) in Colin McLarty, “The Last Mathematician from Hilbert’s G¨ ottingen: Saunders Mac Lane as Philosopher of Mathematics”, The British Journal of Philosophy of Science 58 (2007), no. 1, pp. 77-112. 24 “In the finitistic tradition of D. Hilbert, Herbrand developed among other things a method of reducing the provability of an expression of the predicate calculus to that of a disjunction of propositional expressions (Herbrand’s Theorem). With this method he proved the consistency of a special case of arithmetic as well as decidability theorems. Herbrand laid the groundwork for defining the concept of a recursive function” (Gerd Robbel, entry “Herbrand” in Siegfried Gottwald et al. (eds.), Lexikon bedeutender Mathematiker, Bibliographisches Institut, Leipzig, 1990). 25 Cf. p. 102 of Kurt G¨ odel, Collected Works, Vol. 4, Oxford University Press, Oxford, 2003. 26 Vladimir Tasic reports on Cavaill` es in his celebrated book, Mathematics and the Roots of Postmodern Thought, Oxford University Press, Oxford, 2001. 27 In English translation in Jacques Herbrand, Logical Writings, edited by Warren D. Goldfarb for D. Reidel Publishing Company, Dordrecht, 1971, and Harvard University Press, Cambridge ´ (Mass.), 1971. This volume is a translation from the French Ecrits logiques, edited by Jean van Heijenoort.

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fact superceded the earlier restricted consistency proofs of Ackermann28 (1924) and von Neumann (1927). On the 27th of July Herbrand met his death in the Alps. Helmut Hasse (1898-1979) wrote a short obituary and published Herbrand’s work, which had already been received by the journal on the 14th. Professor R. Courant wrote a letter of condolence to Herbrand’s father. To me it would be astonishing if Gentzen were not personally acquainted with Herbrand in G¨ ottingen. Gentzen learned a lot from Herbrand and, together with P. Bernays, simplified and gave more elegant formulations of some of his results. Gentzen’s Hauptsatz29 is more general than Herbrand’s fundamental theorem. For one thing, Gentzen’s Hauptsatz does not depend on model theoretic assumptions as Herbrand’s result does. Now, on the advice of Paul Bernays,30 Gentzen occupied himself with the ideas of Paul Hertz,31 who was close to the Vienna Circle. He had studied ideas on foundational research, the 1924 Hilbert Programme, Klein, and “G¨ ottingen finitism”. What was the Hilbert Programme and what did Gentzen see as its goals? Matthias Schirn reports, As proclaimed by him in “Neubegr¨ undung der Mathematik” [New foundation of mathematics], Hilbert wanted to reach a metamathematical consistency proof for formalised arithmetic through exclusively finitistic proof means. In essence his proof theoretic programme sets itself the goal of justifying by finite means those non-finitist rules of inference classified as dubious or contestible, such as the application of tertium non datur to infinite totalities, which are applied formally in the arithmetical system under consideration. He held the finite means to be completely safe and reliable because they satisfied the methodological considerations of concrete demonstrability, comprehensibility and controllability. As you know, the goal of a “mastery of the transfinite on the basis of the finite” appears to be unattainable on account of G¨ odel’s Second Incompleteness Theorem. The fashioning of consistency proofs for arithmetic was, in the light of G¨ odel’s incompleteness results, generally viewed as securing, through their constructivity, the reliability of modes of proof used in a metamathematical proof, but not available in the number theoretic formalism. So in 1936 Gentzen arrived at a consistency proof for elementary number theory through the use of a, certainly no longer finite, but constructively permissible proof method, namely the rule of transfinite induction up to the first epsilon number ε0 of Cantor’s so-called second number class.

For many mathematicians, the phrase “Hilbert’s Programme” is not very clear. The most knowledgeable judgement: there is no “Hilbert’s Programme”, but rather 28 Wilhelm Ackermann (29 March 1896-24 December 1962) studied in G¨ ottingen and was promoted in 1924 by Hilbert. A research stipend made possible a short stay in Cambridge. From 1927 to 1961 Ackermann taught in a Gymnasium. He was made an Honorary Professor in M¨ unster and a Corresponding Member of the G¨ ottingen Academy of Science in 1953. 29 Literally: main theorem. It goes by the German name in English. 30 The rˆ ole played by Paul Bernays in Gentzen’s ideas must in my opinion be clarified, but I cannot settle it here. 31 Some biographical details on Paul Hertz are given in footnote 10. For more on him, Bernays, and Weyl, cf. Rudolf Haller, Neopositivismus. Eine historische Einf¨ uhrung in die Philosophie des Wiener Kreises, Wissenschaftlich Buchgesellschaft, Darmstadt, 1993; cf. also: Christian Thiel, “Folgen der Emigration deutscher und ¨ osterreichischer Wissenschaftstheoretiker und Logiker zwischen 1933 und 1945”, Berichte zur Wissenschaftsgeschichte 7 (1984), pp. 227256.

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there are “Hilbert programmes”. There is accordingly a certain openness in Hilbert’s Programme. In his essay on Hilbert’s pragmatism, Volker Peckhaus finds it to be “in parts compatible”—whatever that may mean—with intuitionism, Becker’s phenomenological legitimation, or Nelson’s transcendental-philosophical foundations. All authors aimed at a certain ontological neutrality from Hilbert’s approach. What Hilbert’s programme actually consisted of and how it was destroyed by G¨ odel’s First Incompleteness Theorem is shown in exemplary fashion in Craig Smory´ nski’s work included as an appendix to this book. In the meantime, news of G¨odel’s discovery of the principal difficulty of a consistency proof had gotten around. On 24 December 1930 Paul Bernays had requested a copy of the galley proofs from Kurt G¨ odel, because Courant and Issai Schur had announced its “significant and surprising results” to him. On 18 January 1931, Bernays thanked G¨ odel, confirmed their reception, and characterised G¨ odel’s result as “a considerable step forwards in research on the problem of foundations”. Bernays continued correspondence with Kurt G¨ odel, because Bernays had some difficulties in understanding and didn’t see why the consistency proof of Wilhelm Ackermann couldn’t suffice. G¨ odel clarified this for him and Bernays recognised his errors.32 I am certain that Bernays would also have told his “student” Gentzen about this. On 31 July 1931 Richard Courant wrote to Professor Dr. Hermann Nohl, the representative of the Studienstiftung des deutschen Volkes: Concerning the member Gerhard Gentzen of the Studienstiftung: As arranged, I report to you today on Mr. Gentzen on the basis of his seminar lecture and a personal consultation with him. In his seminar lecture Mr. Gentzen treated an especially difficult theme and demonstrated thereby a superior independence in his outward and intellectual penetration of the material, which shows him definitely to be of the type of a scientifically oriented man. I am confident after this as well as after an oral discussion that Mr. Gentzen can relatively easily be promoted and then do further scientific work. As his inner inclinations obviously push him strongly in this direction, I can with full responsibility give the Studienstiftung the advice, to grant him the promotion.33

Unfortunately, we do not know the subject of Gentzen’s seminar lecture. Courant was definitely in a position to judge Gentzen with regard to foundational research. We even know a written rejoinder by Leonard Nelson to Courant’s criticism and Bernays’ objections from Nelson’s essay, “Kritische Philosophie und mathematische Axiomatik” [Critical philosophy and mathematical axiomatics], which was published in 1927 with a foreword by Hilbert. A side comment: Hilbert’s foreword asserts: All serious endeavours to elevate philosophy into the ranks of the exact sciences have always laid a decisive value on an orientation toward mathematics. Of all the areas of mathematics, foundational research especially claimed the full interest of 32 John

W. Dawson, Jr., “The reception of G¨ odel’s Incompleteness Theorems”, in: S.G. Shanker (ed.), G¨ odel’s Theorem in focus, Croom Helm, London, 1988, pp. 74-95, here: pp. 78ff. 33 Copy from the Mathematical Institute in G¨ ottingen. I heartily thank Prof. Dr. Norbert Schappacher, who looked through the existing records of the Mathematical Institute in February 1988 for Gentzeniana with great success. On the concept of a “scientific man”, compare Britta Scheideler, “Eine besonders edle Gemeinschaft”, offprint 126, Max Planck Institute for History of Science (Berlin).

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the philosophers. Leonard Nelson has followed the path of its progress with highest attention his entire life. He was prepared, by the astuteness of his thinking and the conscientiousness of his search for truth, to establish communication between philosophy and mathematics, because he was convinced of the far-reaching scientific connection of his philosophy with the results of axiomatics. His lecture “Kritische Philosophie und mathematische Axiomatik” bears emphatic witness to this. Ruthless fate has snatched the distinguished researcher from his most successful work. Science has suffered a heavy loss through his early death. May his friends and students further follow the high ideals he set for himself and again develop the great ideas laid down in his work. Hilbert34

Nelson defended the axiomatic method and closed the printed version of his lecture with “Erwiderung auf die von Herren Courant und Bernays erhobenen Einw¨ ande” [Reply to objections raised by Courant and Bernays]. After a critique on Courant and Bernays he criticises the Kantian and Fregean spatial intuition as the source of knowledge and much else. But it obviously does not appear to be a criticism that had disturbed Hilbert or he would have written a different foreword. Gentzen received the promotion stipend in question from the Studienstiftung. If following the promotion he had to be at a school—the times were no better—in any case this raised his social prestige and the stipend. Now and then Gerhard saw his friend Hertha Michælis. Above all Gerhard read scientific treatises, no fiction or poetry. He could sometimes be very quiet; it might happen that the entire day he could say only what was most necessary while he worked out a mathematical formula. What we spoke about—everyday matters as do all young people, and an awful lot on astronomy. . . I wouldn’t describe him as being in love with life; he could be very cheerful, but also pensive. He was shy and modest. . . 35

An additional occurrence from this time: together with Alexandrov, A.N. Kolmogorov visited G¨ ottingen in 1931. He was six years older than Gentzen. Perhaps the three saw each other. Kolmogorov (1903-1987) knew Gentzen’s work. Hans Hermes remembers this period: During a one year residence in G¨ ottingen (in which I had not occupied myself with logic) I occasionally met Gentzen in his rooms after lunch and played a dice game of the type “one robber and several policemen” that presumably he had designed himself and most often won.36

¨ 7. The Decision: Gentzen’s First Publication, “Uber die Existenz unabh¨ angiger Axiomensysteme zu unendlichen Satzsystemen” and His Programme for 1932 From his rooms on the first floor of Schillerstraße 7 in G¨ ottingen, Gentzen wrote to Professor Kneser on 13 December 1932: Very respected Herr Professor! I thank you many times for your friendly card. The work sent you is still not my dissertation. It contains results I obtained (for the most part) in summer 34 Leonard Nelson, “Kritische Philosophie und mathematische Axiomatik. Vortrag, gehalten an der 56. Versammlung Deutscher Philologen und Schulm¨ anner zu G¨ ottingen am 28. September 1927. Mit einen Vorwort von David Hilbert”, special printing from the Unterrichtsbl¨ atter f¨ ur Mathematik und Naturwissenschaften, Otto Salle Verlag, Berlin, 1928. 35 Letter from Hertha Michælis (D¨ usseldorf) to EMT dated 7 February 1988. 36 Letter to the author, 9 August 1988.

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’31, as I occupied myself with those questions at the suggestion of Herr Professor Bernays. In the earlier winter semester I have turned to more general problems of proof theory. My intellectual development has led me almost inevitably to ever more abstract works, and so my entire interest is now concentrated on questions of the character of mathematical inference. Mind you I stop before the frontier of philosophy, which has no appeal for me, i.e. I occupy myself only with problems which themselves are again accessible to a mathematical treatment.37 With these thoughts, I have taken on as a special task the proof of the consistency of the logical inferences in arithmetic. You well know that the antinomies of set theory have given cause for a far reaching skepticism of the reliability of certain logical inferences, and that Hilbert is striving for a justification of these inferences through the proof of their consistency. Through the formalisation of logical deduction, this task becomes a purely mathematical problem. The consistency proof has till now only been carried out in special cases, e.g. for the arithmetic of the integers without the rule of complete induction. I worked almost a year on this and hope soon to arrive at my goal. With this work I will then be promoted (by Professor Bernays). What I will start afterwards is not yet definite. Mr. Collatz has been studying 2 years in Berlin as there he can live near his parents in Potsdam. He is preparing himself for the state exams. We meet occasionally and correspond with one another. He has specialised in applied mathematics. With most devoted greetings, also to your most honoured wife I remain your Gerhard Gentzen

What work had Gentzen sent to Professor Kneser? It concerns itself with the existence of independent axiomatisations for infinite sets of sentences and was received by the editors of Mathematische Annalen on 6 February 1932. This debut was favourably received. Arnold Schmidt gave a very technical review.38 And Thoralf Skolem39 reviewed this, the first of Gentzen’s 11 important publications: ¨ G. Gentzen, Uber die Existenz unabh¨ angiger Axiomensysteme zu unendlichen Satzsystemen [On the existence of independent axiom systems for infinite systems of propositions]. Math. Ann. 107, 329-350. In this paper the question raised by P. Hertz (1929, F.D.M. 55, 627-628), of whether there are closed systems of propositions with no independent systems of axioms is treated. The author proves two main results: First, there are infinite closed sets of propositions for which no independent axiom system exists, and second that every countable infinite closed linear system of propositions has an independent axiomatisation. 37 Cf.

the longer note at the end of this chapter. f¨ ur Mathematik 5 (1933), pp. 338ff. 39 Jahrbuch u ¨ber die Fortschritte der Mathematik 58 (1932). Jens Erik Fenstad, the editor of Skolem’s collected logical papers, says in an e-mail to the author of 18 April 1997, “I doubt that Skolem (1887-1963) had any personal contacts with Gentzen, and I have found only one reference to Gentzen in his published works. See his lecture to the 1950 International Congress of Mathematics at Harvard University, pages 695-704, in the published proceedings. I quote: ‘The great ingenuity in the setting up of these consistency proofs, especially the proofs by Gentzen and Ackermann of the consistency of formalised number theory, must be admired. But it has happened before that the products of great ingenuity have lost their interest, because simpler ways of thinking have been found.’ Skolem’s basic position is rather well indicated (but was never fully spelt out) in a short note—‘A critical remark on foundational research’—which can be found on pp. 581-586 of his Selected Works in Logic, Oslo, 1970.” 38 Zentralblatt

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There follows an additional 34 lines of technical remarks by Skolem. Recent research sees more exactly the relation between the ideas of Hertz and Gentzen. Peter Schroeder-Heister writes:40 In the 1920s, Paul Hertz (1881-1940) developed certain calculi based on structural rules only and established normal form results for proofs. It is shown that he anticipated important techniques and results of general proof theory as well as of resolution theory, if the latter is conceived as a part of structural proof theory. Furthermore it is shown that Gentzen, in his first paper of 1933 [actually: 1932—EMT], which heavily draws on Hertz, proves a normal form result which corresponds to the completeness of propositional SLD-resolution in logic programming.

An anonymous commentator opines: At a first superficial sight it looks like a trivial exercise in a very general theory of consequence. The paper under review shows this impression to be completely wrong. Gentzen’s paper isolates a notion equivalent to SLD-resolution for definite Horn clauses, one of the cornerstones of logic programming. Moreover, the completeness proofs given by Gentzen use constructions similar to saturation of a given set of clauses up to a fixpoint, as well as permutation of rules in a derivation which led later to [his] cut-elimination proof. Both of these ideas are traced back to works of Paul Hertz, whose rˆ ole was completely unclear before the paper under review.

Javier Legris writes:41 Hertz’s original interest was focussed on the formal properties of axiom systems and his goal consisted in developing reduction methods for axiomatic systems from which some sort of minimal and independent system could be obtained. His research led him to the idea of Satzsystem (system of propositions), undoubtedly his main contribution to mathematical logic. With this kind of system Hertz wanted to solve the problem of the deductive closure of formal theories, which was formulated in Hilbert’s program. . . Hertz proposed different interpretations to them, depending on the nature of the basic domain (events or predicates). He established a minimal set of rules for closed systems of propositions, which are an antecedent of the structural rules in Gentzen’s sequent calculus. It must be stressed that Hertz’s approach should be characterized as finitistic. He considered the case of an inference system (Schlusssystem) consisting in proofs for a system of propositions through these rules. In the paper Hertz’s notions of closed system and normal proof are analyzed. Then, the way from Hertz’s propositions to Gentzen’s sequents is also traced and, finally, the idea of reducing logical constants to certain deduction rules is also taken into account and compared with the structural approach due to Gentzen.

Richard Zach writes42 The development of clear and intuitive axioms for propositional logic, and the investigations of the extent to which axioms can be replaced by rules undoubtedly also had great influence on Gentzen’s development of natural deduction and the sequent calculus. Bernays was still teaching in G¨ ottingen at the time when Gentzen was preparing his thesis, and in all likelihood was in close contact with him. Bernays was working closely with Paul Hertz throughout the 1920s, and Hertz’s work on axiom systems is commonly acknowledged to be one of Gentzen’s 40 In

the abstract to his essay, “Resolution and the origin of structural reasoning” (2002). Hertz’s system of propositions and the origins of proof theory”, Internet. 42 “Completeness before Post: Bernays, Hilbert, and the development of propositional logic”, Bulletin of Symbolic Logic 5, no. 3 (1999), pp. 351-364. 41 “Paul

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main sources. The picture is far from complete, however, and it seems well worth filling in the details. In the course of this, in particular in a re-examination of Hertz’s work on logic, it may well be that further important contributions by Bernays may come to light.

Of course Gentzen used ideas of Hilbert, Herbrand, Bernays, and Hertz, but it was he who first developed them further into a systematic, fruitful, and useful whole, and today we can easily return to him as a classic worthy of consideration. In the papers left behind by Paul Hertz, which are stored in Pittsburgh, there are offprints of Gentzen’s first two papers, but there is no hint of any direct and personal contact on his side. However, G¨ ottingen was so small in those days that I really believe that Gentzen had possibly discussed his work with Hertz. Gerhard Gentzen was 22 years old. He viewed mathematics, as the Vienna Circle similarly did, as a special theory of systems of propositions.43 But much more important is the question whether he will be promoted by Paul Bernays. 8. What Do Gentzen’s Intellectual Interests and Attitude in 1931 and 1932 Appear to Be? Let us read the above letter to Professor Kneser more closely. In my opinion, in accordance with his reading of Max Weber’s Science as a Profession, the following attitude says a great deal about his intellectual position at the time and even for his future: Mind you I stop before the frontier of philosophy, which has no appeal for me, i.e. I occupy myself only with problems which themselves are again accessible to a mathematical treatment.

Gentzen laid open his intellectual interests: The consistency proof has till now been carried out only in special cases, e.g., for the arithmetic of the integers without the rule of complete induction.

This was a reference to the work of Ackermann. And he stated his goal: Here I wish to go farther and take care of at least arithmetic with complete induction. I worked almost a year on this and hope soon to arrive at my goal. With this work I will then be promoted (by Professor Bernays). What I will start afterwards is not yet definite.

However, the times were uncertain. The usual choice between work as a teacher and a university career would have to be made, but no possibility for either is indicated. Everything remained up in the air. 43 In his Nachlass he returned several times to his Hertz-work and the concept of a mathematical theory, for example on VIII, 39: “A ‘mathematical theory’ (e.g., number theory, Euclidean geometry, topology) one sees provisionally so: a system of propositions hanging together in this way: certain propositions are accepted beforehand and called ‘axioms’, and others, one after another, are derived from these by certain inferences. Another essential thing is this: definitions, through which new concepts of various sorts are introduced. Concept introductions other than definitions are also conceivable, whence we speak more generally of concept formations. Inferences combine in proofs. . . ”, or “then the general form(?) of a mathematical theory as constructed from an axiom system can be countably infinite via a schema. And this is clear. In the end the entire theory is just a system of propositions of the sort in which the schema for obtaining new theorems is given through formalised inference rules. . . .” I thank Prof. Dr. Christian Thiel for the source and transcription.

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Note the interval. He wrote this letter on 13 December 1932, saying he had been occupied with the consistency proof for almost a year; i.e. he had begun at least as early as the winter semester of 1931. It is thus very likely that on the advice of Paul Bernays he had been busy with the works of Paul Hertz during this time, i.e. Gentzen had worked out his ideas for proof theory inside a year—from his 21st to his 22nd year. But where did his disinclination toward philosophy come from? 9. Political Language in Mathematics in 1918 The past had been and the present was teeming with ideology, foundations, and other cultural illusions. Two examples, which Hilbert had previously skewered: Earlier one let one’s thoughts wander undisturbed until one had an original idea; now as soon as possible the reigning jargon is acquired and as quickly as possible a record is set, to be ahead of one’s colleagues by a nose. The Principle of the Noselength definitively illustrated: Title: On the X-th Proof of a Theorem of Professor Y. One must also attend to a jargon-free written language and write so that the reader can also derive some pleasure.44

And: Mathematics is without prejudice. Mathematics enables us to be unprejudiced and to remain more readily open to pure reason, and thereby to more correctly determine what is probably correct. Mathematicians are thus an unsuitable audience for all sorts of occultism and also for much current modern literary prattle about world decline and cultural annihilation; they see through the often unconscious reasons for this drivel.45

These are undoubtedly reactions to the apocalyptic speculations of the mathematician Oswald Spengler, who, in his Decline of the West (1918), seemed to give an intellectual-historical interpretation of the end of the monarchy and the fall of Germany. On the other hand Spengler explained with Gauss, Cauchy, and Riemann the “inner completion of the mathematical world of forms. The concluding thoughts”—the isolation of the mathematical world in the present period. But via Weyl and Hilbert, Gentzen had also become acquainted with the use of a political and “philosophical” language. 10. Political Language by Hermann Weyl and David Hilbert “. . . and Brouwer—that is the Revolution!”46 In 1944 Hermann Weyl attributed the tone of this remark to the resonance of the turbulent postwar period. Hilbert answered it with . . . no: Brouwer is not, as Weyl believes, the Revolution, but only the repetition of an attempted putsch forcefully undertaken, completely failed and again today, with the State armed and strengthened by Frege, Dedekind and Cantor, is condemned at the outset to failure.47

He warned against Weyl and the establishment of a “Verbotsdiktatur”: 44 Hilbert 45 Ibid.

46 “Uber ¨

605, Nachlass, University of G¨ ottingen.

die neue Grundlagenkrise in der Mathematik”, 1921, p. 226. “Neubegr¨ undung der Mathematik. Erste Mitteilung”, Gesammelte Abhandlungen [Collected Works], Chelsea Publishing Company, Bronx, New York, 1965, 2nd edition, vol. III, pp. 157-177, here p. 159. 47 Hilbert,

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. . . we run the danger of losing a large part of our most valuable treasures if we follow such reforms.48

Frank Plumpton Ramsey spoke of “the Bolshevik menace of Brouwer and Weyl.”49 And indeed Brouwer was seen by his supporters as a representation of the progressive left among his contemporaries, as shown in a citation from a lecture given in Groningen on 20 September 1926 by his friend, the mathematician J. Barrau: But free-axiomaticians50 are manifold and exist in many forms, an extreme rightwing of fanatic and passionate set-theoreticians, a left-wing, almost intuitionists, who, historically seen, pave the way for autonomous intuitionism by their rejection and criticism of the extremism in their own group. And further on there is a broad, massive centre, which above all is attached to the broadening of their field of work which free-axiomatism brought.51

In the period between the wars, Weyl was regarded as a follower of Brouwer’s intuitionism. In Richard Baldus’ Formalism und Intuitionism in der Mathematik 52 one reads As the insights of the intuitionists diverge in particulars, we wish to continue following Brouwer, for whom too the well-known Z¨ urich mathematician Weyl (born 1885) stepped aside, deserting his own path which digressed from Brouwer’s.

I question and speculate how the young Gentzen may have experienced the philosophy and ideology of these years. Only three examples (which can easily be multiplied): Gentzen saw, through Weyl, that ideology in mathematics preaches that what does not redeem itself in works, distances itself from practice, or even the theory position is quickly deserted. A small proof: Hermann Weyl took the language of politics in the service of a treatment of a mathematical problem: The antinomies of set theory would customarily be viewed as border disputes, which concern only the remotest provinces of the mathematical realm and can in no way endanger the inner solidity and security of the empire itself, its own core areas. The explanations, which the authorities on these disturbances would have given (with the intention of denying or settling them), almost always lack the character of clear conviction born from completely investigated evidence, but belong to that sort of half to three-quarters sincere attempts at self-deception, which one encounters so often in political thinking. In fact: every earnest and honest reflection must lead to the insight, that the unhealthiness in the border districts of mathematics must be judged as symptoms: in them comes to light what the superficially gleaming and trouble-free business in the centre hides: the inner lack of security of the foundations on which the construction of the empire rests.53

Is Weyl a frontier guard who recognises in the conditions of the borders of the realm the nature of the realm itself? He sees through the shiny wrappers of a “new” republic. He recognises the dishonest or intellectually clueless mediators! Clarity and freedom beyond chaos and empty meaning are given only through the 48 Ibid.

49 Mathematical

Logic, in Mathematical Gazette 13, no. 184 (October 1926), pp. 185-194. formalists 51 “De onbemindheid der wiskunde”, p. 19, cited after Hesseling, 1994, p. 34. 52 G. Braun, Karlsruhe, 1924, here pp. 23 and following. 53 Page 1 of “Uber ¨ die neue Grundlagenkrise der Mathematik”, 1920; reprinted 1965 by the Wissenschaftliche Buchgesellschaft, Darmstadt. 50 I.e.,

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Brouwerian conception. And following the political metaphor there is, appropriate for the time, an economic one: An existential assertion—say, “There is an even number”—is in general not a judgement in the proper sense, that claims a fact; existential-facts are an empty invention of the logicians. “2 is an even number”: that is a real judgement giving expression to a fact; “There is an even number” is only a verdict won from this judgement. If I think of knowledge as a valuable treasure, then this verdict is a piece of paper, which reports the existence of a treasure without advising where the treasure is. Its sole value lies only in that it drives me to search for the treasure. The paper is worthless so long as it is not accompanied by such a supporting judgement as “2 is an even number”.54

And further: . . . The conception described only gives expression to the meaning which the general and the existential propositions actually possess for us. In their light, mathematics appears as a dreadful “paper economy”. Real value, comparable to groceries in everyday economy, is possessed only by the immediate, the completely singular; all general and existential propositions only indirectly take part therein. But still as mathematicians we rarely think of the redemption of this “paper money”! It is not the existence theorem that is the thing of value, but rather the construction carried out in the proof.55

And then Weyl professed his party affiliation: So I abandon my own attempt and join Brouwer. In the menacing dissolution of the state of analysis, which is in the offing even though it may only be recognised by a few, I tried to secure a solid foundation without forsaking the practice on which it rests, while I implemented its basic principle cleanly and honestly; and I believe it succeeded as far as it could succeed. Because this practice is not tenable in itself, as I have now convinced myself, and Brouwer—that is the revolution!56

Weyl had—and I gladly repeat this—blamed the bombastic language of 1921 on the spirit of the postwar times. His choice of language, speaking of choice sequences—objectively correct but also political—of freedom, of a “new conception” with “radical consequences, which [will] give mathematics a genuinely different appearance”, with choice sequence’s “causing their own deaths (and the annihilation of their prior creation)” through the “termination of the process” could hardly be more politically charged. What’s more, Weyl was ready to take charge and lead the way: We must learn a new modesty. We wanted to storm the heavens and have only piled fog upon fog, which supports no one who would seriously try to stand upon it. What is tenable could at first sight appear so insignificant that the possibility of Analysis is in general placed into question; this pessimism is, however, unfounded.57

But he was still using political language in 1932. In “Topologie und abstrakte Algebra als zwei Wege mathematischen Verst¨ andnisses” [Topology and abstract algebra as two paths to mathematical understanding]58 he wrote: 54 Ibid.,

p. 16. p. 17. 56 Ibid., p. 18. 57 Ibid, p. 32. 58 Unterrichtsbl¨ atter f¨ ur Mathematik und Naturwissenschaften (38), pp 177-188; reprinted in the third volume of his collected works, 1908, p. 349. 55 Ibid.,

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Modern axiomatics, simple nature that it is, loves not (unlike modern politics) such ambiguous mixtures of war and peace; it neatly separates the two parts from one another.

At the opening of the new Mathematical Institute in G¨ ottingen in 1929, in his lecture on Felix Klein, he had said: Mathematics appears to me as a large weapons firm in peacetime. The display window is filled with magnificent pieces, whose useful, artistic and eye-pleasing performance delights the connoisseur. The real origin and purpose of these things, the application to the conquest of the enemy, is almost forgotten in the background.

He added: There is more than a kernel of truth in this but on the whole one generation feels this judgement of its endeavours to be incorrect.

That mathematics was no longer decorative but is seen with respect to battles and war Gentzen saw quite clearly. And how did the new doctrine affect the supporters of those who had “maintained in the abstract formalism of mathematics still some meaning for intuitive actualities”? Like “an awakening from a bad nightmare”. What Gentzen could have noticed anyway is the fact that through the characterisation of mathematics through philosophical basic positions like intuitionism, formalism, logicism, empiricism, and conventionalism,59 mathematical practice was not furthered one iota. And as he could easily see by Karl Menger,60 the lack of clarity in the fundamental concepts of these sorts of presentations was great and the arbitrariness obvious. 11. Whom Did Gentzen Know in G¨ ottingen and What Did He Read? An analysis of all writings and authors referred to by Gentzen shows the fol¨ lowing: Hilbert leads with a work of 1926 entitled “Uber das Unendliche” [On the infinite]. Next comes Arend Heyting with his work “Die formalen Regeln der intuitionistischen Logik und Mathematik” [The formal rules of intuitionistic logic and mathematics] of 1930, in which Glivenko’s work of 1929 is cited, which was significant for Gentzen, although it did not cover intuitionistic logic completely. But too Herbrand and G¨ odel would be cited by von Neumann. In the first (withdrawn) consistency proof of 1935, next to G¨ odel and Finsler for the first time he added Brouwer’s “Intuitionistische Betrachtungen u ¨ber den Formalismus” [Intuitionistic reflections on formalism] of 1928. In his “Stufenlogik” [Type theory] of 1936 Carnap, Behmann’s “Mathematik und Logik” and Russell are referred to. Weyl’s “Neue Grundlagenkrise”, A. Frænkel’s Mengenlehre [Set Theory], Cantor, Church, Skolem, Poincar´e, Brouwer’s “Beweis, dass jede volle Funktion gleichm¨assig stetig ist” [Proof, that every complete function is uniformly continuous] of 1924, Weyl’s “Stufen des Unendlichen” [Degrees of infinity], van der Wærden’s Moderne Algebra, and Gerhard Hessenberg’s Mengenlehre 61 are cited. 59 As,

for example, in W. Ackermann, “Was ist Mathematik?”, Zeitschrift f¨ ur mathematischen und naturwissenschaftlichen Unterricht 58 (1927), pp. 449-455. 60 “Der Intuitionismus”, Bl¨ atter f¨ ur deutsche Philosophie 4 (1930/31), pp. 311-325. 61 Why did he always cite Hessenberg’s Mengenlehre? An answer could be that given by Christian Betsch on pp. 316-324 in Fiktionen in der Mathematik [Fictions in mathematics], Friedrich Frommanns Verlag (H. Kurtz), 1926. This deals with the particular form of an axiomatisation and the property of transfinite sets.

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With whom did he have contact? Who were student comrades or friends? We know that he knew Hermann Weyl, that he spoke and corresponded with Paul Bernays, that he knew Ernst Witt and Heinrich Scholz, that he often played table games with Hans Hermes and saw Kurt Sch¨ utte at least once, and that with Scholz on the occasion of a lecture in M¨ unster in 1936 even became acquainted with Ja´skowski. In 1936 he also spoke with Cavaill`es. In 1937 in Paris he at least saw Brouwer, whom Heinrich Scholz knew, and spoke again with Bernays. He met G¨ odel in December 1939. It is unverified but probable that he was also acquainted with J. Herbrand. His intellectual universe is thus surveyable from the number of persons involved whom he knew or whose writings he read. It is all the more admirable what he made of it. It is easily conjectured that he was acquainted with or at least knew of P. Hertz in the small G¨ ottingen. In a letter to Hugo Dingler of 26 December 1914,62 Hilbert had named his closest co-workers in his research on the foundations of mathematics: Paul Hertz, Bernstein, and Kurt Grelling.63 Paul Bernays would have known this and might possibly have imparted it to Gentzen. 12. Gentzen’s Life in the Early Nazi Period. The Withdrawn ¨ Manuscript “Uber das Verh¨ altnis zwischen intuitionistischer und klassischer Arithmetik” of 15 March 1933 Arend Heyting read over Gentzen’s work and offered some advice. Gentzen responded with a letter: G¨ ottingen, 25 Feb. 33 Schillerstr. 7 Very respected Herr Doctor! I thank you for your letter of 18 Feb. and your assessment of my work. I was very pleased that your views agreed to a large extent with mine. I would like, following your suggestion for the publication of my work to add yet another remark on the relation to Hilbert’s idea of a metamathematical foundation of arithmetic. I think that one can say something like the following in agreement with your explanation: A consistency proof by finite means (what you call propositions [S¨ atze] of first and second order [Stufe]) has till now not been achieved, this original goal of Hilbert thus not reached. Opposed to this, one knows the intuitionistic standpoint already to be on firmer ground, i.e. thus to be consistent, whence through my result the consistency of classical arithmetic is secured. If one wishes to satisfy Hilbert’s requirements, there remains after this, to prove intuitionistic arithmetic consistent. This is now, on account of G¨ odel’s result, together with my proof, not possible with the aid of the formal apparatus of classical arithmetic. Nevertheless I am inclined to think a consistency proof for intuitionistic arithmetic from a still evident standpoint to be possible and appropriate. I hope to be able to make investigations of this in the next year. A further small extension of my work is the following observation, which to my knowledge hasn’t come forth in the literature of intuitionism: One can in intuitionstic arithmetic completely manage without negation, in that one explains ¬a as an abbreviation for a ⊃ 1 = 1, and takes = as a primitive relation for which one assumes the axiom: x = y ⊃ ⊂ ¬x = y (whereby thus ¬x = y only means x = y ⊃ 1 = 1). This conception of negation is correct, for ¬A ⊃ ⊂ .A ⊃ 1 = 1 is intuitionistically provable. Then the axiom 2.12 11 (reductio ad absurdum) is provable, the other negation axiom (2.12 10) becomes 62 In

Dingler’s Nachlass, according to the source cited in the next footnote. from p. 322 of Paolo Mancosu, “Between Russell and Hilbert: Behmann on the foundations of mathematics”, Bulletin of Symbolic Logic 5, no. 3 (1999). 63 Cited

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equivalent to 1 = 1 ⊃ B. One can view this as an arithmetic axiom, or one can even drop it and satisfy oneself that 1 = 1 ⊃ a is always provable so long as a contains no propositional variable. (This follows so: First, 1 = 1 ⊃ .B ⊃ 1 = 1, thus 1 = 1 ⊃ ¬B. In particular: 1 = 1 ⊃ ¬¬x = y, thus (2.148) 1 = 1 ⊃ x = y. Likewise 1 = 1 ⊃ x < y. The rest follows easily (cf. 3.4) by means of 2.33, 2.127, 2.125, 2.22, 2.132.) Yours, faithfully, Respectfully, Gerhard Gentzen64

On 15 March 1933 Gentzen submitted to Mathematischen Annalen his work ¨ “Uber das Verh¨ altnis zwischen intuitionistischer und klassischer Arithmetic” [‘On the relation between intuitionistic and classical arithmetic]. According to Szabo (1969) he withdrew the paper while it was in proof because G¨ odel’s work of 1932 containing the same result became known.65 How did G¨ odel learn about Gentzen’s work? On 15 March 1933 Arend Heyting wrote to G¨ odel: Some time ago Mr. Gerhard Gentzen sent me a manuscript that may perhaps also interest you. He proves among other things the following theorem: “If intuitionistic arithmetic is consistent, classical arithmetic is also consistent.” By “intuitionistic arithmetic” is understood a definite formal system, all of whose correct formulæ can be interpreted by means of intuitionistically correct statements. It thus yields that the grounding of arithmetic in the way entered upon by Hilbert, which, it is highly likely, cannot be carried through by means of the “finitary” resources employed by Hilbert, is possible by drawing upon further parts of intuitionistic mathematics. I don’t know when and where the planned publication of the article is to take place.66

Kurt G¨ odel answered 16 May 1933: That one can interpret classical number theory by means of the intuitionistic theory (whereby of course an intuitionistic consistency proof also results) is known to me and should also already have been known in G¨ ottingen since June 1932. In particular, around that time I spoke about it in the Menger colloquium, and, in fact, in the presence of O. Veblen, who shortly thereafter went to G¨ ottingen.67

And Heyting replied from Enschede on 24 August 1933: I thank you for the offprint of your paper, which agrees throughout in the method with that of Mr. Gentzen. I must assume that the report about your lecture had not sufficiently got through to G¨ ottingen after all.68

Arend Heyting appears to have asked Gentzen where his work would appear in print, for Gentzenwrote back: 64 Archive

A. Heyting, No. B Gen-330225. a version based on the proof sheets, Gentzen’s paper first appeared thanks to Paul Bernays in Szabo (1969) and later in the original German in Arch. math. Logik 16 (1974), pp. 119132. Kurt G¨ odel’s work “Zur intuitionistischen Arithmetik und Zahlentheorie” (“On intuitionistic arithmetic and number theory”) appeared in Ergebnisse eines mathematischen Kolloquium, IV (1933), pp. 34-38; cf. the remarks on this work by Georg Kreisel’s review of Szabo in Journal of Philosophy 68 (1971), pp. 238-265. 66 Kurt G¨ odel, Collected Works, Vol. V, p. 68. 67 Ibid, p. 70. 68 Ibid. 65 In

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G¨ ottingen, 28 Nov. 33 Very respected Herr Doctor, I wish to inform you that my work on the relation between intuitionistic and classical arithmetic, which you at that time were friendly toward, will not appear in print, since I have subsequently learned that Herr G¨ odel had already obtained nearly the same results before me. (As you have probably become aware of, it is given in G¨ odel’s work, “Zur intuitionistischen Arithmetik und Zahlentheorie” in the “Ergebnissen eines mathem. Kolloquiums”, Vol. 4.) You may possibly find it interesting that I have found a decision procedure for intuitionistic propositional logic (which to my knowledge was previously unknown). The relevant work is in press and when it is ready I will of course send you a copy. Most respectfully yours, Your Gerhard Gentzen69

G¨ odel and Gentzen had, independently of one another, shown that the consistency of axiomatic number theory reduces to that of intuitionistic arithmetic. Gentzen named Bernays as co-author of the theorem: Every definite expression of arithmetic, which does not contain the concepts “or” or “there is” and is classically provable, is also intuitionistically provable. Wang summed it up thus: “In 1932 G¨ odel gave a report to establish that classical number theory is as consistent as intuitionistic number theory (an observation also made independently by Bernays and G. Gentzen”.70 Bernays perhaps did not ¨ correctly understand what Gentzen wanted to discuss in his essay “Uber intuitionistische Arithmetik. . . ”. His exposition of intuitionistic ideas could possibly have included a more proper discussion of Brouwer’s approach.71 G¨ odel and Gentzen had independently shown that the consistency of the axiomatic number theory of Dedekind and Peano (the so-called Peano arithmetic, P A) reduced to that of intuitionistic arithmetic (called Heyting arithmetic, HA). G¨ odel proved in 1932 that every theorem of P A that does not contain the symbols ∨ or ∃ is also a theorem of HA. One can read this as showing that the classical axiomatic number theory is actually more restricted than the intuitionistic number theory. And: “G¨ odel and Gentzen showed that the intuitionistic restrictions on existence proofs do not have the effect Hilbert feared as far as theories expressible in first order logic are concerned.”72 At least this work shows that Gentzen was familiar with the ideas of constructive mathematics. And Gentzen was now aware of the existence of Kurt G¨odel. Bernays later wrote about this: A new discovery of G¨ odel and Gentzen leads us to such a more powerful method. They have shown (independently of one another) that the consistency of intuitionistic arithmetic implies the consistency of the axiomatic theory of numbers. This result was obtained by using Heyting’s formalisation of intuitionistic arithmetic and logic. This proof of the consistency of axiomatic number theory shows us,

69 Archive

A. Heyting, No. B Gen-331128. Wang, Reflections on Kurt G¨ odel, MIT Press, Cambridge (Mass.), 1987, p. 57. 71 Cf. Paul Bernays, “Hilbert’s Untersuchungen u ¨ ber die Grundlagen der Arithmetik”, David Hilbert, Gesammelte Abhandlungen [Collected Works], Chelsea Publishing Company, Bronx, New York, 1965, 2nd edition, vol. III, pp. 196-216, here p. 212. 72 Jan von Plato, “Proof theory of classical and intuitionistic logic”, in: L. Haaparanta (ed.), History of Modern Logic, Oxford University Press, Oxford, 2007. 70 Hao

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among other things, that intuitionism, by its abstract arguments, goes essentially beyond elementary combinatorial methods.73

But in fact Bernays wrote on 7 August 1935 to Hermann Weyl: I felt that the prior G¨ odel-Gentzen proof referring back to the consistency of intuitionistic arithmetic—and it may seem the same to you—was not truly satisfactory from the standpoint of intuitive evidence. Mr. Gentzen himself had only viewed it as provisional.74

And Bernays placed Gentzen in Hilbert’s programme so: Starting from this fundamental idea, Hilbert has sketched a detailed programme of a theory of proof, indicating the leading ideas of the arguments (for the main consistency proofs). His intention was to confine himself to intuitive and combinatorial considerations; his “finitary point of view” was restricted to these methods. In this framework, the theory was developed up to a certain point. Several mathematicians have contributed to it: Ackermann, von Neumann, Skolem, Herbrand, G¨ odel, Gentzen.75

13. Gerhard Gentzen’s Dissertation “Untersuchungen u ¨ber das logische Schließen” of 12 July 1933 Gentzen was promoted by Hermann Weyl on 12 July 193376 at the instigation of Paul Bernays. The other examiners were Professor Gustav Herglotz (1885-1953), the lecturer Emil Artin, and Otto Heckmann. From one of Gentzen’s later remarks one can conclude that it wasn’t easy at the time for him to find an advisor for his dissertation. His intellectual advisor could no longer participate in this. Within two days he would be terminated; i.e. he would—together with Landau, Neugebauer, Lewy and Hertz—lose his venia legendi,77 and he held no more functions since 28 April 1933. Paul Bernays, along with Paul Hertz, Hohenemser, Lewy, and Neugebauer, was informed by the dean of the Mathematical and Natural Sciences Faculty on 27 April 1933 he could “not practise [his] venii legendi until a conclusive decision on [his] legal position had been made.”78 Paul Bernays no longer had any teaching functions. Because of his Jewish extraction—a violation of §3 of the Law on the Restoration of the Professional Civil Service—he would be officially released on 31 August. On 21 September 1933 his authority to teach was definitively taken from him by the Ministry. Hermann Weyl arranged, however, that Bernays would be paid through the end of October.79 —An aside: His attempt at juridical compensation would later prove to be a fiasco. He submitted his application on 14 December 1957 to the Lower Saxony Ministry of Culture. The rejection on 23 March 1961 would be justified on the grounds that the application had been submitted too late and he had “disregarded 73 Paul Bernays, “On Platonism in mathematics”, in Paul Benacerraf and Hilary Putnam, eds., Philosophy of Mathematics; Selected Readings, Prentice-Hall, New Jersey, 1964; original in L’Enseignement math´ ematique 34 (1935/36), pp. 52-69. 74 Weyl Nachlass Hs. 91:12a at the Eidgen¨ ossische Technische Hochschule, Z¨ urich. 75 Cf. footnote 73. 76 Day of the oral exam. 77 That is, his license to teach. 78 N. Schappacher 1987, p. 528. 79 Cf. the so-called Weyl Testament in W. Weber, Bundesarchiv Koblenz R 4901 (old: R21), Nr. 10.091.

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the care that could reasonably be expected of him.” The premise of the ministerial bureaucracy was that the “globally organised” Jews had knowledge of all German laws and administrative regulations that had at some time been enacted. Thus the federal republican postwar anti-Semitism in the German ministerial ranks blocked any compensation for Paul Bernays.80 Gentzen received his doctorate on 2 November 1934 for his fundamental dissertation, in which he developed with “his system of natural deduction and with his sequent calculus a most transparent new systematics of predicate logic”.81 It was judged “very good” by co-referee Richard Courant. An improved version appeared in the Mathematische Zeitschrift in 1934/1935, although it had been submitted on 21 July 1933. On 9 June 1933, Hermann Weyl wrote the following report: Report on the dissertation submitted by Gerhard Gentzen: Untersuchungen u ¨ ber das logische Schließen [Research on logical deduction] The work discusses predicate logic, in which the “all” and “there are” apply only to object variables. It encompasses in a suitable way formalised elementary arithmetic without the use of complete induction. Two new logical calculi are developed. The first has the stated goal of better aligning the formalism with the actual deductive procedure of mathematics. Above all this is done in this way, that instead of logical axioms, deductive rules for logical inferences are used and that the components of a derivation can be placed under assumptions which will later be discharged. The second calculus modifies the first one so that with it one obtains deeper-lying theorems on predicate logic. To this end, the operation with “assumptions” must again be eliminated. The inference steps from formula to formula will be represented by sequents, so that the logical consequence symbol representing consequence in a proof must be distinguished from that occurring in the formulæ. (Similarly, the positioning of two formulæ next to each other in a proof is something different from the formula obtained by applying & to the formulæ from which it is constructed.) The researches are carried out at the same time for classical and intuitionistic logic, which distinguish themselves from each other through very simple features. The main content of the work consists in the proof of the theorem that from a derivation of the second calculus that inference can be eliminated which only permits a formula to be derived from formulæ in which the stem-formulæ are no longer contained as parts. Thereby the solution to the problem of decidability is immediately made possible.82 Moreover, by such means one can obtain an amazingly transparent proof of the consistency of elementary arithmetic as delineated above. In a final section the equivalence of Gentzen’s calculi with the familiar logical formalisms of Russell and Hilbert is demonstrated. The work contains a notable contribution to the questions of decidability and consistency. The formulations of logic employed are clearly especially suited for 80 Cf.

the note at the end of this chapter. Sch¨ utte. 82 The main theorem of Gentzen’s work is that a certain rule called the cut rule can be eliminated. As a consequence, each derivation in the predicate calculus can be replaced by one in which the subformula property holds: all the formulæ occurring in the derivation are subformulæ of the formulæ occurring in the endsequent of the derivation. I imagine this is the result Weyl is referring to. It would indeed yield the solution to the decision problem for the predicate calculus were it not for the fact that “subformula” is used here in a broad sense: for any formula φ(v) with free variable v and any term t, φ(t) is considered a subformula of ∀vφ(v) and ∃vφ(v). Thus, quantified formulæ have infinitely many immediate subformulæ.—Trans. 81 Kurt

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these purposes. Through their erection the writer earned his great independence opposite the previous formal formulations of logic. The presentation is of great clarity, reliable in the smallest details and very easily readable. In this area it is already an achievement and proves the value of the formal arguments employed.— In my opinion, the work deserves the rating “very good”.83

On 21 July 1933 the work was already received by the Mathematische Zeitschrift. Gentzen wrote therein: 1. My first point of view was the following: the formalisation of logical deduction, as it has been developed in particular by Frege, Russell, and Hilbert, distances itself rather far from the sort of deduction that mathematical proofs actually use. In doing so considerable formal advantages are obtained. My goal was first of all to set up a formalism that came as close to actual deduction as possible. . . 2. . . The Hauptsatz84 asserts that every proof in pure logic can be brought into a certain, usually in no way unique, normal form. The actual properties of such normal proofs may be described so: they contain no detours. No concepts are introduced in them which are not contained in the end result and thus are not needed in producing the result.

Wilhelm Ackermann reviewed the “Untersuchungen u ¨ber das logische Schließen. Teil I” (Mathematische Zeitschrift 39, Vol. 2/3, pp. 176-210.) in the Jahrbuch u ¨ber die Fortschritte der Mathematik [Yearbook on the advances in mathematics]: The formalisation of logical deduction, as is customary in logic, distances itself rather far from the sort of deduction that mathematical proofs actually use;85 one derives “correct formulæ” from a series of “logical base formulæ” through a few modes of inference and thereby obtains considerable formal advantages. Natural deduction, however, does not generally proceed from logical axioms, but rather from assumptions to which logical inferences are applied. This contentual deduction is reproduced in the “natural calculus” introduced by the author. Here we have no logical base formulæ (axioms), but rather inference rules which indicate which consequences can be drawn from given assumptions, as well as when the dependence of a desired formula on an assumption can be removed. The calculus of natural deduction is set up for classical and intuitionistic logic. A closer examination of the special properties of the natural calculus leads to a noteworthy Hauptsatz which asserts that every proof in pure logic can be brought into a certain (generally not unique) normal form. The characteristic property of this normal form is that it contains no detours. No concepts are introduced in it that are not contained in the final conclusion. To express the Hauptsatz in a convenient form and to be able to prove it, a new calculus is introduced, one that is closely related to the natural calculus, but which again derives formulæ from logical base formulæ. The announced continuation of the work is supposed to bring applications of the Hauptsatz and show the equivalence of the new calculi with the formalisms of Russell and Hilbert, and respectively with that of Heyting in the intuitionistic case.86 83 The original can be found in the institute records of the Mathematical Institute in G¨ ottingen. I thank Herr Prof. Dr. Norbert Schappacher for a copy. The formatting and style of the report suggest to me that it could have been written by Richard Courant or his secretary. It is of the same sort and form as Richard Courant’s report for his recommendation for Gentzen’s scholarship. 84 Cf. footnote 29, above. 85 It is quite common for reviewers to borrow heavily from the author’s introductory remarks! 86 JFM 60/1 (1934), pp. 20ff.

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Another review was given by Arnold Schmidt in Zentralblatt f¨ ur Mathematik 10, no. 4 (1935), pp. 145ff. After citing Gentzen’s foreword, Schmidt discussed the technical details of the natural deduction calculus, sequents, and inference rules. He closed with the remarks: The main thing is the “Hauptsatz” (to state this precisely, as the author emphasises, some dependencies in the calculus L will have to be accepted): From every proof carried out in L all cuts87 can be eliminated; one can give to every provable sentence a “detour-free” proof in which all formulæ are subformulæ of the formulæ occurring in the endsequent. This Hauptsatz is proven by a double finite induction.— If the calculus is extended by two (provable on the trivial interpretation) inference schemata for →, then curiously enough the intuitionistic predicate calculus is characterised by the condition that a succedent is allowed to contain only one member. The Hauptsatz is also valid for this narrower “intuitionistic” calculus. Arnold Schmidt (G¨ ottingen)

Schmidt continued and observed: Gentzen, Gerhard: Untersuchungen u ¨ ber das logische Schließen II. Math Z. 39, 405-431 (1934). The sharpened Hauptsatz yields a consistency proof of “arithmetic without full induction”, which is formulated by adjoining some number axioms to the calculus L. He concludes by proving the equivalence of the Gentzen calculi, namely the classical (resp., intuitionistic) calculus L, with a calculus which “is basically the same as” that of Hilbert (resp., Heyting). Schmidt (G¨ ottingen).

Peter Schroeder-Heister wrote: The sequent calculus of Gerhard Gentzen published in 1934 will correctly be viewed as a fundamental step in the formulation of logical formalisms. By such innovations, which grew out of nothing as it were, the question arises whether there were any forerunners that hinted systematically or historically at the innovation. In this lecture I’d lie to draw attention to a few points which attracted my attention while I was reading the literature on the subject, and which struck me as interesting not only with respect to later developments such as the resolution method:88 1. In some respects a sequent calculus is already present in Frege’s “Grundgesetze der Arithmetik” [Basic laws of arithmetic]. In any event, Frege’s rules may be utte’s interpreted in this sense. The calculus is incidentally closely related to Sch¨ formulation of the sequent calculus. 2. In another place Frege presents a calculus which may be read as a sequentlogical resolution calculus (for propositional resolution, i.e. unification of clauses). 3. Paul Hertz, whose work was decisive for formulating the structural part of the sequent calculus, proved normal forms for structural rule-based derivations.89 4. In his first publication of 1933 immediately following Hertz, Gentzen gave a completeness proof for “propositional SLD-Resolution” (modern terminology).90

In 1938 Alfred Tarski would refer to Gentzen’s work in his paper “Die Aussagenkalk¨ ul und die Topologie” [Propositional calculus and topology].91 87 Cf.

footnote 82, above. resolution method is a mainstay of automated theorem proving. 89 Cf. the longer note at the end of this chapter. 90 This is from an abstract of his essay “Uber ¨ einige historische Hintergr¨ unde der Sequenzenkalk¨ uls” [On some historical backgrounds of the sequent calculus]. 91 Fundamenta Mathematicæ 31 (1938), pp. 103-134; here, p. 133. 88 The

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Gentzen abandoned the Fregean doctrine holding that logic had absolutely nothing to do with the daily, actually practised form of human thought, and oriented his form of mathematical logic toward “the style of inference as it is actually used in mathematical proofs.” This is not a turning away from the antipsychological tradition, but is a pragmatic answer to the question of how to translate most easily and plausibly the sense of forms of proof into a formalised system of derivations. Already in 1932 or 1933 Arend Heyting was familiar with Gentzen’s dissertation, but in 1936 he had not yet received a copy from Gentzen. The latter wrote to him: 23 Jan. 34 Very respected Herr Doctor! Many thanks for your card of 9 Jan. My dissertation appears in the Mathem. Zeitschrift under the title “Untersuchungen u ¨ ber das logische Schliessen”. I prove therein an entirely general theorem on intuitionistic and classical propositional and predicate logic. The decision procedure for the intuitionistic propositional logic yields itself through a simple application of this theorem. One can prove thereby also the intuitionistic unprovability of simple formulæ of predicate logic, as e.g.(x)¬¬Ax. ⊃ ¬¬(x)Ax. How far this actually goes I have not looked into. I am now working on proving the consistency of analysis, which for 2 years has been my actual goal. With most respectful greetings, Your Gerhard Gentzen92

Heyting’s book Mathematische Grundlagenforschung 93 [Mathematical foundational research] appeared in 1934. He had written his book with the aid of the library of Heinrich Scholz in M¨ unster, having visited Scholz because the latter ¨ knew Brouwer well. Gentzen’s then not yet published work “Uber das logische Schließen” is cited on p. 18: On the intuitionistic functional calculus94 there are only a few metamathematical investigations. Gentzen obtained the result: If f (x) is a formula, containing no free variable other than x, and if ∃x f (x) is provable in the intuitionistic functional calculus, then (x) f (x) is also intuitionistically provable. This at first sight paradoxical appearing proposition becomes understandable if one considers that it is no longer valid once one extends the functional calculus by the introduction of definite mathematical objects, so that the proof of ∃x f (x) can be given by exhibiting an example.

Later Heyting went into G¨ odel’s work “Zur intuitionistische Arithmetik und Zahlentheorie”, in which was proved “The (formalised) intuitionistic arithmetic contains the classical, only under a different interpretation; for the intuitionist, however, this interpretation is the real one.” Heyting seems to have had questions with regard to intuitionistic provability, for Gentzen wrote to him: 16 April 34 Very respected Herr Doctor! I can answer your question affirmatively. It indeed follows from my theorem that if A ∨ B is intuitionistically provable, then either A or B is provable, and to be sure not only for the propositional logic, but rather for the entire predicate logic. I have also given this consequence in my dissertation. Unfortunately the work 92 Archive

A. Heyting, No. B Gen-340123. Berlin, 1934. 94 “Functional calculus” is an old way of saying first-order logic, i.e. logic in which one may refer to arbitrary objects of a given domain but not to sets of objects from the domain. 93 Springer-Verlag,

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is not yet printed because it has lain for a long time by the Notgemeinschaft der deutschen Wissenschaft [Emergency association of German science]. If I am not mistaken, from my theorem the following similar consequence may also be drawn: if (Ex)F x is intuitionistically provable (F x an arbitrary formula without free variables [other than x], so is (x)F x intuitionistically provable. In classical logic this is not valid, as e.g. the formula (Ex)(Ax ∨ (y)Ay) shows. With the best greetings Your most respectful Gerhard Gentzen95

14. The Penetration of the Nazis into Mathematical Research at the University in G¨ ottingen 1933 and 1934—Or, Vahlen and Bieberbach vs. Weber and Wegner Because he was of the Jewish religion, Richard Courant took a sabbatical from the Institute business and went to England. In April 1933, on account of the suspensions enforced by the Law of the Restoration of the Professional Civil Service, Weyl assumed responsibility as managing director of the Mathematical Institute. Everyone saw Weyl as a temporary director, and the battles over his succession began. On 18 July, Weyl wrote to Ludwig Bieberbach: Dear Bieberbach, in case of the dismissal of Courant, the local student body had been agitating for Wegner to be named his successor and given the leadership of the Mathematical Institute. In my opinion, such a development would be ruinous for G¨ ottingen.96

And Weyl begged Bieberbach in strictest confidence to make some pronouncement as the unofficial “Reichsf¨ uhrer” in mathematics on the scientific and humane qualifications of Wegner. Udo Wegner had received his Ph.D. in Berlin in 1928 under Issai Schur and Ludwig Bieberbach, habilitating the following year in G¨ottingen, where he was Privatdozent until 1931, when he became ordinary professor in Darmstadt. From 1936 until 1945 he was a professor in Heidelberg, where in fact he was director of the Institute until the end of the Second World War, abusing his position to further a political rather than a mathematical agenda. The students were not alone in wanting Wegner. Werner Weber (1906-1975), a student and longtime assistant to Landau, was now a Privatdozent in G¨ ottingen and an ardent Nazi—a supporting member of the SS since 1932 and a member of the Nazi party and the SA (troop leader) since 1933. In 1940, Weber would report that On the morning of 25 April 1933 I sank into a gloomy brooding over how to save German mathematics.97

According to him, The tradition of Felix Klein that had been destroyed by the Jews. . . could only have been awakened to a new life by only one man: Wegner.98

On 9 October 1933 Gentzen’s current doctoral advisor, Hermann Weyl, explained to the Minister of Science, Art, and Adult Education his reasons for wanting to be 95 Archive

A. Heyting No. B Gen-340416. Weber/Hasse [Weber/Hasse Exchange], Federal Archives Koblenz R4901 (old: R 21), No. 10.091, p. 372; cited in the sequel as “Weber” and page number. 97 Weber, p. 39 98 Ibid., p. 226. 96 Auseinandersetzung

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released from his position as Professor Dr. Weyl, under sacrifice of all claims for salary and pension. Because of the Jewish extraction of his wife he was “out of place” in G¨ ottingen and must on account of the new laws be unwanted by the Ministry. He continued: Also in America I would, as I had earlier in Switzerland, serve Germany and the German spirit to the best of my knowledge and belief. I can no more than wish that the new path on which the government has tread can lead the German people to recovery and rise. As a result of the (in my conviction) unfortunate combination with anti-Semitism I am personally denied direct participation in Germany.99

In April 1937, on account of his Jewish wife, Weyl was transferred to retirement. The formulaton “relinquishing all claims. . . ” would later prevent juridical compensation.100 Weyl was not called back by the Faculty after the war because they were of the opinion that he had left Germany voluntarily. Before departing, Weyl had apportioned the money of the Rockefeller Foundation and that of the Institute for 1933 and 1934 for employment, catalogue work, secretarial help, guest lectures, special teaching positions, assistants, etc., and had even bindingly secured it for 1935. Thus, for example, Weyl had arranged that Paul Bernays would indeed be paid through the end of August 1933 (he was in fact paid through October). With this “Testament” Weyl protected the Institute from immediate disintegration.101 Further, Weyl appointed “the politically handicapped” Arnold Schmidt and Dr. Ulm, whom Weber counted as belonging to the “Courant clique”,102 as replacements for Bernays and Fechtel, respectively. Then Weyl allowed himself to be represented by his assistant Dr. Franz Rellich and went to Switzerland. A permanent director now had to be named. Weber and many students wanted the Nazi Wegner. The decision was made by Theodor Vahlen (1869-1945), who appointed Helmut Hasse, a non-Nazi and the best mathematician of the various candidates. As an ardent nationalist, though not a National Socialist, Hasse was acceptable to the Nazis—except for Werner Weber. When the time came, Weber refused to hand the keys of the Institute over to Hasse. Vahlen instituted proceedings against Weber. The chairman of the fact finding committee, Hillmann, called Hasse by telephone on 5 June 1934 to question the latter about his connection with Courant and received the reply: I met him in G¨ ottingen and recommended to him to allow himself to begin emeritus status.

Hasse, who had organised 14 valuations from colleagues on Emmy Noether’s scientific significance, which were forwarded to the ministry with a petition on her behalf, answered the question of the status of his relationship with her with the words “purely scientific”.103 The whole affair dragged on, and it was as an unscheduled professor in Berlin that on 18 January 1940 Weber sent his written defense to assistant director Prof. 99 Cited from Norbert Schappacher, “Questions politiques dans la vie des math´ ematiques en Allemagne (1918-1935)”, pp. 51-89; in: Josiane Olff-Nathan, ed., La science sous le troisi` eme ´ reich. Victime ou ali´ ee du nazisme?, Editions du seuil, Paris, 1993. 100 Anik´ o Szabo, 2000. 101 With this deed, one no longer needs the assumption that Weyl and Hilbert paid anyone out of their own pockets. 102 Weber, p. 16. 103 Ibid., p. 188.

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Dr. Mentzel of the Reich’s Ministry for Science Education and Adult Education, together with the request to incorporate in it his personal records and “especially the records of the preliminary proceedings then (on 30 May 1934) initiated against me in G¨ ottingen”.104 This document is nearly 400 pages long and is a monument to self-aggrandisement: For a later time it will surely be of interest, anyhow, to be able at least to study a typical example of such a collision,105 and to grasp this together with its apparent inner contradiction.106

Weber characterised himself: Perpetual battle against the Courant clique. Has always held contact with the students and participated half a year in departmental work despite being overloaded with lecturing.107

About Hasse he noted: Great algebraist, German national, apparently incapable of being brought into line. Currently petty and hateful. His appointment here is promised; we cannot approve.108

But his attitude mellowed: With regard to politics, Hasse reveals himself to be thoroughly reliable. He belongs to that group of men who only first acknowledge the new state if they can represent it in some outstanding position. After this was accomplished the Third Reich proved itself in his eyes and he supported it thereafter completely converted, no doubt even against foreigners.109

On politics he noted: But the real opposition today comes not from Hasse, but rather is led by Blaschke.110 But two genuine enemies of the state can no longer be found today in G¨ ottingen.111

And on mathematics he expressed himself thus: From Berlin, acting under the leadership of Bieberbach, a young society of National Socialists have united and taken over the rˆ ole that G¨ ottingen had forfeited.112

The reason that by 1940 G¨ ottingen was no longer leading Germany in mathematics was, of course, Hasse: The tradition of Felix Klein that had been destroyed by the Jews, as I explained already in my letter to Vogt of 10 May 1934, could only have been awakened to a new life by only one man: Wegner. But this tradition did not return—no, even in failure I had in any case so weakened Hasse that he never recovered. G¨ ottingen possessed in him only that which Marburg had lost in him. The Institute at G¨ ottingen no longer held a leading position within the Third Reich. A “G¨ ottingen School” today exists only in Hasse’s narrow specialty; the genius 104 Ibid.,

p. 2. national socialism and the power of the state—EMT 106 Weber, p. 23. 107 Ibid., p. 387. 108 Ibid., p. 385. 109 Ibid., p. 223. 110 Ibid., p. 227. 111 Ibid., p. 227. 112 Ibid., p. 227. 105 Between

¨ 14. NAZIS IN GOTTINGEN

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loci has vanished. To secure himself, Hasse has surrounded himself by-and-by with almost nothing but hangers-on and scientific nullities.113

If Weber’s opinion of Hasse was so negative after it had mellowed, imagine how Weber must have felt back when he was refusing to turn the keys over to him. I had to fight him, even if I were to fall apart. A legal method was no longer a question.114 If the old system in this Institute was to keep the upper strata, then I would be ruthless and fight with all means.115

Young Oswald Teichm¨ uller, ardent Nazi that he was, interceded: On the evening of 29 May I reached an agreement with Teichm¨ uller to the effect that not only at the command of the F¨ uhrer but also already at the command of other leading National Socialists, who had my trust, such as my Sturmf¨ uhrer Heinze Lange, I would give up the resistance.116

In July 1934 Hasse received the keys to the Institute. On 6 September of that year, Erhard Tornier (1899-1982), a Nazi, was appointed ordinary professor and “political leader of the Institute”. But he didn’t hold that post for long. In 1935 he was director of the separate Institute for Mathematical Statistics, and in 1936 he was shipped off to Berlin. Hermann Weyl looked back at the beginnings of the Nazi period in G¨ ottingen on the occasion of his obituary of Emmy Noether at her death on 10 April 1935117 Our generation accuses that time of lacking all moral sincerity, of hiding behind its comfort and bourgeois peacefulness, and of ignoring the profound creative and terrible forces that really shape man’s destiny; moreover of shutting its eyes to the contrast between the spirit of true Christianity which was confessed, and the private and public life as it was actually lived. Nietzsche arose in Germany as a great awakener. It is hardly possible to exaggerate the significance which Nietzsche (whom by the way Noether once met in the Engadin) had in Germany for the thorough change in the moral and mental atmosphere. . . During the wild times after the Revolution of 1918 she did not keep aloof from the political excitement, she sided more or less with the Social Democrats; without being actually in party life she participated intensely in the discussion of the political and social problems of the day. . . It is hardly imaginable nowadays how willing the young generation in Germany was at that time for a fresh start, to try to build up Germany, Europe, society in general, on the foundations of reason, humaneness, and justice. But alas! The mood among the academic youth soon enough veered around; in the struggles that shook Germany during the following years and which took on the form of civil war here and there, we find them mostly on the side of the reactionary and nationalistic forces. Responsible for this above all was the breaking by the Allies of the promise of Wilson’s Fourteen Points, and the fact that Republican Germany came to feel the victor’s fist not less hard than the Imperial Reich could have; in particular, the youth were embittered by the national defamation added to the enforcement of a grim peace treaty. It was then that the great opportunity for the pacification of Europe was lost, and the seed sown for 113 Ibid.,

p. 226. p. 39. 115 Ibid., p. 46. 116 Ibid., p. 115. 117 Scripta Mathematica 3 (1935), pp. 201-220; reprinted in volume 3 of his collected works (1968), pp. 425-444. Here we cite pp. 439ff. 114 Ibid.,

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this disastrous development we are the witnesses of. . . She lived in close communion with her pupils; she loved them, and took interest in their personal affairs. They formed a somewhat noisy and stormy family, “the Noether boys” as we ottingen. Among her pupils proper I may name Grete Hermann, called them in G¨ Krull, H¨ olzer, Grell, Koethe, Deuring, Fitting, Witt, Tsen, Shoda, Levitzki. F.K. Schmidt is strongly influenced by her, chiefly through Krull’s mediation. . . Artin and Hasse stand beside her as two independent minds whose field of production touches on hers closely, though both have a stronger arithmetical texture. With Hasse above all she collaborated very closely during her last years. . . In the spring of 1933 the storm of the National Revolution broke over Germany. The G¨ ottinger Mathematisch-Naturwissenschaftliche Fakult¨ at [G¨ ottingen mathematical-natural scientific faculty], for building up and consolidation of which Klein and Hilbert had worked for decades, was struck at its roots. After an interregnum of one day by Neugebauer, I had to take over the direction of the Mathematical Institute. But Emmy Noether, as well as many others, was prohibited from participation in all academic activities, and finally her venia legendi, as well as her “Lehrauftrag” and the salary going with it, were withdrawn. A stormy time of struggle like this one we spent in G¨ ottingen in the summer of 1933 draws people together, thus I have a particularly vivid recollection of these months. Emmy Noether, her courage, her frankness, her unconcern about her own fate, her conciliatory spirit, were, in the midst of all the hatred and meanness, despair and sorrow surrounding us, a moral solace. It was attempted, of course, to influence the Ministerium and other responsible and irresponsible but powerful bodies so that her position might be saved. I suppose there could hardly have been in any other case a pile of enthusiastic testimonials filed with the Ministerium as was sent in on her behalf.118 At that time we really fought; there was still hope left that the worst could be warded off. It was in vain. . . She harbored no grudge against G¨ ottingen and her fatherland for what hey had done to her. She broke no friendship on account of political dissension.

Weyl continued: Even last summer she returned to G¨ ottingen, and lived and worked there as though all things were as before. She was sincerely glad that Hasse was endeavouring with success to rebuild the old, honourable and proud mathematical tradition of G¨ ottingen even in the changed political circumstances.119 120

118 On 22 June 1933 many mathematicians signed a petition to the registrar of the Albert’s University in K¨ onigsberg in Prussia protesting the suspension of Kurt Reidemeister, which ended his disciplinary transfer to Marburg. Among other signators were W. Blaschke, the organisor of the request, and Hasse in company with Emil Artin, Harald Bohr (Copenhagen), H. Tietze, R. Weitzenb¨ ock (Amsterdam), and Weyl. (P. 67 of Reinhard Siegmund-Schultze, Mathematiker auf der Flucht vor Hitler. Quellen und Studien zur Emigration einer Wissenschaft, Dokumente zur Geschichte der Mathematik volume 10, Vieweg Verlag, Wiesbaden, 1998.) 119 Loc. cit., p. 435. 120 Noether was far from unique in this. Before leaving permanently for New York in the summer of 1934, Courant returned once more from England. The finality of his departure affected him deeply: “I feel so close to my work here, to the surrounding countryside, to so many people and to Germany as a whole that this ‘elimination’ hits me with unbearable force” (Jean Medawar and David Pike, Hitler’s Gift, London, 2001, p. 136). Many emigrants visited their German homeland repeatedly, but unfortunately this has not been systematically researched. In Noether’s case she had an additional, more personal reason for the visit as she bade goodbye to her brother, who emigrated to the Soviet Union—where he would be executed in 1941 for anti-Soviet propaganda.

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15. The State Examination with “Elektronenbahnen in axialsymmetrischen Feldern unter Anwendung auf kosmische Probleme” on 16 November 1933 After the acceptance of his dissertation for publication Gentzen was awarded his doctoral degree: Under the administration of His Magnificence the Rector Professor Dr. phil. Friedrich Neumann, the Mathematical-Natural Scientific Faculty of the Georg August University, through its Dean Professor Dr. phil. Max Reich, appoints Mr. Gerhard Gentzen of Stralsund to Doctor of Philosophy. The dissertation “Untersuchungen u ¨ ber das logische Schließen” submitted by him is judged Very good and the outcome of his oral exam in Mathematical Analysis, Geometry, and Theoretical Physics given on 12 July 1933 is deemed Good. In testimony thereto this certificate is issued with the seal of the Faculty affixed. G¨ ottingen, 2 November 1934 The Dean of the Mathematical-Natural Scientific Faculty121

In November 1933 Gentzen completed his state examination for a license to teach mathematics, physics, and applied mathematics in secondary schools with his work “Elektronenbahnen in axialsymmetrischen Feldern unter Anwendung auf kosmische Probleme” [Electron paths in axialsymmetric fields with application to cosmic problems].122 Slightly paraphrased, the diploma reads as follows: Certificate of the scientific examination for teaching at secondary schools Mr. Gerhard Gentzen, born 24 November 1903 in Greifswald, Lutheran confession, passed the Abitur at the Gymnasium in Stralsund on 29 February 1928 and studied mathematics and natural sciences from Easter 1928 to Easter 1929 in Greifswald, from Easter 1929 to Easter 1930 in G¨ ottingen, from Easter 1930 to Michælis 1930 in Munich, from Michælis 1930 to Easter 1931 in Berlin, from Easter 1931 to Michelis 1933 in G¨ ottingen. On 12 July 1933 he was promoted in G¨ ottingen. According to the report of 26 May 1933 on the scientific examination to be licensed to teach in secondary schools, he was given the problem for written treatment: Electron paths in axialsymmetric fields with application to cosmic problems. As substitute for one of the two written assignments a scientiic publication of his was accepted: Investigations on logical inference. He underwent the oral exam on 13-16 November 1933. Mr. Gerhard Gentzen has passed the exam for teaching at secondary schools. He received for mathematics as main subject the mark “Good” for physics as main subject the mark “Good” for applied mathematics as additional subject the mark “With Distinction”. For the overall result on the written and oral exams the mark “Good” is conferred 121 Bundes-Archiv 122 See

Koblenz: Habilitationsakte Gentzen. the longer note at the end of this chapter.

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on him. Because of the report on teaching practice the regulation on practical education for teaching in secondary schools of 28 July 1917 is referred to. G¨ ottingen, 16 November 1933, Scientific Examination Bureau.123

On 5 November 1933, following the advice of someone yet unidentified, Gentzen joined the (university)-SA as Sturmmann (storm trooper), perhaps in this time not to endanger his university or possible teaching career. The strongly conservative Courant, a one-time social democrat, who saw his stay abroad in “the pure businesslike spreading of German cultural values”, advised Helmut Ulm (1908-1975) on 22 December 1933 not to distance himself on individual grounds if he would not be kept away by force. “Also, e.g., Rellich feels the same and goes with pleasure to its paramilitary sport camp”.124 However, when Rellich arrived in Berlin, on account of his Austrian nationality, he was not accepted and had to travel back to G¨ ottingen. Courant tried to utilise every opportunity he could under National Socialism to further mathematics. Entry into the university-SA was not compulsory in G¨ ottingen.125 16. Why Did Gentzen Join the SA? Why did Gentzen choose a course that could take him from simple storm trooper possibly to a pack leader? Those of Gentzen’s age group were unprepared; there was no shaped, impressive lifestyle: Nazi-archaic stood opposite or even mixed together with the modern. Nazism came over Gentzen—one notes his age—more powerfully and agonisingly than the others. He who even considered music as noise would have fled the public and its noisy manners. One must view Gentzen’s entry into the SA as a joke. For Gentzen more than 3 to 5 men in a room was unbearable. He even fled from the G¨ ottingen University crowds into another realm. All his friends and acquaintances he knew from the time of his youth. He was intimate only with them. His friend Hertha Michælis 123 Bundes-Archiv

Koblenz: Habilitationsakte Gentzen. Siegmund-Schultze, op cit., p. 145. 125 In an e-mail to the author of 16 June 1997, Harold Schellinx (Utrecht) wrote, “At a conference in Utrecht (September 1996) I spoke with Dr. Jervell, and the talk came to Gentzen. Dr. Jervell recited some anecdotes, from which it emerged that Gentzen must have been a fanatic National Socialist, that already in the early 30s in G¨ ottingen with something of pride he decorated himself with the insignia of the party. These ‘facts’ apparently originated with Professor Ketonen, who knew Gentzen personally at this time. The idea that the ‘spiritual father’ of my field might have been a ‘hardboiled’ Nazi shocked me and once again made it painfully clear to me how few biographical facts there are on GG.” Jan von Plato, one-time assistant of Ketonen, strongly doubted this. First, Gentzen himself never spoke much to others, and if he did it was mainly on mathematical problems. Then too Ketonen had not—despite numerous questions by von Plato—spoken about his G¨ ottingen period and Gentzen.— On “external” grounds I don’t believe Gentzen was a “hardboiled Nazi”. Would Bernays have associated with him until 1939? Gentzen first joined the SA in November 1933—and is thus not a “M¨ arzgefallener”, one of those people who immediately sought to join the Nazi party after the Nazis came into power in the beginning of March—and informed Bernays of this by postcard. Gentzen had good contact with Saunders Mac Lane who would certainly have reported something about it if Gentzen had displayed Nazi tendencies to him. Anticipating: In 1937 Gentzen first joined the Nazi party, and in fact before the Parisian Descartes Congress. There he got to know Jean Cavaill`es better. Cavaill`es was already in G¨ ottingen in 1935, and this later leader of the resistance would have informed his friend Albert Lautman by mail or some other means if Gentzen had been a Nazi. Or? Why would one label Gentzen, who is always described as reserved and introverted, as a career Nazi? Because one would otherwise find his death too unbearable and senseless? Sometimes I have the impression that Gentzen is confused with someone else. 124 Reinhard

17. GENTZEN’S POLITICAL POSITION: A CONJECTURE

53

he knew from an early age. He associated with his teachers until the end of his life. His dependence on Bernays is proverbial. Would he otherwise have risked correspondence subject to censor with an exiled Jew living in Switzerland? And one can see with whom he corresponded: only with people whom he knew from childhood and had developed a trust in. And this man should have found it possible to feel comfortable in a mass organisation? Believe it who will! In 1944 or 1945 was there an automatic promotion to or classification of this man as Rottenf¨ uhrer (Group Leader)? It seems there was no automatic taking over one Nazi organisation by another. In fact there were in general no takeovers. To the contrary, there were only admissions. And admissions meant that one had to produce certificates of one’s period of service, for these service certificates, evidence of the fulfillment of one’s service obligations, leadership, ideological inclination, and style of character qualified. Application for admission had to be carefullly filled out and signed.126 As he wrote to Bernays, Gentzen joined the SA “on advice,” but we don’t know why the advice was given or whose advice that was. Gentzen occupied himself with the ideas of Paul Hertz at the advice of Bernays. One can ascertain in Gentzen a certain passive, almost phlegmatic trait in all things which did not concern mathematics. Gentzen even based his career on the advice of Hasse, respectively, Rohrbach. 17. Gentzen’s Political Position: A Conjecture What did Gentzen do socially and politically from 1933 to 1935? From his promotion in 1933 to the beginning of his post as assistant to Hilbert on 1 November 1935, he most likely was only sporadically in G¨ ottingen during semesters and instead was with his mother or sister. Neither in the writings of Werner Weber nor in the newspaper reports does the name Gerhard Gentzen appear. In fact, Weber mentions among others in the series of friends and enemies of National Socialism in the student body only two men whom Gentzen at least knew well—Witt, and Mohr. About Witt Weber wrote: . . . Witt once explained all sciences can reshape themselves in accordance with the spirit of the times; only in mathematics must everything remain as before.127

Mohr, who was supported by Weber,128 was often in the company of Nazis, among them Teichm¨ uller, Kleinsorge, Spengler, Herms, Epmeier, Wachs, Hoge, and Mierow. But even he had his limitations. Tornier, the erstwhile political director, found out that Mohr had said that the establishment of a secret police would be un-German and that everyone would feel his personal security threatened. Neither among the students nor the SA-troops, neither in the Institute affairs nor in peripheral political questions is Gentzen mentioned in Weber’s report of his “Revolution”. My conjecture is that Gentzen had in fact exclusively concerned himself with the consistency of analysis—and that would be at home with his mother in R¨ ugen. First of all he had tackled the consistency proof for pure number theory from 1933 to 1935, and simultaneously, as remarked in a letter from Paul Bernays to Hermann Weyl,129 he had sketched for himself a possible consistency proof for analysis and set it aside. 126 Cf.

Peter Longerich, “Der Fall Martin Broszat”, Die Zeit No. 34, 14 August 2003. p. 4. 128 Ibid., p. 75. 129 Cf. more below. 127 Weber,

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If one looks at the photos of Gentzen, one will notice that Gentzen, in accordance with the times, cut his hair short and wore the appropriate clothing—nothing like that of the “swing boys”. In two photos he wears the party insignia on his lapel: one on the steps of the Sorbonne as an official representative of the German philosophers, but also once in a private photograph taken in Liegnitz. Gentzen furthermore did not come into conflict with the rules of the game of the nazis. Brooding, introverted, thrifty—and modest, and outwardly rather bashful but always friendly, he had put his ideas of beauty, clarity, ease and elegance, and effectiveness of mathematical procedures into his work. In a dictatorship or totalitarian society, it is not an easy matter to make a distinction of biography and character by an assessment of human relations. To the contrary, under a dictatorship, one has to make the immediate distinction between official and private biography. The connection between biography and character, official and private life is so arbitrary that the externalities can hardly capture them. 18. Gentzen in Financial Difficulties In 1934 Gentzen asked Hermann Weyl about a stipend from the Rockefeller Foundation for 1935. As these stipends were cancelled, Gentzen hoped the Institute for Advanced Study would give him an invitation for 1936. From 1 April 1934 to 30 September 1934 he received a research stipend from the German Studienstiftung for the project “Proof of the consistency of analysis” in the amount of 150 Reichsmarks. Gentzen conceived his ideas on the consistency proofs for type logic and number theory in 1933 and the beginning of 1934. On 11 April 1934 he wrote a postcard to Paul Bernays, who was then at Holsteinische Straße 38 in Berlin-Wilmersdorf: Very respected Herr Professor, Even though there is no particular occasion, I’d like to stay in touch. The emergency organisation seems not to be in any hurry in settling my stipend request; in any event I have still not received news from them. After a question from me it was about 6 weeks before I received notification that my documents were still being looked at. To be on the safe side I have also applied for a teaching position, but till now nothing has been decided there either. I have also joined the SA, as has been urgently advised from various quarters.— Incidentally, I have considerably furthered my investigations on consistency: first I treated type logic and introduced a consistency proof for this which follows the known simple consistency proof for predicate logic. Then I added the mathematical axioms with which the real difficulty naturally begins. It appears that the consistency of mathematics is equivalent with the carrying of the Hauptsatz concerning predicate logic over to type logic. I hope soon to have finished a consistency proof “with force”. Then this is to be reshaped so that only reliable modes of inference are used. This I hope to achieve by means of transfinite numbers, analogously to the arithmetic case.— How far are you with your book? Are you coming again to G¨ ottingen for the summer semester? With most respectful greetings, Your G. Gentzen130 130 Hs. 975:1649. I thank Dr. Beat Glaus of the archive of the ETH Z¨ urich for a copy of the correspondence between Bernays and Gentzen.

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I doubt Bernays gave much thought to Gentzen’s financial woes. He had lost his license to teach in 1933. How was he supporting himself in Berlin? And is Gentzen’s query about a return by Bernays of “disarming” naivet´e? Had he not heard of Bernays’ dismissal or did he just not understand it?

19. Financial Straits and Job Hunting From Schillerstraße 35 in Stralsund Gentzen wrote to Hellmuth Kneser131 on 5 December 1934: Very respected Herr Professor! Enclosed I send you my doctoral work. I have in the previous year been promoted in G¨ ottingen (by Professor Weyl) and taken the state exam. Then, with the support of the Notgemeinschaft der deutschen Wissenschaft, I have again taken up my work (the beginnings of which I have, I believe, informed you of) on the consistency of arithmetic. At the moment I am preparing a consistency proof for pure (i.e. no analytic means employed) number theory, which I have finished, for publication in the Mathematische Annalen. What I will do afterwards is not at all clear; the times do not encourage any mental profession! With most respectful greetings Your Gerhard Gentzen132

As of 30 September 1934 he received no more financial support from the Studienstiftung. On 16 February 1935 Gentzen reported from Stralsund: Very respected Herr Professor! From privy councillor Hilbert I received the news that he could give me no assistantship. Further, today I received a letter from Prof. Scholz in M¨ unster to whom I had turned on the same business at the recommendation of Professor Bernays, and he could, if need be, promise me a very uncertain possibility of a tutorship in M¨ unster. In these circumstances I would be most grateful if you could consider me for an assistant position in Greifswald. If a personal response seems desirable to you, I am ready to come over to Greifswald again. Heil Hitler! Your, Gerhard Gentzen133

His difficulties grew more pressing, for on 27 February 1935 he wrote anew to Kneser: Very respected Herr Professor! Yesterday I received a new letter from Professor Weyl. He wrote that the Rockefeller Foundation has recently ceased entirely the giving out of stipends for mathematics, physics, and chemistry, and that also the Institute in Princeton could grant me no stipend this year. He hoped, however, to be able to obtain an invitation from the Institute for next year. If it could be made possible and for this time you have not yet made other arrangements,I would like to take over the

131 On the political views of Kneser, cf. the relevant remarks in Constance Reid, Courant, Springer-Verlag, New York, 1976. 132 The letter is to be found in the archives of Prof. Dr. Martin Kneser (G¨ ottingen). 133 Ibid.

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assistantship in Greifswald for the second semester September 35 to May 36. I could then possibly finish off my “Dr. Habil.” simultaneously. Heil Hitler! Your Gerhard Gentzen134

Already on 2 March 1935 he reported again: Very respected Herr Professor! Many thanks for your letter. It seems to me also then that the best is if I once again apply to the Notgemeinschaft. I had already considered this myself, but wanted to have my current work finished first, to be able to present this. Also I wanted once again to engage in some practical activity between the one-sided intellectual works. Pecuniarily I am certainly in the best position if I hold a stipend, at that time it came to 150M monthly. I should best turn to Hilbert for the application, from whom I received it then. For the summer I have nothing certain planned; if necessary I’ll possibly be able to take up the assistantship already in April, if only the completion of my work followed shortly. I will inform you of the results of my endeavours. With most respectful greetings and Heil Hitler! Your Gerhard Gentzen The letter of Bieberbach is enclosed.135

What did Ludwig Bieberbach write? That he wanted Gentzen as a contributor to his Deutsche Mathematik ? Or had Bieberbach written to Kneser requesting him to enlist authors for Deutsche Mathematik ?136 At this time, Ludwig Bieberbach was a self-important, egoistic “golden pheasant”, a “Berlin Professor”, who even had telephone access to the ministry. That he was only a small midget in the structure of the Nazi party, despite the golden chain of office and position as the Reich’s research councillor, he would soon realise. One should not overlook however that, as a mathematician, he stood up for Hilbert’s position. He defended it immediately against attacks from outside like those of Steck and Dingler. The really excellent mathematician Bieberbach fell flat on his face as soon as he occupied himself “politically”, as he did in 1928 in the battle over the Bologna International Congress of Mathematicians,137 and he would do so again with his declarations on the “national qualifications” of mathematicians. But he knew that after the shattering of the “G¨ ottingen School” in Germany only the “M¨ unster Circle” around H. Scholz carried on Hilbert’s brand of logic. That 134 Ibid. 135 Ibid.

136 “I have no idea what the letter from Bieberbach concerned” (Martin Kneser to the author on 21 February 1997). Earlier, however, on 25 November 1934, Ludwig Bieberbach had written, among others, to Hellmuth Kneser: “Another matter. I am planning the launching of a new journal, the purpose of which is to give in self-evidence and conjoined essays a picture of German mathematics. The journal should definitely place itself on the soil of the National Socialist world view. It should not string treatise upon treatise, but rather give the reader a genuine insight into German creativity. Also questions of instruction, history, philosophy, and biography should find consideration. Occasional reports on foreign works are not excluded, insofar as their significance is justified. The support through the Notgemeinschaft and through the Deutsche Studienschaft [German Students Society] I believe is surely possible. Would you like to participate? Heil Hitler!” But Bieberbach was also involved in matters involving stipends and refereeing. 137 Cf. e.g. the appendix on Hilbert’s programme by Smory´ nski.

20. DISCUSSION WITH PAUL BERNAYS

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Bieberbach had taken part in the founding of Scholz’s chair through recommendation and actions can no longer be verified today because too many records were burned. But that Bieberbach had supported Scholz in this may be accepted as most likely, because Scholz and Bieberbach worked together in publishing the logical works of Gentzen and those of the Scholz school, with Bieberbach taking over the costs of publication. 20. Consistency Proof for Number Theory in Discussion with Paul Bernays From Stralsund, Gentzen wrote Bernays a business letter on 23 June 1935: Very respected Herr Professor! Enclosed I send you the final piece of my “Consistency of pure number theory”. I thank you repeatedly for your attention to the main part. I have adopted the suggested improvements in textual particulars in the final text.— I have remembered to answer your question regarding process 13.6.2, that you raised (as second) in your card. See also 15.2—. . .

The technical discussion continues up to the last paragraph: Do you perhaps know anything about whether Professor Weyl will be again in Europe this summer, or perhaps he already is? I’d like in any case to send him a carbon copy of my work.— From the Notgemeinschaft I now hope to receive information on the stipendium. If it is approved for me, I intend to go in fall or winter to G¨ ottingen and perhaps also once to M¨ unster. I will then take up again the consistency proof of analysis. With most respectful greetings, Your Gerhard Gentzen138

What had happened to the carbon copy? It must have been copied several times. “In September 1935 when I was about to spend a year at the Institute for Advanced Study, Hermann Weyl put in my hands a manuscript by Gerhard Gentzen which he wished me to read. I was prevented from doing so by a telegram offering me a teaching position at the University of Wisconsin.” Stephen C. Kleene (p. 57 in: Stuart Shanker, G¨ odel’s Theorem in Focus, Routledge, London, 1988). On 25 June 1935 he wrote again to Hellmuth Kneser: Very respected Herr Professor! Today I received from the Deutsche Forschungsgemeinschaft the news that at the proposal of privy councillor Hilbert a research stipend for me of 125 Reichsmarks per month has been approved. There is the prospect, on future application, that this will be extended. So for the time being I can get by and continue my work on mathematical foundations. I wish to return again in the winter semester to G¨ ottingen. Also I will try, if it can be reconciled with the stipend and otherwise no other difficulties exist, to make my habilitation. I will report to you occasionally of my further undertakings. Heil Hitler! Your Gerhard Gentzen139

Bernays became Gentzen’s “sparring partner” and “coach” for technical questions. On 17 July 1935 Gentzen wrote Bernays: 138 Hs.

975:1650. Martin Kneser.

139 Archive

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Very respected Herr Professor! I thank you for your letter. Enclosed I send you the final version of the 4th section of my work, and also a couple of amended pages from the 2nd section which are connected with it. All the changes are only minor and concern nothing essential—. . . From the Notgemeinschaft, now called the “Deutsche Forschungsgemeinschaft”, I have in the meantime received news that a stipend from now till the end of the year, with prospect of extension, has been approved for me. With most respectful greetings, Your Gerhard Gentzen140

Mathematics now took centre stage. 21. “Widerspruchsfreiheit der reinen Zahlentheorie” Mirrored in the Correspondence of Bernays and Weyl On 7 August 1935 Hermann Weyl wrote to Bernays from the Palace Hotel Sass Maor in S. Martino Di Castrozza in the Dolomites: Gentzen sent me his manuscript on the consistency of arithmetic; what do you think of it? I hope for an ongoing discussion in Princeton between you, G¨ odel, von Neumann and the foundations people of the Princeton school.141

Bernays wrote from the Institut Montana (Zugerberg) by return mail on 6 August 1935:142 . . . From Gentzen you have presumably received the plan for his new work. (He wrote me that he wished to send it to you.) In it the proof of the consistency of the number theoretic formalism is finally produced in a satisfying manner. (The earlier G¨ odel-Gentzen proof by means of reduction to intuitionistic arithmetic I find—and it possibly fares the same with you—from the standpoint of intuitive evidence as not truly satisfying. Mr. Gentzen himself had only viewed it as provisional.)143

Bernays looked forward in the ensuing text to his trip to Princeton for the academic year.144 Already on 17 January 1935 Weyl thanked Bernays in a letter from Princeton for a copy of his book on foundations145 and remarked: I have already read much in it and am completely bowled over by the content and presentation. In the immediate future it should play the rˆ ole of the standard work on the foundations of mathematics! . . . I wish you the best for the next academic year (October 35 to May 36) at our Institute. I endeavoured besides on Gentzen’s behalf and just yesterday received the news from G¨ odel that he could not come this year and his visit could likewise be pushed back to next year. That would have been a pretty combination! But in your case I haven’t failed.146 140 Hs.

975:1651. 91:12, Nachlass Weyl, ETH Z¨ urich. I thank Frau Dr. Flavia Lanini, who brought the passages concerning Gentzen in the correspondence between Bernays and Weyl and between Bernays and Ackermann to my attention. 142 Evidently one of the two correspondents misdated his letter. 143 Hs. 91:12a, Nachlass Weyl, ETH Z¨ urich. 144 Cf. the longer note at the end of this chapter. 145 This would be the first volume of Hilbert-Bernays, about which cf. footnote 193, below. 146 Hs. 91:10b, Nachlass Weyl, ETH Z¨ urich. 141 Hs.

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At the end of 1935 Gentzen revised his work fundamentally. It appeared in 1936 in Mathematische Annalen. The work was immediately widely reviewed and increased Gentzen’s fame. 22. Difficulties with the “Widerspruchsfreiheit der reinen Zahlentheorie” of 11 August 1935 How were things going with Gentzen’s work? In 1934 Gentzen had written Bernays, “Incidentally, I have considerably furthered my researches on consistency.” On 26 March 1935 he thanked Bernays for suggested textual improvements—thus, not contentual changes. On 11 August 1935 Gentzen’s manuscript of “Widerspruchsfreiheit der reinen Zahlentheorie” [Consistency of pure number theory] was received by Otto Blumenthal, managing editor of Mathematische Annalen. Gentzen must have received some disquieting news from Bernays, for on 4 November, from the third story of Feuerschanzengraben 18, he wrote in answer: Very respected Herr Professor! Many thanks for your letter, which I received today. I believe your worries about my consistency proof are unfounded. The trains of thought which run through your letter are not new to me; I have myself already thought over all these connections even to and including the geometric picture of the Streckenverzweigung.147 You are completely correct therewith, that. . . But that matters not for my proof-ideas! I must admit that passage 14.6 is unclear in that it can create the impression as if it deals with a definite sequence of reductions. . . I have myself now decided to insert the following sentences (and have today sent the corresponding notice to Prof. Blumenthal, as I had already returned to him the corrected sheets a few days ago): . . . I like to think that there is nothing objectionable about this sort of proof-idea. . . The monstrous interconnection of dependencies, in my opinion, is forced on one; this peculiar chain of inductions is necessitated by the consistency proof. I don’t wonder that you, by “eliminating” the sigma-induction, could not reach the goal, because then one inevitably stands before a chaos of ramifications. I have made every attempt via the lemma to clarify the actual form of the production of the reduction rules. . . but I have not succeeded. On this point it seems to me most likely an attack on my proof procedure to be possible, and, to be sure, insofar as the concept “finite” in the proof of the lemma will be somewhat charitably interpreted. On this point you can, please, point out to everyone who wishes to criticise the proof, but I would find it better if one would carry through an improvement in the sense of further finitisation (passage 15.1.1), from the viewpoint of extremely strict finitism. It would interest me very much to have your response to these remarks; unfortunately the exchange of thoughts over the ocean is impeded by the huge loss of time. With respectful greetings—and please to Herr Prof. Weyl, whom you probably see often— Your Gerhard Gentzen148 147 Untranslatable. Strecke translates as stretch or section; Verzweigung as branching. Gentzen’s proof centres on the notion of a reduction of a sequent. (Cf. von Plato’s appendix for some orientation.) Each sequent has a number of reducts, each of which has a number of such and the overall collection is a tree with many branches.—Trans. 148 Hs. 975:1652

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Georg Kreisel holds the objections of those days to be unfounded. Paul Bernays, who was then in Princeton, would have shown the original version of the consistency proof—as referred to here in Gentzen’s letter—to Hermann Weyl. Perhaps Weyl had also shown the carbon copy he had received from Gentzen to G¨ odel and von Neumann. Bernays—so much is certain—had presented some of Gentzen’s results in lectures at the Institute for Advanced Study and written them down in his “Logical Calculus. . . ” of 1935/1936.149 Gentzen’s original proof was posthumously published in 1974 with commentary by Paul Bernays. Possibly, Kreisel speculatively suggests, G¨odel and von Neumann criticised the proof incorrectly.150 This requires clarification, if such is at all possible. In the meantime Gentzen published a review in the Zentralblatt f¨ ur Mathematik 14, No. 9 (1935), p. 385: Malcev, A.: Untersuchungen aus dem Gebiete der mathematischen Logik. Rec. math. Moscou, N.S. 1 (1936), 323-335. A theorem of G¨ odel: “For the satisfiability of a countably infinite system of formulæ of the propositional calculus it is sufficient that every finite part of the system be satisfiable.” (Cf. Mh. Math. Phys. 37, pp. 349-360, Theorem X), is generalised to systems of arbitrary cardinality.— A second theorem (on the extension of an infinite satisfaction domain for a system of formulæ of the predicate calculus), which is incorrectly described as a generalisation of a theorem proven by Skolem (this Zbl. 7, 193 and 10, 49), is, in the form of the author, a triviality. Gerhard Gentzen (G¨ ottingen).

Then Gentzen received a letter from G¨ ottingen, probably from Hasse. 19.10.35, Herr Dr. Gentzen, Stralsund, Schillerstr. 35 Very respected Herr Gentzen! As you know, Herr Dr. Arnold Schmidt is surrendering his position as assistant at the Mathematical Institute at the end of the winter semester, which particular assistance work is planned for Hilbert. Hilbert had in mind after consulting with me to give you this assistantship, in case you are inclined to accept it. Mind you, Hilbert and I would like to consult with you personally beforehand. It would be desirable if you could make it possible to present yourself here as soon as possible, until then naturally there is no commitment on either side. It would then be appropriate if you will shortly let both Hilbert and me know on which day you think to be here. Heil Hitler! The Managing Director of the Mathematical Institute

Gentzen answered immediately from Berlin, where he was probably visiting his sister: Berlin on 21 October 1935 Very respected Herr Professor! I thank you for your letter of 19.10. I am happily prepared to speak with you and Herr Geheimrat Hilbert on the possible assumption of the assistantship. I will travel to G¨ ottingen on Thursday, 24 October, and will be in the Mathematical Institute on Friday around 10 a.m. to learn what time I should see you. 149 Paul Bernays, Logical Calculus (1035/1936). Notes by Prof. Bernays with the assistance of Mr. F. A. Ficken, Institute for Advanced Study, 125 pages. (A copy is in the Fine Hall Library, Princeton University.) 150 Cf. pp. 200ff. in: Georg Kreisel, “Wie die Beweistheorie zu ihren Ordinalzahlen kam und kommt”, in Jahresberichte der Deutschen Mathematiker-Vereinigung 78 (1976), pp. 177223; and, Georg Kreisel, “G¨ odel’s excursions into intuitionistic logic”, in: Paul Weingartner and Leopold Schnetterer (eds.), G¨ odel Remembered, Bibliopolis, Naples, 1987. However, that G¨ odel and von Neumann actually criticised Gentzen’s proof, and not only Bernays and Weyl, is unverified conjecture.

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Heil Hitler! Gerhard Gentzen at the moment at Berlin-Lichterfelde, Gersauerweg 10

The knot of material worries came undone in a letter to H. Kneser of 27 October 1935 from Feuerschanzengraben 18, 3rd story, in G¨ ottingen: Very respected Herr Professor! A sudden change in my future plans has arisen: Hilbert and Hasse have, entirely unsuspected by me, offered me the unscheduled assistantship by Hilbert opening up on 1 November. Naturally I accepted, though I must yet wait for the approval of the rector and the Dozentenschaft.151 A. Schmidt, who till now held the position, wants to go to Marburg to be with Reidemeister because he believes he has no possibility for a speedy habilitation in G¨ ottingen in view. With Hilbert there is no longer much to do, therefore it would be expected that I participate in other Institute business, especially to work in other research areas, outside of logic, as you have also advised. That I would gladly do in the interest of my progress. For habilitation there will perhaps sooner or later be the opportunity here in G¨ ottingen. For my work I have in the meantime received the first proof corrections. Van der Wærden wrote me a letter, in which he spoke very appreciatively of my consistency proof. Please give your wife my greetings. Heil Hitler! Your Gerhard Gentzen

From 1 July 1935 to 31 October he was paid a stipendium by the Deutsche Forschungsgesellschaft (DFG). The question arises why Gentzen received a stipendium of that kind. The award of a stipendium was—in accelerated obedience— characterised by political-nationalistic perspectives since 1933. Already in the spring of 1933 the society had refused to support Jewish scientists. True Nazi Party members received lengthier and more highly remunerative stipends than nonpolitical scientists. With Gentzen it probably had to do with the fact that in mathematics a high percentage of exiled or emigrated scientists were no longer receiving stipends. In the Hoover Institute Archives,152 one can find the following: The representative of the Dozentenschaft in G¨ ottingen could only find out about Gentzen that two scientific works of the applicant exist, which deal with “mathematical logic and similar things” and are deemed “unproductive” by the local specialists. In addition through a little snooping it was found that a few days earlier a letter for the applicant arrived from Jerusalem: “which anyhow allows one to conclude good connections to the representatives of the chosen people.”

This anti-Semitic judgement was ignored, and a month later a handwritten notice reads, “On 24 June 1935, approved: RM 125.- from 1 July 35 to 31 Dec. 35.” Lothar Mertens believes that this would explain why there are no files on Gentzen’s grant available from the DFG (6500 files of the DFG are stored in Koblenz under R73.). The DFG had ceremoniously suspended the support of Jewish scientists and destroyed the relevant records. The dates mentioned above are in agreement with Gentzen’s letter of 25 June 1935 (page 57 above). As seen, Gentzen was informed upon. Through the letter from Jerusalem he possibly came under suspicion of being closely related to the Jews. That could be the reason why the granting of his support, especially that of his subsequent application, was so long delayed. And that he had no prospect for a professorship. 151 A 152 I

National Socialist organisation for docents. Membership was compulsory. thank Dr. Lothar Mertens (9 August 2004) for this reference.

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Gentzen withdrew the application for an extension himself because he had obtained the prospect of a position as Hilbert’s assistant. The leader of the Dozentenschaft was W. Blume.153 But who could the informant have been? And who in Jerusalem could have been interested in Gentzen? Perhaps Adolf Frænkel? Gentzen submitted an application to extend the stipendium on 11 October, but withdrew it in view of the prospect of a position as assistant. On 16 October supplements were requested, which were received on the 19th. The application for extension with supplements went on 24 October “to Prof. Dingler for comment.”154 On 28 October 1935 Hasse wrote to the Dean: Herr Schmidt had the special commission to provide Herr Professor Hilbert’s ongoing scientific assistance. With hindsight on the personality of Hilbert it seems to me an essential requirement to try to find for him a new assistant from the Mathematical Institute well-versed in his current research area. To this end, Hilbert has expressed the desire to have Herr Dr. Gerhard Gentzen appointed.

And he requested the dean to transfer the assistantship vacated by Arnold Schmidt to Gentzen. 23. The Unscheduled Assistantship under Hilbert from 1 November 1935: A Productive Period for Foundational Research Begins On 1 November 1935 Gentzen received the unscheduled assistantship under David Hilbert, as Dr. Arnold Schmidt (who subsequently invited Gentzen to review in Zentralblatt f¨ ur Mathematik ) resigned this post. Arnold Schmidt had, as also Kurt Sch¨ utte in 1933, been promoted by Hilbert—with the aid of Paul Bernays. From 1 October 1934 to 1 July 1935 Gentzen was financially supported by private means. Now he received a monthly salary of RM 190, that is for the duration of this two-year period he had RM 2280 in sight. On this he could live alone. For his position he had to produce certificates. On 28 October 1935 the leader of Sturm 16/42 and senior troop leader Hilgendorff confirmed that the “SA man” G. Gentzen had belonged without interruption to the SA since 5 November 1933. The dean, the Dozentenschaft under W. Blume,155 and the rector agreed to Gentzen’s appointment. On 6 December 1935 he made a written declaration under oath that he had “never belonged to any lodge or lodgelike organisation or a reserve organisation of such”.156 From 1935 to 1938 Gentzen gave individual lectures at the universities of Greifswald, G¨ottingen, Leipzig, M¨ unster and T¨ ubingen. On 1 March 1936 he became a member of the National-Sozialistischen Lehrer Bund (NSLB) [National-Socialist Teacher Union] with member number 335247. An actual task of Gentzen’s was to read Schiller’s poems to Hilbert.157 153 Cf.

footnote 155. Document Centre, personal file “Gentzen, Gerhard”. That even the DFG chose the anti-Semite and Hilbert-opponent Hugo Dingler to evaluate Gentzen is a wicked backstage joke. Cf. Hugo Dingler’s memorandum, cited in note 7 at the end of the chapter. 155 W. Blume was an old comrade-in-arms of Freisler in Marburg and Oberhessen. Freisler was president of the Volksgerichtshof (Peoples’ Court of Justice) and Hitler’s juridicial bloodhound. 156 That is, he was not a freemason or a supporter thereof. 157 Constance Reid, Hilbert, Springer, New York, 1970, p. 212. (This was reported to Reid in a letter of 19 July 1966 from Elisabeth Reidemeister.) Cf. also the longer note at the end of this chapter. 154 Berlin

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24. Consistency of Type Theory In 1936 Gentzen published “Die Widerspruchsfreiheit der Stufenlogik”.158 159 F. Bachmann (1909-1982) reviewed it: One must distinguish the question of the consistency of pure type theory from the question of the consistency of the applied type theory, or, more precisely, the formalisms which arise when one extends the axioms and rules of type theory through the addition of axioms which characterise a given domain of objects. The pure type theory must be consistent, if one can give a domain of individuals so that the type theory in application to this domain is consistent. The author proves the consistency of pure type theory in that he shows that the applied type theory on the simplest non-empty domain of individuals, which consists of a single element, is consistent. (With the same basic ideas Hilbert and Ackermann have proven the consistency of the elementary predicate calculus.) Through modification of the proof the consistency of type theory applied to arbitrary finite (but not yet all (in particular, infinite)) domains of individuals is easily proven. By type theory the author understands (as is usual today) the logic of Principia Mathematica without the ramified type theory and without special class symbols.160

In this paper Gentzen proves the consistency of simple type theory. In it one reads: All the considerations employed in the proof almost yield themselves if one keeps in mind the contentual sense of the γ’s161 (2.1, 2.2), such as the formula and rules of inference (§1). It is then easy to see that the entire proof allows extension to the case in which one takes as basis an arbitrary finite domain of individuals. It remains the case that the number of possible sets at each type level is finite. From this it follows that type theory remains consistent if one adds axioms which assert the existence of a fixed finite number of individuals. If, however, one attempts here to allow infinitely many individuals (“axiom of infinity”), then the situation is entirely different, today not yet clarified. The above consistency proof is, obviously, completely “finite” in the sense of Hilbert’s proof theory; it contains only inferences and concepts of the elementary kind.162

Arnold Schmidt wrote: Gerhard Gentzen: Die Widerspruchsfreiheit der Stufenlogik. Math. Z. 41, 357366 (1936). The author proceeds from a precise formalisation of type theory (with unramified type distinction). Its rules of inference display the extension 158 Mathematische

Zeitschrift 41, No. 3 (1936), pp. 357-366. literally translates to step logic, the word Stufe was introduced in this context by Frege. In English, one follows Russell’s lead and refers, however, to type theory. Type theory is easy to describe. Given a domain of individuals, e.g. real numbers, one labels the elements of the domain as type 0. Sets of objects of type 0 are objects of type 1, sets of objects of type 1 are objects of type 2, etc. In the formal language of type theory there are variables ranging over each of the types. 160 Jahrbuch u ¨ber die Fortschritte der Mathematik 62, No. 1 (1936), pp. 43ff. The type theories of Frege and Gentzen, although related, are not comparable. (Chr. Thiel, Grundlagenkrise und Grundlagenstreit, Verlag Anton Hain, Meisenheim am Glan, 1972, p. 100.) On the rela¨ tion between Frege and Gentzen, cf. in the meantime Schroeder-Heister, “Uber einige historische Hintergr¨ unde des Sequenzenkalk¨ uls”. 161 The γ’s were constants naming everything. γ 0 would be the unique individual of type 0. 1 γ11 and γ21 the two sets { } and {γ10 } of type 1, etc. 162 p. 366. 159 Stufenlogik

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of the rules of inference for predicate logic to arbitrary types; as axioms it contains besides the correct formulæ of propositional logic an extensionality axiom (“Axiom of Identity”), a comprehension axiom . . . and a choice axiom, but no axiom of infinity (by the inclusion of which the consistency problem has not yet been established). For this type theory—in a generalisation of the basic ideas of the consistency proof for the predicate calculus—a finitistic consistency proof is carried out. (The author asks to announce that, as he subsequently became aware of, a consistency proof of a type theory of the same sort has already been given by Tarski in Mh. Math. Phys. 40 pp. 97ff., reviewed in this Zbl. 7, p. 97. Though there the outline of a type theory and the proof are slipped in among more general contemplations.) Arnold Schmidt (Marburg, Lahn).163

Alonzo Church wrote: The system here designated as “Stufenlogik” can be roughly described as the system of Principia mathematica, with omission of the axiom of infinity, but not of the multiplicative axiom (axiom of choice). The original “verzweigte Typentheorie” of Whitehead and Russell is, however, replaced by the simplified theory of types, and the axiom of reducibility therefore dispensed with. An axiom of extensionality, to the effect that equivalent predicates are identical, is included. The author gives for this system a consistency proof which is finite in the sense of Hilbert. The method is a straightforward generalization of that used by Hilbert and Ackermann to establish the consistency of the “engere Funktionenkalk¨ ul”.164

On 28 September 1939 Paul Bernays wrote to Kurt G¨ odel among others: I wanted to inform you that recently Mr. Gentzen has considered that the method of interpreting the classical propositional and predicate logics in an intuitionistic setting easily extends beyond the domain of the number theoretic formalism to simple type theory. Naturally this does not mean an intuitionistic interpretation of simple type theory; because it would not eliminate thereby the intuitionistically unallowable sort of impredicative definitions and existence proofs.165

25. Revising the Proof of the “Widerspruchsfreiheit der reinen Zahlentheorie” On 11 December 1935 Gentzen wrote again to Bernays: Very respected Herr Professor! Many thanks for your letter. I must admit that the critical conclusion in my consistency proof is not satisfactory. I have therefore decided to revise the proof fundamentally, and I hope that I will succeed in completely overcoming the difficulty. The publication is withdrawn for the time being. The possible changes indicated by G¨ odel were known to me, but are in fact inapplicable from the finite standpoint because of their impredicative character. There seems to be little interest in foundational questions here in G¨ ottingen. Only with the Frenchman Cavaill`es,166 who is here for a couple of weeks, have I occasionally discussed such things. I should probably receive the Hilbert assistantship vacated by Herr Schmidt’s departure; it is, however, not yet decided. I send Hilbert’s greetings; he asks that you occasionally write of your condition. 163 Zentralblatt

f¨ ur Mathematik 15, No. 5 (1936), p. 193. of Symbolic Logic 1 (1936), p. 119. 165 Kurt G¨ odel, Collected Works, Vol. IV, pp. 122ff.; here, pp. 127ff. 166 Jean Cavaill` es had, with Emmy Noether, already concluded in March 1933 the preparations for the publication of the correspondence between Georg Cantor and Richard Dedekind. 164 Journal

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With most devoted greetings and many Christmas wishes Your G. Gentzen167

Gentzen reported on 15 January to the very respected Herr Professor Bernays, I have in the last weeks carried out a complete revision of my consistency proof and believe that now the critical point has assumed a more agreeable form. It would mean a great deal to me that you could give your judgement on it, and possibly again make a note of it to Mr. Blumenthal. For this I enclose that part of the proof, which in concise form (and entirely independent of the rest) contains the essential steps for your assessment. The remaining part of the proof is not yet finished in all particulars, but seems to me as good as established, that it is in order; the inferences applied therein are in all cases entirely elementary and are without question finitary.168

The letter ends with the news that he has recently received the assistantship by Hilbert. On 3 March 1936, Gentzen further informed Bernays that “the new version of my proof is completely finished and has lately been sent to Blumenthal.”169 Gentzen further sketches his “thoughts” on the future possibilities for proof theory: I also don’t know if I can claim any “priority” in all particulars; I wish only this once to survey further work on the programme from the now established points, the carrying out of which admittedly can require some years or decades. How far are you with the second volume of your book? It would interest me greatly if you would sometime write me about it. Also, if I can be of any use to you in the preparation, with proof correction or such, I am willing. . .

In Sigmaringen, on 2 May 1936 for the edification of the National Socialists, the memorial plaque for Alfons and Theodor Bilharz was unveiled at “Eckh¨ ausle”, Antonstraße 1, in the presence of local party officials. Gerhard Gentzen, like all family members, was present. 26. “Die Widerspruchsfreiheit der reinen Zahlentheorie” “Die Widerspruchsfreiheit der reinen Zahlentheorie” appeared in Mathematische Annalen 112 (1936), pp. 493-565. Even today Gentzen’s achievement merits mention in the best encyclopædia of the world: The best-known consistency proof is that of the German mathematician Gerhard Gentzen for the system N of classical (or ordinary, in contrast to intuitionistic) number theory. Taking ω (omega) to represent the next number beyond the natural numbers (called the “first transfinite number”), Gentzen’s proof employs an induction in the realm of transfinite numbers (ω + 1, ω + 2, . . . ; 2ω, 2ω + 1, . . . ; ω · 2, ω · 2 + 1, . . .), which is extended to the first epsilon number ε0 (defined as ω the limit of ω ω , ω ω , . . .), which is not formalisable in N. This proof, which has appeared in several variants, has opened up an area of rather extensive work. . . Here, the non-elementary method (not formalizable in N) is an extension of mathematical induction from the natural number sequence to a certain segment of the transfinite ordinal numbers that Cantor had introduced for extending the counting process beyond that provided by natural numbers.170

And Gentzen’s excellent colleague Kurt Sch¨ utte wrote long after the war: 167 Hs.

975:1653. 975:1654. 169 Hs. 975:1655. 170 Encyclopedia Britannica, 1989, 14th edition. 168 Hs.

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The transfinite induction certainly transcended the finite standpoint, as by G¨ odel is necessary, but it proceeds in a completely constructive way, so that the proof of Gentzen is seen as a testimonial for pure number theory in the sense of the extended Hilbert Programme. . . Through the researches of Gentzen it first became known that the proof theoretic strength of a formal system may be indicated by an ordinal number. Today one signifies as the proof theoretic ordinal of a formal system of mathematics the least ordinal α with the property that no recursive well-ordering of ordinal type α may be proven well-ordered in the system in question. As Gentzen proved in 1943 in his Habilitation thesis, ε0 is the proof theoretic ordinal of pure number theory.171

Gentzen wrote in “Die Widerspruchsfreiheit der reinen Zahlentheorie” in 1936: By “pure number theory” I mean the theory of natural numbers without the application of tools of analysis, as for example the use of irrational numbers or infinite series. The goal of the treatise at hand is to prove the consistency of pure number theory, or more correctly stated, to reduce it to a certain foundation of a general form. How such a consistency proof is at all possible and for what reasons it is necessary or at least very reasonable will be discussed in the first section. The whole treatise can be read with no special prior knowledge.

And without thinking, Gentzen cited Weyl, Frænkel, G¨ odel, J. Herbrand, J. von Neumann, A. Church—thus foreigners, Jews, emigrants, and “archenemies”. That would become problematic in 1937. For example, Kurt Schilling attacked Heinrich Scholz on account of his “Was ist Philosophie?” [What is philosophy?] of 1939/40 as follows: None of this will do. If Herr Scholz possesses a certain courage by which in the midst of a war he recommends to the German people, as the only possible philosophy one whose leading and only present exponents cited by him are Poles, Englishmen, emigrants, and Americans, and if he openly declares that he shapes his teaching as a German Professor in Munich 1939/40 “after the Warsaw example” (p. 51), then it seems to me this courage could find better use. (p. 49).172

The contemporary review of F. Bachmann (1936)173 should find a place here because of its easy understandability: Till now consistency proofs using finite means were available only for such fragments of number theory which did not encompass the axiom of complete induction. The consistency of complete number theory could, to be sure, be derived from the consistency of the intuitionistic arithmetic following ideas first announced by G¨ odel; however this solution would not be viewed as satisfactory from the finite standpoint, for from this standpoint reservations against the intuitionistic application of the implication sign174 must be raised. For the first time, the author now proves the consistency of the inference modes of complete pure number theory by finite means. Pure number theory is here thought of (´ a la Hilbert and Bernays) as given by a calculus which encompasses the elementary predicate calculus including the identity axioms, the formalised Peano axioms, 171 Kurt

Sch¨ utte and Helmut Schwichtenberg, Mathematische Logik, in: Gert Fischer, Friedrich Hirzebruch, Winfried Scharlau, and Willi T¨ ornig (eds.), Ein Jahrhundert Mathematik 1890-1990. Festschrift zum Jubil¨ aum der DMV, Vieweg Verlag, Wiesbaden, 1990, p. 724. 172 Cf. Kurt Schilling, “ ‘Zur Frage der sogenannten ‘Grundlagenforschung’. Bemerkungen zu der Abhandlung von Heinrich Scholz: Was ist Philosophie?”, Zeitschrift f¨ ur die gesammte Naturwissenschaft 7 (1941), pp. 44-48. 173 Jahrbuch u ¨ber die Fortschritte der Mathematik 62, No. 1 (1936), pp. 44ff. 174 The intuitionistic interpretation of implication is quite strong and not unproblematic.— Trans.

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and recursive definitions. Such a formalism can prove to be too narrow with respect to the practical needs of research and, besides, it is necessarily incomplete by known mathematical theorems. It is therefore important that the consistency proof of the author be laid out so that it remains applicable under extensions of the formalism. By a theorem of G¨ odel the consistency proof for pure number theory is only possible if some mode of inference is applied which cannot be represented in pure number theory. As such a mode of inference the author applies in his proof a rule of transfinite induction in the domain of ordinal numbers up to Cantor’s first epsilon number. By a remark of Bernays this transfinite induction cannot be replaced in a consistency proof for pure number theory by a weaker mode of inference. The author precedes his consistency proof with sections on the formalisation of pure number theory and on questionable and unquestionable modes of inference in pure number theory, in which he gives a detailed presentation of the Hilbert Programme. Of these details the characterisation in the third section of finitary modes of inference is particularly important, so by the prior detailing of the hallmarks of the finite standpoint special emphasis is placed on the “finite-combinatorial” modes of inference which, as is known, do not suffice for a consistency proof of number theory. The consistency proof is carried out so: a concept of the contentual correctness of the provable formulæ of number theory is introduced and a reduction procedure for the derivations of these formulæ is defined. Through this reduction procedure the question of the contentual correctness of a derivation and thereby of the end-formula of the derivation is reduced to the question of the correctness of simpler derivations, and to be sure (this is characteristic of the applied procedure) in general to the correctness of derivations of a countably infinite set of simpler derivations. In order to show that this reduction procedure always leads after finitely many steps to the simplest derivations, which consist of only one formula the correctness of which one can convince oneself through elementary calculation, the author assigns to the derivations finite decimal fractions which form a well-ordered set of order type the first epsilon-number. The finiteness of the reduction procedure is then deduced with the aid of the transfinite induction rule in the domain of these decimal fractions. The transfinite induction rule is to be interpreted finite-constructively.

Arnold Schmidt wrote in Zentralblatt f¨ ur Mathematik,175 Previous attempts at a consistency proof for arithmetic (Ackermann, HilbertAckermann, Herbrand, etc.) certainly encompass a large part of this discipline, but do not exhaust all modes of inference that one customarily considers as belonging to it, that is to “pure number theory”, but rather find (by certain bound-variable-including recursions) their restrictions. That such is no accident follows from G¨ odel’s Theorem; because the finite modes of inference employed by the named authors are all formalisable within arithmetic, they do not suffice for its consistency proof. The question arises whether all modes of inference reliable from a finite point of view must be contained in a formalism A of arithmetic or whether finite modes of inference can be found which tower above A and by their adoption a consistency proof for A can be obtained. (The simultaneously given proof by G¨ odel and by Bernays and Gentzen of the consistency of arithmetic from the intuitionistic standpoint does not answer this question, for in some essential points intuitionism transcends the finite point of view; but even in this proof—in transcending the finite idea of a consistency proof—the intuitionistic modes of 175 Vol.

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inference are not only metamathematically applied, but moreover the assumption that intuitionistic arithmetic is consistent is explicitly made.)—In the work before us now for the first time a consistency proof for the whole of arithmetic from an extended finite standpoint has been carried out. In this proof the constructive use of a “transfinite induction” singles itself out as the mode of inference used that towers above A. (The carrying over of the familiar term “transfinite” is no cause for alarm.) The consistency of proof-figures from A are deduced by means of a reduction procedure, with regard to which the proofs are divided into classes, which are ordered like numbers of the second number class. The proof that the reduction procedure can be carried out for an arbitrary proof-figure of A leads unavoidably to a “transfinite induction”, which via a detour to a class of decimal fractions works itself out as a stepwise constructively surveyable mode of inference, which (if it extends beyond the framework of what till now has been presented as finitary) to a great extent is geared to finitary thinking. (In the restriction of such a concrete mode of inference the assumed standpoint is clearly distinct from, for example, intuitionism.)— Gentzen precedes his proof with a detailed explanation of the necessity and possibility of consistency proofs; and he finishes with a reflection on the safety of the modes of inference applied.

Arnold Schmidt reviewed Paul Bernays’ “Quelques points essentiels de la m´etamath´ematique”, L’Enseignment Math. 34 (1935), pp. 70-95, in Zentralblatt f¨ ur Mathematik 176 and said, under Point IV: The basic idea of the consistency proof for axiomatic number theory Z simultaneously introduced by Bernays and Gentzen and by G¨ odel by grounding it on odel’s Theorem the consistency of intuitionistic number theory is outlined. By G¨ a consistency proof for Z must be elementary-combinatoric, but not formalisable in Z. The “transfinite induction” is shown as an example of an elementary (and intuitionistically permissible) mode of inference not formalisable in Z.— In the meantime (Math. Ann. 112, pp. 493ff.) Gentzen has carried out a consistency proof for number theory, which from the elementary-combinatorial standpoint transcends the framework of Z only in a (constructively acceptable) application of transfinite induction.

Gentzen’s original proof was first posthumously published and commented on in 1974 by Bernays, who had also reviewed Gentzen’s work in 1936.177 In the presentation of Gentzen’s original section IV of his work, Bernays described the results of their correspondence so:178 Of the constructive proofs of the consistency of classical number theory, which— unlike that by means of the intuitionistic re-interpretation of the logical connectives—do not use the general concept of contentual proof, the first published is that in Gentzen’s treatise “Die Widerspruchsfreiheit der reinen Zahlentheorie” (Math. Annalen 112 (1936)). The consistency proof given here in section IV is however not the original proof conceived by Gentzen but rather a reworking of this. Against his original proof the methodic objection was raised that he implicitly used that principle today mostly referred to as the “fan theorem”, by which any branching figure which branches only finitely many times at any place and

176 Vol.

14, No. 3, p. 97. Paul Bernays, “Review of Gerhard Gentzen, Die Widerspruchsfreiheit der reinen Zahlentheorie”, Journal of Symbolic Logic 1, No. 2 (1936), p. 75. 178 Archiv f¨ ur mathematische Logik und Grundlagenforschung 16, Nos. 3-4 (1974), pp. 97118. 177 Cf.

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in which each path breaks off after finitely many section-pieces has in the whole only a finite extent.179 Gentzen did not oppose the objection, but rather he gave his proof a different form by means of the introduction of constructive ordinal numbers and the application of an ordinal induction, which is the constructive counterpart of the set theoretic transfinite induction up to the first Cantorian epsilon-number. The original proof still exists however in galleys and consideration of this proof shows that the methodic objection mentioned was unjustified. As a genuine aid in this proof, the concept of a “reduction rule” is used. The application of such a reduction rule yields admittedly, on the grounds of a certain freedom of choice, a sort of branching figure. It is not however grounded on a general theorem on the finiteness of branching figures, but rather it belongs more to the concept of the reduction rule that in the given branching figure it always breaks off in finitely many steps, and the proof of the existence of a reduction rule always proceeds in this sense. In the reworking of the proof the concept of a reduction rule enters, but only the concept of a reduction step on a derivation is actually used, which is strictly elementary. Thus the non-elementary is now relegated to the importation of the theory of ordinals, whereby, to be sure, the introduction of ordinal numbers is yet entirely finitistic, but a form of induction is required which can no longer be grounded in the framework of finite considerations. In view of this situation the original version of Gentzen’s consistency proof certainly acquires a methodological interest. Also, this version may be easier to understand than the reworked proof.

Bernays corresponded with G¨odel on this matter180 and reported that he had already spoken in Princeton on the cause he had for opposition to the proof until Gentzen “removed it with the help of induction up to ε0 . Today I no longer believe that Gentzen’s original proof required the fan theorem.” 27. Gentzen Was an Intellectual Independent Gentzen was completely independent intellectually. Bernays’ stance to Gentzen is not unique. It is clear from various pronouncements that Bernays favoured his own platonic approach but worked on Hilbert’s approach and therefore on the works of W. Ackermann. Here however is a self-report by Gentzen.181 We know he wrote it himself, for the material is prefaced by the words: The German mathematical research in 1936. . . In principle only self-reports of the authors will be admitted here.

Thus the author of the following summary is Gentzen. This assumption is supported by the mention of Gentzen in the list of contributors to this issue. Gerhard Gentzen 1. Die Widerspruchsfreiheit der reinen Zahlentheorie, Math. Ann. 112 (1936), pp. 493-565. Serious difficulties in the foundations of mathematics, which are bound to the concept of the infinite, led Hilbert to set up the requirement that the consistency of all subareas of mathematics be proven. In the consistency proofs themselves the concept of the infinite must occur in only the smallest measure and in a form 179 In

simpler terms: A finitely branching tree in which all paths are finite is itself finite. Kurt G¨ odel, Collected Works, Vol. IV, pp. 276ff.; here, p. 278. 181 Deutsche Mathematik 2 (1937), p. 130. 180 Cf.

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that can on the basis of careful considerations be deemed sufficiently unobjectionable (“finite” standpoint, “constructive” conception of the infinite)— In the treatise named such a consistency proof for the domain of pure number theory, i.e. number theory without analytic means, is carried out for the first time. The earlier proofs did not succeed for the full inclusion of complete induction. At this place there were difficulties of a fundamental nature; it was conjectured in connection with a theorem of G¨ odel that Hilbert’s plan could not be carried out this far at all.— In the treatise the reasons are explained further, why a consistency proof appears reasonable, and the “metamathematical” proof methods are developed without assuming prior familiarity. 2. Die Widerspruchsfreiheit der Stufenlogik, Math. Z. 41 (1937), pp. 357-366. The consistency of the so-called type theory is proven by a simple method. As came to light after publication, nearly the same was carried out already in an older work of A. Tarski, “Einige Betrachtungen u ¨ ber die Begriffe der ω-Widerspruchsfreiheit und der ω-Vollst¨ andigkeit”, Mh. Math. Physik 40 (1933), pp. 97-112.

It appears that Gentzen’s first step in tackling the consistency proof for analysis was by means of type theory. It is clear, and is borne out by an unpublished manuscript of his, that Gentzen foresaw great difficulties in this.182 And the novelty of Gentzen’s method would quickly be clear to Bernays as well. Bernays reviewed Gentzen’s work in the Journal of Symbolic Logic: Gerhard Gentzen: “Die Widerspruchsfreiheit der reinen Zahlentheorie”, Mathematische Annalen, Vol. 112 (1936), pp. 493-565. By the name “reine Zahlentheorie” Gentzen denotes the domain of the usual natural number theory, including the classical form of logical reasoning, but not the concept of a set or of a sequence. The formalizing of this domain can be effected on the basis of the logical calculus of first order (“engere Funktionkalk¨ ul” or “Pr¨ adikatenkalk¨ ul”) by adding equality axioms, the formalized Peano axioms, and recursive definitions. For the purpose of proving the consistency of this formalism the elementary combinatorial methods of metamathematics have been found to be insufficient. On the other hand, as first shown by Kurt G¨ odel, the question of consistency for this formalism can be reduced by a rather simple transformation to the corresponding question for the Heyting calculus of “intuitionistic arithmetic”. This leads immediately to a proof of consistency for number theory by interpreting the Heyting calculus. However this method of proof is not quite satisfactory from the intuitive point of view, because the interpretation of the Heyting calculus (which must, of course, be made without applying the tertium non datur ) requires the unanalyzed notion of consequence of an assumption, or instead of that, after Kolmogoroff, the general notion of reducing a problem to another (where each of the problems can again consist in a reducing, and so on). In the present paper Gentzen gives a new consistency proof for number theory. Here the transgressing of the elementary methods is to be found in a transfinite induction up to the first Cantor ε-number. This induction is used to prove that a certain reduction process, if applied to any deduction of the number-theoretic formalism (the deductions being first brought into a certain normal form), comes

182 Cf. the manuscript transcribed by Hans Rohrbach and H. Kneser, now in the possession of Chr. Thiel, Erlangen.

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to an end after a finite number of steps. In this way the deducible formulas are characterized by a property of “reducibility”. The ω-consistency of number theory likewise results.183 The Gentzen treatment does not presuppose anything of metamathematics or of set theory. He gives an account of the problems leading to the Hilbert program and discusses the questions connected with the “finite point of view”. The form of the calculus which he chooses is close to that of his thesis (“Untersuchungen u ¨ ber das logische Schliessen”, Mathematische Zeitschrift, Vol. 39 (1934-35), pp. 176-210, 405-431). The special transfinite induction applied is handled independently of the general theory of ordinals and of the theory of the Cantor second number class, the ordinals below the first epsilon-number being represented by special finite decimals for which the order of magnitude is already a well-ordering. By combining the Gentzen proof with the general G¨ odel theorem on consistency proofs (Monatshefte f¨ ur Mathematik und Physik, Vol. 38 (1931)), the result is gotten that transfinite induction up to the first ε-number cannot be obtained as a derived rule in the formalism of number theory. Also as a consequence of the G¨ odel theorem it can be shown that, for the purpose of proving the consistency of number theory, this transfinite induction cannot be replaced by a lower one. Paul Bernays184

Indeed, Gentzen’s handling of the consistency of pure number theory presupposed nothing from metamathematics or set theory. To the contrary, he pointed out the problem connected with the finite standpoint and sought to overcome it. It is important to bring out this point, because Bernays, in fitting Gentzen in line with Hilbert’s proof theory in Hilbert-Bernays 1939, naturally did not discuss views of that kind. 28. Gentzen Expresses His Thanks to Turing Nothing is known of any deep connection between Alan Mathison Turing and Gerhard Gentzen.185 Among Alan Turing’s possessions, however, were three preprints of Gentzen’s. One of these in the Turing Archive in King’s College in Cambridge is described “Die Widerspruchsfreiheit der reinen Zahlentheorie; from Mathematische Annalen (Vol. 112, No. 4, 1936). Signed A. M. Turing on the cover; inscribed ‘Mit Dank! G. Gentzen’ on the title page.”186 187 To my knowledge this is the only offprint in which Gentzen thanks the recipient, and the question of what he was thanking Turing for arises (Turing’s doctoral work by A. Church from September 1936 to 1938 in Princeton on ordinal logic (‘Systems of logic based on ordinals’, Proceedings of the London Mathematical Society 45 (2) (1939), pp. 161-228), a work which took up Gentzen’s thought and which would possibly have been communicated to him by Weyl, von Neumann, and Bernays. 183 There seems to be an error here: ω-consistency requires induction up to ε , the second 1 epsilon-number.—Trans. 184 Journal of Symbolic Logic 1, No. 2 (1936), p. 75. 185 Gentzen’s name is not mentioned once in Andrew Hodges’ massive biography of Turing. 186 The other two papers are “Unendlichkeitsbegriff und Widerspruchsfreiheit der Mathematik; from Actualit´ es Scientifiques et Industrielles, the Report of the IXth International Congress of Philosophy, Paris 1-6 Aug. 1937. No annotations”, and “Die Widerspruchsfreiheit der Stufenlogik; from Mathematische Zeitschrift (Vol. 41, No. 3, 1936). No annotations.” These are also in the Turing Archive. Nothing else in the Archive concerning Gentzen could be found. 187 On the inside, next to lines 19-22 on p. 545, is an inscription by Robin Gandy, one of Turing’s younger colleagues.

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How did Turing and Gentzen know each other? Perhaps Bernays knew Turing from somewhere earlier or would have heard of him in Princeton in discussions with Alonzo Church or at least in correspondence. Turing was a doctoral student of Church (1903-1995). And Church had worked with Brouwer in Amsterdam. There are thus many links between Bernays and Turing. Bernays apologised to Turing on 24 September 1937 for not having answered a letter earlier. Bernays had found copies of the letters that Turing had lost and was now able to contribute to Turing’s paper “On computable numbers, with an application to the Entscheidungsproblem”, published in 1936, and its corrections published in 1937. He also mentions a letter from Professor Church.188 29. Correspondence between Bernays and Ackermann 1936 to 1940 On 27 November 1936 Bernays wrote to Wilhelm Ackermann: The reading matter of Gentzen’s consistency proof must have stimulated you to a comparison of this proof with your earlier consistency proof. It would interest me very much to hear from you, whether you are of the opinion, that the method of the finiteness proof via transfinite induction lends itself to application to the consistency proof of your dissertation. I would very much welcome it if that is possible. Still, I am made skeptical by the circumstance that by every discovery of an “example” for an inner  the replacement for the outer  once again begins with the 0-replacement.189 190

Ackermann answered on 5 December 1936: . . . Now to the consistency proof for number theory. At your suggestion, which you had made in an earlier letter, I have again attempted to extend my so-called 2nd proof (without complete induction) to the entire number theory by bringing in transfinite methods, but I have broken off this attempt, as the proof idea for an extension of this sort seems to be unsuitable to me. I’d have greater confidence in this respect in the proof ideas of my dissertation. But you must first answer me a question. What do you consider the value of a revision of the proof, after Gentzen’s proof has appeared? You wrote to me then that this had not turned out satisfactory. From Herr Gentzen, whom I have come to know in the meantime, I heard that your criticism only referred to his first version, which has been replaced in proof by another. What would you thus be expecting from a revision of my proof, that would in any way go beyond the Gentzen proof? By the way, it strikes me, now 188 One can look the letter up in the Archive: there are scans of the letter on the Internet: AMT/D/5 image 4 abc. Heinrich Scholz had also inquired about the work in a postcard of 11 February 1937 (image 16 a + b) and expressed his thanks on 15 March 1937 (image 17 a + b) for the works of his that he had requested from Church. In the Turing Archive one can find Turing’s letter of 6 October 1936, in which he declares that, among others, he had become acquainted with von Neumann, Weyl and Church (AMT/K/1.42). On account of this, it seems reasonably certain to me that Bernays advised Gentzen to consult Turing—conversely advised Turing to consult Gentzen, which I consider more likely. It is possible, however, that Alonzo Church drew Turing’s attention to Gentzen. (Cf. Church’s observation on Gentzen in his publications of those years.) 189 Hs. 975:100, Nachlass Bernays, ETH Z¨ urich. 190 The “” referred to here is not one of Cantor’s epsilon numbers, but a choice function used by Hilbert and Ackermann to do away with quantifiers. Ackermann’s consistency proof followed Hilbert’s suggestion of attempting to assign numerical values to -terms in derivations in such a way that the resulting strings of formulæ were again derivations. The nesting of -symbols greatly complicated the procedure: each new substitution required one to go back and make changes. Prior to Gentzen’s work, Ackermann could not prove the procedure stopped after only finitely many steps.

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that I have my dissertation at hand, that there the work with transfinite ordinals will operate in an entirely similar manner as by Gentzen.191

Amplifying, Bernays wrote on 29 December 1936 (typewritten with a handwritten addition): As concerns the consistency proof for number theory, the suggestion given in my previous letter in no way signified a criticism of Gentzen’s proof in its current version. But all the same it could be that the assignment of numbers of the second number class is somewhat more agreeably accomplished with the proof idea of your dissertation; and in any event I would find it very gratifying if through the utilisation of the extended methodic standpoint of the proof procedure of your dissertation and at the same time Hilbert’s first approach to the consistency proof were rehabilitated.192

Bernays’ sympathies were clearly divided. In 1936 Gentzen had also sent Ackermann his corrections and suggestions for improvements for the second edition of Hilbert and Ackermann’s Grundz¨ uge der theoretischen Logik [Foundations of theoretical logic].193 On 29 June 1938 Ackermann, who had also corrected the manuscript of Bernays’ second volume of Hilbert and Bernays’ Grundlagen der Mathematik, wrote that he was interested in the proof of Herbrand’s Theorem. Then Ackermann went into the proof of a possible finiteness theorem, which made a basic statement about the consistency of sets of expressions: I would still like to add a remark to something earlier, namely to page 123, 5th line from the bottom to page 124, 4th line, thus on the possibility also of showing the finiteness of the construction of the total substitution in the general case, i.e. by arbitrary admission of critical formulæ of the second sort. It appears of course as if the procedure would lead after finitely many steps to a resolvent also in this case. But you say correctly that the possible discovery of a finiteness proof would be very difficult because the total substitutions depend on the size of the example found in the course of the procedure. Here I am interested in the question of which requirements would be placed on a finiteness proof, so that it would be satisfactory from the standpoint of proof theory. I have again looked 191 Hs.

975:101, Nachlass Bernays, ETH Z¨ urich. Richard Ackermann, “Aus dem Briefwechsel Wilhelm Ackermann”, History and Philosophy of Logic 4 (1983), pp. 181-202; here, p. 183. 193 The first edition was published in 1928 by Springer Verlag and was written by Wilhelm Ackermann, then Hilbert’s assistant, based on notes of Hilbert’s lectures of a decade earlier. It is a slim volume containing little by way of results—there weren’t many at the time—but it was important in standardising the syntax of mathematical logic. Following this came the massive two volume Grundlagen der Mathematik [Foundations of mathematics] of Hilbert and Bernays, written by Bernays initially on Hilbert’s lines. The first volume came out in 1934, and the second in 1939. It was the most important book on mathematical logic for decades—the only text, for example, to give the full proof of the much discussed G¨ odel’s Second Incompleteness Theorem, whereby a formal system could not prove its own consistency. A second edition appeared in 1968 (volume I) and 1970 (volume II). Bernays was writing the second volume at the time of all this correspondence, and it was devoted to Hilbert’s -Theorems. Ackermann’s attempted consistency proof was along these lines; Gentzen’s successful proof was not. Bernays could either rewrite the book to include Gentzen’s sequent calculi, or he could get Ackermann to adapt Gentzen’s method to the -elimination method for the sake of his exposition. When the volume appeared, it mentioned both a modification of Gentzen’s proof by K´ alm´ ar and a proof being worked out by Ackermann that both fit into the framework of the book; the proofs themselves appeared only in the second edition. These books are often referred to simply as Hilbert-Ackermann and Hilbert-Bernays.—Trans. 192 Hans

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somewhat more closely into Gentzen’s finiteness proof for his reduction procedure. Here, if I correctly understand it, it is only shown that by every reduction a transfinite ordinal is lowered, and from that finiteness is concluded. An explicit instruction on producing the necessary individual reductions for reducing a given proof figure is not provided. I wish to ask if you see this sort of finiteness proof as fully satisfactory from the standpoint of proof theory. If that is the case for Gentzen’s finiteness proof, then no higher standards will be placed on a proof for obtaining a resolvent after finitely many total substitutions. From this point of view it seems to me now the search for a proof of this sort is no longer quite so hopeless. Perhaps even the finiteness proof of my dissertation will somehow allow itself to be used, as it appears to have a similar character. Incidentally, I don’t know if you are aware (I had at the time not felt like transcending the narrow finite standpoint), that in my dissertation transfinite inferences were used. (Cf. for example the remarks in the last section on page 13 and in the immediately following section of my dissertation.)194

And he thanked Bernays for sending him the address of Cavaill`es. It strikes me from the correspondence with Bernays that Ackermann sided with Gentzen before he, after protest, carried out Bernays’ wishes.195 Bernays considered only those aspects of Gentzen’s work that made it appear to conform with the Hilbert Programme ´ a la Bernays. Everything that is constructive in Gentzen but did not fit in the axiomatic method in Bernays’ spirit was ignored or deemed unintuitive. Gentzen’s proof ideas did at least however interest him to the extent that he moved Ackermann to incorporate them into the Hilbert Programme, that is to say, into Hilbert’s “original view” of the finite standpoint. Ackermann realised the intent and became rightfully upset at being misused as a “mining dog”.196 Also, the way Bernays characterised Gentzen with regard to Weyl did Gentzen an injustice. Was Bernays possibly also the reason that Weyl later moved closer to the Hilbert standpoint? Did Weyl’s conversion take place on the occasion of Bernays’ stay in Princeton? Why did Weyl not absolutely support the constructive ideas of Gentzen? Surprising for me is Weyl’s assessment of Gentzen’s life work, which appears inappropriately harsh and, in fact, false: Obviously completeness of a formalism in the absolute sense in which Hilbert had envisaged it was now out of the question. When G. Gentzen later closed the gap in the consistency proof for arithmetics, which G¨ odel’s discovery had revealed to be serious indeed, he succeeded in doing so only by substantially lowering Hilbert’s standard of evidence. . . The boundary of what is intuitively trustworthy once more became vague.197

This sounds like something Bernays would say, even to the choice of words if one reads through his lectures. I am of the opinion that Gentzen was wronged here and he was not judged on what he understood and had published, but rather appears here as a sinner against the Hilbert Programme instead of as the founder of a modern proof theory. And still in 1948 in his lecture “Wissenschaft als symbolische 194 Hs.

975:114, Nachlass Bernays, ETH Z¨ urich. Wilhelm Ackermann, “Zur Widerspruchsfreiheit der Zahlentheorie”, Mathematische Annalen 117 (1940), pp. 162-194. 196 I.e., a miner’s canary. Ackermann was being sent out by Bernays to see if Gentzen was right. 197 H. Weyl, “David Hilbert and his mathematical work”, Bulletin of the American Mathematical Society 50 (1944), pp. 612-614; here, p. 639. 195 Cf.

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Konstruktion” [Science as symbolic construction]198 for the Eranos Circle, Hermann Weyl said: . . . Brouwer surrendered the largest part of mathematics. If one doesn’t wish that, then, as the criticisms of H. Poincar´e, B. Russell, Brouwer and others have shown, one cannot avoid a radical reinterpretation of the meaning of mathematics. Namely, because of that one must sacrifice the attribution of verifiable meaning to individual propositions; one must moreover transform one’s theorems into formulæ constructed from meaningless symbols, and thus they themselves neither mean nor claim anything. In principle mathematics thereby changes from a system of reasonable judgements to a game of symbols and formulæ carried out following fixed rules.

In Hilbert’s metamathematics the mathematics game itself would be made into an object of knowledge: It should be remembered that a contradiction can never occur as the end formula of a proof. What Hilbert would guarantee is only the consistency, not the contentual truth of analysis. Contentual, meaningful thought is required by Hilbert only to achieve this judgement. It is obvious that boundaries of contentual thought erected by Brouwer are to be respected throughout by these contentual considerations. Following initial successes, the implementability of Hilbert’s programme was brought into question by a deep logical discovery of Kurt G¨ odel; it is extremely doubtful if we are capable of securing the consistency of classical analysis along the path envisaged by Hilbert.199

He should have had more confidence in the ideas of Gentzen, Fitch, Lorenzen, Ackermann and Sch¨ utte. In contrast to the prophecies of doom of H. Weyl, they would prove themselves well-founded. 30. The Correspondence between Bernays and Gentzen Merrily Continues On 4 July 1936 Gentzen would continue the argument from Stralsund: Very respected Herr Professor! I thank you for your letter and for sending your Z¨ urich lectures, which I have already read in G¨ ottingen, mind you only after finishing my consistency work. What particularly interested me was that you had already envisaged the possibility of carrying out a consistency proof via transfinite induction up to ε0 . Now to the particulars of the remark of your letter. . . I have lectured on my ideas in G¨ ottingen in May, and in June in Leipzig and M¨ unster.— Do you know more of G¨ odel’s fate, who, I have heard, is supposed to be in mental hospital?200

On 23 August 1936 Gentzen reported from G¨ottingen: Very respected Herr Professor! Through Herr Prof. Hilbert I heard that you wanted perhaps to come through G¨ ottingen. He asked me to tell you that he doesn’t foresee his returning from summer vacation before the 20th of September. If you wish to see him you must therefore come at the end of the month. I myself would be very pleased to be able once again to speak with you. 198 Eranos

Jahrbuch 1948, Rhein Verlag, Z¨ urich, 1944; here, p. 415. pp. 417ff. 200 Hs. 975:1656. An abstract of the lecture of Bernays was published as “Methoden des Nachweises von Widerspruchsfreiheit und ihre Grenzen” [Methods of the proof of consistency and their limits], Verh. d. inst. Math. Kongr. II, Verlag Orell-F¨ ussli, Z¨ urich, 1932, pp. 342-343. 199 Ibid.,

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Below I give you the proof that in a number theoretic deduction all complete inductions can be combined into a single one. . . 201

For at least a year Gentzen received from Bernays the manuscripts of the second volume of Hilbert-Bernays for improvement and correction. In 1936 Friedrich Waismann’s oft-read book Einf¨ uhrung in das mathematische Denken [Introduction to mathematical thought]202 appeared. In Chapter 9, “Die gegenw¨artige Stand der Grundlagenforschung” [The present situation in foundational research], he wrote under the heading of Formalism that on account of the research of G¨odel the result had been obtained that “the proof of the consistency of a logico-mathematical system cannot be attained by means of this system”.203 He continued: With all of this it is not said that the goal set by Hilbert is unreachable. In any case one will have to revise the starting point, the restriction to a primitive part of arithmetic and logic. Whether one could then prove the consistency of classical mathematics (arithmetic, algebra, analysis, function theory), is today debatable. An important advance in this direction is represented in a recently published work of Gentzen, in which the consistency of the whole of arithmetic is proven on the basis of a part of arithmetic (not including the Law of the Excluded Middle) and certain transfinite methods.204

And Gentzen published a review in Zentralblatt f¨ ur Mathematik 15, No. 5 (1936), p. 193: Post, Emil L.: Finite combinatory processes—formulation 1. J. Symbolic Logic 1, 103-105 (1936). The concept of a “finite computation procedure” plays a fundamental rˆ ole in considerations of the possibility of a “decision procedure” for mathematical or logical theorems. A certain precise explanation of this initially quite vague concept was given by Church (this Zbl. 14, 98) following G¨ odel and Herbrand (see also Kleene, this Zbl. 14, 194). The author gives another possible version: the only operations of the computation procedure would be executed on a two-way infinite series, like e.g. the series of integers, and consists therein that one, situated at a position in the series, either has to move one step to the right or to the left, or to place a tag on the place one is situated, or finally to remove an existing tag. A finite number of “rules” of a certain form gives the necessary instructions, which of the possible operations is to be followed at any one time and when the procedure is to come to an end. The sort of executable operation can in particular be made to depend on whether the position at which one is situated is tagged or not.— Such a process is finite, if it comes to an end after finitely many steps (in each individual application). Of what sort of proof of the finiteness of the procedure is to be is, as also by Church, not made precise.— The author expects that his formulation will be proved equal in compass with the concept of “recursivity” of G¨ odel-Church. Gerhard Gentzen (G¨ ottingen)

Through his work Gentzen acquired many contacts, for example A.N. Kolmogorov, L´ aszlo K´alm´ar,205 A.A. Markov, and many others. One of the most important contacts was the friendship with Heinrich Scholz, which was formed 201 Hs.

975:1657; cf. Gentzen’s paper 1944 dedicated to H. Scholz. & Co. Vienna. 203 Ibid., p. 80. 204 Ibid, p. 82. 205 Cf. “Uber ¨ die Axiomatisierbarkeit des Aussagenkalk¨ uls”, Acta scientarium mathematicarum 17 (1935), pp. 222-243. 202 Gerold

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when Scholz visited Hilbert in 1935 to request letters or anything else Frege might have left. 31. Invitation to the Parisian Descartes Congress in August 1937. The Invitation to Lecture to the DMV Conference in Bad Kreuznach on 21 September 1937: “Die gegenw¨ artige Lage in der mathematischen Grundlagenforschung”. The Extension of the Tenure of the Unscheduled Assistantship on 1 October 1937 for a Year During this period lie the recruitment dates of a helpless opportunist. Opportunist because Gentzen never spoke up for National Socialists either orally or in writing, but wished to improve his career prospects: on 1 March 1936 he joined the NSLB (NS-Lehrerbund) [National Socialist Teachers’ League] (Member Number 335247), and on 1 May 1937 he joined the Nazi Party itself (Member Number 4237555). On 1 May 1937 he had filled out the “Parteianw¨ arterkarte” [party candidate card]; on 13 June 1937 he had to fill out anew an application for membership, and then the date of joining would be backdated to the date of submission of the Parteianw¨ arterkarte. The date points to a 1 May impression.206 Perhaps he had applied earlier and fulfilled the conditions, but it cannot be ruled out that the application was carried out “in time” for the Descartes Congress and the entry would be backdated. Probably even the membership was exactly so determined because of his imminent deputation in the German delegation to the 9th International Congress of Philosophy in Paris—the Descartes207 Congress—the leader of which would be Alfred Bæumler. Participation in the congress required special permission. Husserl, for example, was not granted permission possibly because it was feared that he would receive ovations, which could be interpreted as hostile to National Socialism.208 Gentzen was a child of the times, but not its favourite. On 1 January 1941 he became a member of the NS-Dozentenbund [National Socialist University Teachers’ League]. (He himself wrote in his vita for the docentship in Prague the date 1 January 1939.) Perhaps the passage from NS-Leherbund to NS-Dozentenbund was “automatic”? To be sure membership of docents in the NS-Dozentenbund was compulsory, but the date is not plausible. Or was it the case that by the imminent transfer he wished to better his prospects of improving his unscheduled assistantship into a scheduled appointment? On 11 March 1937 Rudolf Carnap and Charles W. Morris wrote to Heinrich Scholz: 206 After G. Leaman, Heidegger im Kontext. Gesamt¨ uberblick zum NS-Engagement der Universit¨ asphilosoph, Argument Verlag, Hamburg, 1993. 207 1937 was the 300th anniversary of the publication of Descarte’s Method, and the Congress celebrated this anniversary. 208 Cf. in this regard George Leaman and Gerd Simon, “Die Kant-Studien im Dritten Reich”, Kant-Studien 85 (1994), pp. 443-464. One should compare the report on the Congress prepared by Gerhard Lehmann for the Reich’s Education Ministry (Bundesarchiv Potsdam 49.01 REM 2940 sheet 240) with the newspaper report by Heinrich Scholz. Lehmann noted that the Congress was attended by numerous emigrants and the Jews had used the occasion to criticise the “new ‘barbarianism’ of National Socialism.” Lehmann and the ministry looked for ways that one could better advance cultural propaganda in German-speaking foreign countries in order to hinder the “machinations of the Jewish emigrants toward the spiritual isolation of Germany” Reichserziehungsministerium (REM) [National Ministry of Education].

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The organising committee of the International Congress for the Unity of Science writes you to participate in the third international conference in Paris 1937. The importance, which befits the works of the group in M¨ unster, suggested to us to ask you to participate in our conference, and to bring your co-worker Dr. Hans Hermes and as well to ask Prof. Dr. A. Kratzer.209

In the Jahrbuch u ¨ber die Fortschritte der Mathematik Gentzen’s publications always received respectable reviews by F. Bachmann, but were mostly well reviewed by W. Ackermann. One of Bachmann’s reviews reads: G. Gentzen, Der Unendlichkeitsbegriff in der Mathematik, Semesterberichte, mat. Sem. M¨ unster 9, 65-80. The author reports, following his comparison of the Hilbertian and Brouwerian standpoints with respect to the foundational questions of mathematics, on the present situation of Hilbert’s proof theory with reference to his work in the Math. Ann. 112 (1936), 439-565 (JFM 62, I, 49, review by Bachmann).210

J. Barkley Rosser also reviewed the paper: This is a non-technical discussion. Gentzen first explains the difference between the Brouwerian view of the infinite as a “becoming” and the customary view of the infinite as a completed whole. Adherence to the first view forces one to reject most of existing mathematics. On the other hand, careless use of the completed infinite leads to known contradictions. A proposal of avoiding these difficulties is to prove, by methods acceptable to Brouwer, that the completed infinite can be used in certain ways without producing contradictions. Gentzen cites his proof of the consistency of number theory (I 75) as an example of such a proof. Gentzen then goes on to explain the use of transfinite induction in his proof, which is the device which makes his proof exempt from the implications of G¨ odel’s theorem. Finally Gentzen states, as a conjecture, his belief in the possibility of a similar proof for analysis.211

During his two-day visits in M¨ unster Gentzen was a house guest of the Scholz family. Erna Scholz wrote, “In each case I remember a very sympathetic, very shy young man.212 Science was constantly the main topic.” On 12 April 1937 Gentzen appears in the agenda of the Faculty of the School of Mathematics of the Institute for Advanced Study in Princeton: “Gentzen is mentioned in a list of ‘candidates (not all of whom have applied), who should be particularly considered in future years’ (1934-39 or later in his case).”213 Gentzen read proofs for the new edition of Theoretische Logik of Hilbert and Ackermann, as well as for volume II of the Grundlagen der Mathematik of Hilbert and Bernays, and held lectures at other universities (M¨ unster). For the annual 209 The letter is in the Institute of Contemporary History (Munich); a copy is also available in the Institute for Mathematical Foundational Research in M¨ unster. 210 JFM 62 (1936), pp. 43ff. 211 Journal of Symbolic Logic 2 (1937), p. 95. 212 Letter to the author of 8 June 1988. 213 “He is again to be found in similar lists in the minutes of three other meetings, the last one on February 3, 1938. . . In particular, no action was taken. The minutes always give the names of the people to whom an invitation is issued or whose application is declared. This leads me to conclude that Gentzen never got an official invitation from the institute and that, in fact, he was not viewed as having applied in an official way” (Armand Borel in a letter to the author of 23 August 1988). In the Courant Institute in New York there is likewise no invitation to be found. I received no information from the Markov family archive in Moscow; from the Archive of the Steklov Institute in Moscow, which also looks after the Kolmogorov archives; and also nothing from the Archives of the Academy of Science.

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meeting of the DMV in Bad Kreuznach he was invited by Hellmuth Kneser to give a lecture. Gentzen wrote to him on 23 April 1937 from Prinz Albrecht Straße 22: Very respected Herr Professor! Many thanks for your letter. I am quite willing to undertake a lecture at the DMV conference, something with the title: “Die gegenw¨ artige Lage in der Grundlagenforschung” [The present situation in foundational research]. I would very much welcome it if also an intuitionist would speak on his direction; in this case I could narrow my topic and title it something like “the present state of proof theory”. Perhaps for the sake of completeness one could also have someone speak about logicism, I think someone like F. Bachmann in Marburg, who had at one time worked with Carnap. I expect to be at the beginning of August at the international philosophers’ congress in Paris, to which I have received an invitation, to participate and give a short lecture on “Unendlichkeitsbegriff und Widerspruchsfreiheit der Mathematik” [The concept of infinity and the consistency of mathematics]. I think it will be very interesting for me, especially as people like Brouwer and Carnap are reported to participate. My health is better, but not so much that I have resumed my own work or preparations for habilitation. I hope that the new circumstances in T¨ ubingen please you. With warm greetings and Heil Hitler! Your Gerhard Gentzen214

In his report on the Congress, “Denken und Erkenntnis des Abendlandes. Der Pariser Descartes-Congress” [Thought and knowledge of the West. The Parisian Descartes-Congress],215 next to Tarski, himself and Hans Hermes, Heinrich Scholz stressed someone else: In the circle of exact logic and mathematical foundational research three members of the German group spoke. In his clear, simple, impressive style the young G¨ ottingen mathematician Gerhard Gentzen reported on the proof he has been able to carry out for the consistency of elementary number theory, i.e. that portion in which analytic proof methods are not used. This proof is a milestone in the history of the endeavour for a proof of the consistency of classical mathematics, such as Hilbert has called for since 1904 on deep-rooted grounds and with a faith which may be called heroic. Today we know that the carrying through of the wonderful Hilbert Programme is bound up with fundamental difficulties, which are so deep that a full realisation of Hilbert’s demand cannot be attained or can only be imagined in a very genuinely restricted sense. But of the big question enough still remains. A first decisive step in the direction of Hilbert’s goal is in any event the solution of the subproblem which Gerhard Gentzen has found. The interest of the listeners was evident; and for all participants it was beautiful to see how solidly in this striking case the honour of the German spirit is bound with the honour of the human spirit in general.

Perhaps the words refer to a statement of Hilbert.216 But the expression “honour of the human spirit” has as its source a citation of the mathematician Jacobi, who had once written that the only purpose of science was the honour of the human 214 The

original letter is in the Archive of Dr. Martin Kneser (G¨ ottingen). Zeitung of 5 September 1937, cultural supplement. Cf. also his reports in the K¨ olnische Zeitung of 19 September and 25 October 1937. 216 He used the expression “honour of the human intellect” in one of Gentzen’s favourite ¨ texts: David Hilbert, “Uber das Unendliche”, Mathematische Annalen 95 (1926); here: p. 163. 215 K¨ olnische

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spirit217 . In his letters to Dr. Mentzel of the Deutsche Forschungsgemeinschaft concerning various grants, Scholz frequently used this set phrase, obviously an appeal to Mentzel to act quickly on the support. But this last sentence in the K¨ olnische Zeitung—it stands in opposition to National Socialism—was a provocation, for in National Socialism, the German spirit is at the pinnacle of the human spirit and goes before all other spirits. The last sentence of Scholz signals a kind of intellectual internationalism. Gentzen’s lecture, “Unendlichkeit und Widerspruchsfreiheit der Mathematik” [Infinity and the consistency of mathematics], at the 9th International Congress of Philosophy, Division VI on logic and mathematics, called the Descartes Congress, appeared in German, the language it was delivered in, in Actualit´es scientifiques et industrielles, no. 535, Hermann, Paris, 1937, pp. 201- 205. The Vienna Circle conference on the International Encyclopedia of Unified Science took place in Paris from the 29th to the 31st of July 1937. It met in connection with the International Philosophers’ Congress. Gentzen could have seen Carnap, Enriques, Heinrich Behmann,218 Paul Bernays,219 Tarski, Scholz, Reichenbach, L  ukasiewicz, Philipp Frank, Ja´skowski, whom he had already met in M¨ unster in 1936, and many others. I believe, however, that he attached himself to Heinrich Scholz and his wife and had very little interaction with others. At the Congress he got to know Mostowski, who apparently disagreed with Tarski on the significance of Gentzen’s consistency proof.220 This is conceivable; however, the successes of 217 C.G.J. Jacobi, letter to A.M. Legendre of 2 July 1830, in Gesammelte Werke, Vol. 1, G. Reimer, Berlin, 1881, pp. 453ff. 218 Heinrich Behmann was shot in the head at the Russian Front in 1915 and was declared permanently unfit for duty, and as 40% severely disabled he was discharged from service with the Iron Cross, Second Class, and the Bremen Hanseatic Cross. He habilitated in G¨ ottingen in 1921 in mathematics. He published his dissertation, “Die Antinomie der transfinite Zahlen und ihre Aufl¨ osung durch die Theorie bei Whitehead und Russell” [The antinomies of the transfinite numbers and their solution by the theory of Whitehead and Russell], which he had completed under Hilbert. From 1922 to 1925 Behmann was a scheduled assistant at the Institute for Applied Mathematics at the University of G¨ ottingen. In 1925 he transferred to Halle, where he assumed a special teaching post in applied mathematics. In 1926/1927 he was in Rome with a stipend from the Rockefeller Foundation. In 1933 he joined several Nazi organisations: the NSLB (cited in section 23 with reference to Gentzen’s membership), NS-Kriegsopferversorgung [a Nazi organisation for victims of the war, initially that being the First World War], and NS-Volkswohlfahrt [a local self-help organisation for the welfare of the “Volk”]; in 1935 he joined the Opferring der NSDAP [The “Victim Ring” or “Donation Ring” of members of the Nazi Party who made donations for those injured in fights with democrats, liberals, Jews, and communists], and the Reichsluftschutzbund [National Air Raid Protection League]; and in 1937 he joined the German Christians. On the strong suggestion of group leader Kr¨ ollwitz he was admitted to the Nazi Party (member number 4047354). (This was the same day as Gentzen’s admission. Would perhaps Gentzen also have been given such a suggestion to join?) First in 1938 Behmann was appointed temporary assistant professor. In 1938 he became Block Leader, and in 1942 he became Cell Leader and joined the Bund Deutscher Osten [Federation of the German East]. In 1945, he was dismissed from his teaching post on account of his Nazism and engaged himself with the management of the library of the Mathematical Institute; however, he very shortly quit and moved to Bremen. (Cited fom the entry “Behmann” in H. Eberle 2002, p. 463.) 219 In 1937 in the Abhandlungen der Fries’schen Schule. Neue Folge Grete Hermann, Minna Specht and Otto Meyerhof still published Paul Bernays (“Grunds¨ atzliche Betrachtungen zur Erkenntnislehre”) along with Adolf Kratzer (“Wissenschaftstheoretische Betrachtungen zur Atomphysik”) and Heinrich Scholz (“Die Wissenschaftslehre Bolzanos”). Though barred from working and teaching there, Bernays could still be published in Germany. 220 Gerd Robbel in a letter to me of 3 July 1988 without further details. I have found the following citation in this regard:

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the second chair in Warsaw, that of Helena Rasiowa and her group, are tied to Gentzen’s results. In no case did the conjectured assessment of Tarski have a negative influence on Gentzen’s reputation in Poland. Above all, however, Gentzen met Jean Cavaill`es again at the Congress. Bernays was also at the Descartes Congress, where he introduced the distinction between “formally” and “concretely” justified elements of a theory: “The concretely justified elements of a theory owe their origin to the facts of the domain of objects concerned; the formally justified elements are of a conventional nature and are introduced for the sake of the elegance, simplicity, and the rounding off of the system”.221 Possibly Gentzen got to know Lautman through Bernays. In 1940 a review of Lautman by Paul Bernays222 appeared. The article reviewed is Lautman’s lecture at the Descartes Congress, where the concepts of structure and existence are understood in a virtually Hegelian manner as “objective spirit”.223 Perhaps the two first got to know each other there. Bernays also spoke at the Descartes Congress.224 Bernays wrote to Weyl on 7 November 1937: Around the change from July to August I took part in the Parisian philosophers congresses (“unity of science” and the Descartes Congress), and I was pleased Furthermore I should like to remark that there seems to be a tendency among mathematical logicians to overemphasise the importance of consistency problems, and that the philosophical value of the results obtained so far in this direction seems somewhat dubious. Gentzen’s proof of the consistency of arithmetic is undoubtedly a very interesting metamathematical result, which may prove very stimulating and fruitful. I cannot say, however, that the consistency of arithmetic is now much more evident to me (at any rate, perhaps, to use the terminology of the differential calculus, more evident than by epsilon) than it was before the proof was given. To clarify a little my reactions: let G be a formulation just adequate for formalizing Gentzen’s proof, and let A be the formalism of arithmetic. It is interesting that the consistency of A can be proved in G; it would perhaps be equally interesting if it should turn out that the consistency of G can be proved in A” [p. 19 in: Alfred Tarski, “Discussion of the address of Alfred Tarski”, Revue internationale de philosophie 8 (1954), pp. 15-21]. Georg Kreisel has popularised Tarski’s skepticism. But Tarski’s judgement may have bounced back on him, for J.A. Robinson wrote: “Some of the greatest logical theorists—Herbrand and Gentzen, to name the two most prominent—refused, on principle, to have truck with the Tarskian extravagancies (as they saw them) and delicately picked their way to important discoveries by using finitist methodology and purely syntactic thought” (p. 288 in: J.A. Robinson, Logic: Form and Function. The Mechanization of Deductive Reasoning, Edinburgh University Press, Edinburgh, 1979). At least Tarski should have felt it so. There remains a problem: models obtained on the basis of a proven or hypothetical consistency are almost inevitably not isomorphic to those from which they derived. A certain reservatio mentalis on the part of model theorists with respect to proof theorists should be kept in mind in considering such discussions. Incidentally, it is easy to prove that Tarski’s suggestion that G and A might prove each other’s consistency cannot occur. 221 Cited by Evert Willem Beth, “Die Stellung der Logik im Geb¨ aude der heutigen Wissenschaft”, Studium Generale 8, No. 7, 1955; here: p. 430. 222 Paul Bernays, “Besprechung zu Albert Lautman: Essai sur les notions de structure et d’existence en math´emathiques”, Actualit´ es scientifiques et industrielles, Paris: Hermann & Cie, 1938, pp. 590-591, Journal of Symbolic Logic 5 (1940), pp. 20-22. 223 Paul Bernays returns to Lautman’s article in 1963 in his correspondence with G¨ odel; cf. Kurt G¨ odel, Collected Works, Vol. IV, pp. 226 and 242. 224 Published in: Travaux du IX. Congr` es International de Philosophie, Hermann & Cie, Paris, 1937, pp. 104-110.

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with the opportunity to see many acquaintances again. Gentzen and I lectured there. During my talk, as I was expressing my opinion that the proof of the consistency of restricted analysis (in the sense of your continuum book)225 presents a task that really goes beyond the consistency proof for number theory, he explained to me that he had laid out a consistency proof for restricted analysis at the same time, in connection with that for number theory. (But he was not prepared to reproduce this in detail.) This winter a Mister Mostowski from Poland, who had studied by Tarski, is here in Z¨ urich. I met him already at the Parisian congress. Last summer in Vienna he attended a lecture of G¨ odel on axiomatic set theory, in which he has proven diverse new results, in particular the theorem that, if axiomatic set theory—he used my version of the set theoretic axioms as basis—is consistent when one omits the axiom of choice, then it is also consistent with this axiom.226

And he asked Weyl whether he couldn’t make possible a second stay in Princeton. 32. Jean Cavaill` es and Gerhard Gentzen On 11 December 1931 Gentzen had written to Bernays that he had discussed foundational questions with the mathematician and logician Jean Cavaill`es (19031944). Both would at least meet once again at the Parisian Descartes Congress, where Cavaill`es was section president. Perhaps they already got to know each other during Cavaill`es’ preparation, in cooperation with Emmy Noether, of the 1937 publication of the correspondence between Cantor and Dedekind,227 which had found him, at the suggestion of Abraham Frænkel, in G¨ ottingen in 1930. But perhaps their connection goes back to a common meeting with J. Herbrand; one simply doesn’t know. Cavaill`es was not only a researcher, but also later he was an energetic resistance fighter, a leader of the French R´esistance, with whom Gentzen had day-long discussions of useful logical things: “I have had long discussions with Gentzen in G¨ ottingen—and am beholden to him for the repair of passages where to him I had oversimplified. He works presently on an analogous proof for analysis—he has thought on it for two years.”228 August for Cavaill`es was described thus: He passed the month of August in Paris—he was president of a section of the Descartes Congress, at which he read a paper—he assisted besides with the invitation committee of the Vienna Circle—and he was pleased to be able to talk of his studies with Frænkel, from Jerusalem, and with Gentzen.229

In September, however—on 3 September 1936 to be exact—he wrote to Albert Lautman (1908-1944) about the Spanish Civil War: 225 Hermann

Weyl, Das Kontinuum, Veit, Leipzig, 1918. 91:17, Nachlass Weyl, ETH Z¨ urich. 227 Emmy Noether and Jean Cavaill` es (eds.), Briefwechsel zwischen Georg Cantor und Richard Dedekind, Hermann & Cie, Paris, 1937. Cf. with respect to this Ivor Grattan-Guinness, “The rediscovery of the Cantor-Dedekind correspondence”, Jahresberichte der Deutschen Mathematiker Vereinigung 76 (1974), pp. 104-139. On Cavaill`es’ connection with Foucault and French structuralism, cf. V. Tasic (2001), particularly G. Canguilhem. 228 J. Cavaill` es to Albert Lautman on 26 August 1936 (cited on pp. 117ff. in: Gabrielle Ferri` eres, Jean Cavaill` es. Philosophe et Combattant 1903-1944. Avec une ´ etude deson oeuvre par Gaston Bachelard, Presses Universitaires de France, Paris, 1950). Cf. also the longer note in the last section of this chapter. 229 G. Ferri` eres, 1950, p. 123. 226 Hs.

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Oui la patience de Gentzen m’a un peu d´ecourag´e—il est d´ej` a dans son poˆele ` a G¨ ottingen. Mais nous sommes un peu jeunes et nous n’avons pas ´et´e a ` la guerre. Les histoires espagnoles, ici toutes proches fournissent ` a r`ever.230

Cavaill`es would certainly later be celebrated by Raymond Aron, but he is hardly known in Germany—perhaps because he was murdered by Germans?231 232 He was one of the participants of the second Davos high school course, the “Locarno de l’intelligence” of 1929, where Heidegger and Cassirer had a well-known disputation, which Emmanuel Levinas, Otto Friedrichs, Bollnow, J. Ritter, Alfred Sohn-Rethel, H. Marcuse, Leo Strauss, Pinder, Brunschvicg, Przywara, Joel, Sauerbruch, Riezler and many others followed with interest. Only Jean Am´ery mentioned him admiringly in an autobiographical sketch. Gentzen might have read with interest Cavaill`es’ two books which appeared in 1938, M´ethode axiomatique et Formalisme [Axiomatic method and formalism] and Remarques sur la formation de la th´eorie abstraite des ensembles [Remarks on the formation of the abstract theory of sets]. Of Gentzen’s contacts with the French world of foundational research I know not much. It seems certain, however, that via Helmut Hasse and Richard Courant he was informed of the works of Jacques Herbrand and his clarification of the Hilbert Programme since 1931—if not earlier.233 That Gentzen has found a solid, appropriate, clear and respected position still dominant today in French foundational

230 “Yes, the patience of Gentzen has me a little discouraged—he is already in his frying pan [or, stove—presumably this is an idiom that would mean something to Lautman] in G¨ ottingen. But we are a bit young and have not been at war. Spanish stories, so close here, provide for dreaming.” The most important thing is that the quote shows Cavaill`es was thinking about Gentzen. It also seems to show that Gentzen was, if not apolitical, not as concerned about politics as many of those around him.—Trans. 231 One can find a quick overview of his achievements in: Paul Cortois, “Paradigm and thematization in Jean Cavaill`es’ analysis of mathematical abstraction”, pp. 343-350 in: Johannes Czermak (ed.), Philosophy of Mathematics, H¨ older-Pichler-Tempski, Vienna, 1993. Philosophers can round out their knowledge of him through: Michel Fichant, “Die Epistemologie in Frankreich, III. Die mathematische Epistemologie: Jean Cavaill`es”, pp. 141-148, in: Francois Chatelet (ed.), Geschichte der Philosophie, VIII. Das XX. Jahrhundert, Ullstein, Berlin, 1975; and M. ¨ Fichant and Michel Recheux, “Uberlegungen zur Wissenschaftsgeschichte, Suhrkamp, Frankfurt am Main, 1997, esp. pp. 131ff. For mathematicians, cf. J. Dieudonn´e (ed.), Geschichte der Mathematik 1700-1900, Vieweg-Verlag, Wiesbaden, 1969; Hourya Sinaceur, Jean Cavaill` es. Philosophie math´ ematique, PUF, Paris, 1994; the same, “Du formalisme ` a la constructivit´e: le finitisme”, Revue internationale de philosophie 4 (1993), pp. 251-283. 232 J. Cavaill` es would be arrested by the Gestapo in 1943 and brought to Fresnes; he could not flee again, as he had done twice before. In prison he wrote “Sur la logique et la th´eorie de la science”[On logic and the theory of science] (published posthumously in Paris in 1960 by Gaston Bachelard). The French mathematician Henri Cartan turned to Heinrich Scholz in May 1944 to rescue Cavaill`es. Scholz turned to Ernst Freiherr von Weizs¨ acker in the Foreign Office, but he could do nothing. H. Behnke also tried to get help from W. S¨ uss in this matter (according to Maria Georgiadou, Constantin Carath´ eodory. Mathematics and Politics in Turbulent Times, Springer-Verlag, 2004; cf. pp. 369 and 581). Cavaill`es was condemned to death by the Germans and assassinated in Arras. He is now buried in the chapel of the Sorbonne near the grave of Descartes. (Cf. Hourya Sinaceur, Jean Cavaill` es. Philosophie math´ ematique, PUF, Paris, 1994; cited by J.W. Dauben, Abraham Robinson, Princeton University Press, 1995.) In 1958 he was commemorated on a French postage stamp of the then yearly issued series of heroes of the French resistance. 233 G. Ferri` eres, 1950, p. 69.

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research shows the clear and extensive appreciation of his accomplishments—an appreciation not yet achieved in Germany.234 Albert Lautman’s works were quite liberally received and reviewed; Arnold Schmidt reviewed two essays in Zentralblatt f¨ ur Mathematik.235 Naumann reported on the three-part work of Cavaill`es, “M´ethode axiomatique et formalisme I. Le probl`eme du fondement des math´ematiques. II. Axiomatique et syst`em formel. III. La non-contradiction de l’arithm´etique”:236 Finally Gentzen succeeded through an extension of the metamathematical domain to produce a complete consistency proof for the entire intuitionistic arithmetic. The finite was indeed transcended, but only through orderly rules for generating new individuals. The radical formalism had not reached its unattained goal; logicism and intuitionism have also proven insufficient.237

It is unclear how Gentzen and Cavaill`es may have influenced each other. 33. Gentzen Becomes an “Associate” of the Publication of Scholz’s Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften The year 1937 also saw the appearance of the Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften. Neue Folge 238 [Researches on logic and on the foundation of the exact sciences. New series]. Heinrich Scholz published these and named Gentzen together with W. Ackermann, F. Bachmann and A. Kratzer as contributors to the edition.239 Gentzen’s name is listed as contributor in all the headings of all the reviews. For example, in Zentralblatt f¨ ur Mathematik 16, No. 1 (1937), p. 1, one reads: Hermes, Hans, and Heinrich Scholz; Ein neuer Vollst¨ andigkeitsbeweis f¨ ur das reduzierte Fregesche Axiomensystem des Aussagenkalk¨ uls. (Forsch. z. Logik u. z. Grundlegung d. exakten Wiss. N.F. Hrsg. v. Heinrich Scholz. Unter Mitwirkung v. W. Ackermann, F. Bachmann, G. Gentzen u. A. Kratzer. No. 1) Deutsche Math. 1, 733-772 (1936) u. Leipzig: S. Hirzel 1937.

One should always keep in mind that the advances in mathematical logic in Germany at this time were really being made by Gentzen, Ackermann and Paul Bernays. W. Ackermann was indeed habilitated, but he was a school teacher, having been condemned by Hilbert to a school teacher’s life. Paul Bernays was in exile and therefore didn’t exist at German universities. And Gentzen had not unster School” was yet habilitated, but was merely an assistant. Only the “M¨ a stronghold of mathematical logic in the academic and official university world. Therefore it is significant that Gentzen received such an honour. In the meantime Gentzen can be found mentioned in the reviews of works of foreign colleagues: Beth, E.W.: Une d`emonstration de la non contradiction de la logique des types au point de vue fini. Nieuw Arch. Wiskunde 19, 59-62 (1936). In Math. Z. 41 (this Zbl. 15, 193) Gentzen gave a finitistic proof of the consistency 234 Marcel

Guillaume, “Logique Math´ematique et sa jeunesse”, pp. 185-365 in: Jean Paul Pier (ed.), Development of Mathematics 1900-1950, Birkh¨ auser, Basel, 1994. 235 Zbl. 21, No. 7 (1939), p. 289. 236 Actual. sci. industr. 608, pp. 1- 75, 609, pp. 76-124, 610, pp. 125-183. 237 Jahrbuch u ¨ber die Fortschritte der Mathematik 64, No. 2 (1938), pp. 930ff. 238 Cf. the longer note at the end of this chapter. 239 A reprinted edition by Verlag Dr. H.A. Gerstenberg, Hildesheim, 1970, exists.

33. GENTZEN AND SCHOLZ

of type theory (without the axiom of infinity). The work before us presents (in essence) the same approach to the proof of consistency, which the author, as he stresses, found independently of Gentzen. Beth declares that his proof goes beyond Gentzen’s insofar as it also includes the ramified type theory, but in fact the data of the ramification are nowhere handled by him. The axioms and rules of the theory to be demonstrated consistent are not formulated at all; there is only a sketch of the basic idea of the approach (representing an extension of a known method); anything further, in particular the working through of the approach with respect to these axioms—which is not trivial, and which is in the foreground by Gentzen—is missing. Arnold Schmidt (Marburg, Lahn)

And there again appeared a review by Gentzen: Ackermann, Wilhelm: Die Widerspruchsfreiheit der allgemeinen Mengenlehre. Math. Ann. 114, 305-315 (1937). The consistency of the “general set theory”, i.e. the Zermelo-Frænkel axioms of set theory with the exclusion of the axiom of infinity, is reduced to the consistency of pure number theory, which was proven by Gentzen (this Zbl. 14, 388). These axioms are, as is shown, already satisfiable through a system of finite sets; and, as is known, one such can be represented in the series of natural numbers.— With the addition of the axiom of infinity, i.e. the claim of the existence of infinite sets, the question of consistency is naturally genuinely more complicated and yet awaits an answer. Gerhard Gentzen (G¨ ottingen).240

On 12 August 1937 Gentzen wrote from G¨ottingen to Hellmuth Kneser: Very respected Herr Professor! Enclosed is the summary for my DMV address; I have sent a copy to Prof. M¨ uller—Paris pleased me very much and seems not to have harmed my state of health. The day before yesterday I arrived here; I hope the summary comes in time for the programme. I will call on Professor Ewald—I have promised it—and will report to you about it. It seems to me that I am at any rate on the path to recovery. With hearty greetings and Heil Hitler! Your G. Gentzen241

And on 28 August he wrote: Very respected Herr Professor! The day before yesterday I was with Professor Ewald, who, thanks to your letter, I found very friendly. Of my illness he could in fact say nothing more than what I had already said myself. Still, he gave me a possibly useful piece of advice. He wanted to be helpful to me, if I wished a rest cure, but agreed with me that in the main one couldn’t do anything other than wait it out. He did not object to it when I said that I couldn’t promise myself a stay at a health resort and most wanted to continue living as before. I have got the impression in recent weeks that my condition on the whole has improved, that a definite recovery lies not too far off. The lecture at the Kreuznach conference will not be a hardship. Till then the heartiest greetings—and many thanks for your friendly support from your Gerhard Gentzen242 240 Zbl.

241 The

242 Ibid.

16, p. 194. original can be found in the archive of Prof. D. Martin Kneser (G¨ ottingen).

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34. “Die gegenw¨ artige Lage in der mathematischen Grundlagenforschung” On Tuesday, the 21st of September 1937, under the chairmanship of Kneser, Gerhard Gentzen gave his lecture, “Die gegenw¨ artige Lage in der Grundlagenforschung” [The present situation in foundational research], which appeared in expanded form in Deutsche Mathematik. In fact, he turned in his article “Die gegenw¨ artige Lage in der mathematischen Grundlagenforschung” together with “Neue Fassung des Widerspruchsfreiheitsbeweises f¨ ur die reine Zahlentheorie” [New version of the consistency proof for pure number theory] over to Ludwig Bieberbach to be published first in Deutsche Mathematik.243 Bieberbach contrasted with other “Aryan” mathematicians, like e.g. the artistic geometer, D¨ urer fan and anti-Semite Max Steck— an assistant of the Jew-hating H. Dingler, who would ground science on life but is today an honoured elder father of the Erlangen and Konstanz schools—and was convinced of the institutional necessity of logic and its research and teaching under National Socialism and therewith its worthiness of funding and being rationed paper.244 Scholz and Gentzen kept in touch with Bieberbach from the beginning to the end of the war; he had influence with the mathematician and Reichs Research Councillor Th. Vahlen. His Deutsche Mathematik, with an initial edition of 3500 quarterly issues, would from the beginning of the war quickly decline in the estimation of mathematicians and National Socialist academic offices.245 From 1 October 1937 the service period of the unscheduled assistant Gentzen was extended another year. As always, the Dozentenschaft, faculty and rector did not oppose. In November 1937 Sturmmann Gentzen transferred to Sturm “Na 1/82”. The editors of Zentralblatt f¨ ur Mathematik proved to have a sense of humour. They published—in English—a review of “Die gegenw¨ artige Lage...” by Haskell Curry, an American logician from Princeton and former assistant to Hilbert: The main theme of this expository article is to contrast the constructive and “an sich” conceptions of infinity, and then to defend the opinion that the Hilbert program makes it possible for the two sides to agree on the retention of classical analysis in its present form. The author begins with a general discussion of the two points of view and the Hilbert program. He then gives an elementary 243 Both appeared in the “Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften. Neue Folge. Herausgegeben von H. Scholz unter Mitwirkung von W. Ackermann, F. Bachmann, G. Gentzen und A. Kratzer”, Vol. 4 (1938). 244 Gentzen’s essay “Die gegenw¨ artige Situation in der Grundlagenforschung” in Deutsche Mathematik and the Forschungen was set in the fraktur typeface. For the younger generation today and the contemporary international intelligentsia this is barely readable. First, on the first and second of June 1941 the typeface of the V¨ olkische Beobachter [National Observer], and thereafter all the other newspapers close to National Socialism, would change over to Antiqua: the conquest of Crete by the Germans had to be internationally easily readable. At just the point in time the Nazis believed they would dominate Europe, an internal party circular denounced the “so-called gothic script” as “Swabian-Jew letters” and decreed the introduction of Antiqua as the normal script. Deutsche Mathematik, as a modernising effect of the German war management, switched over to normal lettering. (Cf. Bundesarchiv Koblenz, R73 (Notgemeinschaft/DFG), Sachbeihilfeakten No. 15934-15936, Zeitschrift “Deutsche Mathematik”; also 14484, 11060, 11094.) 245 “So far as I know, Scholz and his students have published in the volumes of Deutsche Mathematik, because Scholz at that time saw it as the only possibility to print works from ‘logistic’, not yet accepted in Germany.” (Hans Hermes in a letter to me of 22 January 1997.) This is the case; cf. Bundes Archiv Koblenz, R73 (Notgemeinschaft/DFG), Sachbeihilfeakten No. 1593415936, Zeitschrift “Deutsche Mathematik”; also 14484, 11060, 11094.

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account of the principal metatheoretic theorems relating to consistency and completeness, including certain theorems of G¨ odel and Skolem. The upshot of this is that although a constructive consistency proof for classical analysis has not yet been found, yet the author thinks it probable that such a proof will be found, using methods which, like those of the author’s proof for arithmetic (this Zbl. 14, 388) are acceptable from the constructive viewpoint, even though they cannot be formalized within the system. The author devotes a section to the contrast of the two views in regard to the continuum, in which the greater complexity and restrictedness, as well as the greater intuitive evidence of the constructive viewpoint is made clear. In the final section he discusses the unification of the schools. The argument is that the significance of the classical theorems comes from their usefulness in physics, and that they can be interpreted from a constructive viewpoint by the methods of ideal elements (Hilbert). Gentzen makes an interesting comparison with another instance of the simplification which comes through idealization of experience, viz. geometry; here the constructive analysis is compared to the “natural geometry” of Hjelmslev (Hamburg 1923), while the classical analysis is analogous to the ordinary geometry of idealized points and lines.246

Here it is shown that the Deutsche Mathematik was being read abroad247 on account of an article that was penned by a logician: Gerhard Gentzen. In this Gentzen claimed that the significance of classical theorems comes out of their usefulness in physics. And this was no bow to the applied mathematicians or those fit for war work. In Zentralblatt Arnold Schmidt recapitulated for those who could not read English: Gentzen’s report (which in fact corresponds with his address to the mathematicians’ congress in Kreuznach) divides into 4 parts. First G. introduces the antinomies248 starting from the mathematical standpoints on the infinite, in particular the intuitionistic and the proof theoretic (logicism is only briefly touched upon). Then the main problems of axiomatics—consistency, completeness and odel’s Incompleteness Thedecidability—are outlined, and in comparison with G¨ orem, the Skolem-L¨ owenheim Theorem on countable satisfiability is illuminated. After presenting the constructive conception of numbers, into which above all enter the conceptions of Weyl (limitation of means), Church and Turing (“computability”), as well as Brouwer (choice sequences), the author recommends in agreement with Hilbert, that nonconstructive existence be treated as an “ideal element”; this attitude will (after the proof of the consistency of analysis) be suited 246 Zbl.

14 (1938), p. 97. Curry’s book Foundations of Mathematical Logic (McGraw-Hill, New York, 1963) owes a lot to Gentzen’s ideas. Haskell B. Curry, a student of Hilbert, lays down his views of the development of some of Gentzen’s ideas in “Historical Remarks”, e.g. pp. 245-250, 305ff., and other pages. 248 With regard to the antinomies Gentzen was still energetic in his nachlass, thus after 1939: “ ‘Contradictions in logic?’ No. In my opinion that means the concept of logic is taken too liberally. Entering into the Russell antinomy are the concepts set and property, which are too lacking in precision to be counted in this general form as belonging to ‘logic’. Probably logic can speak of sets, but of given assumed sets as objects of a given sort. The general concept includes the real world; that is not a matter of logic. (The Russell antinomy in mathematical set theory is of another sort and here is tied to the concept of infinity. The logical form is not mathematical, for with it far too many unclear concepts are at work for mathematics.)” I thank Herr Prof. Dr. Christian Thiel for the source and transcription. 247 Haskell

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to mediate between the differing standpoints on the foundation of mathematics considering the demands of physics.249 250

Here Gentzen extended the Hilbert programme in his own way. In connection with G¨ odel’s Second Incompleteness Theorem, he wrote: There is first the theorem on consistency proofs, which means that the consistency of a mathematical theory, which includes pure number theory and is indeed consistent, does not allow itself to be proven by the proof methods of the theory, in particular not with a subpart of these proof methods. This theorem is often explained as if Hilbert’s programme is proven impossible to carry out. Namely, one assumes—and this also goes for several viewpoints—that the “finite”, resp. “constructive”, modes of inference permissible in a consistency proof are only to be found in a fragment of pure number theory and may be presented as precisely formulated rules of inference. Were that the case then by G¨ odel’s Theorem the consistency of number theory would naturally not be verifiable. I am, however, of the opinion that there are modes of inference, which are completely in accordance with the constructive conception of the infinite and on the other hand do not belong in the framework of formalised number theory, verily which presumably can anyhow be extended beyond each formally delimited theory. I have given the relevant inferences as far as are necessary for the consistency proof for pure number theory in my treatment. These are connected with the “transfinite induction” of set theory, which is not to say that they share in the theory’s attendant questionability; they are moreover constructively proven in a manner entirely independent of set theory.—G¨ odel’s Theorem naturally retains, entirely regardless of these facts, a great significance as a very valuable result, which, in particular, even lends a great service to the finding of a consistency proof because it tells one with which means one can in any case not reach the goal.251

For one like Gentzen in need of harmony, the word “mediate” was already a sufficient goal. He did not loudly oppose the Hilbert Programme. 35. “Neue Fassung des Widerspruchsfreiheitsbeweises der reinen Zahlentheorie” 1938 In the same issue of Zentralblatt as Schmidt’s review, Arnold Schmidt also reviewed the new version of the consistency proof: It deals with a simplified version of Gentzen’s proof of the consistency of pure number theory, see this Zbl. 14, 388. Instead of the logical calculus N K, the author proceeds from his calculus LK (see this Zbl. 10, 145f ), in which a proof figure already progresses to formulæ of higher degree (number of logical connectives); the complexity of the proof, which is fixed by the Gentzen ordinal, stands thereby in natural connection with the complexity of the end-formula. Further, the concept of the “mathematical basic sequent” is simplified. The consequences of these changes on Gentzen’s reduction procedure are presented.— As domain of proof ordinals a special class of decimal fractions is no longer used, rather, the segment of the 2nd number class determined by the 1st epsilon number is used. The constructivity of the procedure is thereby more transparent, and one realises that this domain is an “economiser”.— The conclusion gives a piece of advice on 249 Zbl.

19, no. 6, p. 241. vast experience in current proof theory is ready-made material which corresponds to the physicist’s experiments and observations. Here we have a body of discoveries, of formal facts by which a theory of proofs can be tested.” (Georg Kreisel) 251 Cf. Appendix C for the complete lecture. 250 “Evidently

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the includability of arbitrary functions and gives a peek at the possibility of a consistency proof for analysis.

Why did Paul Bernays stop in Berlin after his dismissal from the University of G¨ ottingen in 1934?252 His uncle, Gustav Brecher, lived there. The musician was the brother of Sara Bernays-Brecher, the mother of Paul Bernays. In a letter or postcard of 5 September (1938?) Brecher wrote Bernays: I am sending by the same post fifty marks (at the moment I’m afraid I can’t afford more) to Dr. Gentzen.253

On 26 December 1938 Paul Bernays wrote to Hermann Weyl from Eichmattstraße 55: In proof theory the situation at the moment has noticeably improved, since Gentzen’s consistency proof, which in its original form was hardly accessible, has been brought by Gentzen into an understandable and at the same time elalm´ ar has recently so modified this proof ementary form. Incidentally, Herr K´ that it refers immediately to the usual form of the number theoretic formalism, whereby it is simplified still further. Admittedly the final verdict on the fate of proof theory is still to be made, i.e. that with the investigation of analysis. You no doubt know that Mr. Gentzen has worked for some time on this problem. He wrote me recently he has made a good bit of progress, but the result at present is not yet in sight.254

36. Bernays’ View of Gentzen’s Programme (Second Consistency Proof ) ¨ In 1938 Paul Bernays wrote an article, “Uber aktuelle methodologische Fra255 gen der Hilbertschen Beweistheorie” [On current methodological questions of Hilbert’s proof theory]. This was a lecture given by Bernays to a gathering to which Gentzen and Ackermann were also invited, but for reasons unknown today the two did not appear or—probably on account of the times—could not appear. For, on account of authorisations and foreign exchange, even for professors like Heinrich Behnke, foreign travel was made only under the greatest of difficulties. It is possible that Wilhelm Ackermann and Gerhard Gentzen had to refuse their participation because the conference was somehow connected to the League of Nations (cf. p. 357, Karen Hunger Parshall, Adrian Clifford Rice, Mathematics Unbounded : The Evolution of an International Mathematical Research Community, 1800-1945, American Mathematical Society, Providence, RI, 2002). On Gentzen, Bernays reported: Is it necessary with regard to proof theory, to take up all methodic assumptions of intuitionism? For the moment we will at least partially answer this question. Gentzen has in fact carried out a consistency proof for the arithmetic formalism 252 Cf.

the letter on page 54. am indebted to Mr. Ludwig Bernays for this excerpt. It is not clear why Gentzen received the money. Possibly it was for proofreading and assistance with the second volume of Hilbert-Bernays, Grundlagen der Mathematik, which would appear in 1939. 254 Hs. 91:19, Nachlass Weyl, ETH, Z¨ urich. 255 Published pp. 144-152 in F. Gonseth, Entretiens de Z¨ urich 1938, Leemann & Co., Z¨ urich, 1941, as “Sur les questions m´ethodologiques actuelles de la th´eorie hilbertienne de la d´ emonstraton”. Cf. Ivor Grattan-Guinness, “Foundational studies and logics during the 1930s: Gonseth’s Entretiens (1941) and its background”, James Gasser and Henri Volken (eds.), Logic and Set Theory in 20th Century Switzerland, Bern, 2001, for the history, course of events, and content of the four-day event in December 1938. 253 I

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whose methodic assumptions occupy a middle ground between the finite standpoint and that of intuitionism. It is appropriate to refer here to Gentzen’s second proof, which commends itself not only through its emphasis of the central idea, but rather too through the avoidance of certain complicated methodic means. . . To make clear in what kind and way Gentzen transcends finitist methods, we will indicate the logical scaffolding of his proof (with a few small revisions).

Bernays then attempts after a presentation of Gentzen’s proof-idea to see intuitively wherein consists the legitimacy of the transfinite inference principle of transfinite induction and why it represents a proper, appropriate generalisation of ordinary induction, and then summarises: In the end the use of transfinite induction amounts to a certain extension of the methodic framework of proof theory, but in no way to the total assumption of intuitionistic means of inference. . . At the moment no judgement is possible as to whether the addition of a higher induction principle of this kind to the finitist methods can provide a sufficient means for a consistency proof of analysis.

But Bernays is not in favour of “restricting a priori the methodic frames for metamathematical investigations,” because there is no simple and general correspondence between the possibility to formulate a statement and to prove it, and hence likewise little between the formulation and the solution to a problem: Now the question presents itself, of what character the methodic limitation of proof theoretic means has, if this does not consist in the demand for elementary evidence, which distinguishes the finitist standpoint. The answer is the following. The tendency of the methodic limitation remains basically the same; however the evidence and security must not be understood in an entirely absolute manner if one wishes to keep open the possibility to extend the methodic frames. If one proceeds thus, one secures for oneself moreover the advantage of not being obliged to declare the traditional methods of analysis unlawful or dubious. The specific character of Hilbert’s standpoint must be perceived herein: one obliges oneself not to depart from any, in the strictest sense, arithmetic ways of thought by so much as a finger’s breadth, whereas the habitual methods of analysis and set theory feed themselves to a considerable extent on geometric ideas and draw their obvious power therefrom. One can in fact say—and this is certainly the essence of the finitist and intuitionistic criticism of the usual methods of mathematics—that the arithmetisation of the geometry in analysis and set theory is not complete. The methodic orientation of Hilbert’s proof theory can amount to the development and better estimation of the specifically arithmetic thought and to a clearer illumination of the stages of development of arithmetic procedures.

Gentzen had clearly pronounced his position. In Gentzen 1936 (p. 529) he wrote, “But what is finite one must be aware of beforehand,” and that is a criticism of Bernays and Hilbert. Further, he says on p. 557: This [the finite character of the methods for a consistency proof, EMT] does not allow of “proof”, in any case, because the concept “false” is not unambiguously formally delimited and too can securely be delimited. One must look at each individual inference and attempt to be clear about it, or whether it stands in agreement with the finitistic meaning of the concepts found and does not rest on an inadmissible “in-itself” conception of these concepts.

And G¨ odel, who had these thoughts already in the 1940s, agreed in part in his Dialectica essay256 with this: 256 Kurt G¨ ¨ odel, “Uber eine bisher noch nicht ben¨ utzte Erweiterung des finiten Standpunktes”, Dialectica 12 (1958), pp. 280-287.

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Due to the lack of a precise definition of either concrete or abstract evidence there exists today no rigorous proof for the insufficiency (even for the consistency proof of number theory) of finitary mathematics. However, this surprising fact has been made abundantly clear through the examination of induction by ε0 used in Gentzen’s consistency proof of number theory.

But Gentzen wrote already in 1938 (p. 44): It is not essential to prove it [the transfinite induction, EMT]. . . moreover to prove it on a finitistic basis, i.e. to clearly emphasise that it is a mode of inference in agreement with the principle of the constructive conception of the infinite; a no longer purely mathematical matter, which now, however, by the consistency proof, also belongs to the subject.

Gentzen did not resolve the discussion of the legitimacy of transfinite proofs. It continues to this day, and one sees Gentzen only as described by Bernays and Weyl.257 Bernays didn’t give Gentzen a chance to speak because for him it was all about finitistic mathematics, which he defined as the mathematics of “intuitive evidence”, and on that Bernays had a different conception from Gentzen. Bernays tended to view everything in terms of metamathematics, and he would push Gentzen aside to make place for another there: Anyway one must be clear in the judgement of the results of proof theory, that the consistency proof for the arithmetic formalism in no way represents the only advance made in metamathematical investigations in recent years. In particular, on the problems of decidability and effective computability the works of G¨ odel, Church, Turing, Kleene and Rosser have achieved very remarkable results. From now on metamathematics has such significance, that its importance can be appreciated independently of any philosophical doctrine on foundations.

The question here is not whether Bernays made publicity for Gentzen or distanced himself further from this, but rather that Bernays had a very particular view of the problems and that this didn’t agree with Gentzen’s. In contrast to Gentzen, Bernays expressed himself also on general philosophical problems, which in his opinion were connected with the problem of foundations. I’d like merely to suggest about this that one should guard against seeing Gentzen through the restricted and coloured view of Bernays’ glasses. Afterwards, in 1939, Gonseth’s Philosophie Mathematique Avec cinq D´eclarations de MM. A. Church, W. Ackermann, A. Heyting, P. Bernays et L. Chwistek [Mathematical philosophy with five statements by Misters A. Church, W. Ackermann, A. Heyting, P. Bernays and L. Chwistek]258 appeared. In this hundred-page booklet Gentzen is mentioned at least seven times. In chapter II, “Le formalisme Hilbertienne” [Hilbert’s formalism] on p. 50 at number 71, one reads: What is it nowadays? To the Entretiens de Zurich sur les Fondements et la M´ethode des Math´ematiques of December 1938, Mr. Bernays declares, “One is oneself convinced that a theory of proofs in the sense of the first intention of” the finitist point of view “cannot in any measure attain its goal. . . Resuming the original intention, Mr. Gentzen has made a proposal which, although defensible, profoundly modifies the sense of the whole enterprise. The fitness due to the finitist point of view, he says, does not derive so much from the fact that one there prohibits the recourse to an infinity of operations but is due to the fact of 257 Cf.

the longer note at the end of this chapter. & Cie, Paris.

258 Hermann

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its being in conformity with another point of view that comprises and transcends it: the constructionist point of view.”

Alonzo Church reported on the “Die gegenw¨ artige Lage. . . ”, beginning with a remark from Weyl (p. 70): First the carrying out of the consistency proof or the efforts on that account disclose the trickiest logical structure of mathematics, a tangle of circular back associations which do not permit one to assess if they do not lead to spectacular contradictions.

And after this he comes to Gentzen (p. 71): At the present time, however, it seems at least doubtful that such a proof is possible—although it is noteworthy that Gentzen has accomplished a finitary consistency proof of what may be roughly described as a truncated portion of the system of Principia, adequate to (elementary) number theory.

In a footnote he expanded: To describe Gentzen’s proof as transfinite as is sometimes done, because it involves complete induction up to the ordinal ε0 , seems to me misleading. If this method be transfinite, what will be said of induction up to ω—of induction up to ω 2 ?

Afterwards Church went further into the significance of a “finitary consistency proof”. Wilhelm Ackermann discusses Gentzen in his “Bemerkungen zu den logischmathematischen Grundlagenproblemen” [Observations on the logico-mathematical foundational problems]: As far as the results of proof theory in detail are concerned, it has only slowly, step by step, succeeded in including an encompassing formalism in the consistency proofs. The latest phase is represented by Gentzen’s verification of the consistency of pure number theory. (On the difficult question, whether pure number theory can be exhaustively translated into a definite formalism, we prefer not to enter into here; in practice it is in any event the case up to a certain degree.) As an additional goal for proof theory there appears first the proof of the consistency of analysis, perhaps also of the segment of abstract set theory with the inclusion of the Axiom of Infinity. It is usually asserted, on the basis of the previously mentioned investigations of G¨ odel, that the consistency of a formalism, which includes number theory, can no more be carried out only with the means of the original narrow finitist standpoint of Hilbert, which encompasses only a part of the intuitionistcally permissible modes of inference, but rather that for this a stronger borrowing from intuitionism is necessary.

Ackermann also got down to his thoughts on the task of proof theory and consistency proofs. And then he wrote: As far as concerns the conflict between Hilbert’s standpoint and that of the intuitionists, this opposition would be damned to sterility, in case it would only have had the effect of having forced mathematicians to take up sides either for one or the other standpoint and more-or-less ignore the investigations of the opposing side. That is however not the case. To the contrary, there have been interactions between the researches on both sides. The bringing out of the constructive modes of inference by Brouwer and his school also possesses interest for proof theory, as the metamathematical consistency proofs will be conducted only with these modes of inference. Further, after a portion of the intuitionistically permissible modes of inference were laid down in the form of a formal calculus by Heyting, Glivenko and G¨ odel could show that every theorem of formalistic

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mathematics, so far as number theory comes into consideration, can also be so interpreted intuitionistically that it has a meaning. . . At least the fact remains interesting enough that a mathematical statement, which would be proven by means of modes of inference that are questionable from the intuitionistic standpoint, can be assigned an intuitionistically meaningful statement and the proof so changed that this last statement is yielded by means of permissible modes of inference. Incidentally, this also yields, as should not be further carried out here, the consistency of arithmetic. By the way, it seems from remarks of Weyl that on the intuitionistic side the formalistic mathematics is not actually accepted as pure mathematics, which may well be, but in mathematical physics is entitled to a certain right to exist.259

Paul Bernays discussed Gentzen in his “Bemerkungen zur Grundlagenfrage” [Observations on the question of foundations] (pp. 86ff.): Instead of applying the methods of intuitive arithmetic directly to the area of normal mathematics, we can, following Hilbert’s thought, make use of them as methods of axiomatic (“proof theoretic”) investigations of deductive formalisms. For this, in any case, a powerful arrangement of the methods of intuitive arithmetic is required. With that entirely elementary evidence, as Hilbert wanted first to take as the single methodic basis for proof theory, one cannot, as has been shown, get by. This is connected with the fact found by G¨ odel that in the verification of the consistency of a formalism some conceptual tool that is no longer representable within this formalism must always be used. How such a surpassing of a formalism in the sense of an intuitive-arithmetic way of looking at things can be accomplished has been shown for the number theoretic formalism by Gentzen, who has supplied a verification of the consistency of this formalism by which a generalisation of complete induction for a certain domain of the Cantorian second number class has been added to the elementary finite methods as an aid. Whether the consistency of a deductive formalism of analysis allows itself to be obtained by analogous means we cannot yet see clearly.

But Bernays also integrated Gentzen into contemplations of an epistemological kind, which were not Gentzen’s. Bernays recognised three standpoints: the finite, Gentzen’s, and the intuitionistic. 37. Correspondence with Paul Bernays On 15 November 1937 Bernays went into Gentzen’s improvements to HilbertBernays: Dear Mr. Gentzen! Unfortunately the answer to your letter, the one you wrote on the occasion of checking through the forwarded part of my draft, has been considerably delayed. . . Of your various observations—as I already wrote—I have used the majority. I give you further, below on sheet 3, the performed alterations. . . 260

Gentzen answered from G¨ottingen on 30 October 1937:261 Please forgive me that I only now answer to your sending. In the last weeks I had to do with the preparation of a lecture on “the present situation in foundational research” which I have given a week ago at the conference of the DMV and intend later to publish. As to your manuscript I’d like to remark in general that it appears to me very understandably written, and I think it will be easily 259 Cf.

p. 330 or p. 364, below. 975:1658. 261 The dates don’t match. Possibly one of them got the month wrong. 260 Hs.

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readable also for beginners. Sometimes it is somewhat too broad for my taste; I mean it could spare some repetitions, . . . In recent weeks I can for the first time detect a distinct improvement in my state of health. I already make careful attempts to deal again with the consistency proof.262

G¨ odel gave a lecture in Vienna in November 1937 to the members of the Zilsel Circle on consistency questions in logic.263 He maintained therein that a finitary consistency proof for arithmetic is impossible and dedicated the rest of the lecture to the question of how the finitary methods could be extended to obtain a meaningful consistency proof and referred as a third possibility to transfinite induction, as Gentzen had used it, up to certain concretely defined ordinal numbers of the second number class.264 In December 1937 Gentzen informed at least Paul Bernays that he had carried out his consistency proof in a simpler and more thorough form. Since December 1937, Gentzen’s sister, Waltraut, and her husband lived in Liegnitz/Niederschlesien (today: Lignice, Poland). urich that he had On 3 January 1938 Bernays wrote from Besenrain Str. 30 in Z¨ finished §11 of the foundations book for two weeks, but it was not yet typeset: “As soon as the copy is made, I will send it to you.”265 In Zentralblatt f¨ ur Mathematik 17 (1938), p. 242, there appeared: Rosser, Barkley: G¨ odel theorems for non-constructive logics. J. Symbolic Logic 2, 124-137 (1937). The deductive system P treated by G¨ odel (this Zbl. 2, 1), corresponding to the system of Principia Mathematica, is extended through the addition of the following inference rule with infinitely many premises: “If f (0), f (1), f (2), . . . hold, then so does (x)f (x)” (cf. Hilbert, this Zbl. 1, 260, and Carnap, this Zbl. 12, 145). By the degree of nesting of such inferences systems odel P1 , P2 , . . . , Pω , Pω+1 , . . ., etc., can be defined by transfinite recursion. The G¨ result is extended to such systems. . . —If one modifies the inference rule by replacing the infinitely many premises by the single premise: “for each n, f (n) is provable by determined means”, then if one starts first with the proof means of the system P, then permits the same with the addition of this rule, etc., one similarly obtains a transfinite sequence of systems for which analogous theorems are proven.— Further, some investigations are given on the system, which is closed under application of the first rule. Gerhard Gentzen.

On 9 May Bernays sent: the remaining part—grown quite big—of §12. With that the draft of the foundations book should be complete up to yet another pair of appendices. I was sorry to learn from your card of February that you were again so strongly hindered in your work on account of your health. I very much hope that this disagreeable state has passed in the interim. Have you now been able to bring to completion the new version of the consistency proof for number theory?

Following this he gave a few pieces of advice for improvement which had come to him by the study of the old proof.

262 Hs.

975:1659. Works, 1938a. 264 For further contents cf. p. 108 in: John W. Dawson, Jr., Kurt G¨ odel: Leben und Werk, Springer-Verlag, Vienna, 1999. 265 Hs. 975:1660. 263 Collected

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I hope you are in the situation to look over §§11 and 12 soon. (Perhaps you have already come to know §11.) The publisher wants now to begin with the impression of §§9 and 10. . . You will naturally be sent the proof sheets. Incidentally Ackermann has declared himself for participation in proofreading.266

On 12 May 1938 Gentzen answered: Very respected Herr Professor! Many thanks for your letter and the manuscript. In the main I am better again healthwise; the new version of the consistency proof is finished and will now be printed; I have declared that proofs also be sent to you. . . A work by Kolmogoroff, “Sur le principe de tertium non datur”,267 is not easily understood in the particulars by me as it is written in Russian, but it gives me the impression as if it already contains in essence the theorem of G¨ odel and me on not A implying A, so that K. deserves the priority; something similar is said in the big L[ogic] Notes] of the J. Symbolic Logic by the statement of this Russian work.— Then this theorem would thus have three authors!268 Your treatment of transfinite induction in connection with my consistency proof appears to me a quite welcome extension to offer to my publications, insofar as I bring nothing new on transfinite induction in my new version, but rather refer to the old, and particularly because of this, that you prove the fact only mentioned by me without proof of the representability of the proof of transfinite induction up to any number below ε0 within the number theoretic formalism treated. I have incidentally deserted the decimal fractions in the new version and have chosen the set theoretic formulation with powers of ω. . . How I have obtained the consistency proof from the methods of proof in my dissertation is, I believe, now somewhat easy to see in the new version. I have there expounded in detail on the train of thought. Finsler’s objection269 basically persists possibly in not so simple cases like those

266 Hs.

975:1661. Kolmogoroff, “O principe tertium non datur”, Mat´ ematiceskij Sbornik 32 (1924-25), 646-667. English translation in: J. van Heijenoort (ed.), From Frege to G¨ odel, Harvard University Press, 1967. 268 Valery Ivanovich Glivenko (*2 January 1897 in Kiev, †15 February 1940 in Moscow) was employed from 1928 until his death at the Liebknecht University in Moscow and worked on the foundations of mathematics and probability theory. “Stimulated by Kolmogoroff,” Glivenko could “show that every derivation of the classical propositional calculus translated into a derivation acceptable in intuitionistic logic, if one places two negation symbols before each formula. He concluded from this (Sur quelques points de la logique de M. Brouwer, Academie. Royale de Belgique, Bulletin de la Classe des Sciencees, s´erie 5, 15, pp. 183-188, 1929) that both logics accept the same negative formulæ of the same calculi. Thereby one says via an interpretation of a part of classical logic in intuitionistic logic and via the observation (already made by Kolmogoroff) that this part of classical logic is “hidden” in the intuitionistic—assuming that one changes the interpretation. In 1932 G¨ odel showed by applying another interpretation that classical arithmetic may be interpreted in intuitionistic arithmetic and therefore both deserve the same confidence.” (Marcel Guillaume, “Axiomatik und Logik”, pp. 748-882, in: J. Dieudonn´e (ed.), Geschichte der Mathematik 1700-1900, VEB Deutscher Verlag der Wissenschaften, Berlin, 1985; here, p. 849.) 269 Paul Finsler, “Formale Beweise und die Entscheidbarkeit”, Mathematische Zeitschrift 25 (1926), pp. 676-682. The mathematician Finsler, a student of Carath´eodory, had maintained that in every formal system there were propositions which could be seen to be contentually true but are not formally provable. Kurt G¨ odel, who was unaware of Finsler’s essay, proved this result formally. Paul Finsler was known to Paul Bernays. On his biography and scientific accomplishments, cf. Herbart Breger, “A restoration that failed: Paul Finsler’s theory of sets”, pp. 249-264, in: Donald Gillies (ed.), Revolution in Mathematics, Oxford Science Publications, Oxford, 1992. 267 A.N.

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treated by us; I believe something is first to be made of it if one considers the more general domain of higher ordinals.270

On 13 June 1938 Bernays reported to Gentzen: I have received the printed sheets of your new treatment and began reading it immediately. It is in particular good that you operate again with so transparent a formalism and that the ordinal number of a derivation is so simply defined.— Of interest, in comparison with your dissertation, is that while in it the cuts are successively eliminated, your present procedure comes down to successively eliminating the combination-inference figures. . .

And then followed ideas, improvements, suggestions and corrections.271 Gentzen expressed his thanks on 22 June for the corrections, which he had taken into account “so far as they would not cause large changes in typesetting.” Of the proofreading of Bernays’ book, he had looked through galleys 1 to 192, and he gave Bernays an index of typographical errors. He added, “Are you aware that a recent work (more reductions) on the Entscheidungsproblem by Pepis exists (Fund. Math. 30, 1936)?”272 “Die gegenw¨artige Lage in der mathematischen Grundlagenforschung” appeared in Deutsche Mathematik 3 (1935), pp. 255-268. It reappeared with his “Neue Fassung des Widerspruchsfreiheitsbeweises f¨ ur die reine Zahlentheorie” in 1938 as number 4 of the Forschungen zur Logik und Grundlegung der exakten Wissenschaften. One can only speculate on the tactics of the publication sequence. In 1938 the second edition of Hilbert-Ackermann, Grundz¨ uge der theoretischen Logik, appeared. In the foreword to the new edition we read: For various suggestions and comments I also wish to thank G. Gentzen–G¨ ottingen, who also read through the manuscript, Arnold Schmidt–Marburg and H. Scholz– M¨ unster. They all deserve my heartfelt thanks. Burgsteinfurt, in November 1937, W. Ackermann.

Gentzen probably contributed to the standardisation of the terminology, which would be used in Hilbert-Bernays, Grundlagen der Mathematik, and which he had helped improve. In August two reviews by Gentzen appeared in the 18th volume of Zentralblatt f¨ ur Mathematik on p. 337: Frank Jr., Orrin: New algebras of logic, Amer. Math. Monthly 45, 210-219 (1938). Three new propositional systems, namely the many-valued logic of L  ukasiewicz and Tarski (C.R. Soc. Sci. Varsovie UUU 23, 1-21 (1930), the intuitionistic logic of Heyting (S.-B. preuß. Akad. Wiss. 1930, 42-56) and the logic of quantum mechanics of Birkhoff and von Neumann (this Zbl. 15, 196) are compared from the point of view, which logical laws valid in classical boolean algebras fail in the individual systems.— For the L  ukasiewicz-Tarski logic a system of axioms is given (no. 5), from which follow all laws which are valid for an arbitrary number of truth values. Gerhard Gentzen (G¨ ottingen) ¨ Schmidt, Arnold: Uber deduktive Theorien mit mehreren Sorten von Grunddingen, Math. Ann. 115, 481-506 (1938). If one wishes to apply the (lower) predicate calculus to the formal representation of deductions in a theory with several kinds of basic objects—e.g., a geometry with points, lines, and planes—there are two 270 Hs.

975:1662. 975:1663. 272 Hs. 975:1664; cf. Gentzen’s review below on page 97. 271 Hs.

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approaches: 1. one generalises the calculus to a “many-sorted predicate calculus”, by which the inference rules belonging to “for all” and “there is” may only be applied to objects of a given sort.— 2. one sticks with a “single-sorted predicate calculus” and distinguishes the individual sorts through certain predicates, which signify the belongability of an object to a sort.— The author proves the equivalence of the two methods, taking up a proof approach of Herbrand (thesis, Paris/Warsaw 1930), which however was insufficient in some essential points.— Simultaneous with the proof an interpretation of expressions of the many-sorted predicate theory by equivalent expressions of the “single-sorted” theory with sort predicates is given (§3).

Already the following month in Zentralblatt f¨ ur Mathematik 18, p. 385, we find: Pepis, J´ ozef: Untersuchungen u ¨ ber das Entscheidungsproblem der mathematischen Logik. Fundam. Math. 30, 257-348 (1938). The results of an earlier work of the author (this Zbl. 14, 98) on the decision problem of the predicate calculus are sharpened through further development of the methods used [earlier].

And there followed 20 lines of technical material. In a foreword,273 Oiva Ketonen expressed his thanks to Gerhard Gentzen, whom he had visited in 1938. Ketonen studied 1938/39 in G¨ ottingen. He wanted to continue his studies with Bernays, which for other reasons was not possible. My old professor Oiva Ketonen published his doctoral work in 1944, although he studied in 1938-39 in G¨ ottingen by Gentzen. Oiva Ketonen’s work is the best that would be written in proof theory between the works of Gentzen and that of Kleene in the early 50s. This would be recognised, e.g. by Haskell Curry. Ketonen, however, did not continue his logic career. “Headaches”, he said. Then he stated that he never understood why Bernays wrote such a long review of his results (these are three pages in the Journal of Symbolic Logic, 1945, Vol. 10, pp. 127-130): “Perhaps the work was better than I thought.” On the political side he never said much; once, that one had to study a book by Rosenberg.274

On 18 November 1938 the president of the Swiss school inspectors sent an invitation to W. Ackermann to an international conference on the foundations and methods of mathematics under the direction of Prof. Dr. F. Gonseth. In the letter one reads, “With you, for our part, also Herr Prof. Dr. G. Gentzen in G¨ ottingen is invited as a German scholar,” although Gentzen, unlike Ackermann, had not yet habilitated.275 38. Extension of the Unscheduled Position for Another Year Taking Effect 1 October 1938 In November 1938 neither the Dozentenschaft, the rector, nor the faculty had anything against the period of service of the unscheduled assistant Gentzen being “extended from 31 October 1938 for an additional two years.” However, for reasons unexplained, Hasse asked the trustees on 4 November 1938 only for a lengthening of one year. To justify the extension, he enclosed an evaluation that he had written on 3 November 1938: 273 Oiva

Ketonen “Untersuchungen zum Pr¨ adikatenkalk¨ ul”, Sci. Fennica, Serien A, I, Mathm.-phys. 23, Helsinki, 1944. 274 Letter from Jan von Plato to the author, April 1999. 275 Hans Richard Ackermann, “Aus dem Briefwechsel Wilhelm Ackermann”, History and Philosophy of Logic 4 (1983), pp. 181-202; here, pp. 185ff.

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Evaluation of the scientific work of Dr. Gerhard Gentzen Gentzen is a leading researcher in the area of mathematical foundational research. He has above all become known for his consistency proof for pure number theory, which he published first in the year 1936 and in simplified form in the year 1938. In addition to other contributions to mathematical foundational research he has written two clear and widely read summary reports on the present state of foundational research and the work still to be carried out. Presently he is occupied with all available time and power on the largest open problem of this kind, namely the task set by Hilbert, of proving the consistency of analysis. He intends to present this work on completion as his habilitation thesis. Gentzen’s special task here is to be at Hilbert’s disposal for the analysis of unfinished lines of thought. He is irreplaceable in this function. Hilbert no longer possesses the power to carry to the end the securing of the foundations of mathematics begun by him. Gentzen is an excellent expert probably the only one who can succeed in reaching this goal. Hasse

Gentzen was virtually placed under the protection of Hilbert, but he was also being subordinated to an “end”, namely only to be Hilbert’s completer and nothing else. On 14 October 1938 W. S¨ uss wrote to a senior civil servant Dr. Dames of the Reichserziehungsministerium (REM) [National Ministry of Education] concerning Gentzen: G. Gentzen, Special Assistant to Hilbert (Docent?). G¨ ottingen. I’ve known Gentzen already as a young student when he stood out everywhere through his talent. For some years now he has been at the disposal of the Prof. emeritus Hilbert and belongs to the leading foundational researchers in mathematics. His knowledge however extends far beyond this. Through his activity by Hilbert the preference for work on the foundations of mathematics is understandable, as Hilbert has, as is well-known, almost exclusively occupied himself with it since 1920. Here I would now like to suggest that one should now already be thinking about the future of Gentzen. What should become of him if Hilbert dies? We mustn’t allow a power like Gentzen to lie fallow. As a foundational researcher he has only then the possibility of advancement if the state decides to allow in Germany a post in this area, which at the moment possesses secure leadership in that respect. If however this is not to be expected, then one must already today lead Gentzen to another track. If no other way stands open, I would in this case as in that of Magnus be very thankful for an appropriate sign.

Gentzen was recommended by S¨ uss on the basis of his first-rate professional performance. For another “problem child”, for example, professional ability and officer status are joined together: Weiss is certainly an honest man, in any case no opportunist, something some older colleagues have feared appears to be the case because of his collaboration on the Deutsche Mathematik, so that they haven’t recommended him for a position. Weiss is a convinced National Socialist. He is unselfish, practical and idealistic; the last perhaps more than one expects from a man of his age.

For Gentzen, however, that sort of comment is manifestly not brought in at all.276 Gentzen was careless in the matter of his professional advancement. He left such things to be dealt with by others, first by Hasse and then Hans Rohrbach. But 276 Nachlass W. S¨ uss, Bestand C 89/53, University Archives Freiburg im Breisgau.— For a biography of Wilhelm S¨ uss and his relations to Behnke, Hasse, etc., cf. the relevant publications of Volker P. Remmert (see the bibliography), whom I thank for this source.

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Bernays—and also the proofreader Gentzen—was clear that Gentzen was valued in this work on Hilbert-Bernays 1939 exclusively insofar as it fit into Hilbert’s conception. Everything that went beyond that would not be discussed. In the foreword to the second volume of Hilbert-Bernays, Bernays wrote: The second main theme is constituted by the examination of the facts on the grounds of which the necessity has arisen to extend the frame of contentual modes of inference permissible in proof theory, as opposed to the previous delimodel’s discovery of the itation of the “finite standpoint”. In these considerations G¨ deductive openness of every sharply delimited and sufficiently expressive formalism is the focal point. . . The discussion of the extension of the finite standpoint leads to the consideration of Gentzen’s new consistency proof for the number theoretic formalism. Of this proof admittedly only the methodic novelty, namely the application of a special sort of Cantorian “transfinite induction”, is brought to a more exact representation and discussion . . . At present W. Ackermann is in the process of so arranging his earlier consistency proof (presented in §2 of the previous volume) through application of a transfinite induction of the sort that was used by Gentzen, so that it receives validity for the full number theoretic formalism. If this succeeds—for which the prospects are good—then with it the original Hilbertian attempt will be rehabilitated with respect to its effectiveness. In any case the view can already be justified on the basis of Gentzen’s proof, that the temporary failure of proof theory was caused simply through an overloading of the methodic requirement that one had placed on the theory. Admittedly, the decisive verdict on the fate of proof theory will be determined by the task of verifying the consistency of analysis.277

And herewith Gentzen’s problem from now on was named: to find the consistency proof for analysis. Besides, here W. Ackermann is explicitly identified with carrying out Hilbert’s attempt; Gentzen no longer is. Bernays explained the G¨ odelian cause for extending the methodic frame of proof theory and then entered into Gentzen: “We wish not to present the proof in detail here, rather to enter more closely into one of the modes of inference from Cantorian set theory actually used, which too forms the part of the proof in which the formalism (Zμ ) is transcended.” On pp. 360-374 then follows the discussion of transfinite induction. The paragraph ends with the words, “In case this perspective should prove itself, then with Gentzen’s consistency proof a new branch of proof theory would open.” This “new branch” would ever tend to be unrecognised. But even Bernays—if he had held it useful—could have read Gentzen without simultaneously bringing him into line with the Hilbert programme. Naturally Wilhelm Ackermann had recognised this. Thus he gave unconditionally a consistency proof of number theory, which again came much closer to an older Hilbertian intention. Ackermann wrote: A proof of this sort, using a special system of the predicate calculus, was first produced by G. Gentzen. The proof before you goes back to older methods, which I gave at the time for number theory without full induction, following Hilbert’s original attempt at a consistency proof.

The basic idea of the proof is discussed in Hilbert-Bernays volume II (1939), §§2, 4. In this proof a formalism based on Hilbert’s -symbol is applied. At the same time the new Ackermannian proof used ideas of Gentzen. For Ackermann it was clear that Gentzen had distanced himself far from Hilbert’s original insight. 277 P.

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39. Bernays’ Views of Hilbert’s Programme and Gentzen’s Place in It In Paul Bernays’ essay “Hilberts Untersuchungen u ¨ber die Grundlagen der Arithmetik” [Hilbert’s investigations on the foundations of arithmetic] it says that with G¨ odel’s results of 1931 it was clear that the consistency of the formalism of number theory would not be established by the investigations of Ackermann and von Neumann. It was doubtful if a proof in the sense of the “finite standpoint” could be produced at all. What consequences of G¨ odel’s results did Bernays see for proof theory? G¨ odel had nothing immediate to say on the possibility of a finitary consistency proof: all the same a criterion arises which each proof of the consistency of the formalism of number theory of a more encompassing formalism must satisfy: some reasoning must come forward in the proof that is not—on the basis of the arithmetic translation—representable in the formalism in question. With this criterion one would become aware that the existing consistency proofs do not suffice for the full formalism of number theory. [Footnote: The proof of von Neumann dealt from the start with a narrow formalism, yet it seemed that the extension to the entire formalism of number theory would not be difficult.] From this the conjecture would even be raised that in the frame of the elementary, intuitive considerations, like those corresponding to the “finite standpoint” on which Hilbert based proof theory, a consistency proof for the number theoretic formalism could not be produced. This conjecture has till now not been disproved. However, K. G¨ odel and G. Gentzen have observed [Footnote: K. G¨ odel, “Zur intuitionistischen Arithmetik und Zahlentheorie”, Erg. Math. Kolloqu., Vienna, 1933, no. 4. G. Gentzen withdrew his paper on the subject, which was already in press, on account of the appearance of G¨ odel’s note of similar content.] that under the assumption of the consistency of the intuitionistic arithmetic formalised by A. Heyting (A. Heyting, “Die formalen Regeln der intuitionistischen Logik” and “Die formalen Regeln der intuitionistischen Mathematik”, S.-B. preuß. Akad. Wiss. Phys.-math. Klasse, 1930, II) the consistency of the usual formalism of number theory is rather easily proven. [Footnote: It can namely be shown that each derivable formula of the ordinary formalism of number theory, which contains no formula variables, no disjunction, and no existential quantifier, is also derivable in Heyting’s formalism.]. . . While this paper was being typeset, the proof of the consistency for the full arithmetic formalism was produced by G. Gentzen [Footnote: This proof will be brought to publication shortly in the Math. Ann.], through a method which complies throughout with the fundamental demands of the finite standpoint. Thereby at the same time the mentioned conjecture concerning the scope of finitary methods (p. 212) finds its refutation.278

Otto Blumenthal summarised Hilbert’s “Lebensgeschichte” [“life story”, i.e. biography] in the same volume of Hilbert’s collected works (p. 425): A presentation of Hilbert’s arrangement of the logical calculus came out already in 1928: Hilbert-Ackermann, Grundz¨ uge der theoretischen Logik. The questions are still in progress; there would be, particularly by K. G¨ odel, weighty objections raised against proof theory. But Hilbert is imbued with the conviction of its final success, and in any case it is an achievement of such high originality and power that this work of his old age will count among his greatest. [Footnote: (Additional remarks added in proof, Sept. 1935.) The complete proof of the 278 Paul Bernays, “Hilberts Untersuchungen u ¨ ber die Grundlagen der Arithmetik”, in: David Hilbert, Gesammelte Abhandlungen, III, Springer, Berlin, 1935, pp. 196-216; here, p. 216.

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consistency of number theory has recently been obtained following Hilbert’s plan by G. Gentzen (Math. Ann. 112, pp. 211ff.).]

40. Closing Thoughts on Paul Bernays Let me make a few remarks about Paul Bernays at the time of his friendship with Gerhard Gentzen from 1931 to 1939. His professional status was that of an extraordinary professor at G¨ ottingen from 1922 to 1933. In the Winter Semester 1930/31 P. Bernays, K. Friedrichs, and H. Weyl officially received teaching duties. Gentzen came to Bernays already in the Summer Semester of 1931. Following Hilbert’s retirement with emeritus status in 1930, Weyl was appointed his successor. With the enactment of laws against the Jews by the Nazi regime, Paul Bernays (together with Paul Hohenemser, Lewy, and Otto Neugebauer) had to leave the university on 27 April 1933. Weyl arranged that Bernays was paid until October 1933, when Weyl himself emigrated. Bernays went into exile. Bernays never considered himself a supporter of Hilbert’s Programme, at least not philosophically. In a letter to Kurt G¨ odel of 7 September 1942 he wrote, among other things: These conceptions do not correspond at all with a strict “formalist” standpoint, but I have never taken up such a one; in particular, in my essay “Die Philosophie der Mathematik und die Hilbertsche Beweistheorie” [The philosophy of mathematics and Hilbert’s proof theory] (written summer 1930) I distanced myself clearly from this, and even more so in the lecture (probably known to you) “Sur le platonisme dans les math´ematiques” [On platonism in mathematics].279

Despite the fact that Bernays was exiled and no longer being paid to do so and that he did not accept the philosophy underlying it, he felt obligated to complete Hilbert’s Programme through publishing the second volume of Hilbert-Bernays and marketing the programme at conferences. His main duties for the Hilbert Programme were • to maintain the programme not only conceptually but by giving concrete proofs to fulfill the demands of the Programme (audience: specialists), • to report on progress at conferences and in journals (audience: general mathematicians). We must, however, go beyond these duties and occupy ourselves with Paul Bernays and his situation. One can hardly imagine it. Bernays conceived and wrote the first volume of the Grundlagen der Mathematik (1934). Before that volume was published, Bernays not only lost his position, but as a Jew was no longer welcome in Germany and thus had emigrated to Switzerland. Nonetheless the foreword to the book bears the inscription, “G¨ ottingen, March 1934. Paul Bernays.” He simulated “normalcy”, which had long disappeared. His book would furthermore be sold in Germany, and it served as an adornment of German mathematics. But Bernays was unemployed, with no permanent position and pursued doubtfully a professorship, for which reason he even applied to H. Weyl. Bernays sat in Z¨ urich and tried there to make contacts. Because it hardly succeeded, he attended every accessible congress. He was offered a position in Quito, Ecuador, but wanted to go to Princeton to the Institute for Advanced Study. He was given a visiting assistant professorship there, but he had to vacate the position at war’s end, when the Swiss 279 Kurt

G¨ odel, Collected Works, IV, p. 134.

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offered him a professorship at the ETH Z¨ urich. In such uncertain times, it is hard to believe he conceived the second volume of the Grundlagen der Mathematik and therein first is the inscription, “Z¨ urich, February 1939, Paul Bernays”. But again the volume appeared in Germany, where, as a Jew, he had to take care not to enter. We look with astonishment at the unshakeable discipline and devotion to the Hilbert-Bernays Programme. To be sure one can ask why Bernays didn’t have greater confidence in Gentzen’s work, instead of viewing him only from the HilbertBernays point of view, but these are not realistic considerations. We should be glad that Bernays had dealt with highly talented individuals like Ackermann, Gentzen, Schmidt and Scholz, who sought to associate with people like themselves and so could show the world that the ideas of Hilbert and Bernays were still fruitful even in dismal times. The existential difficulties of Bernays are dramatic, but were never born in public by him. On the contrary, in the most distinguished fashion he lived only for the work in the Hilbert-Bernays Programme. But during the war he occupied himself ever more with questions of set theory and model theory. That is an area where the representatives in Germany would all be forced to emigrate. 41. Longer Notes 5

Presumably Gentzen used the edition, Max Weber, Wissenschaft als Beruf, Wissenschafliche Abhandlungen und Reden zur Philosophie, Politik und Geistesgeschichte, VIII, third edition, Duncker und Humblot, Munich and Leipzig, 1930. This volume was found among the few things Gerhard left behind in Rottweil. The following passages are marked. p. 12: “And who thus doesn’t possess the ability to put on blinkers and manage to convince himself that the fate of his soul depends on this: whether he gets this, especially this conjecture at this place in his manuscript right; he remains remote from science.” p. 13: “It is in fact true that the best things occur to one so,. . . by taking a walk on a slowly ascending street, in any case, however, when one doesn’t expect it, and not during the brooding and searching at the writing desk.” p. 14: “ ‘Personality’ in the scientific domain is had only by he who serves the purity of the field.” p. 16: “Scientifically to become outdated—it must be repeated—is not merely our universal fate, but rather our universal goal.” p. 21: “Who—except some big children, as they turn up especially in the natural sciences, still believes today that findings of astronomy or of biology or of physics or chemistry can teach us something about the meaning of the world, and so only something about that: in what way can one track down such a ‘meaning’—if it exists?” p. 22: “Natural sciences such as physics, chemistry, astronomy presuppose as self-evident that the (so far as science is concerned: constructible) final laws of cosmic events are worthy of discovery.” p. 26: “I offer to furnish proof in the works of our historians that whenever the man of science has his own value judgement, the full understanding of the facts stops.” “Indeed science on its part does not know ‘wonder’ and ‘revelation’. These would be untrue to its own assumptions. . . And any assumption-free science claims for itself no less—but also no more—as the acknowledgment: that if, for an empirical explanation, a course of events is to be explained without appeal to supernatural causal elements, then so, as desired, must science explain it. But this it cannot do without being untrue to its beliefs. . . If one is a decent teacher, then it is his first task to teach his students to acknowledge uncomfortable facts, such, in my opinion, which are inconvenient for his party-beliefs. . . ” p. 36: “It is neither accidental that our highest art is intimate and not monumental, nor that today only within the smallest circle, from man to man in quiet does something

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pulsate that corresponds to that which in earlier days boomed like a prophetic blast of stormy fire through the great community and welded it together.” This is an expression of the objectivity of the value-free specialisation, and the freedom from assumption of science, and of its dedication—in private circles. And this expression and lifestyle would help the Nazis to overpower the science and high schools. 7

Much of the credit for G¨ ottingen’s leading position in mathematics at the time must go to Felix Klein. Weyl’s lecture “Kleins Stellung in der mathematischen Gegenwart” [Klein’s position in the mathematical present] is of particular interest here, especially in light of later remarks by Hugo Dingler. The penetration of National Socialism really becomes apparent if one reads a memorandum—a sort of quasi-commentary—of Dingler’s of 1933. I gladly present a few passages so that the reader may verify the stylistic and contentual penetration of National Socialism: . . . As the possibility now opened up for entry into a state funded academic position, and soon thereafter, through Germany’s upturn after the war of 1870/71, a strong increase in docent positions at the higher schools came about, to which the earlier employments of the bourgeoisie began to be considered as prejudice by the Jewish co-inhabitants in Germany and from the progressively loosening liberal trend of the time, there resulted, as said, a brisk influx of Jewish teachers into careers in the exact sciences. Now here is a man to be remembered; significant events in the public relations of university mathematical studies in Germany are tied to this person. Felix Klein would be born in 1899 in D¨ usseldorf as the son of a personal secretary of the government president and former military under tax officials and is of Jewish ancestry on the side of at least one of his parents. He took up the suggestion of his teacher at Bonn, the important physicist and mathematician Pl¨ ucker, and combined this via Cayley with the entirely new noneuclidean geometry, which he knew to fashion into a sensational success. In person he was a confident manifestation of the dictatorial nature, to which already in his youth his surroundings bowed. He represented the type of scholar of the exact sciences who resembles more a government minister than a researcher. In pure research he achieved nothing fundamental which did not already exist in his basic ideas, however he repeatedly had a lucky catch in his programmatically presented theories. Klein’s inner striving was for governance and power over men, for which science only supplied the background and the possibility. So, after suffering a scientific defeat in a race with the famous Frenchman Henri Poincar´e to solve a problem, he had a nervous breakdown (1862) and turned, particularly since the commencement of his position in G¨ ottingen (1886), exclusively to “organisational” activity. He had the peculiar conception that one can “organise” a science, which yet must rest on freely creative powers. So he sent out (probably c. 1896) a circular to the full professors of mathematics in Germany in which was suggested that one in a central position (i.e. G¨ ottingen and he himself) would assign the mathematical problems which must be solved and the person suitable at the time that must be engaged with the solution to the problem. As if mathematical problems could be solved on command! (Personal communication from privy councillor F. Lindemann—Munich.) As he met with no approval with his suggestion, he conceived the plan (c. 1898) to codify the entire existing mathematics in a colossal work. As this plan (the later Enzyklop¨ adie der mathematischen Wissenschaften [Encyclopædia of the mathematical sciences]) met with approval, he had obtained the instrument to subjugate the “whole of mathematics”, i.e. the totality of those persons at official positions in Germany reflecting his business mentality. Everyone who dared to hope “to be somebody” must now collaborate

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here or in some other way in Klein’s mathematical workhouse. Now, what can be taken from this, in investigations of problems, will be recognised; everything else falls under the table. Bye and bye all mathematical journals worth naming ottingen, then the became directly or indirectly dependent upon him, upon G¨ main publisher. In the end no one in Germany could hold more than the smallest position without Klein’s consent. Soon nearly all the mathematical chairs in Germany were occupied by his people. The persons whom Klein chose for these were exclusively such who proved themselves worthy as collaborators in his “Betrieb” [enterprise] (a favourite word of his). But his ambition reached further. He wished also to subjugate mathematics abroad. So he drew foreigners as students to G¨ ottingen, where each arbitrary foreign student finds the highest interest, while the young German, in case he is not Jewish, must lead a modest existence. The atmosphere is not only somewhat international and pacifistic, but is already since the 90s outspokenly anti-German. A hint of nationalist convictions by a young German leads automatically to a professional transfer to the provinces [Kaltstellung]. The Jewish students, on account of their skillfulness, quickness, and rapid absorption, are everywhere in the foreground. (This agility in the intellectual handling of the— for the moment—immediately vital and important facts exists only because of a lack of depth; the non-Jewish mentality therefore only appears slow because many more secondary associations will be made—associations which, to be sure, can appear less important for the purely superficial material consideration in the moment, but in fact carries with it a, in many ways, larger breadth of perspective and of possibilities which are preconditions to the extraction of new inventions.); the young Germans are so impressed thereby that they, not to be too different, emulate them in speech, bearing, and gesture with all eagerness, so that a specific G¨ ottingen mathematical behavioural style has formed and become widespread German mathematical fashion. Only those German non-Jews who have adapted themselves in nature and attitude as completely as possible have a modest prospect to arrive in the general struggle for a permanent job. Through Cauchy’s reform an entirely analogous situation has developed in mathematics as in physics. With the turn to pure approximations, the incentive is lost to search for large lines and categories in functionals. This situation is exceedingly favourable for purely casuistically oriented mentalities like the Jews have been ever since their classical literature, and so there developed then, especially in the so-called function theory and its neighbouring areas (analytic number theory, set theory), a vast, extensive casuistic research, which lies mainly in Jewish hands. Countless small and insignificant results were inflationarily complicated, and thereby published in high scientific appearing presentation, and were accepted by the like-striving with the greatest expenditure of enthusiasm and praise that previously was reserved only for genuinely fundamental advances of science. So there soon developed a uniform protected public opinion in mathematics, which soon became the more uniform and could be the more uniformly controlled, ever more the Klein dictatorship resulted in the creation of a uniform-minded mathematicianship, which finally became a heavily visibly closed organisation. (Already in the first decade of the 20th century there was at every German high school a, usually Jewish, “G¨ ottingen envoy”, who reported occurrences involving the mathematical and physical personnel to G¨ ottingen, and through the G¨ ottingen influence almost ruled.) Through this thus-created public opinion this then appeared: Sheaves of such little theorems soon acquired larger than life significance and would work for the given candidate for an open position, whereby the G¨ ottingen organisation acquired further increase in power.

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A very far-reaching side effect of this development was the following: the outstanding mathematically gifted German non-Jews are, in the direction of their talent, intent mostly very intuitively on form-ideas, as these shaped the nature of all pure science since the Greeks and make up the proper nature of this science. Through the casuistic type of treatment, however, such talents no longer find a proper field of work. Opposed to that, this circle focuses more narrowly on the dialectic mastery of hundreds of casuistic special cases, which must not seem meaningful enough to them (and correctly so) to devote one’s life to these. The Jewish powers were, through the casuistic gift, an advantage here. At the same time it seems for them the lack of an inner sense is less of a hindrance there, where a biological-utilitarian sense of an occupation is given. Through the ever more widening casuistic treatment of mathematics in books and lectures the appearance arose that the Jewish ability at mathematics was a greater “gift” than the non-Jewish. This in turn automatically had an effect on the selection of the supposedly gifted, where the lectures and exams would be held by casuistically oriented teachers. Thus the Jewish-casuistic element penetrated ever further into mathematics, so that finally only Jewish teachers appeared suitable to occupy the higher mathematical positions. Which then occurred. Another characteristic of the Jewish talent comes in, namely the denial of all that is intuitive, which so to speak is a corollary to the predilection for the casuisticdialectic treatment of pure word-laws. (“You should make yourself no image nor any likeness at all.”) So it came to pass that in German mathematics all that is intuitive (on which in the shape of form-ideas all pure science rests), thus in particular pure geometry, would be pushed aside ever more. This circumstance has worked out so that today it is very difficult to find by the above methods among the younger generation selected representatives who are suitable to occupy a chair in pure geometry. Almost only at the technical high schools, where an absolute material necessity forces intuition, there lingers yet a remnant of real geometry. But through the above treatment of public opinion this remnant leads only a patient existence. Its works and problems will, through the general cheap propaganda, be considered as of little value and they have difficulties with publication in professional journals, which are entirely in the hands of the non-intuitive direction. The invisible organisation created by Klein has full power in its hands, uses it mercilessly, and admits only such non-Jews who have subjugated themselves body and soul. Everything that exists in the talents in German non-Jewish circles has from the beginning with all means been put off and attempted to be kept aside, whereby the mechanism described proves itself restlessly useful and effective. Now let us turn to physics. . . (Memorandum by Professor Dr. Hugo Dingler of the Technische Hochschule Darmstadt on “Abwegige Entwicklungen im Gebiete und in P¨ adagogik der Mathematik und der exakten Naturwissenschaften im letzten halben Jahrhundert” [Absurd developments in the area and pædagogy of mathematics and the exact natural sciences in the last half century], selections from two larger memoranda of April and September 1933, combined on 12 November 1933 by Philipp Lenard and sent on 21 December 1933 and in January 1934 to the Bavarian State Ministry for Teaching and Culture, Minister Hans Schemm, other culture ministers, and the Reichsminister of the Interior as denunciation of the G¨ ottingen School. Source: Bundesarchiv Koblenz, former Zentrales Staatsarchiv Merseburg, Rep. 76 Va Sekt. 1, Tit. VII. no. 14. See David E. Rowe, “ ‘Jewish Mathematics’ at G¨ ottingen in the era of Felix Klein”, Isis 77 (1986), pp. 422-449; here, p. 422.)— The apologetic recruitment for Dingler, O. Becker and others is still a strong tradition in Germany. Kurt Wuchterl (Handbuch der analytischen Philosophie und Grundlagenforschung. Von Frege zu Wittgenstein, Paul Haupt, Verlag, Bern, 2002) holds Dingler’s “attempts at conformity and concessions to those in power, which do not shy away from anti-Semitic formulations”

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(p. 285) to be merely “obscure” [undurchsichtig]. The offer of denunciations and acts are never mentioned. One is almost tempted to say that Joseph Goebbels had it right: his ideology continues on. On Felix Klein, Hermann Weyl reported differently in his lecture at the opening of the new institute building: But I do not wish to describe the powerful and imaginative organiser that he was, who had worked so fruitfully leading mathematics out of its cultural isolation, for its ties to applications in physics and technology, for the stimulation and modernisation of mathematical and natural scientific instruction of middle and high schools; I will not paint a picture of his effectiveness as a high school teacher, which in the field of mathematics had hardly seen the like of him either before or after, rather I wish to dedicate these hours to reflect on the position Klein holds in the construction of contemporary mathematics through the methods and results of his own scientific research. And too I may be permitted to restrict myself to pure mathematics. I am aware how much I offend at the very least against the Kleinian ideal which constantly places and views mathematics in interaction with its applications. In his historical lectures he had awarded Gauss the crown among mathematicians, because “he combined the greatest individual works in each area taken up with the greatest versatility and because [in him] mathematical inventiveness, rigour of execution, and a practical feeling for applications even to including carefully executed observation and measurement” balanced each other. Klein’s relation to applications was—it may be frankly expressed and justifies to some extent the restriction imposed by me—genuinely looser than that of Gauss. It is only here occasionally, and then it stood out more as organisational than creative. . . In mathematics the danger is particularly great to accept only the first, the objective side; the mathematician tends to make things absolute. Klein was free of this blindness. The lectures he held on the history of mathematics in the 19th century during wartime, reveal his sense of history. . . Klein has complained that “it seems to be that the German society refused to develop a uniform cultural atmosphere that would embrace the exact scientific element as a peculiar and natural component. . . The striking texture in Klein’s scientific personality is the passion for mixing, for bending together, to allow the various disciplines to penetrate each other. . . Herein, in capturing inner connections and correspondences, he is quite ingenious. . . I should in conclusion enumerate what appear to me as Klein’s most important and most potently continuing accomplishments in pure mathematics, so I would name: He has replaced a narrow concept of geometry, like that the projective school had erected, by a very free and encompassing view of the nature of geometry. . . Klein’s idea of geometry is nothing other than the relativity theory in its general, mathematically formulated version. He has understood the group as an organising and illuminating principle universally in algebra, geometry, and analysis, and brought it to application. He has shown the advantage of the basic ideas of the Riemannian theory of functions, accompanied by suggestive physical intuitions in a liberated and creative manner. With the theory of automorphic functions and their applications to uniformisation he has ascended to the real summit of the Riemannian theory of functions and in this way on the paths of understanding and discovery of topology developed a problem area that today is function theoreti-

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cally far from exhausted and has scarcely begun to be clarified from the standpoint of abstract algebra. Thus Klein, who stamped an entire epoch with the seal of his genius, powerfully and vividly in the present worked his effect on developing mathematics in signs of group theory, topology, and abstract algebra. The Nazi denunciation of Klein was not universal, as shown by Werner Weber’s remark cited on page 48, above. 10 Paul Hertz was born on 29 July 1881 in Hamburg. He studied mathematics and physics in Heidelberg, G¨ ottingen, Leipzig, and then again in G¨ ottingen, where he earned his Ph.D. in 1904 as a student of Max Abraham. In 1909 he earned his Habilitation at ottingen in 1912. Between 1915 and 1919 he performed his Heidelberg and returned to G¨ military duty. On 4 March 1918 he was awarded the title “Professor”. He was professor extraordinarius (i.e. unsalaried) there from 3 August 1921 till 1933. From 1 September 1922 until 31 January 1925 he was an unscheduled assistant and later came also a modest special teaching post compensation. So he was, to be sure, an “unchaired extraordinary professor of physics”, with the status of professor, but with only a very, very small income as assistant. He taught “Methods of the Exact Sciences”. Hertz began his scientific career as a theoretical physicist and, together with Moritz Schlick, edited Helmholtz’s epistemological papers. He belonged to that group of scientists and philosophers who published the journals Annalen der Philosophie and Erkenntnis. Later on, he made significant contributions in the foundations of mathematics and in axiomatics. His approach to research was characterized by thorough treatment of conceptual issues and close investigation of methodological problems. He introduced new, methodologically important concepts and achieved results significant for the foundations of mathematical logic. His investigations were the precursor to various later researches in this area. Gerhard Gentzen’s sequent calculus in particular took as its starting point Hertz’s observations about systems of propositions. Eventually, his interests led Hertz to philosophical-logical questions. On 14 September 1933 his teaching authority was officially taken away in accordance with section 3—“‘Jewish extraction”—of the Law for the Restoration of the Professional Civil Service and the Reich Citizenship Law, and his assistantship was terminated in October of the same year. He moved with his wife to Hamburg and attempted to find work abroad. But he was regarded as careless, impractical and didactically incompetent. To make up for this his wife, Dr. Helene Hertz, was a good pædagogue. Political circumstances, the Nazis, caused him to emigrate first to Switzerland, where he lectured in Geneva (1934-1935), then to Prague (1936-1938). He always received only small stipends, which were financed by research grants from American aid committees. His children, Hans (*1915), Rudolf (*1917), and Elisabeth (*1924) emigrated already in 1936 and 1937 to the USA. In 1938 Hertz and his wife also moved there. Paul Hertz couldn’t establish himself professionally. He died on 24 March 1940, at the age of 58 in Philadelphia. His wife, Helene, died there in 1973. Paul Hertz was an important representative of the philosophy of science at a time when it received virtually no recognition in German-speaking lands. This information is cited from V. Michele Abrusci (“Paul Hertz’ Logical Works. Contents and Relevance”, Atti del Convegno Internationale di Storia della Logica, San Gimignano, 4-8 Dicembre 1982, Clueb, Bologna, 1983, pp. 369-374), the Hertz-Archive in Pittsburgh, and Anik´ o Szab´ o.

37 In a note left behind in 1939, Gentzen reiterates, “Speaking of various formal possibilities, philosophical treatment: doesn’t belong here.” I thank Dr. Christian Thiel for the source and transcription.— I hold this decision to be one of the important ones Gentzen made. He quasi-withdrew from the fuzzy area between philosophy and mathematics of the

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19th century, where supposedly the “reworking of the prior aristotelian-scholastic formal logic with the aims of foundations of mathematics” (Volker Peckhaus, Logic, Mathesis universalis und allgemeine Wissenschaft. Leibniz und die Wiederentdeckung der formalen Logik in 19. Jahrhundert, Akademie Verlag, Berlin, 1997, pp. 307ff.) would be carried out and at the same time with the mathematical instrumentalisation of logic, mathematicians would stake a claim on logic. Beyond a mathematically grounded logic or logical foundations of mathematics, Gentzen aimed at a mathematical analysis of the practice of mathematical proofs and attempted to give philosophy of mathematics the least room possible in mathematics. With this reduction another national or sociological or contextual interpretation and historiography of the ground within mathematics should be drawn. Paradoxically, the opposite occurred. Where the gods would be driven away, demons come calling.— Gentzen mentions this standpoint in every lecture, once again in “Die gegenw¨ artige Lage in der mathematischen Grundlagenforschung”, which is reproduced in Appendix C: It seems to me that a key feature of Hilbert’s standpoint is the attempt to remove the mathematical foundational problem from philosophy and handle it so far as possible with the proprietary methods of mathematics. Clearly one cannot solve the problem entirely without extramathematical presuppositions. . . In the following I will not go into all the philosophical disputes, the responses to which have no influence on mathematical practice, and which make the problem situation appear unnecessarily complicated and difficult. This doesn’t mean, however, that Gentzen identified with the Hilbert standpoint. To the contrary, he had distanced himself from it, so that he explained what he took over from the Hilbert standpoint, for otherwise he wouldn’t have had to be explicit about this. 80

In addition to Paul Bernays, Paul Hertz and Emmy Noether lost their positions in accordance with §3 of the Law of the Restoration of the Professional Civil Service, which section dealt with extraordinary (i.e. assistant) professors. Following §6, which dealt with full professors, Professor Felix Bernstein was dismissed, and, at his own request, again under §6, Professor Edmund Landau was likewise dismissed (cited from Sybille Gerstengarbe, “Die erste Entlassungswelle von Hochschullehreren deutscher Hochschulen”, Berichte zur Wissenschaftsgeschichte 17 (1994), pp. 17-39; here: p. 24. For Emmy Noether, cf. Lisa ae, No. 12 Glagow-Schiche, “Die Mathematikerin Emmy Noether (1882-1935)”, Koryph¨ (1992), pp. 57ff.). On the other hand, the leaves of absence of Noether, Courant, and Bernstein of 25 April 1933 were already announced on 26 April 1933 in the G¨ ottinger Tageblatt (cf. page 547 of Rolf Schaper, “Mathematiker im Exil”, in: Edith B¨ ohne and Wolfgang Motzkau-Valeton (eds.), Die K¨ unste und die Wissenschaften im Exil 1933-1945, Verlag Lamber Schneider, Gerlingen, 1992).— I don’t know whether, and if so, how often—it happened elsewhere—one was deprived of an already awarded doctor’s degree. The University of Marburg, for example, revoked the doctorates of 46 degree recipients (cf. the accompanying book of the exposition “Unworthy of a German Academic Degree” of the Marburg Municipal Archives, 2002). Some of those affected had to wait until the year 2002—all having died in the meantime—for rehabilitation and renewed rewarding of their degrees.— The “Gebrauchs-Universit¨ at [untranslatable: “Gebrauch” means “usage”; roughly, the University of Halle was of use to the Nazis] of the National Socialist city” Halle removed the degrees of 51 people for various reasons, among which loss of citizenship, homosexual behaviour, financial offenses including forgery of documents, acts of resistance, or “racial disgrace” (i.e. sexual relations with non-Aryans) (H. Eberle 2002, p. 183.). For the awful o Szab´ o, Vertreibung, R¨ uckkehr, Wiedergutmachung, prevention of compensation, cf. Anik´ Wallstein Verlag, G¨ ottingen, 2000.

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Along with these forerunners one should also mention Stanislaw Ja´skowksi (19061965), who, independently of Gentzen, developed a system of natural deduction. Ja´skowski published his “On the rules of supposition in formal logic” in Studia Logica I (1934), pp. 532. He had gotten his ideas from L  ukasiewicz already in 1926. Should something be made of this? Ja´skowski’s style was rather tedious, so no one noticed his relevant results. Even in the Russian literature only Gentzen is referred to. In the collection Matematiceskaia teoria logiceskovo vyvoda [Mathematical theory of logical inference]), Nauka, 1967, only Gentzen’s writings are translated and printed. Gentzen’s priority for natural deduction is recognised in Polish logic—cf. pp. 71ff. in Witold Marciszewski, ed., Mala Encyclopedia Logik, PWN, Warsaw, 1987 (2nd edition 1988). I thank Dr. Krzystof Ciesielski for this information. However, Ja´skowski was an influence on Willard Van Orman Quine, who wrote in his autobiography, The Time of My Life, MIT Press, Cambridge (Mass.), 1985: I began working toward Methods of Logic and mimeographed a draft of it for my spring course under the title A Short Course in Logic. I handled the logic of quantification in somewhat the style that Gentzen had called natural deduction, a formal method deducing consequences from the premises along lines attractively akin to ordinary formal reasoning. I had got the idea not from Gentzen but from Ja´skowski, when in Warsaw in 1933. Cooley had used a form of natural deduction in his Primer of Formal Logic, 1942, and Rosser, unbeknownst, had used variants of it in mimeographed teaching aids. Gentzen’s rules had been uncomfortably asymmetrical; Cooley brought more symmetry, and this required increased delicacy in framing the rules lest they conflict. I worked on it. [p. 195] Gentzen met Ja´skowski in M¨ unster in 1936 and read Ja´skowski’s works in 1939. In his Nachlass one reads: Stipulation of assumptions following Ja´skowski: Numbering the same . . . natural calculus, without the constraining form of a family tree. . . Incidentally, I have also worked with numbering of assumptions in my N -calculus: It is only one step to the appropriate spelling with liberation from the family tree form. Jan von Plato, in “Proof theory of classical and intuitionistic logic” (in: L. Haaparanta (ed.), History of Modern Logic, Oxford University Press, Oxford, 2007), sums up the situation thus: Before turning to Gentzen’s sequent calculus, we note the independent development of systems of natural deduction by Stanislaw Ja´skowski (1934). This work contains no profound results on the structure of derivations, in contrast to Gentzen. On the relation between Hertz, Ja´skowski, and Gentzen, cf. Marcel Guillaume, “La Logique Math´ematique en sa jeunesse”, in: J.-P. Pier, ed., Development of Mathematics 1900-1950, Birkh¨ auser, Basel, 1994; here: pp. 250ff. 122

I haven’t been able to find any investigation on the development of applied mathematics in G¨ ottingen. Perhaps this is clarified in terms of attitude, why within mathematics in G¨ ottingen occupation with a philosophy of mathematics was frowned upon if this occupation was not itself mathematically accessible and thus remained pure speculation. Richard Courant reported (in: Die Naturwissenschaften 15 (1927), p. 230) how Felix Klein, to the dismay of his mathematical colleagues, had “desired and promoted the separation of a chair of applied mathematics from the mathematical science.” The first such chair was held by Carl Runge. Felix Klein sponsored the fluid mechanic and gas dynamicist Ludwig Prandtl. The successor of Runge would be, likewise through Felix Klein, Gustav Herglotz. Both Prandtl and Theodor von K´ arm´ an were averse to any philosophical

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contemplations in the usual style, and for them facility with words could not substitute for “honest hard work” in mathematics. About Prandtl H. G¨ ortler said (“Ludwig Prandtl”, Zeitschrift f¨ ur Flugwissenschaften 23 (1975) no. 5, p. 158), “Certainly he was convinced of a great harmony behind the appearances of nature and especially also behind the not yet understood things. The question, however, of why therefore mathematics is generally applicable to the description of the appearances, especially in his fluid mechanics—this question probably never presented itself to him. That the phenomenon of the applicability of mathematics must be a central problem for mathematicians and natural scientists and that an ontology of mathematics must treat these phenomena fairly hardly disturbed Prandtl at all. . . What Prandtl looked at as ‘intuitive and useful mathematics’ he had presented in a course of lectures in the 1931/32 winter semester. The notes of these lectures by G. Mesmer were privately published in 1937.” Before and during the First World War Prandtl developed, incidentally, a difference engine, which is described and depicted in G¨ ortler’s article. Moreover, the lifestyle of the G¨ ottingen mathematician permits itself to be described as modest, plain, simple, undemanding and often humorous—even when everyone knew the significance and weight of his achievement. Could this have had its effect on Gerhard Gentzen?— Mathematics as subject history is always a question of attitude. 144 On 20 September 1935 Bernays embarked with G¨ odel on a ship to New York (John R. Dawson, Kurt G¨ odel. Leben und Werk, Springer-Verlag, Wien, 1999, p. 109). G¨ odel explained to Bernays his Second Incompleteness Theorem. It is highly probable that Gentzen’s work was also discussed on this occasion. After this it is possible and indeed probable that Bernays and G¨ odel, together with Weyl and von Neumann, discussed this, though it is not yet certain. In a review of G¨ odel’s correspondence, Jan von Plato writes (Bulletin of Symbolic Logic 10, 2004): In fact, from correspondence between Bernays and Gentzen it can be seen that G¨ odel had made a suggestion of his own for clarifying Gentzen’s proof, which later had been considered by Bernays. Gentzen writes on 11 December 1935 that he had known the possibility of modification suggested by G¨ odel, but it cannot be used from a finitary point of view, due to its impredicativity. G¨ odel’s correspondence contains also a letter by Bernays (7 January 1970) in which the latter recollects his talks with G¨ odel about Gentzen’s original proof. Bernays held lectures at the Institute for Advanced Study that covered Gentzen’s proof. These lectures were distributed in a mimeographed form, with the title Logical Calculus. In his 1938 “Zilsel” lectures (see volume III of G¨ odel’s works), G¨ odel was wrestling with Gentzen’s proof. By 1941, he had found his alternative consistency proof, as is shown by his paper “In what sense is intuitionistic logic constructive?” published in volume III. . . With Gentzen’s paper on the provable initial segments of transfinite induction, a complete clarification of the second incompleteness theorem was achieved. Another perspective (Charles Parsons, “Platonism and mathematical intuition in Kurt G¨ odel’s thought”, Bulletin of Symbolic Logic 1, No. 1, 1995): There is another more global and intangible consideration that could lead one to doubt that G¨ odel’s views of the 1930s were the same as those he avowed later. This is the evidence of engagement with the problems of proof theory, in odel the form in which the subject evolved after the incompleteness theorems. G¨ addresses questions concerning this program in the MAA lecture 1933 and more thoroughly and deeply in the remarkable lecture 1938a given in early 1938 to a circle organized by Edgar Zilsel. This lecture shows that he had already begun to think about a theory of primitive recursive functionals of finite type as something relative to which the consistency of arithmetic might be proved; it is now well known hat he obtained this proof in 1941 after coming to the United States.

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The lecture of Zilsel’s also contains a quite remarkable analysis of Gentzen’s 1936 consistency proof, including the no-counterexample interpretation obtained later by Kreisel. . . What he says about the philosophical significance of consistency proofs such as Gentzen’s is not far from what was being said about the same time by Bernays and Gentzen, in spite of somewhat polemical remarks about the Hilbert school in this text and in others. I owe this observation to Wilfried Sieg. Cf. our introductory note to 1938a in G¨ odel’s Collected Works, Vol. III, at p. 85. 157

In fact, I found some pages of Schiller’s poem “Die K¨ unstler” (“The Artist”) torn from a book of Schiller’s poetry in an offprint of Gentzen’s “Widerspruchsfreiheit der reinen Zahlentheorie”, which Gentzen had sent with other papers from Liegnitz to Sigmaringen in 1943. Among others, the following passage was marked in pencil: “Mankind’s dignity is delivered into your hands.— Perfect it! It sinks with you! With you it raises itself!” (A well-commented edition of Schiller’s poems is Georg Kurscheidt (ed.), Friedrich Schiller: Gedichte, Deutscher Klassiker Verlag, Frankfurt am Main, 1992. Cf. also Franz Berger, “ ‘Die K¨ unstler’ in Friedrich Schiller. Entstehungsgeschichte und Interpretation” (Dissertation by Emil Steiger), Juris Verlag, Z¨ urich, 1904.) In the first part Kant is transposed to a poetic plane. Nature will not be governed by reason, but the feminine Nature loves the fetters. Man is enabled only through thought. The progress of knowledge of Nature also occurs in Art! The history of civilisation is the history of emancipation; on the other hand we remain in the prison of Nature. Art and science are the consolation within the dungeon walls. If one wishes, one can see this outstanding programme-poem also as justification of sensuality and desire, where a harmony of Art and Nature appear fictitious. The poem, I believe, can also be read as an entry into the inner exile. A further motif of these readings is given by the physicist Marie Goeppert (1906-1972), who on 24 April 1935 reported in a letter to Courant on her visit to the sick 73-year-old Hilbert, “We were all struck by his continuously increasing weakness of memory, which occasionally made one sad. He doesn’t want to discuss the recent times: he consciously shoves it away and no longer reads the newspaper” (p. 133 in Reinhold Siegmund-Schultze, Mathematiker auf der Flucht von Hitler, Quellen und Studien von Emigration einer Wissenschaft. Dokumente zur Geschichte der Mathematik, Vol. 10. Published by the German Mathematicians Union, Vieweg-Verlag, Wiesbaden, 1998.) 228

Born in 1908, the philosopher and mathematician Albert Lautman would be imprisoned by the Germans and, on account of his activities for the resistance, shot on 1 August 1944. For the then current concept of mathematics cf. F. LeLionnais, Great Currents of Mathematical Thought, Vol. 1, Dover, New York, 1971 (French first edition, 1962). F. LeLionais worked in Mittelbau Dora on V-2 production and would be punished on account of his writing on a piece of paper the names of those who could be possible contributors to this book—a punishable use of a German pencil on a paper of the German Reich. That—thank heaven—did not prevent the planned book from coming to fruition. One finds in it, for example, Albert Lautman’s “Symmetry and dissymmetry in mathematics and physics” (pp. 44-56). Unabridged in the original language the beautiful works are collected in Albert Lautman, Essai sur l’unit´e des math´ematiques et divers ´ ecrits, Prefaces de Costa de Beauregard, Jean Dieudonn´e, membre de l’Acad´emie des sciences et Mau´ rice Loi du S´eminaire de Philosophie et de Math´ematiques de l’Ecole normale sup´erieure, Union G´en`erale d’´editions, Paris, 1977. Cf. also the entry “Lautman, Albert” in: J. Mittelstraß (ed.), Enzyklop¨ adie Philosophie und Wissenschaftstheorie 2, Bibliographisches Institut, Mannheim, 1984.— Jean Cavaill`es, Albert Lautman and Claude Chevalley—he was also known to laymen through his article “Math´ematique” in the Encyclop`edie francaise—were friends with Jacques Herbrand and through him became acquainted with each other’s works. I am convinced that through his contact with Cavaill`es, Gentzen as well was informed of the mathematical ideas of the other three. Cf. Lautman’s remarks on

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Gentzen (loc. cit., pp. 93, 141). Albert Lautman gave a paper, “De la r´ealit´e inh´erente aux th´eories math´ematiques”, at the Descartes Congress. I consider it entirely possible that Lautman, Cavaill`es, and Gentzen saw and spoke to each other there.— What did Lautman think of Gentzen? The facts show that it is nevertheless difficult to display purely structurally the consistency of a theory. In this regard the most authentically structural attempt for a proof of the consistency of arithmetic is that of Mr. Gentzen. We do not ourselves propose to expound on it here in any detail, but refer for this to the thesis of Mr. Cavaill`es; we shall only recall the essential element. Distant from the search to find models of the arithmetical axioms, Mr. Gentzen remains faithful to the conception of the proof theory of Hilbert and he proposes to attend to each of the inferences that intervene in a proof, to demonstrate that at each instant no contradiction whatsoever has insinuated itself. For this Mr. Gentzen attempts, as Herbrand had already tried, to reduce each inference of the proof to progressively simpler inferences until the final formula of the proof has been put into the form of an evidently true expression. For this, Mr. Gentzen assigns to each proof an ordinal number and proves an essential theorem which roughly comes to this: the reductions terminate because the ordinal numbers are “well ordered” in the sense of the theory of sets. The proof therefore does not rest on the existence of an interpretation of the system; it does not in any way resort to the consideration of a set of numbers whose existence is equivalent to the completion of the proof of unprovability. (Pp. 93ff. of: Albert Lautman, Essai sur les notions de structure et d’existence en math´ ematique, dissertation published originally by Hermann & Cie, Paris, 1937, and ecrits reprinted pp. 21-154 in Lautman’s Essai sur l’unit´e des math´ematiques et divers ´ cited above.) Mrs. Hourya Benis-Sinaceur gives reliable information on the discussion connections between Gentzen, Cavaill`es and Lautman in “Structure et concept dans l’´epist´emology math´ematique de Jean Cavaill`es”, Rev. D’hist. Sci. 40, No. 1 (1987), pp. 5-30 and 117-129. At the same time, a short biography by Mrs. Hourya Sinaceur (pp. 344ff.) is in Joseph W. Dauben and Christoph J. Scriba (eds.), Writing the History of Mathematics: Its Historical Development, Birkh¨ auser Verlag, Basel, 2002. 238

In the prospectus of the Felix Meiner Verlag in Leipzig for the Forschungen zur Logistik und zur Grundlegung der exakten Wissenschaften for the fall of 1934(!) one reads among other things: The logistic has made such genuine progress in the postwar period that today it may be regarded as a fundamental area of research in its own right. Nonetheless logistic research in Germany has till now actually been accomplished by mathematicians. The philosophers have either generally considered it with a cool reserve or have expressly rejected it. As opposed to this, in other countries, above all in Poland, Austria, and the United States, a decided transposition has asserted itself. In these countries the logistic is first of all a philosophical research area, with the beautiful success that within a few years epoch-making accomplishments have come out of philosophical issues—accomplishments by which the Leibnizian ideal of philosophy as a strict science is for the first time tangibly realised. In the investigations announced here a study group emerging from German soil would like for the first time to join in rigorously in the service of logistic research. We wish to present a series of strictly verifiable and meticulous works in which problems of modern logic, the theory of science and the history of science, which only a generation ago would not have been deemed accessible, are discussed, solved, or at least moved forward. We do not wish to be locally restricted. If we should

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succeed to find like-minded collaborators outside our circle, they will be truly welcome. The collection of researches will appear as “Forschungen zur Logistik” [Researches on logistic]. They have as their subject a system of a modern, exact, symbolic logic. These researches should be linked with “Forschungen zur Grundlegung der exakten Wissenschaften” [Researches on the foundation of the exact sciences]. Such a link suggests itself and is suitable because through its unsurpassed precision the logistic symbolism has today become a necessary condition for all exact investigations of meaning-interpretation [Sinndeutung] and especially for axiomatics of the rigorous sciences. Finally, there should be investigations of a still more general sort, thus also investigations of the character of the histories of thought and science; investigations are permitted among these researches if they deal with important questions which stand in a sufficiently narrow relation to the given problem circles. With particular care attention will be paid that a style of presentation will be found for the published works through which they will be accessible to the largest possible circle of readers interested in the subject. . . The first works planned are: (1) logical investigations of the foundations of Euclidean Geometry (2) investigations on the theory of definitions through the construction of abstraction classes (3) the real numbers in logistical constitution (4) critical investigations of Frege’s foundation of logic (5) the Frege-Russell theory of metrics and its applications in mathematics and physics (6) a logistic commentary on Dedekind, Was sind und was sollen die Zahlen? (7) logistic studies of Aristotle and Euclid (8) logistic studies of space and time (9) logistic investigations on the foundations of mechanics Afterwards, the first volume was announced: Friedrich Bachmann, Untersuchungen zur Grundlegung der Arithmetik, mit besonderer Beziehung auf Dedekind, Frege und Russell [Investigations on the foundation of arithmetic, with special attention to Dedekind, Frege and Russell]—78 pages, price RM 3.- (for subscribers to the journal Erkenntnis RM 2.25). The subject of these investigations is the problem of the derivability of arithmetic from logic or, more precisely: the problem of derivability of an interpretation of arithmetic from logic. The solution to this problem consists of the solutions to two subproblems: (1) a logical characterisation of the models of arithmetic (given an axiom system for arithmetic), (2) the logical constitution of a model of arithmetic (given a system’s logical constants and a proof that this system of constants suffices for an axiom system of arithmetic). The answer to the question, whether the derivability of an interpretation of arithmetic from logic is provable depends on which logic one chooses as a basis. A.N. Whitehead and B. Russell did not obtain a derivability proof with the logic presented in Principia Mathematica; they had to adjoin a new axiom, the “Axiom of Infinity”. The present work uses a logic, which can be described in a nutshell as the logic of Principia Mathematica without the (ramified or unramified) type theories. This logic is based on the analysis of the paradoxes by H. Behmann and was developed by H. Scholz in his Vorlesungen u ¨ber Logistik [Lectures on logistic]. With its aid derivability proofs are obtained for various interpretations of arithmetic.

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In the first part of the work a solution to both subproblems will be carried out as singlemindedly as possible. The solution to the second subproblem rests on those thought-things [Gedankendingen] that G. Frege has sketched as a programme for the verifcation that “arithmetic is a part of logic” in his Grundlagen der Arithmetik [Foundations of arithmetic] and developed in the first volume of his Grundgesetzen der Arithmetik [Basic laws of arithmetic] to a point of great rigour and exactness. The attempt is made on this basis to replace the lengthy and confused Fregean proof by a clearer one. In the second part, in connection with R. Dedekind, to whom is owed the first system of propositional forms that could be used as an axiom system of arithmetic, other solutions of the first subproblem are offered and some general investigations on the axiomatics of arithmetic, in particular the “axiom of complete induction”, are carried out. Further, the second part is devoted to the assessment of the Fregean constitution of arithmetic ` a la Principia Mathematica. The work closes by proving relative to the underlying logic the identity of the interpretations of arithmetic in the Principia Mathematica with the interpretation presented in the first part. ¨ [The word “Gedankending” probably stems from Hilbert, who used it in “Uber die Grundlagen der Logik und Arithmetik” (pp. 174-185, in: A. Kratzer (ed.), Verhandlungen des dritten internationalen Mathematiker Kongresses in Heidelberg vom 8. bis 13. August 1904, B.G. Teubner, Leipzig, 1905) and also mentioned it in his third notebook (Cod. Ms. 600, 3, p. 101).] The Forschungen was a mixture: books like that advertised, a journal, and even an inlay in Bieberbach’s Deutsche Mathematik. H. Scholz began the publication before the financing by the Deutsche Forschungsgemeinschaft (DFG) [German Research Council] was in place. The series Forschungen was continued through 1943 and amounts to eight volumes. Thereafter Scholz no doubt received neither financing nor paper for it. 257

It strikes one immediately that no historian of mathematics has taken up the history of modern proof theory but that they incessantly spout many senseless and uninformed prejudicial statements against him. One can read even today: And, yet more, that in particular the statement that expresses the consistency of a given theory belongs among the unprovable sentences of G¨ odel (G¨ odel 1931). The goal of a proof of the consistency of arithmetic or of set theory was thereby only sensible on the basis of a very fundamental modification of the idea of a formal theory, resp., of the permissible proof procedures. In fact Gentzen then succeeded in 1936 to “prove” the consistency of ordinary arithmetic under assumption of a “transfinite” inference procedure which, mind you, has till today not found general acceptance. [Footnote: The discussions of Gentzen’s argument show as well as those of impredicative definitions, that actually not once was there a real consensus among mathematicians and logicians on the question of what a correct proof or a correct definition is.] (Moritz Epple, p. 52, “Das Ende der Gr¨ oßenlehre”, 1996 (offprint. Mathematics Department, J. Gutenberg University, Mainz). In the altered, printed version (p. 408 or “Das Ende der Gr¨ oßenlehre: Grundlagen der Analysis 1860-1910”, in: Hans Niels Jahnke (ed.), Geschichte der Analysis, Spektrum Akademsicher Verlag, Heidelberg, 1999) the boldness of the claim is weakened through a new footnote: The exchange concerning the permissibility of impredicative definitions in mathematics continues to this day. A good survey of the course of the discussion is given by Thiel 1972.

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It seems to me that again this all stems from H. Weyl. In 1946, for example, he wrote (“Mathematics and logic. A brief survey serving as a preface to a review of The Philosophy of Bertrand Russell ”, American Mathematical Monthly 53, pp. 2-13; here, p. 13): It is likely that all mathematicians ultimately would have accepted Hilbert’s approach had he been able to carry it out successfully. The first steps were inspiring and promising. But then G¨ odel dealt it a terrific blow (1931), from which it has not yet recovered. G¨ odel enumerated the symbols, formulas, and sequences of formulas in Hilbert’s formalism in a certain way, and thus transformed the assertion of consistency into an arithmetic proposition. He could show that this proposition can neither be proved nor disproved within the formalism. This can mean only two things: either the reasoning by which a proof of consistency is given must contain some argument that has no formal counterpart within the system, i.e., we have not succeeded in completely formalizing the procedure of mathematical induction; or hope for a strictly “finitistic” proof of consistency must be given up altogether. When G. Gentzen finally succeeded in proving the consistency of arithmetic he transcended those limits indeed by claiming as evident a type of reasoning that penetrates into Cantor’s “second class of ordinal numbers”. From this history one thing should be clear: we are less certain than ever about the ultimate foundations of (logic and) mathematics. Like everybody and everything in the world today, we have our “crisis”. We have had it for nearly fifty years. By “all mathematicians” Weyl probably meant himself, for he wrote in his “Randbemerkungen zu Hauptproblemen der Mathematik” [Marginal notes on the main problems of mathematics] (Mathematische Zeitschrift 20 (1924), pp. 131-150; here, p. 150): “But next to the Brouwerian one must follow the Hilbertian path”— And we will continue to have the “crisis” so long as Weyl’s dictum—which in all probability he simply took over from P. Bernays—is unconsciously repeated by “historians”. Epple cites no contemporary proof theorist, has obviously conducted no interviews with such or asked them for infornski characterised the mation or advice. His solution however is a problem. Craig Smory´ problem on the occasion of a review of S. Shapiro’s Foundations without Foundationalism. A Case for Second-Order Logic, in which he classifies the various views of logic or the foundations of mathematics so: “Mathematics (M), Foundations of Mathematics (FM), Mathematical Logic (ML), Philosophical Logic (PL), Philosophy of Mathematics as practiced by Philosophers (PMP), Philosophy of Logic (PoL), Foundations of Logic (FL), and Epistemology (E).” Smory´ nski goes on to say: “While there are relations among these subjects (e.g., PMP, E), no two of them coincide (except possibly PoL and FL), and what may be inadequate for one may be perfectly adequate for another.” Historians should thus make their assumptions clear insofar as they explain the perspectives they think to write from. Perhaps new results are too suspect or inaccessible to them to want to make note of them. But modern proof theory has established itself upon Gentzen as a basic discipline of mathematical logic. And the field of investigation, “Proof Theory in Computer Science”, grows steadily in its significance. A seminar announcement for Oberwolfach for a workshop for 2004 reads, for example: A recently discovered document has shown that Hilbert had considered adding a 24th problem as the last one to his famous list of mathematical problems of the year 1900: to develop a general theory of proofs in mathematics, and more specifically, to study the complexity of proofs. An attempt at the first part was made in Hilbert’s old proof theory that formalised the general principles of mathematical proof by the use of a system of logical axioms and just two rules of proof. From the 1930s on, structural proof theory developed by Gerhard Gentzen has gradually replaced this cumbersome machinery and can be considered an answer to the general part of Hilbert’s last problem. Steps of inference are represented directly

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as instances of rules and no axioms at all. Formal derivations constructed by the rules are objects in tree form that can be analyzed combinatorially in various ways. The first remarkably successful analysis resulted in Gentzen’s proof of the consistency of arithmetic in 1936, five years after G¨ odel’s incompleteness results. Proof analysis is very sensitive to the specific formulation of the rules used. The main task of foundational studies in structural proof theory is to explore different systems of rules, their interrelations and extensions. The original applications of structural proof theory were in the foundational questions of mathematics, such as the questions of consistency and completeness and the existence of a decision method for mathematical theories. This is still an important field of research in which continuous progress is made. The main effort has been to extend consistency proofs to proof-theoretically strong systems of analysis and set theory. In another direction, theories with much lesser deductive strength keep posing challenges to proof analysis, often with the aim of giving (at least partial) positive solutions to the decision problem. Newer applications of structural proof theory concern proof search: Proof-theoretical calculi have been developed in which the search of a formal derivation can be carried through in a systematical way starting from the desired end result. For many classes of problems, it can be shown that proof search terminates, thus either giving a proof or a positive proof of unprovability. Proof search leads naturally to the question of proof complexity. For some theories and/or classes of problems, it is known that there exist computationally feasible decision algorithms. For other theories, such algorithms grow exponentially and are useful only theoretically, not for actual computations. Logical calculi that support proof search are the theoretical basis of many applications of structural proof theory in computer science. Foremost among these is logic programming (Prolog etc.), but there is a host of other applications and a lot of interaction between researcher in logic and in computer science. A historian no longer needs to go into this.

https://doi.org/10.1090/hmath/033/03

CHAPTER 3

1939-1942—From the Beginning of the War to Dismissal from the Wehrmacht and the Wartime Habilitation under Helmut Hasse 1. 1939: At the Highpoint of Reputation Gentzen read Tarski’s “Der Wahrheitsbegriff in den formalisierten Sprachen” [The concept of truth in formalised languages], Studia Philosophica 1 (1935), pp. 261405, and noticed, c. 1941 for example, with reference to p. 311: The super-theoretic character of the concept of truth. . . in essence is that the truth definition as super-theoretic method contains a bound variable over a relation between all the individuals already occurring in the theory on the one hand, and the formulæ, i.e., in practice: natural numbers, on the other hand. Thereby one transcends the theory minimally.1

In the meantime in G¨ ottingen one was thinking about a scheduled position to give to Gentzen: Mathematical Institute of the University of G¨ ottingen, the 4th of March 39, Bunsenstraße 3/5 To the Registrar of the University of G¨ ottingen via the Dean of the Mathematical-Natural Scientific Faculty With reference to the writings of the Registrar No. 621 II of 9 February 39 for the filling of the 6 unscheduled assistantships to be converted into scheduled assistantships as of 1 April 1939 I suggest: a.) For the Mathematical Institute 1. Dr. Martin Eichler (present holder) 2. Dr. Uwe Timm B¨ odewadt (ditto) 3. Dr. H. Braun (special application follows, new occupation) b.) For the mathematical-physical seminar 1. Dr. G. Gentzen (present holder) 2. Dr. R. Kochend¨ orffer (ditto) 3. Dr. Th. Schneider (new occupation, special application follows) Heil Hitler! The managing director of the Mathematical Institute signed Hasse.2

Perhaps this explains why Hasse had earlier only requested a one-year extension of Gentzen’s unscheduled assistantship. At the beginning of 1939 Gentzen’s mother moved to Liegnitz to live with the Student family. Thenceforth Gentzen frequently visited his mother and sister there. 1 From

the Nachlass. I thank Herr Prof. Thiel for the transcription. File III D, 335 (52 II) of the Mathematical Institute, G¨ ottingen.

2 Secretarial

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On 5 January 1939 he himself moved in G¨ ottingen from Prinz Albrecht Straße 22 to Reinholdstraße 6. He was a member of the DMV [German Mathematicians-Union] from April 1939 at the latest. In February, another of his reviews appeared: Ono, Katudi:3 Untersuchungen u ¨ ber die Grundlagen der Mathematik, J. Fac. Sci. Univ. Tokyo, Sect. I 3, 329-389 (1938). The author bases predicate logic in the formulation as a sequent calculus (following Gentzen, this Zbl. 10, 145). Further, one thinks of a basic domain of individuals with finitely many specific atomic predicates given, characterised by an axiom rule, i.e. a rule which for each statement-expression built up from the atomic predicate symbols and the logical connective symbols decides whether it represents an axiom or not. It is assumed not to be possible to derive a contradiction from these axioms by means of predicate logic. The author now gives a series of extensions through additional axioms and shows, starting from Gentzen’s theorems on sequent calculi, that the consistency is preserved [under these extensions]. The additional axioms are Comprehension Axioms, i.e. they assert the existence of individual objects which stand in for the represented (assembled) predicates. Even predicates of predicates and so on are introduced so; thereby each “impredicative” action (“circulus vitiosus”) is strictly avoided; rather a hierarchy of types results, which corresponds somewhat to the original “ramified type theory” of Principia Mathematica, but without the “Axiom of Reducibility”. In addition to this the types are continued so to speak a bit beyond epsilon, in that statements about the types themselves are formed and corresponding new predicates can be introduced.— In the second part of the work, using the previous results, the consistency of a portion of pure number theory is proven, which contains all the essential components of the same, however the inference rule of complete induction occurs only in a restricted version. The sort of restriction is complicated and its scope difficult to survey. The author opines that almost all instances of full induction actually occurring in number theory fall under it; this is plausible, as full induction on statements of somewhat complicated logical structure practically never occur. Gerhard Gentzen (G¨ ottingen)4

Saunders Mac Lane, in reviewing the work, opened with the remarks: In a consistency proof for arithmetic the chief difficulty lies in analyzing proofs involving many successive mathematical inductions. For example, Gentzen (I 75 and IV 31) has found a consistency proof which uses certain transfinite numbers 3 Cf. also the review of Hermes on page 124. Born in 1909, Katzui Ono died at the age of 92 on 18 August 2001 after several years of serious illness. He was one of the fathers of mathematical logic in Japan. From 1943 to 1969 he was professor of mathematics at Nagoya University. In 1954 he was awarded a prize from the Japanese Academy of Sciences for his research on relays in computers. Beginning in 1969 he spent three years as president of Shizuoka University. From 1972 to 1974 he was president of the Japanese Society for Operations Research. His scientific works concerned applied mathematics as well as mathematical logic. In 1938 he published a work on the consistency of mathematics in which he followed Gentzen’s works but used a restricted form of mathematical induction. (Cf. Gentzen’s review.) This work he signed with the name Katudi Ono. His dissertation developed from it. His son, Professor Hiroakira Ono, who works at the Advanced Institute of Science and Technology in Japan in the area of mathematical logic, knows from discussions with his father that the latter admired Gentzen and had even corresponded a few times with him. In addition to this Ono was familiar with Gerhard M¨ uller, Haskell B. Curry, and Alfred Tarski. Since his 60s he directed his attention to the philosophical foundations of mathematics and mathematical modes of argumentation, and published a book on these subjects at age 80. 4 Zentralblatt f¨ ur Mathematik 19 (1939), p. 242.

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to enumerate the inductions metamathematically. In the present paper, Ono restricts himself to the finite metamathematical methods and establishes the consistency of a form of arithmetic in which the use of the induction axiom is restricted.5

Gentzen also read a treatment by Hugo Dingler—a correspondence between the two has not been verified—and wrote on 14 March 1939: At the wish of Professor Heinrich Scholz I have read the manuscript of H. Dingler, ¨ “Uber das Verh¨ altnis meines Beweises der Widerspruchslosigkeit der Arithmetik ¨ zu den G¨ odelschen Uberlegungen” [On the relation of my proof of the consistency of arithmetic to G¨ odel’s considerations] and observe the following: The treatment rests in fact on a misunderstanding of G¨ odel’s results. The consistency proof of Dingler’s concerns, as he himself remarks on pp. 10-11, only a narrow fragment of number theory, which in particular does not yet contain “transfinite” modes of inference (which for example in Hilbert’s Programme are first in line to be proven consistent). G¨ odel’s Theorem however does not apply to so narrow a system. (If it is conjecturally already valid for this, it is however not yet proven.)6

On 1 April 1939 Gentzen’s unscheduled assistantship was transformed into a scheduled appointment. The faculty, Dozentenschaft, rector, and registrar again brought forth no grounds against it. On 27 April he swore the oath of office to Adolf Hitler: I swear: I will be true and obedient to the leader of the German empire and people, Adolf Hitler, to obey the laws and to fulfill my official duties conscientiously, so help me God.

He had to sign a “declaration of commitment on foreign appointments”. Because the “deployment of German scientists abroad” fell under the responsibility and discretion of the Reichsminister, he must immediately report to the latter, in case he envisioned accepting any appointments abroad, as such required authorisation. Officially he remained assistant in G¨ ottingen despite the interruption of the war from 1 April 1939 to 31 March 1944, and was thereafter docent in Prague until his death. 2. The Second Volume of Grundlagen der Mathematik of Hilbert and Bernays Appears Only a few months before the beginning of the Second World War, the second volume of Hilbert-Bernays, Grundlagen der Mathematik, appeared. The war hindered a worldwide distribution of the book to all relevant libraries. Thus the book was, for example, for a long time even after the war the only book in which a detailed proof of G¨ odel’s Second Incompleteness Theorem was to be found.7 5 Journal

of Symbolic Logic 4 (1939), pp. 89-90. original of this letter can be found in the Institute for Foundational Research in M¨ unster. I thank Enno Folkerts for this finding. 7 Indeed, the first major American text on logic by Stephen Kleene (Introduction to Metamathematics, D. van Nostrand Company, Princeton, 1952), who apparently did not understand all the nuances involved, did not even attempt to outline the proof. Most textbooks presented the proof of the First Incompleteness Theorem (if a theory is consistent and contains enough arithmetic, then it is incomplete) and then stated that formalising the proof given yields the Second Incompleteness Theorem (said theory cannot prove its own consistency). Often, however, the given proof of the First Theorem does not formalise. The earliest exception in American texts that I am aware of is Joseph Shoenfield, Mathematical Logic, Addison Wesley, Reading (Mass.), 1967. The proof given therein of the First Incompleteness Theorem does indeed formalise.—Trans. 6 The

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Gentzen’s longest review was of a book, on the ideas of which he participated and in which some of his ideas were incorporated, and of which he had read the first 200 pages in proof. The Grundlagen der Mathematik had originally been conceived by Hilbert as a detailed presentation of his worked-out lectures. Paul Bernays, as the sole author, however, had worked in the latest results of proof theory, so that content and compass already went beyond Hilbert’s intentions: Hilbert, D., and P. Bernays: Grundlagen der Mathematik, vol. 2. (Die Grundlehren d. math. Wiss. in Einzeldarstell. mit besonderer Ber¨ ucksichtigung d. Anwendungsgeb. vol. 50.) Berlin: Julius Springer, 1939. XII, 498 pp. RM 42.-. In this second volume the comprehensive presentation of the proof theory founded by Hilbert is brought to a close at the current state of research (cf. the review of volume 1, this Zbl. 9, 145). The work has again been written by Bernays. It contains a richness of details, many first-time published results, through which run the various approaches to consistency proofs in the domain of number theory as connecting threads. Contents in detail: §1-3 deal with the -symbol. Proof of the “First -Theorem”, which says: If with the help of predicate logic including , a consequence can be derived from any axioms (without formula variables), where the axioms and consequence contain no bound variables (thus no ), then the bound variables can be eliminated entirely from the derivation. Application for consistency proofs: From any axioms of the indicated kind no contradiction can be derived with the aid of predicate logic, if the axioms are “verifiable”, i.e. “correct” in the elementary sense of the calculation of truth values. As examples certain axiom systems of geometry are treated; the truth values are here obtained from arithmetic models of the usual sort.— §2. The same is repeated for the consistency of number theory, under the restriction of “complete induction” to elementary applications. Then the first -theorem is extended to the case in which the general identity axiom, a = b implies (A(a) implies A(b)), is included. There follows a presentation of the original Hilbert attempt (in the arrangement given by Ackermann) to prove the consistency of number theory, as presented by means of the -symbol. The result is not really more difficult to obtain than the first -theorem.— §3. There follows a “Second -Theorem”, which asserts: If with the aid of predicate logic, including , a consequence can be derived from arbitrary axioms (without formula variables), where the axioms and consequences contain no , then the -symbol can be eliminated entirely from the derivation. The proof bases itself upon the first -theorem. It yields at the same time a normal form for derivations of predicate logic, which in fact goes back to Herbrand (his “Fundamental Theorem”).— There follow the proofs of the L¨ owenheim Theorem (satisfiability in a countable domain) and G¨ odel’s Completeness Theorem for predicate logic as well as a view of the “Entscheidungsproblem”.— §4. A detailed presentation of the arithmetisation of metamathematics. Application of the same to the proof of a certain finitary counterpart of G¨ odel’s Incompleteness Theorem.— §5 first treats “limitations on the presentability and derivability in deductive formalisms”: The impossibility of defining truth for a system within itself (Tarski) odel), and further incompleteness as of proving the system’s own consistency (G¨ results are proven under far-reaching restrictions on the assumptions made about the systems. It is shown in detail that in particular the system of pure number odel’s Theorem has theory satisfies the assumptions (a fully detailed proof of G¨ not previously been produced), and a “truth definition” for this is given, which transcends the frame of the system via the use of a non-finitary recursive definition schema, and which makes possible a non-finitary, formal consistency proof for this system. There follow contemplations on the formalisability of the modes of inference of proof theory and of the question to what extent these count as

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“finitary”. In this connection G¨ odel’s proof of the eliminability of the Law of the Excluded Third from number theoretic derivations, with re-interpretation of the ∨- and exist-signs, is presented and then the consistency proof of Gentzen is reviewed, particularly the use therein of a special transfinite induction. It is shown that a formal proof of this only entails a modest transcending of the proof means of pure number theory.— As an appendix four “supplements” are added; they are: I. A compilation of basic concepts of predicate logic and simple theorems about it. II. The concept of a computable function; Theorem of Church on the impossibility of a general solution to the Entscheidungsproblem. III. The completeness of formulations of fragments of propositional logic. IV. Formalisms for analysis; development of the theory of real numbers as well as of numbers of the second number class. (Gerhard Gentzen)8

Another review followed: Rosser, Barkley: On the consistency of Quine’s new foundations for mathematical logic. J. Symbolic Logic 4, 15-24 (1939). Quine made the attempt (this Zbl. 16, 193) to remove the ambiguity of essential concepts of (simple) type theory, through eliminating the type distinctions, however, only allowing formulæ in the Comprehension (= abstraction) Axiom that can be assigned type indices in such a way as to meet the demands of type theory. There is the fear that this system may be inconsistent. The author examines this question. Quine’s system becomes more manageable when extended by the ι-operator (the thing which), which is shown eliminable. Further all the primitive symbols and thereby all the formulæ and terms can be written in such a way that only the numerals 0 and 1 are used. Through this every formula can without further ado be assigned a natural number (in dyadic representation) for a convenient arithmetisation. It is further shown how the natural numbers, sums and products of such as well as arbitrary recursive functions are contained within the system. Finally, a rule of inference previously treated by the author (this Zbl. 17, 242) is added, which makes possible a certain rounding off of the system without being capable of introducing consequences that, viewed contentually, are incorrect.— The author then informs us that all attempts to derive a contradiction in the full system by known methods— in particular by those procedures applied with success to other systems by Kleene and Rosser (this Zbl. 12, 146)—have had no success. Gerhard Gentzen (G¨ ottingen)9

On 16 July 1939 Gerhard Gentzen wrote to the very respected Professor Paul Bernays “regarding the elimination of ‘not not A implies A’ ” and clarified it to him following his “interpretation” of all derivative forms: You expressed reservations on account of the case that tertium non datur could come forth as an implicit component of a definition schema. I cannot imagine that a difficulty could lie here. Because: In analysis one can (following Dedekind, if I am not mistaken) transform recursive definitions into explicit definitions. Then the tnd no longer enters as a component of the definition, but rather as a 8 Zentralblatt

f¨ ur Mathematik 20, pp. 193ff. f¨ ur Mathematik 20 (1939), p. 194. There followed yet 8 lines of reviews of two very technical works of E.V. Huntington, “Note on a recent set of postulates for the calculus of propositions”, J. Symbolic Logic 4 (1939), pp. 10-14, and L´ aszl´ o K´ alm´ ar, “On the reduction of the decision problem, I. Ackermann prefix, a single binary predicate”, J. Symbolic Logic 4 (1939), pp. 1-9. Afterwards one work each of Thomas Greenwood, George D. Birkhoff (“Intuition, reason and faith in science”, Science 88 (1938), pp. 601-609), and Ignacio M. Azevedo do Amaral are named. 9 Zentralblatt

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conclusion in a proof (for instance in the existence proof for the definiendum), and falls under the treatment of the modes of inference. . . 10

Gentzen also published a very short review: A. Padoa. Come si deduce. Periodico Mat. (4) 18, pp. 228-236. Reflections on basic concepts of logical deduction and its formalisation. Gentzen11

And Gentzen himself was reviewed: G. Gentzen. Die gegenw¨ artige Lage in der mathematischen Grundlagenforschung. Deutsche Mathematik 4, 255-286. The author reports in detail on the present status of Hilbert’s proof theory and problems connected with it. The intuitionistic standpoint regarding mathematics and the in-itself conception of Hilbert are compared and elucidated mainly in their various conceptions of the continuum. The consequences of G¨ odel’s Incompleteness Theorem for the carrying out of a consistency proof are more closely discussed. The paper concludes by going into the question to what extent does there exist the possibility of a union of the two opposing conceptions. Ackermann12 G. Gentzen. Neue Fassung des Widerspruchsfreiheitsbeweises f¨ ur die reine Zahlentheorie. 44 pages. Leipzig. S. Hirzel (Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, New Series, vol. 4). The author gives a new, more transparent version of the proof of the consistency for arithmetic delivered in section IV of his work “Die Widerspruchsfreiheit der reinen Zahlentheorie” (Math. Ann. 112 (1936), 493-565; F.d.M. 62, 44). Ackermann13

In 1940 Max Black is reviewing both Gentzen’s work “Die gegenw¨ artige Lage in der mathematischen Grundlagenforschung” and “Neue Fassung des Widerspruchsfreiheitsbeweises f¨ ur die reine Zahlentheorie”, together with “Grundlagen der Mathematik”, volume 2, by David Hilbert and Paul Bernays (in Mind, vol. 49, No. 194, pp. 239-248), and finds that the tremendous range of topics in Gentzen make a breathless and tantalisingly brief appearance. He thinks that the ground covered coincides very nearly with the second chapter of Heyting’s useful report, which appeared in 1934 (Mathematische Grundlagenforschung, Intuitionismus, Beweistheorie), while Gentzen’s pamphlet, for all its compression, contains a few novelties of emphasis (i.e. Skolem’s theorem). He finds in Gentzen a “striking contribution to the formalist programme” (p. 240). Max Black then characterizes the “Grundlagen der Mathematik”, outlines the method in Gentzen’s proof of consistency and underlines the permanent importance of the acquaintance with Gentzen’s work—and its comprehensive setting in the treatise of Hilbert and Bernays. Gentzen’s contribution is exclusively seen by Max Black through the eyes of Bernays’ “Grundlagen der Mathematik”. But he remarks: “Is it then permissible to regard Gentzen’s leading idea as ‘finitist’ ? I believe anybody who tries to replace a specific pattern, constructed in the way described above, by a series of steadily preceding patterns will receive a vivid impression of the elementary and intuitively convincing character of the type of argument involved.It may be noticed that a rather similar extension

10 Hs.

975:1665. u ¨ber die Fortschritte der Mathematik 64/2 (1939), p. 923. 12 Ibid. 13 Ibid. 11 Jahrbuch

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of the strictest finitist method is involved in the definition of Brouwer’s free-choice sequences (Wahlfolgen)” (p. 247). J. Barkley Rosser, reviewed Die gegenw¨ artige Lage in der mathematischen Grundlagenforschung. Neue Fassung des Widerspruchsfreiheitsbeweises f¨ ur die reine Zahlentheorie in the Bulletin of the American Mathematical Society (Vol. 45, No. 11, 1939, pp. 812-813): This is really two books printed in one volume. The first one, “Die gegenw¨ artige Lage in der mathematischen Grundlagenforschung”, has also appeared in Deutsche Mathematik, vol. 3 (1938), pp. 255-268. The second one, “Neue Fassung des Widerspruchsfreiheitsbeweises f¨ ur die reine Zahlentheorie”, is a revision of Gentzen’s famous paper in the Mathematische Annalen, vol. 112 (1936), pp. 493-565. The first book is a well written summary of the present status of foundations, and contains one of the most lucid accounts of the Brouwer viewpoint that the present reviewer has seen. The distinction between the Brouwer and Hilbert schools is presented from the point of view of their treatment of the infinite. For Brouwer, who always insists on finite constructibility, the infinite exists only in the sense that he can at any time take a larger (finite) set than any which he has taken hitherto. Hilbert would treat infinite sets by the same methods used for finite sets, as if he could comprehend them in their entirety. Gentzen refers to this point of view as the “as if” point of view. He presents various paradoxes which arise when the “as if” method is used without proper care. This of course opens the question of what is “proper care”. In the nature of things, the Brouwer method must fail to produce a paradox, since it never leaves the domain of the constructive finite. However the Brouwer method does not produce sufficient mathematical theory for physical and engineering uses. So Brouwer’s method must be described as “excessive care”. A proposed way out of the difficulty is to base the “as if” method on an appropriate formal system, and use the Brouwer method to prove that the formal system is without a contradiction. For none of the various formal systems so far proposed has such a proof of freedom from contradiction been given. More serious still, a well known theorem of G¨ odel says that if a logic L1 is used to prove the freedom from contradiction of a Logic L2 , then L1 must in some respects be stronger than L2 . So the above program will fall through unless one can point out some respect in which the Brouwer method is stronger than the “as if” method. Gentzen thinks he has found it. His idea is to use the Brouwer method, involving the use of transfinite induction up to a certain ordinal α, to prove the freedom from contradiction of that part of the “as if” method which involves transfinite induction only up to an appropriate smaller ordinal β. If β is fairly large, the resulting “as if” method, though restricted, should be adequate for physics and engineering. In the second book, Gentzen illustrates the above proposal by using the Brouwer method, with induction up to 0 to prove the freedom from contradiction of number theory with induction up to any ordinal less than 0 . An important gap in the proof is the absence of a constructive proof that induction is valid up to 0 . Gentzen himself comments on this gap, and expresses the belief that it will shortly be filled. The present proof of freedom from contradiction is made considerably simpler than the earlier proof (in Mathematische Annalen—see above) by using Gentzen’s LK-calculus, rather than his NK-calculus. The proof is too complicated to be sketched here. However it is worth saying that what Gentzen does is to describe a means of attaching an ordinal number (less than 0 ) to any proof of number theory. He then describes how, if one had a

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proof of a contradiction, one could find a second proof of a contradiction having a smaller ordinal number than the first proof.14

And Hans Hermes reviewed Gentzen via K. Ono: K. Ono. Logische Untersuchungen u ¨ ber die Grundlagen der Mathematik. J. Fac. Sci. Univ. Tokyo I 3, 329-389. The author proceeds from inference rules of the predicate calculus in the form of G. Gentzen (Math. Z. 39 (1934), 176-210, 405-431; F.d.M. 60, 846) and shows that any consistent axiom system remains consistent, if certain “set- (resp., relation-) formation axioms” are added. . . On this basis detailed investigations on complete induction are made. Finally, a not simple condition is given which the “kernel” of an induction axiom must satisfy, by which the addition of this axiom to an axiom system of number theory (with the Identity Axiom, but not the Induction Axiom) is consistently possible. (This condition is satisfied by almost all complete inductions that actually occur in number theory.) Hermes15

It happens that the same authors always appear as reviewers: Gentzen, Scholz, Behmann, Hermes, Ackermann, Skolem, K´alm´ar, Hornich, Aumann. It also plainly happens that, at the latest since 1935, those colleagues who had emigrated or were deceased or unable or unwilling to work are missing. Thus develops the impression of an incestuous citation- and commentary-cartel. But as only a few specialists concerned themselves with the foundations of mathematics, this image is all too understandable, for it is correct. Scholz fought hard for the recognition of “logistics” within the humanities. Gentzen stood for the backing of the “mathematician”, because neither Schr¨ oter, Hermes nor others stood in the centre of mathematical attention. The unity of the reviews kept the quality high. The book and offprint exchange with other lands, libraries, and researchers function without problems; Scholz himself could still in 1941 review a paper of Alonzo Church of the same year. From then until 1945 this exchange was completely interrupted if one did not organise this via neutral countries. Already at the beginning of the 1930s the field of mathematical logic was claimed in the review journals by the G¨ ottingen finitists and the M¨ unster logicists. The isolation in National Socialism allowed the M¨ unster school a yet firmer occupation of the field of mathematical logic. Gentzen could participate because of his connections with M¨ unster. From outside—for example by the National Socialist thinkers Steck and Dingler—that could easily be viewed as a “clique”. The logicians mutually reviewed themselves, arranged reviews and stuck to this subject in all the leading journals. On the one hand there is the attempt to keep German logical research high at the international level under difficult circumstances. On the other hand reviewing raised one’s own level of fame and possibly one’s reputation (but it also is taken as a sign of degeneration and inbreeding). Scholz, Schmidt, and Gentzen intended a consistent quality of the reviews (imagine the crazed anti-Semite Hugo Dingler as reviewer). The others regarded this as manic activity which should distract from a certain isolation. Instead of international solidarity this public desired a concentration on “German mathematics” and above all the abolition of mathematical logic and logistic. Why did Gentzen succeed in venturing into the narrow circle around the logician Heinrich Scholz? Because one could produce him internationally. Gentzen moved in the “right” circles, had support in M¨ unster and G¨ ottingen, and had the right 14 Bull.

Amer. Math. Soc. 45/11 (1939), pp. 812-813. u ¨ber die Fortschritte der Mathematik 4/2 (1939), pp. 26ff.

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contacts. They helped him scientifically, but in the longed-for help in the art of living, not a bit. 3. Active Military Service at the Homefront as Radio Operator by the Flugwachkommando On 28 September, disgusted, bored, with extreme reluctance and the fear of never again being able to think productively, Gentzen was drawn into the war. In Zentralblatt 1939,16 Gentzen mentioned the mathematical survey by Max Bense of the Geist der Mathematik. Abschnitte aus der Philosophie der Arithmetik und Geometrie 17 [Spirit of mathematics. Selections from the philosophy of arithmetic and geometry] because of its popularity. As Bense’s main problems he cited a few chapter headings: “The irrational in mathematics”, “Mathematics and æsthetics”, “Intuitionism, logicism and formalism”.18 There are indications that from April 1939 Gentzen himself wanted to write a popular presentation of mathematical foundational research.19 16 Vol.

21, p. 97. Munich, 1939. 18 Cf. the lovely polemic by J¨ urgen von Kempski, “Max Bense als Philosoph”, Archiv f¨ ur Philosophie 4, no. 3 (1952), pp. 270-280. J. von Kempski, who was silent about his own life under National Socialism, didn’t do justice to Bense’s performance under National Socialism. Where did this hatred of Bense come from? Max Bense was promoted by Oskar Becker in 1937 for his “Quantenmechanik und Daseinsrelativit¨ at” [Quantum mechanics and the relativity of existence], but for a while he stood close to the Nazis (cf. the letters of Bense to Gottfried Benn, 1935). Certainly Max Bense showed in an article in the K¨ olnische Zeitung that he was decidedly opposed to Deutsche Physik [German physics, a movement to purely Aryan physics similar to that in mathematics]. A detailed clarification of what Bense and von Kempski initially did and thought under National Socialism is yet lacking. 19 Christian Thiel has noted this since 1988: “However, do not go into ‘philosophy’. To acknowledge the elegance of the methods, speak of various formal possibilities, philosophical treatments: do not belong here.”— Why did Gentzen mention the books of Max Bense? First, he perhaps liked the idea of a popular, nevertheless scientifically grounded book on logic. Possibly the book served as a model for Gentzen’s project of a popular book on proof theory.— Second, Max Bense was one of the few who defended mathematical logic under National Socialism, although he was not a professional mathematician or philosopher. He wrote, for example, Aufstand des Geistes. Eine Verteidigung der Erkenntnis [Revolt of the intellect. A defense of knowledge] (DVA, Stuttgart and Berlin, 1935). What were the contents? Knowledge is life protection. “The highest objectivity can only happen through the highest subjectivity” (p. 18); “Life must meet the intellect, and the measure of knowledge is a measure of the securing of existence” (p. 33); “One reproaches physics and mathematics in one breath for crisis and height. But the physics and mathematics of Heisenberg, Minkowski, Weyl and Bohr still had ideas and did not sink in reflection. . . Because the eternal, stale reflection, the unending boredom of the intellect can only be killed through a great idea, through a realisation. We have exchanged intuition for reflection, and thereby clarity for confusion” (p. 36); “Hilbert knows of the secret means of mathematics and places the whole of mathematical methodology and axiomatics in the service of the task of grasping the infinite in its transparency. He casts the infinite so to speak in symbols, to get hold of them, and so creates the ‘symbolic logic’ ” (pp. 38ff.). Max Bense’s concern that blood be not played against intellect was certainly a courageous, but nonetheless unsuccessful, tortuous manœuvre, because he had to stick to Nazi ideas, mentioning only these, to achieve a certain credibility and attractiveness. It was however a noteworthy attempt to grant a philosophy the right to exist alongside the Nazi philosophy. “I believe I remember that Max Bense was helpful to H. Scholz with the restoration of evacuated (seminar or personal) properties in the later period of the Soviet Occupation Zone, which however did not preclude that Scholz would express himself critically about later publications” (Prof. Dr. Gisbert Hasenjæger to me on 11 March 1997). 17 Oldenbourg,

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He had the good fortune to be stationed by the Luftwaffe on the “home theatre” between M¨ unster and Brunswick: “. . . have a relatively bearable service, admittedly, thereby no time for any other activity.” Nonetheless he began to write ideas and sketches for a textbook on mathematical foundational research in shorthand on slips of paper, including the certificate of registration of the Flugwachkommando [Flight-Watch-Command]. On 12 October 1939 the “Soldier Gentzen, 12th Flugm. Res. Komp. / 6, Brunswick, Hospital” wrote a postcard to the registrar of the university: In extension of my communication of 27 Sept. concerning my call-up to active military service. I inform you of the following details: Date of entry into service was 28 Sept. 1939; my service rank: soldier first rank [i. Mannschaftsgrade]; salary from the Wehrmacht: monthly 30.- RM military pay. Heil Hitler Dr. phil. G. Gentzen scheduled assistant at the Mathematical Institute of the University of G¨ ottingen20

From the university he immediately received in accordance with the Armed Forces Allowances Law [Einsatz-Wehrmachtsgeb¨ uhrnisgesetz] of 28 August 1939, an estimate of his “Peace Service” income of RM 229.02. Gentzen objected by postcard to the adjusted sum and after an official confirmation of the military pay of RM 30.-, the “radio operator” Gentzen received RM 260.18 monthly from 1 November 1939. At the beginning of the war he visited his longtime friend Hertha Michælis in the labour service camp Gailsdorf in Thuringia and proposed marriage to her. He feared being sent to the Front and wanted to have someone in Germany whom he could think of with love. Hertha Michælis, who was 9 years younger, promised she would write regularly. They had known each other as long as they could remember.21 Hans Student was likewise called up, and so Gentzen’s mother and sister lived with the children, Barbara and Hans Lothar, in Liegnitz, the city in which E.E. Kummer had once taught. On 15 December 1939 Gentzen met Kurt G¨ odel for the first time. G¨odel, on his way to Berlin to procure a visa for his American trip, stopped off in G¨ ottingen to give a lecture. About this, Gentzen wrote: The question of the confirmation of his lectureship in Vienna by the Ministry was unsettled at the time; I have not heard more of him since.

G¨ odel himself wrote only: I visited G¨ ottingen in 1932 and talked about my work. Siegel and Noether talked with me afterwards. I saw Gentzen only once. I had a public discussion with Zermelo in 1931.22 20 The

field post number 38650 was assigned as follows: 1) by mobilisation: 12th Aircraft Reporting Reserve Company / Air District Message Regiment 6; 2) from summer 1940: 12th Company / Air District Message Regiment 6; 3) from winter 1942/43: 12th Company / Air District Regiment 11 (information from the Military Archive of Freiburg, 13 January 1997). Its activity consisted of monitoring the air district by radio and reporting back by telephone and telegraph to the flak corps. 21 Letter from Hertha Michælis (D¨ usseldorf) to me of 2 March 1988. 22 Hao Wang, A Logical Journey; From G¨ odel to Philosophy, MIT Press, Cambridge (Mass.), 1996, p. 84.

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On 5 December G¨odel had written to Helmut Hasse, then managing director of the Mathematical Institute in G¨ ottingen: I have [things] to do in the immediate future in Berlin and intend. . . to stop over in G¨ ottingen on the return trip. I could use the opportunity to give a report in a lecture on my proof of the consistency of Cantor’s continuum hypothesis, if there is interest in it.23

He sent his greetings to Gerhard Gentzen. Hasse wrote two days later that he had scheduled G¨ odel’s lecture for Friday, 15 December, at 8:30 p.m. and had informed Gentzen, who at the time was performing military service in Brunswick, in order that he could arrange a leave.24 Hasse was obviously, despite his known political, national and patriotic views, also an acceptable contact for G¨odel. The lecture was the only time he spoke about his result on Cantor’s continuum hypothesis before a European audience, and he carefully outlined it in a shorthand manuscript. After odel a brief survey of the historical background on the continuum hypothesis, G¨ presented the most important steps of his proof. He avoided an excessive amount of technical detail, but brought however a well-motivated presentation of the basic ideas. As was appropriate on this occasion, he paid tribute to Hilbert in that he stressed certain analogies between his own arguments and those Hilbert had put ¨ forth in 1925 in his lecture “Uber das Unendliche” [On the infinite]. In particular, G¨ odel recalled that Hilbert “had singled out a class of functions of natural numbers, namely those recursively defined.” Corresponding to this, G¨ odel said, he himself had defined a certain class of sets (the constructibles) and proven two analogous facts about them: The set of constructible sets of natural numbers has cardinality less than or equal to ℵ1 , and the class of constructible sets is closed under the definition methods applied in mathematics (including the nonpredicative). In addition the proof of the first fact applied the same idea Hilbert attempted to use in order to demonstrate his first lemma, namely “the avoidability of higher variable types in the definition of constructible sets.”25 G¨ odel departed from G¨ ottingen on the morning of the 17th of December and arrived a few hours later in Berlin.26 G¨ odel then gave some lectures at Yale, about which he said: I obtained my interpretation of intuitionistic arithmetic and lectured on it at Princeton and Yale in 1942 or so. Artin was present at the Yale lecture. Nobody was interested. The consistency proof of arithmetic through this interpretation is more evident than Gentzen’s.27 23 Kurt

G¨ odel to Helmut Hasse, 5 December 1939, G¨ odel’s Nachlass 010807, 4 Folder 01/68. p. 127, Chapter 7: Heimkehr und Vertreibung, in: John W. Dawson, Jr., Kurt G¨ odel: Leben und Werk, Springer-Verlag, Vienna, 1999. 25 G¨ odel’s lecture would be published posthumously in volume III of his Collected Works, pp. 126-155. For a detailed analysis of the contents, see the introductory remarks thereto by Robert M. Solovay, pp. 114ff. and 120-127. 26 Dawson, op. cit., p. 127. 27 Hao Wang, op. cit., p. 86. The interpretation G¨ odel is referring to is not the common interpretation of classical arithmetic within intuitionistic arithmetic obtained independently by G¨ odel and Gentzen, but a new proof of the consistency of the intuitionistic (and hence classical) arithmetic by using functionals of higher type to interpret arithmetical formulæ. G¨ odel finally ¨ published the result in a philosophical journal in 1958: “Uber eine bisher noch nicht ben¨ utzte Erweiterung des finiten Standpunktes”, Dialectica 12 (1958), pp. 280-287. Cf. Solomon Feferman, “G¨ odel’s ‘Dialectica’ interpretation and its two-way stretch”, in: S. Feferman, In the Light of Logic, Oxford University Press, 1998. 24 Cf.

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Gentzen was already known in the U.S. since 1935/36 through Bernays and then through the works of H. Curry. The significant American logicians like Church, Kleene, Rosser and Post all knew Gentzen’s works. In the 30 December 1939 issue of Zentralblatt three more reviews by Gentzen appeared: Moisil, Gr. C.: Recherches sur le syllogisme. Ann. Sci. Univ. Jassy, 1: Math. 25, 341-384 (1939): The Aristotelian syllogistic of the “calculus of classes” is investigated from various perspectives and generalised. So in particular for the case that one assumes the classes occurring are not empty, respectively, not universal; further for more than two premises; then too with the modification that the concept “for all” is replaced by “almost all”, in various senses which arise from connections to the notion of probability among others. There follow applications to propositional calculus, especially the calculus of Heyting; further there are some derivations carried out in its predicate calculus and it is shown that the combination-concepts of syllogistic can be defined in a number of ways herein (where the Law of the Excluded Middle is not at one’s disposal). Novikoff, P.S.: Sur quelques th´eor`emes d’existence. C. R. Acad. Sci. URSS, N.a. 23, 438-440 (1939). The classical propositional calculus is extended by allowing disjunctions and conjunctions with a countably infinite number of terms; for this the concept of a “true formula” is defined and it is shown that if an infinite disjunction of finite formulæ is true, then already a finite part of the same must be true. Menger, Karl: A logic of the doubtful. On optative and imperative logic. Rep. Math. Colloq., Publ. Univ. Notre Dame, II. s., No. 1, 53-64 (1939): 1. The author develops the beginnings of a propositional logic, for which a third modality “doubtful” is added to the truth values “true” and “false”. Unlike the manyvalued systems of Post (Amer. J. Math. 43, 180ff.) and L  ukasiewicz (C. R. Soc. Sci. Varsovie, DL III, 23), the modality of a combination will, by Reichenbach (this Zbl. 10, 364; §74), depend not only on the modalities of the two propositions, but rather it turns out—if one wishes to comply with the linguistic use—for the case that both subexpressions are “doubtful”, as depending on a certain coupling of both expressions (cf. the “coupling degree” of Reichenbach, loc. cit.); it turns out 7 different levels of coupling are possible.— 2. The author then sketches, following a criticism of a system of Mally (Grundgesetze des Sollens, Graz, 1926), the foundations of a system of commands and desires. These are naturally connected to “doubtful” propositions. Desired occurrences may be ordered by their degree of desiredness. Further desires are frequently dependent upon the validity of certain conditional expressions, or are mutually subordinate. These possibilities are thoroughly discussed for the case of one or two propositions.28

4. 1939/40: Preparation for Habilitation. “Beweisbarkeit und Unbeweisbarkeit von Anfangsf¨ allen der transfiniten Induktion in der reinen Zahlentheorie” Gentzen’s habilitation thesis was laid before the dean of the University of G¨ ottingen in mid-1939. The dean wrote on 26 September 1939 to the secondary school teacher W. Ackermann: Very respected Herr colleague! After consulting with Herr colleague Hasse I ask you to evaluate, on technical grounds, the habilitation thesis of Herr Dr. 28 Zentralblatt

f¨ ur Mathematik 21 (1939), p. 290.

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Gentzen, “Beweisbarkeit und Unbeweisbarkeit von Anfangsf¨ allen der transfiniten Induktion in der reinen Zahlentheorie” [Provability and unprovability of initial cases of transfinite induction in pure number theory]. At the same time I ask you to present the work to Herr Professor Dr. Heinrich Scholz, Logical Seminar of the University of M¨ unster i. W.. . . Postscript: Because the work falls into none of the work areas of our local colleagues, I permit myself to trouble you for advice.29

Scholz received the habilitation work of Gentzen for evaluation in February 1940 at the latest, for the dean of the mathematical-natural scientific faculty confirmed the delivery of the corresponding signed receipt of Scholz on 27 February 1940 with the words, “You have thereby done the faculty a valuable service.” At the request of the dean and Heinrich Scholz, Ackermann wrote an evaluation of Gentzen’s habilitation thesis: From the investigations of the foundations of mathematics in the sense of Hilbert’s proof theory, on the basis of a theorem proven by G¨ odel in 1931, it came to light that one could not deduce the consistency of pure number theory by the proof means of the same theory, but rather that higher aids are necessary for this purpose. From the consistency proof for number theory, delivered by the author in 1936, it was revealed that the addition of transfinite induction in domains of segments of the second number class reaching up to ε0 (following the Hessenberg notation) suffices in order to secure the necessary extension of the modes of inference. One can conclude from this that the transfinite induction up to ε0 cannot be derived via the number theoretic modes of inference. The author gives a new, direct roof of this result in the present work, one that makes no use of the G¨ odel Theorem or of G¨ odel’s arithmetisation of the metamathematics. To understand the significance of the mentioned theorem, for which a new proof is here delivered, a few observations are essential. At first sight, namely, the theorem appears to be trivial. For the system of pure number theory contains in and for itself no transfinite ordinal numbers at all and therefore also no theorems about such, thus also no transfinite induction. However, here the system of pure number theory is to be understood so that in the natural numbers as further objects transfinite ordinal numbers as well as certain functions and predicates and mathematical axioms that belong thereto (cf. §1, p. 10 of the work) will be adjoined. This adjunction of transfinite ordinals is, as the author remarks, fundamentally of the same sort as the adjunction of negative numbers, functions and such like, so that through it no inappropriate element comes in. The author refers to his own above cited work in which it was shown that, in place of ordinal numbers, one can just as well use certain decimal fractions. The newly adjoined axioms are incidentally only such that contain the definitions of the predicates and functions introduced for the ordinal numbers. It is then shown that the formula which expresses transfinite induction up to ε0 cannot be the end formula of a proof. On the other hand, however, as is likewise shown, transfinite induction up to any fixed, but arbitrary ordinal number less than ε0 , can be reduced in the frame of the system to ordinary induction. For this second theorem another proof (also given by the author) is already available (Hilbert-Bernays, Grundlagen der Mathematik, II, §5, 3c). To bring out clearly the significance of the author’s attained results, it would have been desirable if the author had anyhow done without the introduction of the transfinite ordinal numbers. One is of course too inclined to see a foreign 29 Hans Richard Ackermann, “Aus dem Briefwechsel Wilhelm Ackermann”, History and Philosophy of Logic 4 (1983), pp. 181-202; here, p. 193.

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element in the bare introduction of notation schemata for ordinal numbers in number theory. The avoidance is possible because the natural numbers themselves can be represented by a proper correspondence with a certain segment of the second number class. The ordering predicate for the segment up to ε0 can be defined by the usual recursion in the frame of pure number theory. Transfinite induction can itself then be formulated as a purely number theoretic proposition, in which symbols for ordinal numbers do not appear. The verification of the unprovability of this theorem can be achieved in exactly the manner given by the author. In this way it would be more clearly expressed that the incompleteness of the proof means of the pure system does not concern any supplements. As far as the proof given by the author is concerned, he serves up reflections similar to those of his earlier consistency proof. Every derivation (proof figure) is assigned a certain ordinal number less than ε0 as a value. Further it is shown that one can derive from a derivation of value α at most a formula that symbolises transfinite induction up to α, but not such that goes further. From this the desired result then follows.— The inferences are entirely correct and clearly worked out. Summarising, on the assessment of this work the following should be said: For the theorem first proven by G¨ odel that a formal system of axioms and inference rules of pure number theory is necessarily incomplete in that doubtlessly correct number theoretic propositions may be formulated but not proven by the means of the system, the author has given a new direct proof which does not use G¨ odel’s arithmetisation. By the importance of the named proposition, which among other things is right, to throw some light on the fact that certain number theoretic conjectures have until now resisted all proof attempts, this new proof is a scientifically thoroughly significant achievement.30

And Heinrich Scholz echoed Ackermann’s remarks on the new, original proof of G¨ odel’s Incompleteness Theorem in his “Urteil u ¨ ber die Habilitationsschrift Gentzens” [Verdict on Gentzen’s habilitation thesis], and always with reference to the second volume of Hilbert-Bernays: Verdict on Gentzen’s habilitation thesis, “Provability and unprovability of initial cases of transfinite induction in pure number theory”. The foregoing opinion of Herr Ackermann is so exhaustive and clear that I may be permitted to restrict myself to the following supplementary points: (1) Herr A. has himself worked in the direction in which Herr Gentzen has ventured. He not only mastered the difficult problem, but rather he is an acknowledged authority in this area. I therefore have taken it as necessary and held it appropriate to make the verdict entrusted to me dependent upon the opinion of Herr A. Now I can emphatically, without any lingering doubts, identify myself with his opinion. (2) G¨ odel’s Incompleteness Theorem, to which Gentzen’s work refers, is a theorem of the first order. It is by far the deepest lying theorem of all of modern foundational research. Hilbert’s original approach to a proof of the consistency of classical mathematics with the transfinite use of the “tertium non datur” was upset by this theorem. Cf. Hilbert-Bernays, Grundlagen der Mathematik, vol. 2, 1939, §5. A new approach became necessary. In the sense of this approach 30 The original may be found in “Logistischen Seminar der Universit¨ at M¨ unster i./W. Prof. Scholz”, which is today in the Institute for Foundational Research. I express my thanks for this finding to Herr Dr. Enno Folkerts. In the copy of 9 February 1943 for Gentzen’s lectureship in Prague, in place of the word “achievement” [Leistung] is simply “matter” [Angelegenheit]; moreover, the last paragraph is underlined (personal file).

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Herr Gentzen attained through his proof of the consistency of pure number theory (cf. remarks 2 and 4 of the coming habilitation thesis) decisive results. G¨ odel’s theorem for some years continues to be the focal point of significant discussion. It has had a second essential result as a consequence. With the aid of original methods which G¨ odel discovered for the proof of this theorem, in the course of the last few years one has succeeded in surprisingly different ways, first to make precise the unsolvability of the so-called Entscheidungsproblem that it could be mathematically discussed at all, and then to prove the thus made precise unsolvability of the proof theoretic (in Carnap’s language: syntactic) decision problem already for the so-called narrow predicate calculus of the first order (without the Axiom of Identity). This is the restricted logical calculus, on which in the present state of things a strict formalisation of the arithmetic of natural numbers can be based. To this complex, above all, the second supplement of the second volume of Hilbert-Bernays is to be consulted. A new, original proof of a theorem of this significance is a result one has fought for. Herr Gentzen develops such a proof in his habilitation thesis. (3) Already Herr A. has indicated that the inclusion of a certain segment of the ordinals of the second number class in elementary number theory is at first very disconcerting to mathematicians. He has, however, said how this disconcertment can be made to vanish. I’d like here expressly to agree with him. Also this point is now so clarified in the second volume of Hilbert-Bernays, pp. 360ff., that one can look the essentials up there. One should see in particular what is said here on p. 362 on a generalised principle of the least number. M¨ unster i.W., 24 February 1940. Heinrich Scholz31

In a written reply of Ackermann of 27 February 1940 to the no longer available cover letter by Scholz, one reads in the first paragraph: I thank you for sending the report on the work of Mr. Gentzen. After I read through my valuation once again, it struck me incidentally that in the formulation of the conclusion I have not expressed the following completely clearly, although it does yield itself from the preceding. Gentzen’s work gives a new proof that the usual system of arithmetic is incomplete in that the formula expressing transfinite induction up to ε0 is unprovable. That is the most important special case of G¨ odel’s Theorem, out of which also the consequences regarding the conducting of the consistency proof yield themselves. To this special case of G¨ odel’s Theorem you have also referred in your assessment. . . 32

On 28 September Gentzen’s deputised company commander and first lieutenant informed the registrar of the university (“Concerns: Settlement Claim”) that company member Gerhard Gentzen effective 1 October 1940 would be promoted to private first class and from this day on would receive a monthly military salary of RM 36.-. His university salary rose thereby to RM 289.22.— In December 1940 Gentzen was requested by the mathematician Gustav Herglotz (1881-1953) to give a scientific talk to complete the habilitation process. Private First Class G. Gentzen (L 38650 Lg. Postamt M¨ unster) wrote on 7 December 1940: 31 Ibid.

In the copy for the lectureship at Prague it also reads, “With the authorization agreed. 20 Feb. 1940 signed Kaluza.” Like sounding remarks by Herglotz and Hasse. Additional remarks are “Seen by Siegel 1940 February 24. . . 5 March 1940 signed R. Becker” and “Herr Dr. Gentzen is a recognised authority in the area of foundational research and I place value in the fact that the G¨ ottingen lecture staff has him. I assume today’s absent colleagues Hasse and Kaluza share the same opinion. 13 Dec. 1940 signed Dr. G. Herglotz.” 32 Hans Richard Ackermann, op. cit., p. 193.

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To the Herr Director of the Math. Institute, G¨ ottingen From the office of the Dean I received the request for a scientific talk on 12 December, 6:00 p.m. sine tempore. I have received leave from my company from ottingen that afternoon, and immedimidday 11 Dec. and intend to arrive in G¨ ately, probably between 4 and 5 p.m., to call on the Math. Institute. I don’t know to whom I am addressing this note; I assume, to Herr Professor Herglotz? In any case I would like to call on you, respected Herr Professor, possibly already on 11 Dec. I am, as things stand, poorly prepared for the scientific talk; I have occupied myself rather one-sidedly with the area of foundations and have through the war’s outbreak been prevented from extending my scientific knowledge with regard to the habilitation in other directions. I hope, however, that it will be possible to consider these special circumstances. With most respectful greeting and Heil Hitler! Your Gerhard Gentzen33

The examiners for this talk were Herglotz, Hasse34 and Theodor Kaluza (18551954), famous for Kaluza-Klein theory. Although exhausted by the war, Gentzen acquitted himself brilliantly with a talk in which in consideration of my uniform my lack of an all round mathematical education would be disregarded, not exactly with the same willingness from all the masters—but I had a clear conscience on that.35 33 I

thank Dr. Ulrich Hunger of the University Archives in G¨ ottingen for this letter. 15 March 1939 the number theorist had defended the Nazi measure against the Jews to the American mathematician Marshall Stone (“Looking at the situation from a practical point of view, one must admit that there is a state of war between the Germans and the Jews. . . ”) and would be employed as lieutenant commander of the Supreme Command of the Navy in Paris. Cited from p. 164, R. Siegmund-Schultze, Mathematische Berichterstattung in Hitler Deutschland, Vandenhoeck & Ruprecht, G¨ ottingen, 1993 (cited from p. 331, Nathan Reingold, “Refugee mathematicians in the United States of America (1933-1941): reception and reaction”, Annals of Science 38 (1981), pp. 313-338. Likewise cited in: Rolf Schaper, “Mathematiker im Exil”, and Andreas Kamlah, “Die philosophiegeschichtliche Bedeutung des Exils nicht-marxistischer Philosophen zur Zeit des Dritten Reiches”, in: Edith B¨ ohne and Wolfgang Motzkau-Valeton (eds.), Die K¨ unste und die Wissenschaften im Exil 1933-1945, Verlag Lambert Schneider, Gerlingen, 1992.) One likes to take the view of a “Jewish Conspiracy” as criminal or negligent, but it had many adherents in the right wing, like for example the lawyer Professor Dr. Friedrich Grimm (Mit offenem Visier. Aus dem Lebenserrinnerungen eines deutschen Rechtsanwalts, Druffel Verlag, Leoni, 1961). Grimm himself stood up for Jewish lawyers and in 1933 was the advocate of the synagogue patrons against the Hitler Youth in Essen. On p. 272 he depicts a verbal exchange with his prosecutor during the Nuremberg Trials and said to him in 1947: “One cannot lie though, that it was a German-Jew war; not in the sense of international law. For, there is no Jewish state, but there was a Jewish power, with which a de facto state of war prevailed.” An always excellent source on conspiracy theories is Johannes Rogalla von Bieberstein, Die These von der Verschw¨ orung 1776-1945. Philosophen, Freimaurer, Juden, Liberale und Sozialisten als Verschw¨ orer gegen die Sozialordnung, Flensburger Hefte Verlag, Flensburg, 1992. 35 Gustav Herglotz appears to have been a witty examiner with whom one would likely have few difficulties. An anecdote: 34 On

As I stood before the assembled G¨ ottingen natural sciences faculty for my habilitation colloquium and the question of the physicist Joos on the extreme principle in physics was twisted around by the mathematician Herglotz to the request to me to express myself on the so-called ontological proof of the existence of God, Prandtl angrily intervened with the comment, “That really doesn’t belong here! Perhaps he only wanted to protect me, because he didn’t know that I had an extensive study of philosophy behind me and was rather pleased with the question of Herglotz. H. G¨ ortler, “Ludwig Prandtl, Zeitschrift f¨ ur Flugwissenschaften 23, no. 5 (1975); here, p. 159.— Herglotz was considered politically unreliable. He was supposed to be a corresponding

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One of the perhaps not entirely willing masters may have been the naval soldier Hasse, who had precisely circumscribed the area of logic within mathematics. On orders from Reichsminister Rust—as with all officially chosen representatives of the German sciences—on the occasion of Hitler’s 50th birthday in 1939 Hasse wrote an article for an anthology. There he again practised his political language: Finally the wonderful idea of Hilbert, to protect mathematics against unfounded attacks on its rigour through a deep going logical analysis, was led to a beautiful success. It was one of his students who proved the consistency of pure number theory.36

This student was, of course, Gentzen. But we see too to which task Gentzen is always exclusively assigned and that only G¨ odel can be meant as the one who leads “unfounded attacks on the rigour of mathematics.37 That is the justification for the spirit of research, which is carried out from a high idealism, bound with a genuine realism, which from German researches would never be understood in the sense of a naked usefulness, but rather by the logical-epistemological researches in like manner possessing life like that of those that are directed toward the applications.38

twice justified the logical-epistemological work in mathematics. What disturbed him was the “accumulation of individual facts without [those] inner ties” which it seems the “final goal of research should be.” That happened “under the influence of the rationalistic efforts of clarification and overemphasis tied thereto of the unrestrained intellect.” Hasse desired complete orientation, organic depth and many-sidedness, greatness and tradition, living form of the “free creation” instead of the counterfeit of content of mathematical research through “rigorous modes of inference and logical hairsplitting of an unrestrained intellect.”39 It seems to me Hasse cited too often what he saw as stock themes, which probably in his view should correspond to what he took for National Socialism: Mathematics is the doctrine of the laws in the world of numbers, of functions, and of geometric images. . . Also the Faustian urge to understand what holds the world together in the innermost, that is deeply rooted within us, extends itself to those worlds of thought. The further mathematical knowledge advances, so much more powerful mathematics becomes in the hand of man as a tool for the tremendous tasks that will be placed by the natural sciences and technology, so much more thoroughly and widely can they serve human life. In the striving for such powerful ability and proud mastery lies a driving incentive to mathematical research. To this comes the wonderment before the beauty and elevation of the mathematical world, before its crystalline clarity and inviolable rigour. member of the mathematical natural scientific class of the Bavarian Academy of Science. His election in 1943 was not confirmed by the Minister, the Gauleiter Wagner, on political grounds (F.L. Bauer, “Fritz Hartogs. Das Schicksal eines j¨ udischen Mathematikers in M¨ unchen, Zeitschrift Aviso 1 (2004), pp. 34-41). 36 P. 151 of Deutsche Wissenschaft. Arbeit und Aufgabe, S. Hirzel Verlag, 1939.— On Hasse cf. also S.L. Segal, “Helmut Hasse, Historia Mathematica 7 (1980). A proper biography of Hasse is wanting. 37 This was, of course, the view of the common German working mathematician who had no deep understanding of foundational issues. Gentzen was seen as the fulfiller of Hilbert’s Programme, Hilbert as the one who succeeded in real rigour, and G¨ odel as having launched a major assault on Hilbert, providing nothing but chaos. 38 Ibid., p. 149. 39 Ibid.

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A German mathematics—the main areas being number theory (G¨ottingen), function theory (M¨ unster), and geometry (Hamburg)—“multiply linked to one another”—was presented.40 The themes of world mastery, beauty, and clarity of mathematics, whereby the mathematician stood in a tradition, which could only be artistically grasped, were often repeated by Hasse. The currying favour, even the subordination to natural sciences and technology, like the pathos laden language, is genuflection to the spirit of the times, if not a conviction to the marrow. Nonetheless, we see here why mathematical logic was held so high. It was mentioned for the three main areas as security, as control, and verification of the rigour of mathematical argumentation. To this Hasse set the ascent of the succession in the traditional sequence of German mathematicians differently from Max Steck or Hugo Dingler: “Leibniz, Gauss, Dirichlet, Riemann, Weierstrass, Felix Klein and Hilbert.”41

40 No mention of Bieberbach or Vahlen.— The trotting out of mathematics in Festschrifts was supposed to document the fact that scientific life without the Jews functioned at least as well as before. And Hasse complied with this desire. For Gentzen however it meant that he had to fill the empty place of Hilbert-Bernays and others. 41 Deutsche Wissenschaft, p. 150. After the war Hasse represented similar views, a bit more cautiously expressed, in a lecture fragment he frequently repeated: “Tragedy of the mathematician through striving and formalisation”. He represented therein a Steckian form-theory:

By each of these steps [from the intuitively conceived idea, outline, multiple revision, improvement, extension, reorganisation into final form—EMT] there is something of the natural life, of the intuitive seizability, of the whole of the inherent dynamics, shortly before the beauty is lost in the large, while the beauty in the small is perhaps attained. If one carries this process to the extreme, there will remain in the end a bloodless, dead image of the emaciated structure: Definition, Theorem, Proof in a multiple repetition. And it is thus in this flagrant manner that the already named life law of mathematics is broken. And after Hasse presents the Elements of Euclid as a prime example of a “lifeless final state”, he continues: The attraction to something of the sort, complete in external form, but of internal bloodless composition, following formalisation and axiomatisation. . . has since Euclid’s times first been brought back to life in this century. As already said, this endeavour flowers the greatest at present in North America. But it has also taken hold of us, partly as a consequence of the life work of our great Hilbert, who in his Grundlagen der Geometrie takes up Euclid directly, partly too as a repercussion of the American taste on European mathematics, more and more about itself. I see therein a tragedy, living inside each mathematician: The, in itself healthy urge to form what is seen and recognised as perfectly structured, ends, if one works it out to the last consequence, with the complete mortification of all that lived in the original proper work that flowed through with effervescent life. And how “life-aborters” were proceeded against, part of the population would very precisely know after the war. Mathematicians like Hasse saw their power in the “power over the concept world of our thought” (Helmut Hasse, Mathematik als Wissenschaft, Kunst und Macht, Verlag f¨ ur angewandte Wissenschaften, Wiesbaden, 1952). In his inaugural address in Hamburg in 1951 Helmut Hasse elevated the idea and image of the pure mathematician as opposed to the industrial mathematician drilled only on applications, and defended it aristocratically. This naturally had its basis in the political experiences Hasse must have had under the Third Reich. And who knows what would have happened to Gentzen if Hasse had not had his positive attitude toward socially withdrawn mathematicians, to the purely intellectually directed man with his knowlege won from purest idealism. The stars certainly smiled down on him and his habilitation.

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5. 1941: Encouragement from Hellmuth Kneser On 1 January 1941 Gentzen joined the Dozentenbund.42 The habilitation thesis, “Provability and unprovability of initial cases of transfinite induction in pure number theory”, would be published in 1943 in Mathematische Annalen. The work was received in 1942 and would be favourably reviewed by Ackermann in Zentralblatt: Gentzen, Gerhard: Beweisbarkeit und Unbeweisbarkeit von Anfangsf¨ allen der transfiniten Induktion in der reinen Zahlentheorie, Math. Ann. 119, 140-161 (1943): The idea of pure number theory, as the author has developed in earlier works (cf. this Zbl. 14, 388), is extended here in such a way that to the natural numbers transfinite ordinals are added as further objects, as well as the functions and predicates belonging thereto which can be decidably defined. In the system so constructed the inference mode of transfinite induction can be formalised and it is explained what is to be understood as a derivation of this mode of inference. The following remarkable result is then given: Transfinite induction up to each ordinal number below ε0 is provable in pure number theory; in comparison, it is not provable up to ε0 itself or to any higher ordinal number. The theorem is in agreement with the incompleteness of each sufficiently expressive consistent formalism proven by G¨ odel. That it actually deals with an incompleteness of the number theoretic formalism would come to a yet clearer expression if the introduction of the transfinite ordinals were entirely omitted. That is possible, because the natural numbers themselves represent the ordinal numbers in question by a suitably chosen recursively defined order relation. The inference rule of transfinite induction then refers only to natural numbers. This possibility is indicated by the author himself. Ackermann43

Gentzen himself wrote in this work: That transfinite induction up to the ordinal ε0 cannot be proven by pure number theoretic proof methods may be inferred indirectly from the following two facts: 1. G¨ odel’s Theorem: The consistency of pure number theory cannot be proven by the proof means of the theory itself. 2. The consistency of pure number theory can be proven by application of transfinite induction up to ε0 and the usual purely number theoretic means of proof. In the following I give a direct proof of the unprovability of transfinite induction up to ε0 in pure number theory. The procedure moreover permits the verification that further initial cases of transfinite induction up to ordinals below ε0 are not derivable in certain subdomains of the number theoretic formalism. On the other hand, it is known that transfinite induction up to any arbitrary ordinal less than ε0 is provable purely number theoretically.44

Paul Bernays reviewed the thesis in the Journal of Symbolic Logic—a pity Gentzen could not read the manuscript beforehand— and thereby brought the paper to international attention: 42 According

to the self-entry in the Prague files; in the G¨ ottingen papers the date is 1 January

1939.

43 Zentralblatt

f¨ ur Mathematik 28 (1943), p. 102. an offprint of his habilitation thesis from Mathematische Annalen 119 (1943), pp. 140161, there is on the cover page (p. 140) a handwritten “Mit ergebensten Gruss G. Gentzen” [With most respectful greeting, G. Gentzen]. This shows that in sending his works he dispensed with the “Heil Hitler!”. That too is a sign that he stood opposed to Nazi ideology and even after Stalingrad did not indulge in a motto like “Und dennoch!” [And yet!]. The author purchased the offprint on 3 February 2004 from Antiquariat Renner. It could no longer be determined who the recipient of Gentzen’s greeting was. 44 On

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Gerhard Gentzen Beweisbareit und Unbeweisbarkeit von Anfangsf¨ allen der transfiniten Induktion in der reinen Zahlentheorie Mathematische Annalen, vol. 119, no. 1 (1943), pp. 140-161. As Gentzen stated in I 75—it is one of the final observations he adds in §16 to his proof of the consistency of “reine Zahlentheorie”—this proof can be formalized itself within “reine Zahlentheorie” by the G¨ odel method of arithmetizing metamathematics, with the exception only of the the transfinite induction used for it. Concerning the way of formalizing “reine Zahlentheorie” it is necessary to notice (what was not expressly mentioned in the review I 75) that Gentzen admits, as a means of introducing symbols of numerical predicates or numerical functions, any schema which, like that of primitive recursion, has a constructive interpretation under which the predicate is decidable or the function is computable, likewise that he allows to be chosen as axioms any formulas without bound variables which (in the sense defined in Grundlagen der Mathematik, cf. vol. 2, pp. 35-36) are verifiable with respect to the constructive interpretation of the arithmetical symbols occurring, or else are formally equivalent (“deductionsgleich”) to such a formula. As to the transfinite induction in question, this is the restricted form extending only over the ordinals below ε0 —i.e. the first Cantor epsilon-number, which is the limit of the sequence ω0 , ω1 ,. . . defined recursively by the equations ω0 = 0, ωk+1 = ω ωk . (In I 75 the ordinal numbers below ε0 are represented by certain finite decimal fractions; in the newer version IV 31 of the consistency proof Gentzen goes back to the usual notation for ordinals.) It follows by the second G¨ odel theorem on formally undecidable propositions that it is impossible to replace transfinite induction up to ε0 (as we may call it briefly) by an inference formalizable in a system of “reine Zahlentheorie”. This way of obtaining the indicated impossibility is very indirect and includes several complicated discussions. In the present paper Gentzen gives a more straightforward demonstration of it by strengthening the method of proof used in IV 31; and at the same time he obtains in this way some further results. These results refer to an enlarged formal system of number theory which surpasses “reine Zahlentheorie” by including among the “Zahlterme”, or numerical terms (substitutable for the numerical variables), expressions for transfinite numbers representing the elements of a certain segment of the Cantor second number class, extending at least as far as ε0 (for the ordinal numbers below ε0 Gentzen refers to the symbolism used already in IV 31, which without difficulty can be continued quite as elementarily somewhat beyond ε0 ). In connection with the new numerical terms, new symbols of predicates and functions and also new formal axioms are allowed to be introduced, but these must satisfy the above indicated conditions with respect to a constructive interpretation. . . 45

On 11 January 1941, the registrar was concerned about Gentzen because his position expired on 31 March. He inquired of the Ministry and requested a statement on the matter, whether in view of the regulation of clause 3 of the ministerial decree of 27 January 1940 the discharge through cancellation during the term of Gentzen’s conscription should not happen: “By the return of the assistant Dr. Gentzen it would then require a special request for the express lengthening of the contract.” On 10 May the registrar noted, “The extension of the contract is to be effected on Dr. Gentzen’s return from military service.”

45 Journal

of Symbolic Logic 9 (1944), pp. 70-72.

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On 11-14 May 1941 Gentzen wrote from B(raunschweig?) to Hellmuth Kneser, who had posed a couple of questions on which he perhaps wanted to write in his planned lecture “Notio et notatio”: Very respected Herr Professor! I thank you for your letter, which to me seemed almost like news from another, past world. You have given me no small pleasure thereby. Today I finally find the time to answer you in peace. First to your first question. I know in Hilbert and Bernays no place of a reference to Leibniz, while on the other hand the “logicists” happily refer to him as an initiator of their science, but with regard to his thoughts of a universal language of science of a mathematical nature. For Hilbert the “formalisation” is more a tool, which can just as well count as such as anticipated in principle by Leibniz’s thinking in the manner indicated by them. (Logistic references to Leibniz are presumably to be found in Scholz, Geschichte der Logik, or with certainty in the volumes of the periodical Erkenntnis; also by Carnap, Der logische Aufbau der Welt, or perhaps by Couturat; by Frege and Russell I wouldn’t know.)46 Now to the question of the future of math. foundational research in Germany. That, at the time, all “applicable” branches of science stand in the foreground of public support is thoroughly understandable. I very much hope, however, that after the war a balance herein may succeed. And even math. foundational research and mathematical logic have through their major upswing in recent times achieved a significance that in some countries abroad, especially in the USA, is more widely recognised than up till now in Germany, this although the starting point of these researches was originally via Frege and Hilbert,—and a scope that an individual must already devote his entire work energy to survey and work out. One or two chairs for this subject would naturally be a huge advance, but also necessary, to again fairly level the existing imbalance. G¨ ottingen recommends itself as a place for this through its appeal to mathematics students and through its extensive library, in which the subject of foundations is represented with almost all relevant publications. In M¨ unster to be sure there lacks an ample number of mathematics students, though Herr Scholz has anyhow managed to unite a certain circle of young co-workers and so at least to create a place in Germany where with some difficulty a regular lecture schedule and an ongoing series of publications from the foundational area is being maintained. (During the war the lectures have temporarily been discontinued.) He has in fact succeeded in having his chair transformed into one for “Philosophy of Mathematics and Natural Sciences”—which admittedly is only a compromise. Further, M¨ unster possesses likewise an encompassing collection of relevant literature. Concerning the question of personnel, naturally the name G¨ odel stands by far at the top. I don’t know if you have heard of it, that already before the war G. has returned to Vienna from America. He was in December ’39 shortly in G¨ ottingen and I had there the opportunity to become acquainted with him. The question of the confirmation of his lectureship in Vienna was then undecided; I have not heard more of him since. Of the remaining young mathematicians, who stand out through their publications on the foundations industry, like Ackermann, Behmann,47 Bachmann, Hermes, Arnold Schmidt, the Dutchman Heyting, etc., 46 At least in the English translations, references to Leibniz by Scholz and Carnap are minimal. A quick check of the indices of these books finds merely “passim” in Scholz, and 3 references in Carnap. These latter are mere asides, as are the two references to Leibniz quickly found in footnotes in Scholz. Louis Couturat, however, wrote a book on the logic of Leibniz. 47 Cf. Gerrit Haas and Elke Stemmler, “Der Nachlaß Heinrich Behmanns (1891-1970). Gesammtverzeichnis. Heft 1” (42 pages) of the Aachener Schriften zur Wissenschaftstheorie,

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Ackermann would probably be placed in the first position. In conclusion, yet a couple of words on my own state: Since September ’39 I am a soldier, have a relatively bearable service, admittedly no time thereby for any ottingen, other activity. I completed my “scientific talk” in December 1940 in G¨ whereby in consideration of my uniform, my lack of all round math. education was ignored, not exactly with equal willingness by all the masters—but I had a clear conscience in the matter. With warm greetings, also to your respected wife Your G. Gentzen48

One imagines that Gentzen held it in the domain of the possible and probable that Kurt G¨ odel took a lectureship at the University of Vienna and was available for filling a position in Germany. Perhaps he had heard of the temporary visits of Noether and Courant in 1933. But that he could hold possible a return by G¨ odel in 1941 astonishes me greatly. I ask myself really, how Gentzen had seen National Socialism and its political and spiritual effect on mathematical logic. On 20 June 1941 the university set Gentzen’s salary to RM 387.47. L´ aszl´o K´ alm´ar published a paper in which Gentzen was cited: K´ alm´ ar, L´ aszl´ o: Zielsetzungen, Methoden und Ergebnisse der Hilbertschen Beweistheorie. Mat. Sz. Lap. 48, 65-116 and German summary 116-119 (1941) (Hungarian). For Hilbert’s proof theory, a summary overview with respect to its reasons for existence, goals and methods is given. Following a sketch of the foundational crisis and an overview of the axiomatic method in general and of mathematical logic, the oldest methods of consistency proof (construction of a model, i.e. the reduction of the consistency of one axiom system to another) for the well-known classical applications (consistency of noneuclidean geometry, etc.) is discussed. There follows the presentation of Hilbert’s proof theory in its original sense, which had the goal of an absolute consistency proof. Of the methods belonging here, first is discussed the valuation method which goes back to J. K¨ onig, which would be used in an extended form, as the so-called partial valuation method of Hilbert, Ackermann, von Neumann and Herbrand, to show the consistency of arithmetic under certain restrictions with regard to the use of complete induction. There follows further an explanation of the method of proof transformation with the aid of which Gentzen first succeeded in showing the consistency of arithmetic without any restriction. That also the older partial valuation method makes possible the complete consistency proof for arithmetic, as the referee has recently shown, is likewise mentioned. In conclusion a short comparison of Hilbert’s proof theory and intuitionism is given. Ackermann. (Burgsteinfurt)49

In general Gentzen’s genius was praised from Ackermann and Bernays to Scholz and Schr¨ oder. Rarely had a young logician like Gentzen received so much praise from colleagues and equally rarely had it counted so little for him and his life chances. What should the soldier have gotten from it? In 1943 Hellmuth Kneser’s “Notio und notatio. Auszeichnung zu einem Vertrag in Heidelberg im Juni 1941”50 [Notio and notatio. Sketch of a lecture in Heidelberg in June 1941] appeared. In it we read: Logik und Logikgeschichte, ed. by Christian Thiel, Aachen: Lehrstuhl f¨ ur Philosophie und Wissenschaftstheore der RWTH Aachen: July 1981. 48 The original letter can be found in the Archive of Dr. Martin Kneser. 49 Zentralblatt f¨ ur Mathematik 24 (1941), p. 241. 50 Jahresbericht der Deutschen Mathematische Vereinigung 53, 2nd Division, No. 1 (1943), pp. 9-21.

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I will here attempt to present two major principles, to show how they stand opposite each other and amplify one another.

And it continues, because Gentzen is celebrated as the astute executor of Hilbert’s bold plan. G¨ odel’s results are mentioned and Gentzen’s proof compared. The results of metamathematics are put up and defended as valuable and meaningful cultural achievements. G¨ odel’s work shows that possibly the consideration of the calculus itself leads to trains of thought which can point beyond it and cope with tasks to which it has not grown.51

The review by Eugen L¨ offler gives good report of Gentzen: The author subtitles this work a “sketch of a lecture in Heidelberg in June 1941”. Concept and notation are two major principles of scientific advance which in mathematics stand opposite to and amplify one another. . . Finally, the art of replacing difficult thought by mechanistic manipulation with the aid of an appropriate symbolic apparatus is acknowledged. As an example there is the problem of the consistency of the axioms of number theory raised by Hilbert, which has recently come a major step closer to solution through the socalled Incompleteness Theorem of G¨ odel and the verification of the consistency of the theory of the whole numbers by G. Gentzen.— Thus the work at hand establishes in the most concise form not only the great value of a suitably chosen notation, but it also shows that the great researchers of our science were always conscious of the combination of notio and notatio, but also of the priority of the thinking mind over the applied force of the formula.52

But again the completely senseless confrontation between G¨odel and Gentzen is played up. 6. 1942: Discharge from Military Service On 21 January 1942 Gentzen suffered a “state of nervous exhaustion”53 and was placed in sick bay the following day. From Brunswick Gentzen wrote to Hellmuth Kneser on 23 April 1942: Very respected Herr Professor! My days in sick bay are soon numbered; then it is off to M¨ unster to the demobilisation place, where anyhow I can probably count on a stay of another couple of weeks until all the formalities are taken care of. From there I hope to go to G¨ ottingen, and then I thought of seeking out a sanatorium. In G¨ ottingen I will once again visit Prof. Ewald, who will at the least have to confirm my need of rest to the University and perhaps can also give me some advice in regards to the choice of a village; today it is not so easy to find good accommodations anywhere. The financial side makes no difficulties for me; I assume that the University would grant me leave with pay, and moreover I have saved some in the war and have come into a not entirely insignificant inheritance.— My state of health is good, so long as I can live peacefully, as is predominantly the case here. The 51 Ibid.,

p. 20.

52 Zentralblatt

f¨ ur Mathematik 28 (1943). Gentzen in the curriculum vita of 25 August 1943, Bundesarchiv R 31/377 folio 1—, the registry of the German scientific high schools in Prague, habilitation files of Docent Dr. Gerhard Gentzen; partially available under Sen. Prot. Nr. 876/1943 in the archive of the Charles University of Prague. 53 Gerhard

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permanent state of affairs from intellectual exhaustion and daze changes itself barely noticeably for weeks. With most respectful greetings Your G. Gentzen54

He lay in hospital until 5 June 1942. On 8 June he was discharged from military service as a veteran (intact) without reserved status.

54 The

original is in the archive of Prof. Dr. Martin Kneser.

https://doi.org/10.1090/hmath/033/04

CHAPTER 4

The Fight over “German Logic” from 1940 to 1945: A Battle between Amateurs A Short Preface The fight over a “German logic” took place within the bounds of the “German mathematics” founded by the outstanding mathematician Ludwig Bieberbach.1 The mathematicians Max Steck and Hugo Dingler, joined by the pedagogue Friedrich Requard and the “malaria expert” Eduard May, polemicised against the university mathematical logic or “logistic”, most of all against Hilbert and Heinrich Scholz. None of the four were mathematical logicians and as such were not respected by their mathematics colleagues. To the university mathematical logicians, they were amateurs and the contemporary reviews of the books by Steck and Dingler were equally negative.2 All four amateur polemicists represented the point of view of the “folkish mathematics”. And they used quotes from Gentzen’s works as “witnesses” against mathematical foundations. Ludwig Bieberbach, on the other hand, helped Gentzen by financing and publishing his important lecture “The current situation in mathematical foundations” of 1937, and he gave Heinrich Scholz, whose publications in the Scholzian “M¨ unster School” he had also financed, the opportunity to respond to Steck’s unfounded allegations. By excluding “Dinglerism” from mathematical logic and from mathematics in general, Bieberbach guaranteed an international connection for mathematics under National Socialism.3 1 Cf.

the longer note at the end of this chapter. unsalaried lecturer, Dingler published an unconventional and incorrect sketch of a formal analysis; what that led to was that Dingler after that was practically excluded from the ranks of mathematicians.” (Kurt Wuchterl, Handbuch der analytischen Philosophie und Grundlagenforschung, Von Frege zu Wittgenstein, Verlag Paul Haupt, Bern, 2002; here, p. 283.) 3 The ideas of Max Steck and Hugo Dingler are still influential in Germany today. Hugo Dingler is the spiritual father of the “Erlangen School” in the theory of science and logic in Germany that spread out into many universities, among which Konstanz, Marburg, Paderborn, and Jena. The theories are expressly professed by Peter Janich, J¨ urgen Mittelstraß, Kuno Lorenz, and many more. Cf. the views of Kurt Wuchterl, Handbuch der analytischen Philosophie und Grundlagenforschung. Von Frege to Wittgenstein (Verlag Paul Haupt, Bern, 2002), on Dingler’s classification of Brouwer, Weyl, Becker, and Paul Lorenzen. Cf. the material-rich chapter “Der Streit um den Wissenschaftsbegriff w¨ ahrend des Nationalsozialismus” in Wilhelm Schemus, Verfahrensweisen historischer Forschung, Dissertation, Hamburg, 2005, for more on the intellectual terrorists Hugo Dingler, Eduard May (who with the anatomist Hirt met for “natural scientific work conferences”, at which Hirt reported on his numerous experiments on men as in weapons experiments), SS-Hauptsturmf¨ uhrer [literally: main storm leader], Tratz (the third in the alliance, whose “House of Nature” still stands today in Salzburg and is readily visited), the race theorist Bruno Th¨ uring (director of Vienna’s University Observatory from 1940 to 1945, who saw a parallel between relativistic physics and Talmudic thought), the incompetent Lothar Tirala (who saw a “logic appropriate to [typical of?] non-Aryan scholars” in Hans Reichenbach’s “Wahrscheinlichkeitslogik” [probability logic]), F. Requard (who taught physics for six years at the Tung-chi 2 “As

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1. Ludwig Bieberbach as Supporter of the Idea That the Validity of Mathematics Be Decided through World Views Ludwig Bieberbach writes at the end of his public lecture4 “Two hundred fifty years of Differential Calculus”:5 The foundational crisis also had its good side, for it opened our eyes to the assumptions6 that are a basis within the realm of world views also for grasping mathematical truth; opened our eyes to the last anchor of the validity of mathematical truth in philosophical strivings. Thus the issue, e.g. the conflict between formalism and intuitionism which we today regard as in general the core of the crisis of foundations, was whether the demonstration of the consistent conceivability of a mathematical object is a proof of its existence or not. Existence means here the textual justification. Intuitionism declares: the textual justification of formalistic mathematics through the demonstration of its consistency is based on a vicious circle, for this justification is based on the textual correctness of the statement that the correctness of a proposition follows from the consistency of that proposition, i.e., from the textual correctness of the law of the excluded middle. Thus philosophical concerns stand among the assumptions of mathematical science, they define their own basis for validity. In order to explain what’s said through a popular example: Nothing that I’m aware of contradicts the fact that a fly was squashed between the next two pages of this pamphlet. Anyone would rather check it out themselves, including the intuitionist. He wants to construct mathematical objects in order to believe in their existence. Or another example. If Mr. Hopfenstang explains, he wasn’t aware for a moment that he has a Jewish background, then we are far from being able to draw the conclusion that he is an Aryan as a result. Our laws treat him from that moment on as a non-Aryan, from that moment forward that a Jewish ancestor is found: mathematically speaking: the correctness of a claim does not follow from the proof of the absurdity of a contrary claim.

Ludwig Bieberbach is no longer an intuitionist, this much is clarified by the content and style of this essay, but rather he is firmly rooted in folk ideology. This is a folk ideology that was however missing from his mathematical works. Bieberbach remained a good teacher of everything from differential calculus to the theory of functions in Germany until after the War. However his political talk encouraged others to appeal to similar forms of thought. And this can have fatal consequences. First of all we determine that after L. Bieberbach the foundation of mathematical truth would be dependent upon philosophy, more precisely on a world view and its popular foundation. However we ask to what purpose is the empowerment of a world view, of what a dominance of popular foundations over mathematics is University in Shanghai and referred to Dingler), and many others in connection with the National Socialist conception of science. [A small note of explanation: “Wahrscheinlichkeitslogik” is Tirala’s name for Reichenbach’s book Wahrscheinlichkeitslehre (A.W. Sijthoffs Uitgeversmij M.V., Leiden, 1935), Chapter 10 of which was titled “Wahrscheinlichkeitslogik” and dealt with probability and multi-valued logic.] Cf. also http://de.wikipedia.org.wiki/Hugo Dingler. 4 Organised by the Berliner Verein zur F¨ orderung des math. Nat. Unterrichtes, 13 November 1934. 5 Zeitschrift f¨ ur die gesamte Naturwissenschaft 4/5 (1935), pp. 171-177; here: p. 177. 6 Voraussetzungen. This can be variously translated as prerequisites, preconditions, assumptions, etc. I have uniformly chosen the last in translating it. This may lose some nuances in the sequel, but it does seem to fit.—Trans.

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supposed to mean? Does it have any meaning? Will the foundational crisis be overcome by putting mathematics on safer “popular” foundations? 2. An Advocate of Racial Purity Sets the Standards in German Mathematical Logic, Where Attempts Are Made to Confuse Scientific Results with Matters of Race The gynecologist Tirala said: The point of view that even within the bounds of logic, where, as we’re to believe, it is a question of pure thought, the thinkers take the position of thinking according to their own teachings. This is of course a question of purely formal thinking. The high level conceptual training is characteristic of the northern race. . . The laws of logic depend upon the laws of linguistic constructions and they hold, as comparative linguistics has shown, only for certain races and languages. . . It is a characteristic of the northern race that it is not led by its thoughts alone and, even if it is strictly logical, never loses control over its thinking. The northern race does not let itself be tyrannised by pure thought alone. And finally it is merged into the consciousness of the northern-race thinker that logical thinking, despite all respect paid it, cannot do without creativity. This Nordic value can best be summarised in the statement: “All the thought in the world doesn’t benefit thinking.”7

Thus at the point where mathematical thought reaches its limits it must be assured through the race standpoint. Many philosophers like Steck thought that axiomatic systems within mathematical logic, which may be restricted also in the case of Gentzen through certain assumptions, are there to clarify the one philosophy of mathematics. Here it is a question of at least three assumptions: a logical assumption; the metaphysical assumptions, thus the question how it is at all possible to create mathematical objects; and finally assumptions of the goals and purposes of axiomatic systems, because these are not purposes in themselves, but rather are aimed at solving the problem. All three assumptions are to be explained in terms of the racial point of view. Lothar G. Tirala became the leader of the Munich Institute for Racial Purity of the Ludwig-Maximilian University from 1933 to 1936 and summarised the “Vienna Circle” in his lecture, “Nordic Race and Natural Science”, at the inauguration ceremony for the Lenard Institut on 13 and 14 December 1935 as follows: Also within the realm of logic as the foundation of all science, differences must be observed that will force the thinker and researcher, not just to think or research this or that way, to take a stand, but to work differently in a purely formal way. Logic rests on the construction of the Indo-Germanic languages and applies only within their range. The others follow it, because our logic adapts best to nature, and the successes of the Nordic life force all others to follow. Already in the Latin language there are important and substantial differences in relation to the Germanic languages. The preference for a passive construction is just as characteristic for the Latin language as the preference for the active tense is for the German. And those are two languages of two racially rather nearly related peoples! The so-called Vienna Circle, a combination of a, to a large extent, foreign people, to a large extent of near-Asian and eastern race, announces a new logic, which differs radically from the Aryan logic. This “Vienna Circle”, to which also Einstein is close, stated that for them there is no fixed logic; formalist computational 7 Lothar

Tirala, “Rasse und Weltanschauung”, NS-Monatshefte 5 (1934), p. 944.

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thinking is the primary, logic the secondary. One listens formally to the Near Asian, who calculates until reality disappears. Hugo Dingler showed that spokesmen of this Vienna Circle, Reichenbach, Russel (sic), Wittgenstein and Schlick, represent a different logic from that which the scientific world so far has meant, that that one is universally valid. This European, in reality, Aryan logic, says Reichenbach, one of the members of the Vienna Circle, must be replaced by a new logic, which, as one of the leaders of the Vienna Circle does, can be described as probabilistic logic. If we grant these men that they seriously represent their opinion, in fact driven by their innermost thoughts, then there cannot be more beautiful proof for my statement of the racial bond of logic. These men are trying if only after their own formalistic manner of thinking and willing, supported by, from opinion, utterly independent mathematical secret teachings, to disclose a new, type-specific logic of a non-Aryan scholar of the scientific world.— For us this odd attempt by Einstein and the Vienna Circle, to destroy the clarity of Nordic thinking by an excess of mathematics and to eliminate the simple basis of our logical thinking, is of the highest possible interest. . . 8 If it is objected that precisely the Jews are unusually capable in the area of mathematics and have developed special abilities for sharp logical thinking, then I must point out that among the great creative mathematicians there are no Jews, but rather that compared with a Gauss and Euler the Jewish mathematicians are in greatness at best of second or third rank. With the Germanic natural scientists however mathematics is only a subsidiary science. They are also conscious of the fact that mathematics is only to be used, if it applies, to represent relationships in nature that they have already understood. Gauss himself only used mathematical representation if he had completely understood the features of nature. He never used mathematics in order to advance into the depths of the experience. Observation and logic come first, then mathematics. Mathematics is also just an instrument of the human spirit. . . 9

And the “blond long-cranial” Tirala summarises: From these perspectives we reject the Jewish “science”, whose top group wants our logic to just be considered as a part of mathematics and to advance a new logic.10

However this racial purity point of view did not have, as far as I can see, any followers.11 3. Gentzen as a “Witness” for a National-Racial Interpretation of the Mathematical Foundational Research through Steck and Requard Gentzen was in the war and couldn’t do anything to prevent his being used by supporters of German mathematics as a witness against the Hilbert program. They used Gentzen in order to be able to proceed against Heinrich Scholz. Heinrich 8 Ibid.,

p. 29 Tirala, “Nordische Rasse und Naturwissenschaft. Gek¨ urzte Wiedergabe des Vortrags (der Herausgeber) Von Dr. phil. et med. Lothar G. Tirala o.¨ o Univ. Professor, M¨ unchen”, pp. 27-38, in: August Becker (ed.), Naturforschung im Aufbruch. Reden und Vortr¨ age zur Einweihungsfeier des Philipp Lenard-Instituts der Universit¨ at Heidelberg am 13. und 14. Dezember 1935, J.F. Lehmanns Verlag, Munich, 1936; here, p. 29. Cited from Christian Thiel, “Folgen der Emigration deutscher und ¨ osterreichischer Wissenschaftstheoretiker und Logiker zwischen 1933 und 1945”, Berichte zur Wissenschaftsgeschichte 7 (1984), pp. 227-256. 10 Ibid., p. 31. 11 Cf. the longer note at the end of this chapter. 9 Lothar

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Scholz had held the only chair of logic and mathematical foundational research in Germany since 1938. He, they felt, needed to be eliminated. The glowing agitator of “German mathematics”, Docent Dr. habil. Max Steck of the Technical University of Munich, attempted to drive the “logistic”, “mathematical foundational research” from the universities, in particular from the Universities of M¨ unster and G¨ ottingen. Max Steck was a student of the Heidelberg mathematician Heinrich Liebmann (1874-1939), who worked particularly in the areas of the differential geometry and non-Euclidean geometry. But Max Steck, just as May and Dingler, was no specialist in mathematical logic. He was in fact a stranger to the field. But that did not prevent him from portraying himself as one of the primary researchers in the area of the mathematical foundations. One must remain conscious. . . of the fact that scientific mathematics without a sound grassroots philosophical point of view is impossible and therefore all consideration of the mathematical spirit has to culminate absolutely in a consideration of meaning.12

Gentzen was considered as a “formalist”. It was mentioned eight times by Steck in his Das Hauptproblem der Mathematik [The main problem of mathematics]— Steck had the book printed especially for Hilbert’s eightieth birthday on 23 January 1942—and Steck quoted from Gentzen’s paper13 “The present situation of research in mathematical foundations” (1938): G. Gentzen characterised this aspect of the Hilbert point of view, which all formalists—and most mathematicians are formalists—more or less share, and he, as a typical representative of logistic, is surely not suspect for having taken the position against the formalist position: “One of the primary features of the Hilbert point of view seems to me to be the tendency to extract the mathematical foundational problem from philosophy and to examine it as much as it is at all possible with mathematics’ own tools. One can certainly not solve the problem completely without extra-mathematical assumptions.” The Hilbert plan limits these to a minimum. We can only ask: Where has Hilbert indicated these extra-mathematical assumptions, without which one cannot solve the problem, and where is this minimum treated systematically and axiomatically; where do these “extra-mathematical” assumptions enter into his construction? Nothing has come to our attention about this, apart from those items of Hilbert himself that we will use and introduce later in order to show that though he may really have the right intention, that this doesn’t come out at all, from which the narrow-mindedness of the formalist effort can be seen, a formalism that may give emotional credence to the second problem area B but nowhere tries to investigate it, to take it into account in its construction of mathematics and to let it enter into the “foundations” itself.14

The extra-mathematical assumptions are of special interest to the Nazis, because that would be their very own area. And again Gentzen is here quoted as a witness: G. Gentzen, who belongs directly in the logistic camp, says of the logicism the following: “We should briefly mention the so-called ‘logicism’, which is usually named together with intuitionism and the Hilbert view as the third substantial point of view for the foundation of mathematics. Its theses are justified in terms 12 Max Steck, “Mathematik”, pp. 10-21, in: Das Studium der Naturwissenschaft und der Mathematik. Einf¨ uhrungsband, bearbeitet von Fritz Kubach, Studienf¨ uhrer, Gruppe III: Naturwissenschaften und Mathematik, C. Winter, Heidelberg, 1943; here, p. 10. 13 The full texts of Gentzen’s lectures appear in Appendix C. 14 Ibid., p. 65.

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of certain philosophical opinions (read: logical empiricism), which I don’t want to go into(!). This school has taken a cautious or indecisive position relative to the antinomies and problems with infinity that are of primary importance for practical mathematics, and has contributed little to their resolution, since its interest has been primarily in other questions, e.g., the foundations of the concept of number.” Nevertheless logistics announces repeatedly its claims on the (mathematical and philosophical!) foundation of the exact sciences, while it was limited nevertheless by one of its own representatives in their field. The confession of H. Scholz is similar; he even avoids the actual difficulties when he writes: “On the metaphysical and epistemological assumptions of the Russellian logic—for of course the new logic is also burdened with such assumptions, only to a very much smaller extent than any earlier formal logic. . . : (Geschichte der Logik, Berlin, 1931, p. 74).15

Afterwards Gentzen becomes the witness of the alleged destruction of classical mathematics by antinomies16 and by unreasonable “consistency proofs”: G. Gentzen characterises the point of view of the formalism in this connection as follows: “In this way I come to the view of Hilbert. He sets up the programme to save the whole of classical mathematics as far as possible from its precarious situation by proving its consistency in a mathematically exact way. The execution of this programme is unfortunately to a large extent still pending. The difficulties of such consistency proofs have been shown to be greater than one thought them to be (G¨ odel’s Theorem).” G¨ odel was able to prove a series of theorems in the context of a purely formalistic theory of mathematics, theorems that Gentzen discusses in the following way and suggests their consequences: “Then there is the theorem about consistency proofs that says that the consistency of a mathematical theory, which contains pure number theory and is really consistent, cannot be proven using the methods of proof of this theory itself, in particular naturally not with a fraction of these methods. This theorem has been frequently presented in such a way as if thereby the Hilbert programme had been shown to be impracticable. . . Another of the G¨ odel theorems concerns the decision problem, and in particular for the ‘predicate calculus’. It states that certain theorems of this system cannot be decided even with certain very powerful mathematical methods. The theorem was. . . sharpened substantially. . . by Church. . . to state that there cannot be any general decision procedure for the predicate calculus. . . A third of the G¨ odel results concerns the completeness problem. It states that each formally defined consistent mathematical theory is incomplete in the sense that one can provide numbertheoretic statements that are correct but cannot be proven in this theory. . . The theorem exposes naturally a certain weakness of the axiomatic method.’ Gentzen then continues in his description of the Hilbert point of view as follows: “. . . the—for practice most important—[consistency] proof for analysis is of course still pending. “In order to carry out a consistency proof, one needs of course certain mathematical means of proof ” (thus “without assumptions” one can obviously not do anything!) “whose safety one must presuppose” (a glaring admission!) “and that cannot be justified further in this way. An absolute, i.e. conditionless, consistency proof is of course impossible. . . It seems however that one does need

15 Ibid., 16 Ibid.,

p. 127. p. 131.

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somewhat stronger means for consistency proofs than Hilbert had supposed and had understood under the concept of ‘finitary proof methods’.”17

Thus Gentzen was accused by Steck of this poorly thought through foundational problem. Gentzen’s political neutrality ran parallel to Gentzen’s world-descriptive neutrality in mathematical foundational research. In a discussion of any philosophical reasons for the conflict over mathematical foundations, Gentzen had no interest, but he read however the logicists Leibniz, Carnap and Couturat. He agreed with Hilbert’s point of view only with regard to certain questions and, for example, “for Hilbert ‘formalisation’ is much more an aid.” Gentzen wrote “before the borders of philosophy, which is not pleasant to me, stop, i.e. I concern myself only with problems which are accessible even again to a mathematical treatment.” What does this mean? First Gentzen avoids delving into the myths of the origin of mathematics, of the ontology of mathematical logic. Second however, and this is where it gets dangerous, he leaves this to the idiots Steck, Dingler, May, M¨ uller and Bieberbach. Even today certain representatives of the “Erlangen School” and their followers still regard Steck and Dingler as the greatest thinkers of the foundational movement. 4. The Somewhat Sharper Point of View of Friedrich Requard The Cologne professor Friedrich Requard was also of the opinion that one had to put the contingency of exact natural science in place of the fiction of assumptionlessness through the race-folk condition. He joined Hugo Dingler in opposition to E. Cassirer: The physical concept is no “substance concept” of traditional conceptual theory, which, as is common between a group of things, is only an extract and an image of reality. And even the characterisation as “functional concept” reflects only the purely formal logical side of the physical concept as it can be understood only through mathematical function. The meaning of our concepts is not created solely by their simple function in judgement; the contexts are not exclusively of a formal logical nature. Further the concept becomes really physical only through the heroic spiritual-soul felt conviction.18

Where do we find concepts as “effectual forms” of the heroic spiritual-soul self conviction of Nordic man in mathematics? Racial cause of mathematics as the will of mathematics is demonstrated in the lack of clarity in the foundations of mathematics. Russellian logicism, Brouwerian intuitionism and Hilbert’s formalism (struggle) in vain for a, from the beginning, strictly unambiguous foundation of mathematics.19 If the procedures that give mathematics its own rigour are to really be substantiated, then you cannot use these methods themselves to do it. For this reason the mathematician cannot avoid his becoming familiar with the thought that logical 17 Ibid.,

pp. 142ff. Requard, “Heldische Weltanschauung und sch¨ opferische Leistung in der exakten Naturwissenschaft. Eine Untersuchung u ¨ ber das Wesen der national politischen Ausrichtung des Unterrichts in der exakten Naturwissenschaft”, pp. 4-7, in: Der deutsche Erzieher, Klett, Stuttgart, 1940. 19 F. Requard, “Strenge Mathematik und Rasse”, Rasse. Monatsschrift der nordischen Bewegung IX (1942), pp. 11-22. 18 F.

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methods have given this preliminary area where these methods cannot yet be used because they have first to be shown correct there.20

The thinkers before Requard did not recognise what the real problem was and lamented the “old” mathematics: The scientific strictness consisted only in connecting the theorems of mathematics one after the other. With this a new meta-mathematics took the place of the old mathematics, an assumption-based theory of relations, a theory of logical schemata without any relation to reality. Truth was then nothing more than pure logical consistency.21 Also the “Calculation Rules” of the so-called logistics (purely formula-based view of the logical) are by their nature nothing other than procedural operating instructions.22

However these operating instructions, which lead to certainty, are symbols of the northern race’s science: To recognise mathematics as racially dependent means nothing more than understanding the production of strictly unambiguous relations as racial, i.e., the form, characteristic, expression of a race’s psychological conviction. Hugo Dingler’s great contribution was to have completely penetrated the inner nature and possibility of strictly unambiguous relations for the first time and in so doing finally to have found a completely satisfactory answer to the question of the “origin of axioms”.23

Gentzen’s rˆole seems clear: he is to be held as witness to the vast contradictions of the various schools with regard to investigations into the foundations of mathematics. That Gentzen does commit himself sufficiently does not add to his honour. Friedrich Requard begins his essay “The problems of strictly mathematical thinking in light of the theory of inherited traits”24 with a “notable result in the recently published, meritorious and astute investigation” by Gerhard Gentzen. “The current situation in the foundations of mathematics”—he mentions the product of “research” and not the source of “German mathematics”: This contrast [the “constructive” and the “in-itself point of view”—EMT] went so far that the followers of constructive mathematics, for example, declared all the theorems of the in-itself understanding of infinity within mathematics to be senseless, and its teachings to be an empty game with symbols without any meaning.”25

Gentzen’s attempt toward harmony is blamed: If Gentzen makes this apparently worthy attempt in his research to accommodate these two different points of view on how to retain classical analysis in its original form, then one should not ignore the fact that both conceptions are deeply anchored in the innermost essence of the researcher’s personality and in their final basic position do not allow for reconciliation.26 20 Ibid.,

p. 12. p. 12. 22 Ibid., p. 17. 23 Ibid., p. 1. 24 F. Requard, “Probleme streng-mathematischen Denkens im Licht der Erbcharakterkunde”, Zeitschrift f¨ ur angewandte Psychologie und Charakterkunde 59 (1940), pp. 351-370. 25 Ibid., p. 351. 26 Ibid., p. 352. 21 Ibid.,

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And so: Gentzen emphasises in his investigation again and again that without extra mathematical assumptions the important foundational problem cannot be solved at all, that derivation in the constructive sense has an essentially larger degree of certainty associated with it, that a constructive existence proof has more meaning than an indirect in-itself proof and that the constructive point of view deserves greater consideration because of the particular significance of its results for the whole of mathematics.

Requard continues: The fundamental response of the constructive point of view of the living bodysoul-organism in the world and life creates not only the really disciplined mathematics but also the truly exact natural science.27

As a result of a “folkish” reading of Gentzen, Weyl, and Dingler, Requard holds firm: The constructive and the in-itself points of view are two fundamentally different kinds of thought that rule far beyond the working of mathematics over the whole scientific activity of men and are rooted deeply in the character and so in the inherited and immutable essence of the individual.28

This is then a lesson for men “in the middle”. One sees here clearly that Gentzen is being misused by Dingler-follower Requard and that the purpose of mentioning Gentzen resembles that of Steck’s mention of Weyl.29 Steck himself points out that Gentzen has to know as pars pro toto about these questions, but he doesn’t consider them. For Steck there lies in this fact a form of self-destruction of mathematics. Steck dons the counter-programme of an archaic origin-genealogy of German mathematics. Gentzen however ignored this whole form of romantic, nationalistically dominated ideology of mathematics. This is on the one hand a relief, but the—from a professional perspective obvious—separation of mathematics and philosophical foundational research allows for the Nazi ideologues to fight even on this question. He could not win a power struggle. But the retreat to his discipline left the border between the two with a depleted rule of rationality, and this terrain did not remain untouched like a wilderness that one later could reclaim but became instead a parade ground for the political ideas of mathematics. But because Gentzen withdrew, he deflected a defence of Hilbert’s programme and his programme from a historical, genealogically dominated discussion of ideology to the logical-systematic realm of an examination and securing of true mathematical theorems. Gentzen did not balance out the tension between philosophy and ideology 27 Ibid.,

p. 353. p. 270. 29 The relationship of science, lifeworld [lifeworld is a philosophical term coined by Edmund Husserl. It is supposed to mean something like “life as lived” prior to our theorising about it, i.e. the perceptual, unintellectualised world] and their assumptions or their lack of assumptions can also be answered in a completely different manner. One reads in this regard Max Planck, “Die Frage an die exakte Wissenschaft”, Deutsche Luftwacht / Ausgabe Luftwelt 10 (1943), pp. 362ff. Herein the starting point, “the firmness of which no sceptic can dare to approach,” will be found in that which we with our physicality through our sense organs can receive of the outer world. And so goes the way from the received sense world via experiences to representations, out of which we make a practical, useful world image that will change and continually correct itself, and thus is not closed. The road from everyday life to science thus also goes without the use of the race concept. 28 Ibid.,

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on the one side and mathematical treatability on the other in foundational research. He decided in favour of one side. In doing so he bisected the discussion and helped to lead the discussion into the sphere of “subjectlessness”. The real disruptions of the subject, for which Steck and Dingler used concepts like will, nation, struggle, history among others, had no correspondence in mathematical logic. To what degree he had headaches during this tear up as representative for the dissolution of subject and nation, self and species, I cannot judge. It was not true that the political discussions were mirrored in the mathematical battles over intuitionism, logicism and formalism. Gentzen clarified the inner mathematical discussions in his lectures, made them explicit, explainable and understandable for others. And in these discussions Gentzen put value in the mathematical parts and in this way defended himself against every historical fascination with origins that each of these schools at one time or another maintained, since each considered itself at some point in time to be the origin, to be the sufficient foundation of mathematics. Steck, however, did not follow the line of “professional mathematics” that typically in meager times is a defence of pretenses of conviction that are brought to bear from outside—and wanted as a result that Hilbert would err, with his programme and his students, including Gentzen, on the side of irrelevance and relativism and simply be un-German. The problems that are decisive for the German line are trivial for formalism.30

And Steck quotes from Hilbert’s letter to Frege: Each and every theory can always be transformed into infinitely many systems of Urelements,

and draws the conclusion, But this signifies, as E. May has shown convincingly, the relativism of truth.

5. Ludwig Bieberbach and his Deutsche Mathematik On 14 October 1934 Ludwig Bieberbach applied for financial assistance, a “credit” of RM 2500.-, for the first three volumes of a “German Journal of Mathematics”31 : It is to be a collection point for German mathematical work at the universities— consequently it is to give a summary of the progress of German mathematical research and, at the same time, contribute actively at the very forefront creatively. The Journal will not attempt to compete with other already existing mathematical journals by following them work by work, but rather it will strive through scientific exposition to give a complete and true organic view of progress in German mathematical research and of the lives of researchers. In its student part it will provide information about the professional work on collaborations and the scientific teams of mathematics departments, and so be responsible for a mutual exchange of current results within the individual universities. At the same time scientific works by students should be taken up in this part, chosen primarily in accordance with the guidelines of the office of the national leader of the Deutschen Studentenschaft [German Student Organisation] on the scientific work of the Deutschen Studentenschaft. On this basis all available efforts of students and lecturers should be committed to joint collaboration. The Journal will in this way pave the way for the creation of a German mathematical research society. It already supports itself, with reference to the organisation of 30 Ibid.,

p. 159. the sequel this will be cited from the DFG files, “Deutsche Mathematik” (Bundesarchiv, Koblenz, R 73/15934). 31 In

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the student mathematical departments, which exist at all universities and technical high schools, on a network spread over the entire Reich. With its plan the Journal has no other already existing competition. Moreover, it fulfills an already long felt need, in that it moves men and science, folkish affiliation and scientific achievement in common effort of lecturers and students closer to one another. . . Currently it is unfortunately not possible to enlist the organisation of the German Mathematicians -Union for the above purpose, as it has been clearly proven in its Pyrmont conference that it does not wish at present to be the bearer of national interests.

The thought is to print 1000 issues.32 The DFG [German Research Council] decided that only 2000 copies would be printed as a promotion, i.e., all together 3000 copies. In 1937, when the the first issue of Deutsche Mathematik had a printing of 1000 copies, the DFG noted in alarm: instead of 40 pages they had printed 44 pages, and there had been a disproportionate amount of royalties paid out. Bieberbach defended himself on 17 February 1937. To limit the yearly scope (of the journal) would hinder achieving the task of building community, since then good contributions would have to be passed on to other journals, by whom the prevailing conditions are more or less as follows: One of them (Mathematische Annalen) is edited by a Jew. And another one (Mathematische Zeitschrift) publishes among other things works that are dedicated to the Jewish communists. In a third (Crelle’s Journal) they print works by immigrants. A fourth (Quellen und Studien) is jointly controlled by a Jew and a half breed emigrant. The rejection of a per se good piece of work by us would force the German in question to turn to one of the journals briefly described here. Also a rejection or waitlisting of in principle good works would keep us from achieving our community building task. . . For this reason we prefer issuing smaller honoraria, since the scientific worker is in any case worth his pay and we’ve discovered that these small honoraria contribute considerably, advance the cause of building community. We also believe it proper that the three gentlemen Tornier, Weber, Weiss, who are a good professional complement to us, and who support us in editing and publication by proofreading and evaluation, receive an appropriate honorarium for their efforts.

On 2 March 1937 the president Rudolf Mentzel of the DFG and Ludwig Bieberbach signed an agreement. Under 1) it states: “The editorial staff will forgo any honorarium. The journal will be run in accordance with its purpose entirely through the political will and sacrifice of those involved.” Under section 4) it is confirmed that Deutsche Mathematik will hold itself to 40 pages. The following passage is important: 32 The

DFG obtained estimates by the Heine Press and the publishing houses Vieweg (contemplated 600 copies, of which 500 would be subscribers), Teubner (should estimate for the DFG an edition of 1000 copies), Hirzel (contemplated 700 to 800 subscriptions). The lot went to the trade publisher S. Hirzel in Leipzig; printing was by Heine in Gr¨ afenheinichen. Between the DFG and the editorial staff of the journal and between the DFG and Hirzel a publishing contract would be closed on 25 November 1935 (Bundesarchiv R 73/15935). Bieberbach received an editor’s honorarium from the DFG of RM 2000 yearly, “payable in 4 quarterly installments in advance.” The DFG advertised the journal at the high schools, colleges, and universities via the Reich’s and Prussian Ministries for Science, Education, and National Education. The German Student Society also put up posters for this journal. The Heine Press confirmed on 20 December 1935 that in addition to the normal edition of 1000 copies a further 3500 copies were to be printed as promotional issues. The reader should pay only 12 marks for 240 pages of mathematical composition, thus disproportionately little in comparison with other scientific journals.

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The journal will also continue within its framework to steward logic, but in such a way that the works in logic will appear without exception as special issues and not also as a part of the journal. The norm for such works will in the future consist of 2 12 sheets33 per year, whereby the current publication of the Roth work existing in press will run as an exception. With regard to works in logic there will be conceded an additional 5% relative to the scope of the full year’s issues.

There followed, despite protests by L. Bieberbach, a “drastic reduction” of the budget of RM 25,000 to “twelve thousand Reichsmark”. This amount was debited the physics department’s budget.34 Deutsche Mathematik had 294 subscribers in the year 1938.35 The number of issues ordered to be printed of Deutsche Mathematik was reduced by half by the Hirzel Verlag at the request of the DFG.36 In 1942 printing of Deutsche Mathematik sunk to 500 copies, where the printings remained until 1943. And one more thing was important: many mathematicians had already been driven away or killed—the Nazis wanted to close the ranks of the other university scientists behind it. Every other discriminating and as a result decimating criterion would have produced further losers and outsiders, and one could no longer afford that in the current personnel situation. And that was the point that first of all Hitler’s Mein Kampf was not to be drawn on, neither should it be interpreted by others even as a source of official Nationalist Socialist opinion. And so the opinions of Steck, May and Dingler remained always only “private opinions”. That is also why there never occurred a break up within mathematics in the National Socialist sense. And so the finding of functionaries and critics, from A. Rosenberg to M. Steck: “Everything remained as it was before!” 6. NS Ideology in Mathematics through Bieberbach Receives Negative Resonance Even within His Own Camp “Ludwig Georg Elias Moses Bieberbach came from a pietistic family. Their given names were usually taken from the Old Testament. Bieberbach was a ‘good republican’ as a young lecturer who wasn’t at all ‘leftist’ or ‘progressive’.”37 What was the scandal with Bieberbach? That he played around with the terminology of Jænsch38 and drew analogous conclusions about mathematics? No, but he did it at a time when that sort of game, if it were connected with names, would be life threatening and could result in expulsion, deportation, imprisonment and murder. In a bulletin from the April 1933 issue of the Prussian Academy of Sciences regarding a Weierstrass-Edition, Bieberbach notes according to Issai Schur: I am amazed that Jews still belong to the academic commission.

Knowing full well that his Minister Vahlen would read it, he continues: I request a change. sheet = 16 pages. Thus 2 12 sheets would be 40 pages or enough material to fill one issue. to the honorable council representative Prof. Dr. Esau on 3 May 1938 (Payment record Deutsche Mathematik, Bundesarchiv Koblenz R73/15936). 35 Letter from Hirzel to the DFG on July 8, 1938. 36 Cf. the longer note at the end of the chapter. 37 Cited from Rudolf Fritsch and Gerda Fritsch, “Arthur Schoenflies (1853-1928)” (p. 151), in: Menso Folkerts, Stefan Kirschner, Theodor Schmidt-Kaler (eds.), Florilegium Astronomicum. Festschrift f¨ ur Felix Schmeidler, Institut f¨ ur Geschichte der Naturwissenschaften, Munich, 2001. 38 Jænsch and his theory of personality types are mentioned further on pages 156ff. 33 1

34 Fischer

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I noticed this anticipatory obedience from Bieberbach “towards authority” in his letter to H. Grunsky.39 There he writes among other things: . . . I emphasise again that you must assemble your advisory staff according to the guidelines that have been required for every German since 30 January 1933. You are running the risk that your actions are represented as a lack of political instinct. What am I supposed to say in the case of the recent attack on the Academy that I received, where one accuses it (the Academy) of still employing Jewish advisors for the annual abstracts? I can only say about this that for years I have taken pains to get you to reduce the number of Jews, but my efforts have still had no success, because time after time you find reasons to enlist one Jew or another. You see how you damage the well-understood reputation of the Academy through your restraint.

But of course Bieberbach was the only one hurt and misrepresented. In particular he knows nothing of reports assigned to Jews: I cannot corroborate the Jew in question is the only one with this subject matter expertise. In most cases I would have myself been able to write the report without problem.

Bieberbach felt insulted. If necessary he would—like Werner Weber in G¨ ottingen— probably take all of German mathematics on his shoulders and master it. Bieberbach was overanxious in his task to make mathematicians in bodies, organisations and administrative offices “Jew-free”. But his letter betrayed his fear, pride and desire to please his superiors through denunciation, persecution and inhumanity towards those below him. And his behaviour was acknowledged partially by the party and the Ministry of Education. This is what made his theories so dangerous. That they were taken note of by the Nazis like Dingler’s writings of denunciation. 7. Ludwig Bieberbach: Representative of “German Mathematics” What is the story with the chief representative of “German Mathematics”, Ludwig Bieberbach? Alexander Dinghas tells us: After 1935 Bieberbach took over primary or, better said, the secondary responsibility as the leader of the seminar. The primary position was held no doubt by the department chairman for natural science, Karl-Heinz Boseck, a small version of Robespierre and later “Sonderf¨ uhrer” in the SS. . . So I saw Bieberbach very seldom until 1935, but heard a great deal about him. Especially after 1933 the “legend” about him spread and his transition into the sect of the big guys was talked about everywhere. Bieberbach, the friend of Landau’s (I have myself read a letter from Bieberbach to Landau from 1915 from Basel), Ostrowski, Schur and many others. . . Traitors!. . . The rapid flight of Saul from the persecutors of Christians on the contrary frequently came to mind. That was also the reason I rejected the opinion, expressed to me and my friend D. Schoch on a trip to Babelsberg, of the mathematics historian Hofmann, that Bieberbach’s political opinion depended upon the well-being of his family. Later Hofmann was more of my opinion;. . . For I’m sure: If Boseck had been a member of the convent in the French revolution, then he would have been no less than Robespierre or Saint Just, since he possessed all the properties through which a fanatic reaches great power: narrow views, lust for power, desire to dominate other human beings and the blind belief in ideas. . . My position was weak. I could only stand 39 Printed on p. 44 in: Jochen Br¨ uning, Dirk Ferus, Reinhard Siegmund-Schultze, Terror and Exile. Persecution and Expulsion of Mathematicians from Berlin between 1933 and 1945, Springer-Verlag, Berlin, 2000.

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my ground with the support of Boseck, since one word from Boseck would have been sufficient to remove me from my position. That this impression was correct became apparent later in 1944 when it only required a few negative comments to Bieberbach to separate me from him. More on this point with evidence later [In the incomplete manuscript no more was said about this—EMT]. Boseck supported me without reservation until 1943. In 1939 he called the Ministry and demanded that my assignment to teaching be hurried along. I only relate this in order to show what kind of power Boseck had during these years. Later his power increased and Bieberbach sank ever lower. It is sad that Bieberbach never tried to free himself of Boseck. I believed for a long time that he wanted to and, as we often must, just grinned and bore it, until I finally convinced myself in 1944 of the opposite: Bieberbach respected Boseck. But I had to put aside this incident that almost cost me my position, right to the end.40

But nothing more was said even about this “incident”. Hofmann, who later worked with Oskar Becker and published a history of mathematics with him, had shown ¨ in his article “Uber Ziele und Wege mathematikgeschichtlicher Forschung” [On the goals and ways of research on the history of mathematics]41 his skill relative to the Nazi talk of peoples and races and should have had sufficient sensibility to have evaluated the situation correctly. Where Perron showed no hesitation, Bieberbach appeared the fearful and respectful servant of another master. And to me it seems that the statement of his contemporary Dinghas corrected what had been a one-dimensional picture of Ludwig Bieberbach that had been passed on to us historians from the beginning. At last breaks and prejudice that are not apparent in the literature become visible. There he was portrayed as a bad guy, even with Vahlen and Hamel, and then come the less “evil” about whom one remains silent because they still had careers in the Federal Republic and East Germany. I think that precisely people like Dingler, May and Steck were much worse. Accordingly I regard Bieberbach as more powerful naturally than Steck, May and Dingler in the area of mathematics in general, within the area of mathematical logic, and that is my only topic of course. Herbert Mehrtens himself described the importance of Bieberbach and his “German mathematics” realistically: Bieberbach attempted, so to speak, a scientific-counter-revolution with the help of the new posers. His attempt to dominate mathematics in Germany failed, however. The Mathematicians’ Association managed to exclude him in early 1935. He started a journal Deutsche Mathematik to create his own, competing organisation, but did not find the support he needed from the Nazi Minister of Education. By 1937 Bieberbach and his group were an ideological residue in the system of mathematics without substantial influence.42

It is well remarked that Bieberbach and his so-called group remained unsupported by the appropriate ministry. But was this really true? And, if so, why? Financially he received support at least from the DFG (German Research Council). Why did neither party nor minister nor cultural bureaucracy award him any 40 Alexander

Dinghas, “Erinnerungen aus den letzten Jahren des Mathematischen Instituts der Universit¨ at Berlin”, in: Heinrich Begehr (ed.), Mathematik in Berlin. Geschichte und Dokumentation. Zweiter Halbband, Shaker Verlag, Aachen, 1998. 41 Deutsche Mathematik 5 (1940), pp. 150-157; here, p. 153. 42 Herbert Mehrtens, “Irresponsible purity: the political and moral structure of mathematical sciences in the National Socialist state”, in: Monika Renneberg and Mark Walker (eds.), Science, Technology, and National Socialism, Cambridge University Press, Cambridge, 1994; here, p. 329.

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support? And if he was already isolated in 1937, why did they make such a big deal about him and his political opinion? I would like to pose the question whether a paradigm shift was begun with Bieberbach’s articles and lectures. But it is clear that the paradigm shift of mathematics to a “German mathematics” sought from the side of the Nazis did not succeed. And that, although murder of Jews, expulsion of undesirables and the enforced conformity of professional groups are beyond a doubt. But mathematics was not pursued or taught in the spirit of the Nazis.43 In the winter semester 1933/34 he was part of a working group on great German mathematicians, the results of which were summarised in a lecture with the title “Personality Structure and Mathematical Creation”. Bieberbach published this lecture as the editor of DMV in 1933 without the knowledge of the two other co-editors and in it led a polemic against the Danish mathematician Harald Bohr (1887-1951), out of which there arose a controversy and Bieberbach lost his appointment as managing editor of this journal.44

His articles—essentially reworks of his lectures, “Styles of mathematical creation” (1934) and “The nationalistic roots of science” (1940), do not appear in the journal Deutsche Mathematik. Why not? Here a little “publication history”: On the 36th General Assembly of Mathematicians and Scientists, 2-7 April 1934, in Berlin the Geistige Arbeit reports:45 In the series of mathematical lectures Professor Dr. Bieberbach spoke about personality structures and mathematical creativity. To this end he points out that the G¨ ottingen student league had rejected the activity of the number theoretician E. Landau because his presentations were strange, and justified this with the fact that the mathematics itself is international in its results but not in the manner of its thinking and representation.

After that there come reviews of the lectures by Hamel and Dubislav. Bieberbach’s lecture appears in the Unterrichtsbl¨ atter f¨ ur Mathematik und Naturwissenschaften.46 On 8 April 1934 there appeared a summary of Bieberbach’s lecture in the journal Deutsche Zukunft. On 1 May 1934 Harald Bohr heavily criticised the reputed views of Bieberbach in the Danish newspaper Berlingske Aften. Bieberbach responded with “Die Kunst des Zitierens. Ein offener Brief an Harald Bohr in Kobenhavn” [The art of citation. An open letter to Harald Bohr in Copenhagen].47 For the first time a political polemic appeared in a professional journal. And the indignation of professional mathematicians was great. This had a consequence: In 1936 a new journal, called Deutsche Mathematik, had to be launched in order that there be at least one Jew-free mathematical journal.48

And Bieberbach would stick to his position that no political discussion would be allowed in “his” mathematical journal. 43 I in no way wish to rewrite the “Bieberbach” publication history, but rather only to emphasise a few points differently. 44 Hans Hofer, p. 45. 45 5 May 1934, p. 6. 46 Vol. 40 (1934), pp. 236-243. 47 Jahresbericht der Deutschen Mathematiker-Vereinigung 34 (1934), p. 2, divisions 1-3. 48 H.J. Fischer, “V¨ olkische Bedingtheit von Mathematik und Naturwissenschaften”, Zeitschrift f¨ ur die gesamte Naturwissenschaft 3 (1937/38), pp. 422-426; here, p. 424.

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There must have been turbulent objections from the readers because from that point onward Eva Manger tried to explain the operational situation. Again “New Mathematics:”49 With the recognition of the national dependence also of the mathematical researcher in his creative activity, as revealed in style and choice of problems, Bieberbach claims in no way that mathematics bases its validity as science on its roots in race and dependence.

She tried to inform the readers that Bieberbach was not against the lectures of Jews or French writers, that it was only a question of difference between problem, content and form, hence style or presentation. On 20 June 1934 a summary of this discourse appeared in Forschungen und Fortschritte.50 Dripping with irony, the J¨ udische Rundschau 51 reported on Bieberbach’s lecture, bearing the subtitle “The effect of blood and race on assured results of mathematics”. Richard von Mises (1883-1953), who resigned his professorship at the University of Berlin in 1933 and left Germany for Istanbul (Turkey), satirically attacked Bieberbach’s doctrines by resorting to statistical parody. He published in 1934 in Matematichesky Sbornik (41), pp. 359–374, “Probl´eme de deux races”. On March 9, 1938, S¨ uss wrote a letter to Helmut Hasse, Conrad M¨ uller, and Emanuel Sperner, three of his colleagues in the DMV (Deutsche Mathematiker-Vereinigung), noting that “My attention has been drawn to v. Mises, who allegedly published a statistical proof in Moscow, according to which Jews are intellectually superior to Nordic people.” (For an analysis of the satire and its political context and its serious mathematical content, cf. Reinhard Siegmund Schultze, Sandy Zabell, Richard von Mises and the “Problem of two races”: A statistical satire in 1934, Historia Mathematica 34 (2007), pp. 206-220). On 18 August 1934 G. H. Hardy’s famous letter to the editor “The S-type and the J-type among the mathematicians” appeared in the journal Nature (No. 134). In an article, undated and without place of publication present but apparently of that time, Professor Dr. Oskar Perron wrote under the title “Falsification of science” in opposition to Lenard, M¨ uller, Th¨ uring, Dingler, and above all against Bieberbach, Vahlen and Tornier. The fearless, clear, often dispatching tone is clearly there. The American Journal of Psychology 52 published as well, on pages 1-22, the conception of Bieberbach in an article by the theory’s originator: Erich Rudolf Jænsch, “Wege und Ziele der Psychologie in Deutschland” [Methods and goals of psychology in Germany]. On 22 November 1934 H. Kneser wrote to Bieberbach: Dear Mr. Bieberbach!. . . You wrote once that you were interested in coming to know my objections to your “Personality structure etc.” I never elaborated in written form to you—it would have required a letter on the order of a small dissertation or multiple exchanges—but rather was postponed and became a message in response to a verbal statement. I could only have been able to express my opinion publicly if I were able to attribute something more positive to what I doubt, or better, attack in your case. Now a good deal of what I had to say 49 Deutsche

Zukunft, Sunday, 13 May 1934, p. 15. 10, No. 18, pp. 235-237. 51 Berlin, No. 55, p. 4, 10 July 1934. 52 Vol. 1, Nos. 1-4, November 1937. 50 Vol.

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has already been said by Bavink in the reported part of “Unserer Welt” [Of Our World] (November issue). Perhaps I would have added something about Jacobi and the rights and merit of such challenges as your lecture gives them. I only regret that the merit, in my opinion, had not been given so many openings for attack from the beginning. By the way I will be working in a very detailed way with your work together with my students this semester and am exhausted at the moment by the burden of the Jænsch teachings as I’m sure you must have been in your time. Heil Hitler! Yours, H. Kneser.

Bieberbach replied on 25 November: Dear Mr. Kneser. Thanks for your reference to the report by Mr. Bavink that I had missed. After the second paragraph of his presentation he, Bavink, is of the opinion that my claim was to the effect that there were only two types when I have only said that Jænsch had only worked out two major types. With this I leave open the possibility that there are more types, not only those that appear for Jænsch as subtypes of J and S, but further types have also been worked out. In this regard the claim by Mr. Bavink that appeared on page 344, that I had tried to break the world’s mathematicians into two types, also omitted this. On the contrary I only considered a few examples of great mathematicians in order to point out in their cases the presence of different styles. When Mr. B. claims on the top on the right of page 344 that in the case of this concept of types of mathematicians that it is a question of the concept of statistical averages and that I had made the mistake of wanting to turn the individual case into a rule, to this I would remark that I had searched out these great individuals in order to avoid this mistake. For there will never be sufficiently many outstanding mathematicians at our disposal that one would be able to arrive at a statistical average. I chose the greater ones because in their case certain characteristics are particularly evident, and since with the race and type concepts there are also certain characteristics that are strongly exposed and because, further, these great men have clear ancestry, in particular geographical areas, and that one can arguably assume that their character is connected with their race and stock. This is most certainly a hypothesis and something to determine with scientific measure, where strong character is present. I think that no one will dispute the fact that there is a strong well-defined character in the examples I’ve chosen. For this reason I cannot use less defined personalities because then I wouldn’t know whether their ability is as great as they purport themselves to be. It could be the case, and will be, that the character of weak personalities remains hidden beneath the influence of teachers and upbringing, especially with respect to intellectual performance. In my opinion one can only pass judgement on the character of Gauss if one has worked intensively with Gauss. I believe that he came to his opinions demonstratively and deductively and after that worked them out logically. This indicates a sense that is certainly critical but constructive just as one finds a Nordic component in all system builders. This was the case with Lagrange. When it comes to complex numbers, I don’t think that Gauss simply wanted to address his opponents with his reference to the demonstrative meaning, for in his paper on bi-quadratic residues it is very clear how the demonstrative explanation is used to gain insight, and Gauss’s first proof of the Fundamental Theorem of Algebra survived as a result of its demonstrative character. With regard to Weierstrass I would like to point out, by way of support to my thesis, the deep opposition between Weierstrass and Cantor. It is the persistentconstructive criticism as opposed to the persistent-degrading one of Cantor. I haven’t at all made a detailed statement about the character of Cauchy but rather only mentioned him in one place as characteristic of a certain sort. In

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any case it is one of Bavink’s errors to claim that Cauchy was a southerner. His family appears to come from northern France. Concerning Hilbert I refer the reader to a well-known place in the preface to Hilbert’s lectures on deductive geometry. I have only mentioned Poincar´e in one or two places without going into his type in any detail. The image that Bavink relates to is however that of the president’s cousin and is also described by [Karl Friedrich] G¨ unther as east Nordic and not Nordic as Bavink claims. I also have the impression of a strong eastern component from the writings of Poincar´e which can also be seen clearly in pictures of Henri Poincar´e. By the way, given the opportunity, an in depth, long and complete psychological study of Poincar´e would have a lot to say. But up to now I haven’t gone into any detail with this example. In addition I cannot ignore what Weierstrass himself has said in his academic acceptance speech, even when it doesn’t fit the thesis of Mr. Bavink. If Bavink thinks I want to sow conflict between the different sorts of mathematical activity, then he his mistaken. I am simply of the opinion that we should take care of our own because then we will perform better. These few examples of mine make it clear that there are also differences between Germans. That there should only be one single mathematics sounds strange to those who know that intuitionism rejects large parts of so-called classical mathematics. I don’t find much in B. that I have ever considered.

I will also leave open whether the critics and interpreters of Ludwig Bieberbach have understood him correctly. He had himself represented the view that he has not been correctly represented. One cannot reach individuality from traits; Frege himself had made this remark in his diary. But this much can be said: Hilbert embodied for him the best German character. At the same time it is just as clear that he had made a distinction between German and un-German, thus Jewish and foreign mathematics, and added that the delusional ideas of a militant science in the sense of E. R. Jænsch also became officially accepted and disputable in mathematics. This certainly damaged his image, his views even doubted by his politically rightist colleagues. Which mathematicians took a positive stance relative to his views and even defended him? I don’t know of any that did so openly. His lectures and essays sent out shock waves, even internationally, and this damaged the reputation of German mathematics. Since then Ludwig Bieberbach bore the stigma of those who had tried to create a delusional idea of an NS-“Race Research” within mathematics. He learned however that there was no point in having his ideological views appear in mathematical journals. Mind and mathematical culture get bogged down in the sands of the desert of tyranny. The new is afraid to burgeon, everything living persists in peace and withers, for to take for oneself the right to freedom means death.53 With Bieberbach the phrase has almost gone the way to murder. This cannot be seen with Goethe’s words, “You may see knowledgeable people err/especially in things they do not understand.”54 Rather this dilettantism, to want to view mathematics as “race conflict”, ends for him who wants to see it this way predictably in demarcation, selection and murder.55 53 Cf.

the longer note at the end of this chapter. from J.J. Sylvester, The Study That Knows Nothing of Observation, 1869. 55 And also insofar as this is something different from the “Jewish examples” of Lewis Carroll used in Das Spiel der Logik [The game of logic] (Tropen Verlag, K¨ oln, 1988, pp. 57ff., 82, 87, 90, 97, etc.), which also today seem to disturb nobody, not even the editor, Paul Good. 54 Cited

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8. A Contemporary of Bieberbach’s in Exile: Johann L. Schmidt I ask myself curiously, what did Bieberbach’s contemporaries in exile think? Dr. Johann L. Schmidt held a lecture under the title “The National Socialist Science and the Rˆ ole of free German Research”56 at the Free German University in Paris. He concludes: There is no part of the life of the mind that will not be used by National Socialism, in its phraseology, its entirety, and in its individual elements, to found, to justify, to sanction, to glorify, to derive historically, etc., etc.”57

And further: National Socialism needs science for the foundation, justification of its phraseology to the enhancement of its effect. It requires biological arguments for its race phraseology, economic, sociological, legal, moral and other arguments for its social phraseology and its leader phraseology, philosophical, historical, economic, biological and other arguments for its national phraseology, etc., etc. . . . None of these scientific theories of National Socialism contains creatively new elements, since in general the whole intellectual production of National Socialism cannot create anything new in any area as a result of the character of the economic and political assumptions of its rule.

Everything is borrowed from the 19th century. And National Socialism also places science in the service of the immediate preparation of the war. . . For this reason the National Socialist state must destroy all conditions and possibilities of a free scientific research, and create its own science with its own theory of science and with an organisation of scientific work specific to it. The complete destruction of academic freedom, the complete centralisation of all scientific institutes, academies and other research centres, the removal of all Jewish and politically inappropriate scientists, as well as the appointment of new professors solely according to political views; the strict censorship of all scientific publications—all of these measures provide the material and organisational framework of scientific work in National Socialist Germany.58

Rust and Rosenberg were against value freedom and unconditionality in science; Prof. Helmut Hasse was against any objectivity in science. The foundations of science would have the root validity of the soldier’s spirit, which is grounded in the Germanic theory of values. One Prof. Schulze-S¨olde wrote: Such peoples’ predilection carries its certainty in itself, and not in the logical truth criteria, because they don’t reach the metalogical levels such as symbol, mythos, fate, heroism or destiny. These are clear limits of scientific productivity; there is at the same time however the limitlessness of the highest veracity.59

In this regard we discuss the German Physics of Professor Philipp Lenard, Heisenberg’s objection and Professor Stark’s reply in the spirit of National Socialism.60 Science becomes an instrument for the development of National Socialistic repression and war technology. A free German science will for this reason continue the 56 Reprinted

in Zeitschrift f¨ ur freie deutsche Forschung, 1, no. 1 (1938), pp. 109-121. p. 110. 58 Ibid., pp. 112ff. 59 Prof. Walter Schulze-S¨ olde, Politik und Wissenschaft, Junker and D¨ unnhaupt Verlag, Berlin, 1934. 60 Cf. J.L. Schmidt, Denkschrift “Die Internationalit¨ at der Naturwissenschaften und der Nationalsozialismus”. Internationale oder Nationalsozialistische Physik, International Natural Scientists Congress, Paris, 1937. 57 Ibid.,

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tradition of free, impartial research. The task of this science will be the “protection of the existing body of knowledge from attacks by pseudo-science.”61 The Journal for Free German Research (Paris, since July 1938) became, with the singular publication of E.J. Gumbel’s “Free Science” (Strassburg, 1938), the only consideration of the scientific situation from the point of view of emigrated scientists. Scientists were able to continue to write in already existing journals, while the German writers, literati and journalists had to, for the moment in foreign countries, put in place their own organisations in the German language.62 That’s the reason why there was no “Emigre Science”. It is amazing how the emigrated scientists adapted themselves to their host countries. 9. Bieberbach and Intuitionism Since 1930, Heyting had been trying to promote a sort of neutral intuitionism in arithmetic. For this reason it is not surprising that many mean by intuitionism solely the Heyting variant (and neither the Weyl nor the Brouwer variant). Did Steck really know what intuitionism was? What did Steck or Bieberbach understand of intuitionism? One guess is: Bieberbach meant Klein; Steck meant the metamathematical principles. With Bieberbach it became clear, especially in his extant works, that he understood nothing of intuitionism. Where does one find Bieberbach’s “Brouwerism”? Bieberbach did not work according to Brouwer’s principles in any of his mathematical books. One uses speeches by Bieberbach that he liked to give as a professor, for example the speech from 1926(!). Why should this party speech—the vain Bieberbach liked to celebrate himself in speeches like all the other German “mandarins”—acquire validity for all of Bieberbach’s life? Isn’t Bieberbach an example of how a person, seduced by power and professorial position, can modify his political opinion as career demands? “Post it higher, so everybody can see it better!” is what I think. One should not attribute to things a timelessness that they don’t have and don’t rightfully deserve. It is pointed out that even Hilbert regarded Bieberbach as a proponent of Brouwer. Then they claim that the “battles” of the Weimar Republic were foreplay for the Nazis and that Brouwer was the actual father of the Nazi-mathematicians. Why add any further argument? Unfortunately it was only Bieberbach who supported, among the Nazis above all and only, Hilbert’s formalism and those he considered Hilbert’s students. 10. Formalism and Proof Theory “Formalism” is a slogan that was coined in 1912 by L.E.J. Brouwer. One can just as correctly say that “intuitionism = ideology of National Socialism in mathematics” as one can say, though just as far from the truth, that “formalism = ideology of National Socialism”. Formalism can harmlessly and neutrally be taken to be the linguistic superstructure of intuitionism, for one can certainly have doubts whether intuitionism legitimises formalism in a technical sense. Does one have to concede that every mathematical practice must have a “foundational position” where the ontological, i.e., epistemic, status of mathematical concepts, axioms, rules, calculi, 61 Ibid.,

p. 117. Walter A. Berendsohn, Die Humanistische Front. Einf¨ uhrung in die deutsche Emigranten-Literatur. Erster Teil. Von 1933 bis zum Kriegsausbruch 1939, Europa-Verlag, Z¨ urich, 1946. 62 Cf.

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etc., are explained, as a quasifinal justification? Does Steck want philosophy always to rule over mathematics and to answer, for example, the question, “What are the objects of and what is their status in Gentzen’s thought?” Should mathematics always be tied to a “foundational position” as Gentzen himself admitted in 1937? One may regard the separation of mathematical strivings from the general epistomological questions as illusory, but Gentzen regarded it as pointless to want to solve specifically mathematical problems in proof theory by general philosophicalepistemological considerations. Contentual reasoning in mathematics and logical calculi constrain one another, even if logic doesn’t justifiy mathematics and conversely: the autonomy of both areas is outside our discussion. Even further removed however are epistomological-philosophical speculations about the foundations of both areas. The new foundation of mathematics that Hilbert called proof theory sees itself as having done away with questions of the foundations of mathematics once and for all by associating every mathematical proposition with a concretely demonstrable formula and by doing so to have shifted the whole complex of questions into the domain of pure mathematics. With this the question of foundations is finally withdrawn from adjudication by philosophers and historians of mathematics. Proof theory is not a philosophy; it is (as the word means) a theory about the structure of formal proofs in mathematics. What Steck calls formalism is the old school of Thomas, Schubert, etc. After 1922 Hilbert’s formalism was a new programme (with finitism, etc.). If there existed any reduction for Hilbert it was limited to finitism. Bernays, Gentzen and others had already (definitely after G¨ odel, but partially already earlier) shown that finitism was not sufficient. For this reason one considered exactly—Bernays did certainly from 1934 onward—how to ultimately support oneself using the methods of intuitionism. It was just that Mr. Steck could neither know nor understand this. Haskell Curry had already defended formalism as a foundational position in his 1951 Outlines of Formalist Philosophy of Mathematics. Upshot: During the Nazi times none of the mathematicians or philosophers involved understood intuitionism as the bearing ideology of “German mathematics” or of National Socialism, let alone imagined and correspondingly cited or defended it. Whatever “formalism” or “Hilbert’s Programme” was or may have been in the eyes of many historians: Gentzen set himself apart from Hilbert without being an intuitionist. Gentzen and Hilbert shared common convictions without promoting the same views in “questions of foundations”. But this doesn’t mean that philosophers defined “questions of foundations” of mathematics or questions of mathematical logic and decided them. The competency of philosophy for this must first be verified. Until now philosophy and related disciplines have failed utterly. Because the Hilbert Programme can distinguish mathematics with induction as its own method and thereby becomes a pure science, and as Hilbert is celebrated as a German master, there are no problems here. Gentzen is left alone here. Bieberbach was thoroughly sympathetic to the further development of Hilbert’s Programme, as were the mathematicians that were close to National Socialism.63 Even Georg Hamel worked in the area of the axiomatisation of physics and had an impact on the Hilbert Programme.64 The Hilbert Programme was well known to the Nazi 63 Cf.

the longer note at the end of the chapter. pure intuition is the soil of mathematics, so is the axiomatic method the concrete foundation and logic its iron construction. Thus is the work of Hilbert not only desirable, rather 64 “As

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mathematicians and it was—and this is by far more important!—recognised by the Nazis. It was clear to all from Vahlen and Bieberbach to Hasse or Scholz—even if he also saw a “Hilbert crisis” after the “Frege crisis”65 —up to Tornier or Teichm¨ uller that Hilbert was under no circumstances to be “leased off ”!66 Hilbert was the common foundation of all professional mathematicians in National Socialism. 11. Applied Mathematics as Folkish Mathematics Naturally there are rhetorical attractions to National Socialism. In the book Mathematik im Dienste der nationalpolitischen Erziehung [Mathematics in the Service of National Political Education]67 it is stated: Every responsible mathematician recognises that the most important task in the education of a German to be a political human being lies in the goal of having each German judge in all questions that affect his people and fatherland, just as other peoples have done from time immemorial. And in this we help. A political mathematics exists!. . . The fundamental facts that give direction in the management of government can be hammered into the students from the very beginning.

But “Folkish Mathematics” is very seldom encountered.68 There are a few books on “Mathematics and Defence Sport”, Peoples’ Education, Statistics, Inheritance Statistics, Air Defence, Biology, Population Science, Terrain Mathematics, Aerial and Earth Photomeasurement, and National Political Exercise Book for Mathematics Instruction by Otto K¨ ohler and Ulrich Graf.69 And certainly there exist racist mathematics that practices “Folkish Mathematics” with the help of innocent mathematics where the Quarter-Jews were calculated. But this affected neither mathematics as science nor mathematics as teaching subject.70 Heinrich D¨ orrie thanked his publisher in 1941 in the preface of his great book Vektoren:71 I would like to take this opportunity to express my deepest thanks to the publisher R. Oldenbourg Press, Munich, who in this difficult time despite the extreme effort ¨ it is even necessary.” (P. 15, in: Georg Hamel, Uber die Philosophische Stellung der Mathematik, Technische Hochschule Berlin, Berlin, 1928.) 65 “Now the Frege crisis was thus extended via a Hilbert crisis of really the same rank. More thought-worthy is the fact that this crisis could also have been overcome. This is accomplished through the discovery by Gentzen that the finitary character of a mode of inference—and what matters is this—is independent of whether or not it falls under the modes of inference of a formalised number theory. Only the Hilbert-restricted postulate can no longer be held in its original sense. Hilbert, however, is still not exhausted” [because of the Entscheidungsproblem—EMT], Heinrich Scholz, “Zum gegenw¨ artigen Stande der mathematischen Grundlagenforschung (Review of Stephen Cole Kleene, Introduction to Metamathematics, North-Holland, Amsterdam, 1952), Archiv f¨ ur Philosophie 5, no. 3 (1955), pp. 322-331; here, p. 327. 66 Should a farmer of a hereditary farm lose his respectability or his economic capability, he could be “leased off ”; i.e. the management and right of usage, under some circumstances even the property on the farm, taken from him. Cf. the Reichserbhofgesetz of 29 September 1933 and its extensions. 67 Edited by A. Dorner on orders from the Reichsverband Deutscher mathematischer Gesellschaften und Vereine [Reich’s association of German mathematical societies and organisations], 3rd edition, Moritz Diesterweg, Frankfurt am Main, 1935/36. 68 E.g. arithmetic books for elementary schools. 69 Ehlermanns, Dresden, 1937. 70 Cf. the longer note at the end of this chapter. 71 “Dedicated to the memory of my wife,” Verlag von R. Oldenbourg, Munich and Berlin.

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required has published the difficult composition of this book with continual care and untiring ambition.

He criticises in contrast the fact that the vector calculus was not yet standard fare in all mathematically interested circles, although it had done valuable service for both technician and engineer alike in their theoretical work, and this he connects with a critique of Felix Klein: It is clear that the thing that stood in the way of expansion of the vector calculus, strangely enough, was the underestimation shown it by many mathematicians. The genial and visionary mathematician Felix Klein couldn’t grasp its endeavor (Elementarmathematik vom h¨ oheren Standpunkt aus, Bd. II ).

But this kind of ingratiation to “applied” mathematics was quite rare. 12. Nazis Criticised the Lack of Support from Mathematics What did it really look like? How many docents and professors published in Deutsche Mathematik ? How many mathematicians were active in the NSD Docent organisation and actively organised? What did they actually accomplish? What were the results of their work? The national leader of docents, Professor Dr. Walther Schultze from Munich, made a fool of himself: The political battle, which was actually fought for naked existence, was seen at the front with only few exceptions by university faculty in general. Either they rejected participation based on pseudo-scientific grounds or they were indifferent or they allied themselves with the ranks of opponents of the people’s Germany. At the very least it is a question of a lack of understanding that is difficult to compete with compared to the most pressing necessity of a people’s political battle that was started for no more or less than for the life of one’s own people. . . These historical times have not been recognised by the intelligentsia, or let’s say, only to a small degree. They devoted themselves to their little work group, without taking part in the powerful events of the time. This criticism is hard, but doubtless correct. And when 30 January arrived, the German university was honestly surprised.72

The fundamental goal, as the NSD Docent Organisation had understood it, of securing the philosophical and spiritual unity of the German people still had a big obstacle to face in 1938: There was no new blood! At the same time the NSD Docent Society defended the unity of research and teaching, seeing clearly the secondrate nature of the current applied research compared with foundational research and reaffirmed a common basis of all science against the complete demoralisation, professional fragmentation and torpidity. And this is the greatest task of the NSD Docent Society, which has been appointed by the F¨ uhrer as the trustee of the party to see to it that the National Socialist body of thought is also implemented at the universities. . . At the same time we must always keep in mind that the universities stand or fall with the kind of ready for the call, political- and community-bound, and professionally high class lecturers that don’t isolate themselves from the needs of the German national community, but rather are in the thick of things. 72 Walter Schultze, “Die Hochschule im deutschen Lebensraum”, in: Werden und Wachsen des Deutschen Volkes. Wiedergabe der auf der Hochschultagung des NSD-Dozentenbundes Gau Berlin am 24. und 25. Februar 1938 gehaltenen Ansprachen und Vortr¨ age, NSD-Dozentenbund Gau Berlin, Berlin, 1938; here, p. 12.

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But mathematicians were already in 1934 the lively ones, even if there were contrived examples of students who belonged to “the good old days”: A paradigm for the liberal free students. Free: without principles, without philosophy, without duties or obligations and therefore also without strength, joy and commitment!73

In Faber’s book there is a Berlin student on the way to railway work in Schlesian. There he did his voluntary work service during the university break. He meets another student in his railway car: He really studies mathematics and has tumultuous hair. He feels in powers and thinks in roots.74

The mathematics student doesn’t think much of the inner duty of the volunteer work service during the semester break. He recommends work service to the unemployed, claims that his job is at the university and he doesn’t want to know anything about the solidarity of workers and students. The work supervisor reprimands him sharply: But why do you work at the university? For your own enjoyment? For its own sake? Or for the people? This has always been the problem with academics, that they miss the connection to the world of thoughts and feelings of working people. I cannot collaborate with him whom I don’t know or esteem. Leaders only come from action and sacrifice! We have certainly not become leaders with student books in our pockets! We have to fight for this position. Their seminars or their libraries inside strike me as a bad battleground.

The mathematician shows to his parents primarily through his professional success his thanks for the possibility of studying and strives to serve his people by extending his thoughts’ following outside the country. The test is thus what is currently important. The worker requires accountability before the people and fights against objectivity: Impractical book learning is worthless if it never acquires meaning through empathy and application to the events of everyday life and justification. We should be vital and not outmoded!75 76

But the ideology is not convincing, hence the mathematician is disposed of: They want to become educators of youth without sacrifice? To lead young people into life with sacrifice? Beautiful! They can recite their logarithm tables by memory! But in response to the question from the youngest of their pupils, what their position is, when Germany called on all of us, their response was: “I had something else to do at the time!”

And so the “mathematician”, who in 1934 is in the sixth semester, is already abandoned. Gentzen, I then add jestingly, had at this point in time already completed his doctorate. The NS student president Andreas Feickert said in 1934: The biggest problem today for the universities is that we don’t have any National Socialist docents. 73 Gustav Faber, Schippe, Hacke, Hoi! Erlebnisse, Gestalten, Bilder aus dem freiwilligen Arbeitsdienst. Mit 35 munteren Zeichnungen vom Verfasser, Verlag Kulturpolitik, Berlin, 1934; here, p. 28. 74 Ibid., p. 24. 75 Ibid., p. 27. 76 Cf. the longer note at the end of this chapter.

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The lack of scientific offspring became virulent in 1937.77 For Gentzen’s generation it was less a question of promoting an “observant” science with “creative intuition” and more a question of achieving a goal-oriented science within a National Socialist four-year plan.78 This met with no success, and it was Adolf Hitler’s criminally offensive war that first compelled science anew: “to struggle with the bare existence of the people.” Adolf Hitler despised the professor uprising,79 and for that reason no scientific politician was ever empowered to promote a scientific program in the name of the Nazis. His book Mein Kampf could be quoted, but not interpreted for one’s own purposes. Private factionalism and disputing groups became possible and promoted as a result. Adolf Hitler himself knew that the intelligentsia did not stand behind the Nazi Party. On 11 November 1938 he emphasised in his “Speech to the German Press” that the “Conviction of the Leadership” was to be thanked for the success of 1938 of the “immense education effort” that National Socialism had taken up for the German people and he continued: For it is of course not the case that the whole nation, in particular in its intellectual groups, stood behind these decisions; instead there were naturally very many clever people—at least they imagine that they are clever—that brought more doubts than agreement to these decisions. It was that much more important to persevere and implement, against all resistance, a fortiori with iron will the once composed and to May predated decisions.

And he proceeded later to go into details regarding those “overbred intellectuals” who were unable psychologically to be prepared for the war. What we need is a consolidated, strong public opinion, if possible even within our intellectual circles. [movement and laughter]

And he made fun of the dignitary politics of the upper Ten Thousand that sent him self-contradictory memoranda. He describes these people as hysterical “Chickenpeople”, who behave in a detestable and disgusting fashion. Opposite these “futureshy” people Hitler placed those with unconditional belief in themselves and the Nazi Party.80 And in Mein Kampf he asserts one criterion for the effect of a speech: The speech of a statesman to his people is not to be measured by the impression that it makes with university professors, but rather by the effect that it has upon the people. And this alone provides a measure of the brilliance of the speaker.81

It appears as though certain professional scientists courted particularly the favour of the Nazi Party, because it hardly took them seriously. The National Socialists despised intellectuals if they didn’t become absorbed by the movement completely and without reservation. On the other hand frequently no more occurred to them 77 Cf. p. 247 in: Notker Hammerstein, Die Deutsche Forschungsgemeinschaft in der Weimarer Republik und im Dritten Reich. Wissenschaftspolitik in Republik und Diktatur, C.H. Beck, Munich, 1999. 78 Cf. on this Professor Bach´ er, “Naturerkenntnis und ihre Auswirkung auf das t¨ atige Leben”, in: Werden und Wachsen des Deutschen Volkes. Wiedergabe der auf der Hochschultagung des NSD-Dozentenbundes Gau Berlin am 24. und 25. Februar 1938 gehaltenen Ansprachen und Vortr¨ age, NSD-Dozentenbund Gau Berlin, Berlin, 1938. 79 Notker Hammerstein, op. cit., p. 119. 80 Adolf Hitler, “Rede vor der deutschen Presse”, Vierteljahreshefte f¨ ur Zeitgeschichte 6, No. 2 (1958). Further references are given there. 81 Cited in: Uwe Jens Kruse (Brother Christiansen), Die Redeschule, 5th reworked edition, Philipp Reclam, Leipzig, 1939.

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than the “purification” of German science through the exclusion of Jews, communists and socialists. And the “folkish” science did not mean primarily natural science. At the same time however ambitious scientists could satisfy their vanity and resentment there. Only this mustn’t be directly attributed to the Nazi Party, though it often provided the career-minded, often in anti-bourgeois clothing, the opportunity and the means to do so. After 1943 the school conflicts were completely irrelevant, since every German had to fight on the homefront, and the scientists also on the front of their special sciences. The discussion of “self-mobilisation” or of the “self-management of mathematics” is a more odd and obscure concept, one that doesn’t correspond with the facts. It is true that individual mathematicians and those who had positions of power in the organisational leadership had shown themselves to be anxiously obedient and were tolerant in relation to the terror and persecution. If you consider individual professional organisations like doctors, veterinarians or teachers, the way they unconditionally, rapidly and fanatically lined up behind the goals of the Nazi Party, then mathematicians seem a far cry in comparison. It is a more reasonable assumption that the majority of the German university teachers within mathematics were more than surprised by the “Law for the Restoration of the Professional Civil Service” of 7 April 1933 and the ensuing ordinances to implement this law. From the files of the Technical University in Munich, for example, one can gather that the “race argument” against a further employment of this sort was new, that one still endeavoured to use other arguments.82 How far the reservatio mentalis went in physics, for example, is shown in the following correspondence: The outstanding spectroscopist Heinrich Kayser—he owed his education, profession, extensive freedom in research, a new institute, high social prestige as professor, financial security and international reputation to the imperial era—wrote on 13 July 1933 to the American spectroscopist William Meggers: Daily one is forced to watch while the universities and the sciences are destroyed systematically and according to plan. A professor is no longer someone who knows and has performed, but rather one who has fervently engaged himself politically. You know from the physical technical federal institution that Paschen has been replaced as president and that Stark has taken his place, a blackguard through and through, ignorant, impertinent, spiteful. The informer organisation is blooming; law and order doesn’t exist anymore.— But I don’t want to talk about it anymore; it doesn’t help. A disease of the spirit has poured out over the people like a plague.

And addressed to the same recipient it continues on 13 March 1936: Have you heard from that idiot Lenard, the one that published a textbook on “German Physics”? It differs from Jewish physics, which includes relativity theory, wave mechanics, also quantum mechanics that are all supposed to be Jewish nonsense. His friend Stark praises him for that reason to the skies. Hopefully someone exists who will discharge him thoroughly; the two of them are always

82 Ulrich Wengenroth (ed.), Die Technische Universit¨ at M¨ unchen. Geschichte, Technische Hochschule M¨ unchen, Munich, 1993, pp. 224ff.

Ann¨ aherung an ihre

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gloating: “We two Nobel Prize winners.” Only a “Germane”,83 a National Socialist with a swastika can understand real physics.84

13. Mathematical Foundational Research Remains Unmolested by the Nazis How was it going with logical foundational research? It would not be noticed as long as it stayed within the realm of mathematical seminars, for it was already “Jew-free”. Why did Bieberbach support Gentzen, in fact, the area of mathematical logic driven by Hilbert and Scholz, and not those who were doing mathematics in the sense of Brouwer? In the beginning it was Hilbert who was the uncontested leader of German mathematics; the foundational problem was internationally recognised just like the area of mathematical foundations, and the possibility was provided to attain international recognition with the completion of Hilbert’s Programme. The area of mathematical logic was so conceptually delimited and sharply methodologically defined that the Party had no reason to be afraid that this form of rationality— chains of reasoning, provability, etc.—could be introduced into other areas except in metaphorical form. For this reason mathematical logic could not be a danger uring, however, wanted, for National Socialism. The physicists Dingler and Th¨ like the biologist May, an entirely different mathematics, one which would neither be capable of an international connection nor have any uses for the purpose of a German Mathematics: they would actually destroy mathematics. And Bieberbach opposed the ideologues.85 Here the Nazi Party accepted too the entirely “modern”, as it always did if it appeared important to the Party. With the studies by Reinhard Siegmund-Schultze, Mathematiker auf der Flucht vor Hitler [Mathematicians in flight from Hitler],86 we finally have a work that documents qualitatively and quantitatively the consequences of the forced emigration, transport and murder of German mathematicians under the Nazis. Nonetheless I’m still not clear about the scope and numbers. Who was responsible for which measures; i.e. I still don’t have a list of those responsible. The persecution definitely proceeded from the ministries; the student boycotts against individual professors like Landau were on the other hand also unnecessary, rather largely supportive, but in no way decisive measures. They increased the suffering of lecturers, in no way decisive measures themselves. Expulsion from the society was however an individual matter of the mathematicians themselves. In this regard the names that show up are the usual ones like Bieberbach, Teichm¨ uller, Tornier, Vahlen, etc. But they are very seldom connected with the actual decisions and measures. Bieberbach’s denunciation of Issai Schur or Eduard Rembs and his defence of the student boycott 83 Lost in translation: He uses here the fancy Latin “Germane” in place of the usual “Deutscher”. 84 Heinrich Keyser in: Matthias D¨ orries and Klaus Henschel (eds.), Erinnerungen aus meinem Leben. Annotierte wissenschaftshistorische Edition des Originaltyposkriptes aus dem Jahr 1936, Institut f¨ ur Geschichte der Naturwissenschaften, Munich, 1996; here, p. viii. 85 Cf. on cooperation and opposition of the various science-politically active state and party departments, Carsten Klingemann, “Social-scientific experts. No ideologues: sociology and social research in the Third Reich”, in: Stephen Turner and Dirk K¨ asler (eds.), Sociology Responds to Fascism, London, 1992. 86 Vieweg Verlag, Wiesbaden, 1998.

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of E. Landau is reviewed. One possible motive for this turned up in a letter from Karl L¨ owner (Prague) from 2 August 1933 to Professor Silverman (USA): It was evident that Bieberbach was afraid for himself since he had earlier denounced the Nazis.87

Isn’t there a better way of quantitatively representing and accounting for the “atmosphere of arbitrary accusations and denunciations through student boycotts”? For example, how did it happen that nationally conservative Courant, who was mistreated by Hasse, living and teaching in the USA, was still entertaining the idea in 1935 of visiting Alwin Walther in Darmstadt? How differently were the various cases of persecution, which Siegmund-Schultze set forth excellently, actually perceived? The clear guilt of mathematicians for the persecution and emigration of the Jewish, socialist, feminist or otherwise oppositional colleagues strikes me, from the short-range perspective of those that emigrated, to be insufficiently documented. What I’m trying to say is that for a professional history that would like to take into account both the institutional and ministerial history, it is important to incorporate the point of view, from which I see the development of mathematical logic under National Socialism, as well as to regard solely the points of view of those persecuted and murdered. The web of individual mathematicians, professional organisations and political and administrative centers of power has not yet been sufficiently worked through. It lacks the synthesis that would expose who did what and with which consequences: put simply, how it was and weighted according to importance. The professional history of mathematics of this period is still in its infancy. What was done by the state through ministers or party organisations, through the professional organisation, individual functionaries or mathematicians in the persecution, emigration and murder of the Jewish and opposition mathematicians and what was their rˆ ole? For me this is still unclear even if we can imagine. This is why lists of the guilty must be compiled or even of all the mathematicians that stayed in Germany or even emigrated and returned there, to put it simply. There was a multifaceted opposition to the Nazis. It came from the labour movement, the conservative Catholic milieu, and there were among them youths, (very few) scientists, artists and intellectuals, Jehovah’s Witnesses, Jews, evangelical churches, prisoners of war and forced workers, draft dodgers and deserters, and there was also a spontaneous and individual protest in everyday life.88 There was a good deal of contemporary literature that warned about the Nazis that one can read. “He who abandons the victims, kills them once more” (Ignaz Bubis), “Forgetting leads to captivity, remembering is the secret of deliverance” (Baal Schem Tov). Today the warnings and resistance documents are all available. To get over, to ignore, to not want to know is either a surrender to the ideology of Goebbels and Hitler or finally the determined will to hate historical truth. He who does not want to know anything about “this” is an unscrupulous ideologue and falsifier of history. Putative ignorance is not allowed in the sciences.

87 Ibid.,

p. 312. for example, Marion Detjen, “Zum Staatsfeind ernannt. . . ” Widerstand, Resistenz und Verweigerung gegen das NS-Regime in M¨ unchen, Buchendorfer Verlag, Munich, 1998. 88 Cf.,

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14. Attacks on Mathematical Logic from Without: Dingler, Steck, and May Dingler and Steck promoted a political “intuitionism” that didn’t deserve its name since it was supposed to blow away the hated “formalism” as well as the Scholzian “logicism”. But Gentzen took the intuitionism of Brouwer, Kolmogorov and Markov seriously. He followed them and wanted to “transcend” it with an enhanced finitism, in order in this way—but beyond G¨ odel—to return to an integral mathematical theory with a method characteristic of it alone. Gentzen was very impressed by the riches of intuitionistic logic. But he regarded “constructivity” as mathematical practice and was probably opposed to the “foundational illusion” as a philosophical movement of a “mathematical constructivism” of Dingler and his worshippers. Dingler’s constructivism required by the way that one imitated everything and only afterwards reflected upon it. To judge it is only possible if one has understood the essence of its inner form and is familiar with its proper application. For this, experience is necessary (Pirmin Stekeler-Weithofer).

Only Dingler thought of his “philosophy” as an initiation for those willing who must first obey in order to be allowed to understand. The Nazi Party also argued from this fundamental point of view, where everyone had to participate and fulfill the rules of construction through execution—in clear contradiction to the schematic deduction of formalists—in order to be able to understand. Isn’t a bit of “anticipation” necessary? Whoever positioned himself outside was immediately categorised as an opponent to be fought or in the best of cases, if he placed himself at the mercy of the others, as ignorant, one who would not receive higher consecration. H. Dingler has always had a difficult time with reviewers in mathematical logic. In 1915 his work Das Prinzip der logischen Unabh¨ angigkeit in der Mathematik, zugleich als Einf¨ uhrung in die Axiomatik [The principle of logical independence in mathematics, an introduction to axiomatics] (Th. Ackermann, Munich) appeared. It was reviewed in the Jahrbuch u ¨ber die Fortschritte der Mathematik 89 by “Lw”, who wrote after noting that the choice of axioms is a voluntary act of will and the conceptions of Hilbert and J. Thomæ are to be favoured: “Just how much the author lacks mathematical discipline is also apparent at the end of the book where Dirichlet’s Theorem on the infinity of prime numbers in arithmetical series is proven in a few pages and without calculation.” Hugo Dingler’s Metaphysik als Wissenschaft vom Letzten [Metaphysics as the last science]90 got an ironic critique from Kurt Grelling;91 his Das Experiment;92 was demolished in the same journal by P.P. Ewald.93 Heinrich Scholz admits willingly in his note on Dingler’s Philosophie der Logik und Arithmetik [Philosophy of logic and arithmetic]:94 “As a logician I myself am much too accustomed to the formalism that is opposed continually in this new foundation, to enable me to come to a judgement.”95 And he compared it to the “recently published dissertation of 89 45

(1914/15), p. 100 (published 1922). Reinhardt, Munich, 1929. 91 Deutsche Literaturzeitung, 1931, Issue 37, Cols. 1735-1738. 92 Ernst Reinhardt, Munich, 1928. 93 Deutsche Literaturzeitung, 1931, Issue 42, Cols. 2006-2012. 94 Ernst Reinhardt, Munich, 1931. 95 Deutsche Literaturzeitung, 1933, Issue 13, 26 March, Cols. 617-620. 90 Ernst

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¨ the Polish logician St. Le´sniewski, ‘Uber Definitionen in der sogenannten Theorie der Deduktion’ ” [On Definitions of the so-called Theory of Deduction], on which he remarks: “The conceptual content of this dissertation is something completely original, to the extent that the reader is challenged to the highest degree.” One could not be publicly attacked in a greater or more resounding fashion. The Marburg philosopher Dietrich Mahnke, toward whom the Nazis were not exactly hostile—the battlefront soldier had given his “commitment” relative to Hitler “officially” in November 1933 (with Joachim Ritter, Otto Bollnow, Martin Heidegger, Erich Jænsch, Arnold Gehlen, Hans-Georg Gadamer and many others), and belonged since 1934 to the SA Reserves II—distanced himself on the other hand from Hugo Dingler. In “Untergang der abendl¨ andischen Wissenschaft” [The demise of Western science]96 Mahnke goes into his “Zusammenbruch der Wissenschaft und der Primat der Philosophie” [Dissolution of science and the primacy of philosophy] and acknowledges a “deep difference” between Dingler and the necessary “GalileiNewton-Method” of other sciences. And so it is also in the reviews of Dingler’s writings. But he never gets tired of claiming that his active logic, based on his logical capacity, is that the foundation of the system is the self justifying will. This is the story in the case of “Strahl” in the review of Dingler’s “Das System, das philosophisch-rationale Grundproblem und die exakte Methode der Philosophie” [The system, the philosophico-rational fundamental problem and the exact method of philosophy]:97 The system appears as an important instrument, embedded in life, but not the converse, life is not a part of the system. The system is to a certain degree the most complete world view. Being itself is completely irrational. Values are the goals of free will. The whole is according to the author a detour from a “passive” to an “active” position. The scientist is an enabled acting being who is alive also in his theory definition.

In just this way Dingler levels the road to a Nazi science. And everywhere Dingler resounded the authoritative “untenable, false, disproven, senseless.” Ackermann certified Dingler’s Philosophy of Logic and Arithmetic 98 a “fundamental error,” in case the author believes himself to have done away with the contradictions of arithmetic.99 Perhaps Dingler was thinking, when he turned to the Nazis, that he then—as official Nazi-scientist—could no longer be overcome by the “formalism people” (Grelling, Scholz, Ackermann et al.), whom he himself criticised, because his ideas about Nazi-ideology would be closer, so he could embed them in it that much better. He didn’t later oppose their critique. Did he want to simply get rid of those unwelcome reviewers? In the first year of National Socialism there was a back biting battle over the true teachings of “German physics”, and each little group fought for power, influence and money. Of course Dingler agitated, since he viewed himself as the spiritual leader of German physics, even before the National Socialistic physicist Pascual Jordan. In a review of Jordan’s Physik des 20. Jahrhunderts [Physics in the 20th century],100 he accused Jordan of involvement in the theoretical speculation of Bohr 96 Archiv

f¨ ur Geschichte (n.s.) 10 (1927), pp. 216-232. u ¨ber die Fortschritte der Mathematik 56 (1930), pp. 82ff. 98 Ernst Reinhardt, Munich, 1931. 99 Jahrbuch u ¨ber die Fortschritte der Mathematik 57, p. 1312. 100 Zeitschrift f¨ ur die gesamte Naturwissenschaft 3 (1937/38), pp. 321-335; review of the second edition of the book, the same, 4 (1938/39), pp. 389-393. 97 Jahrbuch

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as a Jewish alliance that wanted to sabotage the logical clarity and the great Aryan tradition of meticulous science with pseudo-religious fantasies. And the dean of the Rostock University spoke out against the philosophical defamation of a loyal party member and informed the NS Docent Society. On the other hand Jordan had read Dingler’s “Die Kultur der Juden” [The culture of the Jews] and commented with quotations on Dingler’s “philosophical metamorphosis” to the Docent Society leader, Professor Dr. Gissel. Finally he explained to the Rostock rector that it was paradoxical that he, a party member and SA man,101 would be philosophically corrected and censored by a fanatical propagandist of Jewry. It would not serve the National Socialist cultural work if a scientific zero and proponent of the Jewish race were to be named staff member of the Zeitschrift f¨ ur die gesamte Naturwissenschaft of national professional group leader Dr. Kubach. And Dingler’s having denounced Jordan was the reason for his dismissal. Dingler got involved with Ludwig Mach, who had tried to show, with the help of forgeries, that his father, Ernst Mach, was against Einstein’s relativity. Certification for this was provided by the German physicists Philipp Lenard and Ernst Gehrcke. Hugo Dingler, who wanted to distinguish himself through the battle of the SS and Nazi student leadership against Werner Heisenberg, was supposed to be rewarded in that the Munich docent society tried to make him honorary professor of physics above Th¨ uring.102 In the third 103 volume of Nazi-Meyer of 1937 it states: Dingler, Hugo, natural philosopher and theoretician of natural science, influenced by Wilhelm Ostwald, *7 July 1881 Munich, since 1933 Professor at the Darmstadt Technical University, opposed both the mechanistic as well as the relativistic view of nature to the favor of the teachings of the highest logical and natural laws, wrote. . .

In an article published in 1942 on the “Jewish” relativity theory Dingler was no longer mentioned.104 Sommerfeld and his colleagues successfully opposed this plan.105 Hugo Dingler was actually supposed to be rewarded with a professorship in Vienna. The astronomer B. Th¨ uring proposed to Dean Christian in the winter semester 1941/42 a national professorship of Methodology and the History of Exact Natural Science for Hugo Dingler, which was planned for Vienna. Dean Christian passed this on to Dr. F¨ uhrer of the REM (National Ministry of Education) on October 22, 1941. Dr. F¨ uhrer replied on 5 November 1941: Strassburg University had received, as one of the few German universities, a professorship for Methodology and the History of Natural Science. Since the Faculty of Natural Science of the University of Strassburg does not want to fill this professorship at this time, I’ve been considering making this professorship available to the University of Vienna in order to, if it’s possible, appoint Professor Dingler to the position. 101 Later he worked for Adenauer’s Christian Democratic Union and Christian Social Union Defence Minister Franz-Josef Strauß on the reply to the Manifesto of the G¨ ottingen 18 on the renunciation of German atomic weaponry, and directly Heisenberg wrote out two Persilscheine [clean bills of health nicknamed after a popular detergent] for Jordan. 102 Th¨ uring was a “German physicist” and opponent of Carl Friedrich von Weizs¨ acker during the Munich “religion discussion” of 15 November 1941. 103 Meyers Lexikon is one of the standard German reference works. 104 Bruno Th¨ uring proudly published in the Federal Republic: Die Gravitation und die philosophischen Grundlagen der Physik, Duncker & Humblot, Berlin, 1967. 105 D. Cassidy, Heisenberg. Leben und Werk, Spektrum, Heidelberg, 1995, p. 749.

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The faculty dealt with the basis for this appointment at the committee meeting on 29 January 1942. The Vienna faculty opposed the opinion of B. Th¨ uring and determined that Dingler was not qualified: Wichmann summarised this best in his last communication: “Dingler has no knowledge of biology, in physics and mathematics not very original; he’s always been subject to the eternal trends of the moment which puts the validity of his philosophy in doubt.”

In the minutes of the meeting it is expressed more strongly: The representative must know a great deal in order to be able to enter into meaningful discussion with the various professional representatives, which is not true of Dingler.

Vahlen [President of the Prussian Academy of the Sciences—EMT] takes a position relative to this judgement in a letter to the leader of the docent society, A. Marchet: With regard to Professor Dr. H. Dingler my opinion is that he is well suited for the subject of Methodoloy and History of Natural Science. In the lecture circuit in Vienna, where I was present [Dingler spoke at the invitation of B. Th¨ uring in the university observatory in Vienna—EMT], he was however attacked many times, but these attacks always came when he thought he could go beyond his own area into the foundations of geometry, mechanics and physics and when he was exposed to attacks from the more qualified experts in that area, he never succeeded in meeting their objections.106

As we know, Dingler was never appointed in Vienna despite the intervention of Vahlen. It was funny though that B. Th¨ uring used the same arguments as his opponents in the search committee, that Dingler was an opportunist, in defence of Dingler again after the war when it came to appraising his philosophy anew. Here by way of salvation one relied on the victim mentality in order to console Dingler after the fact: a weak human being gives in to the trends of the day. Nothing was said for Dingler’s victims. For the second time they were held to silence through the efforts of professorial, scientific reason. Dingler was supposed—with Fritz Kubach and Pascual Jordan and C.F. v. Weizs¨ acker—to publish a ‘Zeitschrift f¨ ur systematische Philosophie [Journal of systematic philosophy], but it never materialised.107 Max Steck was on Dingler’s side in his determined battle against Hilbert and his foundations of mathematics. A good example of the cooperation of the group around Dingler, Th¨ uring, M¨ uller, May, Steck and Requard is the volume of the Zeitschrift f¨ ur die gesamte Naturwissenschaft from May/June 1941,108 where they had gathered together the above mentioned on the occasion of Hugo Dingler’s 60th birthday, to celebrate the call to battle and the rise of German Science. Hilbert was relegated to conceptual formalism and on the side of Hugo Dingler stood the good values of the contentual material, of sensible meaning and of the determinacy

106 All

citations from Hans Karl Hofer, Deutsche Mathematik. Versuch einer Begriffsbestimmung, dissertation, Vienna, 1987. 107 George Leaman and Gerd Simon, “Die Kant Studien im Dritten Reich”, Kant-Studien 85 (1994), pp. 443-469; here, p. 461. 108 7th Volume, Series 5-6, published by Ernst Bergdolt, Fritz Kubach and Bruno Th¨ uring, jointly with among others Hugo Dingler, Wilhelm M¨ uller, Ernst R¨ udin, Johannes Stark, Rudolf Tomascheck and Theodor Vahlen, Ahnenerbe-Stiftung Verlag, Berlin-Dahlem.

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of the formula. This support for Dingler109 was discussed in this way by W. Ackermann.110 Unbelievable, but true: an attraction to Dingler was also present among some “leftist” intellectuals in the Federal Republic of Germany. Not only did Paul Feyerabend regard Dingler’s theories as fascinating, but also Bodo von Greiff. It is surprising that even Hans Albert, later a follower of Karl Popper, used Dingler’s theories in his dissertation “Rationalit¨ at und Existenz” [Rationality and Existence] 1952 (reissued 2006). And this resulted through not only lack of understanding but also purely through the opposition to Hilbert’s formalism. This discomfort was expressed—as with Steck—for example contemporarily also by Ernst Bloch 1944-45:111 Of course nothing was raised against the pronounced more than self-referential mathematical formalism (most consciously with Hilbert), no appeal from what was real, no “rightful purpose” (as with Ihering against the conceptual attorneys).

Between reality and mathematical calculus there was however no tension, because “still nothing of the most sublime thought construction of new numbers and manifolds had found any application in the mathematics of natural science.” But freedom and development were not possible. In stark contrast to their expression there was always the predeterminateness of progress made through this approach, in general, the foundations and, in the end, the axiom system. A mathematical axiom system demands that it be developed in never ending consequences of thought, but in this way you only characterise those objects that can be formed through the axioms—the axioms are “implicit definitions” for everything that can be developed using them. In this regard the quantum stands alone, as a mathematical area of study, the paradigm of the new and qualitative.

Mathematics in the Hilbert style is thus a conservative theory of relations, from which nothing new can develop. Albeit the dialectic process can also be modelled. The left collectively remains as the position of adversaries of mathematics. Though I look not without smiling at the tracts and pamphlets of the Roten Zellen Mathematik [Red cell mathematics] of the Berlin Technical University, where mathematics—as by Ernst Bloch—would first be interpreted metaphorically. Both of these amateur philosophers wanted to forge mathematics in the popularphilosophical way of the Nazis and to establish its race-based foundation.112 This is 109 Max Steck, “Mathematik als Problem des Formalismus und der Realisierung”, Zeitschrift f¨ ur die gesamte Naturwissenschaften 7, pp. 156-163. 110 In Jahrbuch u ¨ber die Fortschritte der Mathematik 67 (1941), p. 30. 111 Now in: Ernst Bloch, Logos der Materie. Eine Logik im Werden. Aus dem Nachlass 1923-1949, Suhrkamp Verlag, Frankfurt am Main, 2001, p. 407. 112 An analysis of the Zeitschrift f¨ ur die gesamte Naturwissenschaft is urgently needed, because therein Th¨ uring, May and M¨ uller sing the praises of H. Dingler as master of National Socialist natural philosophy. Cf. the essays in volume 7 (1941) of the Zeitschrift: Bruno Th¨ uring, ¨ “Hugo Dingler’s Zeitschrift f¨ ur die Naturwissenschaft”; E. May, “Dingler und die Uberwindung des Relativismus”; Wilhelm M¨ uller, “Dinglers Bedeutung f¨ ur die Physik”; and Max Steck, “Mathematik als Problem des Formalismus und der Realisierung”. In the same number— I am tempted to take this for a coordinated, concentrated action—the freeloader Kurt Schilling published his essay, “Zur Frage der sogenannten ‘Grundlagenforschung’. Bemerkung zu der Abhandlung von Heinrich Scholz ‘Was ist Philosophie?’ ” (pp. 44-48), which ends with the words:

This isn’t allowed. If Herr Scholz also possesses a certain courage, by which in the middle of a war he recommends to the German people as the only philosophy, the leading–and by him only cited—supporters of which today are only Poles, Englishmen, emigrants and Americans, and if he openly expresses that he fashions (p. 51) his lectures as a

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ideology and for that reason a battleground. The “Munich School”—it led the way in establishing the “German physics”: M¨ uller, Dingler, May, Steck and Th¨ uring— viewed logicism and the philosophical point of view of the Scholz School as the successors of Carnap, Popper, Dubislav, etc., i.e. of those who were exiled or murdered. Dingler and his Munich School would like to discuss the popular assumptions of the theory of science and of mathematics.Philosophy was obsessed however with the rivals Krieck and Bæumler: Dingler, Steck and May rejected Bæumler. At the same time Bieberbach warned jealously that physicists, biologists and astronomers didn’t have any place in the discussion of the race-political assumptions or the structure of mathematics in “his” German mathematics. When Bieberbach presented his “own kind of ” mathematics, he experienced every incursion into this area as an afront. He knew that he didn’t want to discuss the assumptions underlying German mathematics in his own journal, nor could he avoid it. His reputation depended upon the fact that many well-known, powerful and competent mathematicians expressed themselves in his journal professionally and not ideologically.113 In this regard Bieberbach had the support of Vahlen, but this could be destroyed by a rival from the “Rosenberg Office” or anywhere else: no initiative, not even the phrase “German mathematics”, was officially recognised or published by the party. For this reason he turned over to Heinrich Scholz—who was counted as a German national and also knew Hasse through their joint publication114 during his time in Kiel—space to respond to the attacks that Steck published in his book Das Hauptproblem [The main problem]. However Scholz had to fight on two fronts. Heinrich Scholz’s Geschichte der Logik [History of Logic]115 was reviewed by E. Ahrends in the annual report of the German ordinarius in M¨ unster 1939/40 “after the Warsaw example,” so it seems to me this courage ought to find itself a better subject (p. 48). It would be interesting, however, if the Munich Professor Schilling meant in his “Studienf¨ uhrer zur Geschichte der Philosophie” (Heidelberg, Winter, 1949) that philosophy can be expected of every man, if “inner human value, sympathy and interest, tidiness and authenticity of one’s own lifestyle, honesty and clarity of questions and answers” are present. The propriety of this enumeration of cardinal virtues, which Himmler so admired, eludes me. 113 Hilbert was a man of the 19th century, who consciously used the position of a professor, of a “mandarin” (Fritz Ringer), in the Weimar Republic and his position as privy councillor also for his professional-political interests and remained intact with all good properties like a sense of honour and sincerity. A position of that sort and power—but free of the virtues attached to it—verily the glory of the position and of the influence is what the vain Bieberbach wanted to reproduce and increase in the Academy, the Reichsforschungsrat and ministerial contacts, and for this it came at the right time for him that all those who could be dangerous to him in this regard–from the profession and from the scientific-political position—would be driven from their offices by the Nazis. Bieberbach had certainly been liberal in his youth, but the inner drive of career for the sake of career left him without scruples. With the publication of his views on racial typology, however, he had gone too far—this was shown to him by the worldwide reaction to them—and this had an influence on his professional reputation. He published nothing further of the sort, and one might ask what this episode signified for his life. And yet Bieberbach showed himself in 1936 still to be a follower of Hilbert: One finds a discussion of the famous problems outlined over 30 years earlier ¨ by Hilbert and their influence on the development of mathematics in Bieberbach’s paper “Uber den Einfluss von Hilberts Pariser Vortrag u ¨ ber ‘Mathematische Probleme’ und die Entwicklung der Mathematik in den letzten dreißig Jahren”, Naturwissenschaften 18 (1936), pp. 1101-1111. 114 Helmut Hasse and Heinrich Scholz, Die Grundlagenkrisis der griechischen Mathematik, Metzner, 1928. 115 Juncker & D¨ unnhaupt, Berlin, 1933.

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DMV.116 After Scholz was praised for his view that logical thinking was important for the strictness and sharpness of philosophical thought, it was remarked critically that: The criticism that Scholz attributed into this section to the anti-metaphysical efforts of the Vienna Circle, . . . appears to me not to have substantial justification

since for Ahrends tenets of metaphysics—and it’s especially these that H. Scholz would like to “preserve”—are not part of science, because they don’t have any substance. However Scholz would like to build up metaphysics as a strict science; i.e. he uses formal logic as that part of scientific method that formulates the rules of thought needed for establishing any science, and, of course, all of this provides what is necessary for an exact formulation of these rules. Scholz is sure that one can also construct an unassailable theology. His foundation finished however also with “evidence”—and the animosity of Hugo Dingler was also aimed at this evidence.117 But the mastery of “logistic”118 is not all of philosophy for Scholz. Logistic is a necessary, but not a sufficient, condition to be able to philosophise. Scholz takes a clear position against the Vienna Circle and its goals. He is an anti-positivist. In 1943 Scholz defended himself against Steck, Dingler, May and consorts in his lecture “Logik, Grammatik, Metaphysik” with the statement: . . . it’s clear that I am just as indebted to the, in its own way, masterful work of Mr. Carnap [Logische Syntax der Sprache [Logical syntax of language], Vienna 1934—EMT] as anyone who has studied it thoroughly. But the modesty of his demands for a logical calculus is not for me. I wouldn’t call myself a logician at all, even if that were the whole of magnificence. And I would be able to do so even less if I had to adhere to the philosophy of Mr. Carnap in order to consider myself a logician. If things were really as Mr. Carnap presents them, then for anyone to be viewed seriously as a logician he would have to be a positivist in the style of Mr. Carnap. For that I am utterly inappropriate.119

This balancing act, having to defend logistic against the Nazi science philosophies and to advertise among mathematicians the applicability of logic to a value-bound metaphysics, was held by Scholz throughout the Nazi period and well into the spiritual atmosphere of the postwar period. This was especially difficult in contacts with critical philosophers. Karl Jaspers, who in 1931 titled his volume of essays (it appeared as Volume 1000 of the G¨ oschen Collection) Die geistige Situation der Zeit [The intellectual atmosphere of the times], didn’t think much of “formal logic”.120 He viewed it of course as helpful in overcoming willful disappointment and error, but considered the continual explanations of sophisms to be a waste of time that distanced one from the matter at hand.121 He had a very low opinion of logistic, whatever that might mean: 116 42

(1933), p. 136. the longer note at the end of this chapter 118 In 1904 L. Couturat, Itelson and Lalande proposed the word “logistic” as the designation for mathematical logic at the international congress of philosophy in Geneva. It didn’t catch on. 119 Heinrich Scholz, “Logik, Grammatik, Metaphysik”, Archiv f¨ ur Rechts- und Sozialphilosophie 36 (1943), pp. 393-433; here, p. 423. This has been reprinted with minor changes in Archiv f¨ ur Philosophie 1 (1947), pp. 39-80. 120 Cf. the appropriate pages in Hans Saner and Marc H¨ anggi (eds.), Karl Jaspers. Nachlaß zur Philosophischen Logik, Piper, Munich, 1991. 121 Ibid., p. 132. 117 Cf.

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The pleasure with which one is continually reading Euler’s algebra—the discomfort, as if one got little or nothing from the study of logistic, foundations of mathematics, theoretical physics of quanta.122

And this resentment was shared by many who were of another political opinion than Karl Jaspers. Contemporary political opponents of the Nazis didn’t even have to agree on the means used for intellectual analysis or for combating stupid ideology. Scholz defended his institute—it had been counted among the mathematical departments within “German mathematics” since 1936—and this initially ensured its survival—and attracted Hilbert students—this criterion always attracted—and helped the mathematician Gentzen, although he didn’t think much of the G¨ ottingen Finitism, but certainly tolerated it. Gentzen was classified by Hasse as a Hilbert proponent and, for this reason, received his label: he was no longer regarded as an independent entity, as his own person; rather as a supporter of Hilbert, as a means to an end—no more. For this reason he was seen as a sort of “Anti-G¨ odel” in the publications by Hasse and Scholz, who knew one another since their article “Die Grundlagenkrisis der griechischen Mathematik” [The foundational crisis of Greek mathematics].123 When filling chairs in philosophy Scholz was regarded as a philosopher and much sought after. 15. The Attacked: Heinrich Scholz (1884-1956) Heinrich Scholz was born in Berlin on 17 December 1884.124 His father was for a long time the senior minister at the Protestant Marien Church. From 1903 to 1907 he studied philosophy with Friedrich Paulsen and Aloys Riehl and theology with the church historian and promoter of science A. von Harnack. On 28 July 1910 he was promoted to Professor in the Philosophy of Religion and Systematic Philosophy and worked after that as a lecturer. During World War I annexationist Scholz participated as a writer of propaganda: Der Idealismus als Tr¨ ager des Kriegsgedankens 125 [Idealism as the bearer of the thought of war]. There was never any doubt about his patriotism to Germany. The unseducible, strong human being is celebrated with Fichte, Clausewitz, Moltke, Kant and B¨ ohme, for whom psychophysical self-assertion is a moral goal, as far as it serves the ethical self-preservation: The premises of the will to make war will always remain as long as we are human beings.126

“We felt like Prussians right to the bone,” H. Scholz remembered and surely this is much more than well-fortifiedness, industriousness, ambitiousness of will and the fruits of self-control like obedience, punctuality, endurance, self-deception and 122 Ibid.,

p. 432.

123 Kant-Studien

33 (1928), pp. 1-27. us H. Scholz merely plays a rˆ ole as a scientific organiser and a defender of the conception of “logistic”. On his biography and philosophy of mathematics, Arie L. Molendijk, Aus dem Dunklen ins Helle. Wissenschaft und Theologie in Denken von Heinrich Scholz. Mit unver¨ offentlichten Thesenreihen von Heinrich Scholz und Karl Barth, Rodopi, Amsterdam, 1991, is especially detailed. Cf. the review by Volker Peckhaus, “Essay review”, History and Philosophy of Logic 14 (1993), pp. 101-107. 125 Friedrich Perthes, Gotha, 1915. 126 Ibid., p. 29. 124 For

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effectiveness. The archetype is the Prussian officer.127 In 1917 he moved to Breslau as full professor. At the request of W. Jæger he was appointed successor of Paul Deussen in Kiel. In 1921 his “Religionsphilosophie” appeared. Also in 1921 he came across Principia Mathematica. He immediately wanted to familiarise himself with mathematics and theoretical physics and he read Frege and Bolzano. From 1922 to 1928 he regularly attended lectures in mathematics by Professors Toeplitz, Hasse and Steinitz and theoretical physics by Kossel. During the summer term of 1922 he announced a seminar with Moritz Schlick on the philosophical analysis of relativity theory. He also published a paper on foundational research with Helmut Hasse (1898-1979). In October of 1928 he was appointed in philosophy in M¨ unster.128 In 1929 he came to know the theologian Karl Barth. Already in 1936 the philosopher received a teaching contract for mathematical logic and foundational research.129 Scholz had received in 1936 an additional grant for the printing costs for 3 volumes on research in logistic from the DFG130 and for editing of the Frege Archives. He wanted to prepare a Frege monograph and knew, since 21 April 1936, that Frege was politically to the far right, although not a “Nazi”. Heinrich Scholz had returned by mail to Gentzen on 20 April 1936 the letters from Frege to Hilbert that he needed for the publication of the Frege-Hilbert correspondence. In March Scholz had presumably sought Leopold L¨ owenheim in Berlin. L¨ owenheim had been laid off due to “non-Aryan heritage” and was put into “retirement”. In 1936 Kurt Grelling set up a regular colloquium in which J¨ urgen v. Kempski and Leopold L¨ owenheim took part.131 In the conversations between Scholz and Gentzen politics played apparently no rˆ ole. Heirich Behnke—somewhat jealous of the globality and the impressive correspondence of Scholz with the great thinkers of his time—regarded his nationalism as closed minded and his Prussian ways as too arrogant. 127 Arie L. Molendijk, Aus dem Dunklen ins Helle. Wissenschaft und Theologie im Denken von Heinrich Scholz, Rodopi, Amsterdam, 1991, p. 39. 128 Cf. the longer note at the end of the chapter. 129 Which traditional prejudices Heinrich Scholz himself had to fight against in M¨ unster comes out in the critique of the M¨ unster logician Wilhelm Koppelmann (cf. the chapter “Logistik” in Wilhelm Koppelmann, Logik als Lehre vom wissenschaftlichen Denken. Zweiter Band: Formale Logik, Pan-Verlaggesellschaft, Berlin, 1933). Here the derivation of logic from an axiom system is judged as impossible as relapse in the calling for evidence: “Because how can logic be an organon, a tool for assessment of the deductions of mathematics and possibly too of other sciences, if the axioms from which it proceeds are ‘admitted without proof’, and consequently also another axiom system and with it another logic were possible as a tool for the assessment of the correctness of scientific deductions.” (p. 109)— That would not be any better through a “birth of logic from metaphysics,” as Hans Heyse liked to put it in “Idee und Existenz” (1935).— In the Nachlass of Richard H¨ onigswald, who was sent to the Dachau concentration camp in 1938, one finds the following contemporary citation: “Logistic does not found its conduct; what it offers is a mathematical natural history ‘with respect to’ relations. It does not concern itself with the meaning that comes into consideration thereby and the right of the ‘fact’ of relations.” (Richard H¨ onigswald, “Philosophie und Kultur”. Herausgegeben von G¨ unter Schaper und Gerd Wolandt. Nachlaß. Band 6, Bouvier, Bonn, 1957, p. 26.) Foundations were understood—in contrast to the Jew H¨ onigswald denounced by Heidegger—by the Nazis and their “philosophers” as origin-myth. And concepts like folk, evidence and precision-engineering entered into unholy alliances. 130 Cf. the longer note at the end of the chapter. 131 Volker Peckhaus, “Von Nelson zu Reichenbach: Kurt Grelling in G¨ ottingen und Berlin”, in: Lutz Dannenberg (ed.), Reichenbach und sein Kreis, Vieweg Verlag, 1995; here, p. 63.

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The grandeur of the university was lost with the shift in power. One day— it was already apparent at the end of the Winter semester—there were plain old Nazi Party flyers on the empty seats of the ASTA hall. Kratzer and I spontaneously criticised this. Scholz wanted to defend it, but then hesitated. For me it was queer. Heinrich Scholz took on an attitude of extreme political correctness. Otherwise he was used to reacting very quickly particularly when it appeared to him that his academic space was threatened. But in this case what a lame duck! That was a sign.132

Heinrich Behnke’s words ring today perhaps more pithily than he himself had behaved at that time. Peter Thullen (1907-1996) recalled133 in his notes from the year of the “take-over” for example on 7 May: The air of Behnke disgusts me. I’m ashamed to see him. Hopefully I’ll soon be independent of him. His fear borders on cowardice. While others speak openly about the exaggerated Jewish hate, he doesn’t dare say a word—not even to us.

Behnke’s first wife was a Jew. She died while giving birth and left behind a son who, by the measure of the times, was a “Half-Jew”. On 22 May he wrote: The cowardice of the professors is really despicable. Kratzer (my physics professor) refuses to sign a petition for Courant. Scholz (philosopher, my teacher and friend) invites the new student leaders.

Adolf Kratzer was who was meant (1893-1983). On 20 July he wrote: The professors, who had celebrated with utter enthusiasm the “national independence”, have now been “awoken”: H. Scholz, St¨ ahlin, Sa.? Even Herrmann. Today they notice which way the new esprit, or rather lack of esprit, is blowing.

H. Behnke allied himself with W. S¨ uss and began a collaboration with him that lasted into the post-war period.134 Perhaps Scholz would have also gotten the socially and politically na¨ıve Gentzen to join the SA (Storm Troopers). Scholz was in the beginning a strong supporter of the Nazis, like his school friends Eduard Spranger and Werner Jæger. But with good reason I can claim that his political opinions converged with those that carried out the attack of 20 July 1944, or made it possible or created the intellectual climate for it. Precisely many of those who were German-nationalists in a conservative manner were, because of the later open criminal acts, important opponents 132 Heinrich

Behnke, Semesterberichte, Vandenhoeck & Ruprecht, G¨ ottingen, 1978, p. 116. f¨ ur meine Kinder”, Exil 20 (2000), pp. 44-57. 134 Cf. Volker Remmert, “Ungleiche Partner in der Mathematik im ‘Dritten Reich’: Heinrich Behnke und Wilhelm S¨ uss”, Mathematische Semesterberichte 49 (2002), pp. 11-27. The “Aryaniser” of mathematics under National Socialism, W. S¨ uss, was numbered in the years 1936 to 1940 among the editorial body of Deutsche Mathematik. (Cf. Volker R. Remmert, “Die Deutsche Mathematiker-Vereinigung im ‘Dritten Reich’: Krisenjahre und Konsolidierung”, DMVMitteilungen 12, no. 3 (2004), pp. 159-177.) At the beginning of April 1939 W. S¨ uss wrote in “Kassen- und Rasseangelegenheiten” [cash and race affairs—S¨ uss was treasurer] to his fellow board members in the German Mathematical Society that Paul Bernays “on account of his reputation [could] not be removed without consequences for the DMV” as he was firmly ensconced in Switzerland. (Cf. Volker R. Remmert, “Die Deutsche Mathematiker-Vereinigung im ‘Dritten Reich’: Fach- und Parteipolitik”, DMV-Mitteilungen 12, no. 4 (2004), pp. 223-245.) S¨ uss recognised the humble position of mathematics under National Socialism in that the most famous German mathematician, David Hilbert, would not be awarded on his 80th birthday in January 1942 the “Eagle’s Shield” as had been received by Ph. Lenard in 1933—but rather like “hundreds of worthy and competent conscientious professors [he would have to settle for] the Goethe medal” (cf. the last cited reference of Remmert). During and after the war W. S¨ uss was the founder of the still renowned Mathematisches Forschungsinstitut in Oberwolfach. 133 “Erinnerungsbericht

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of the Nazis. While a National Socialist critical circle formed around Professor Behnke and the physicist and scientist Kratzer,135 it was true at the same time: “Professor St¨ ahlin commented in his memoirs”136 that one could speak “in trusting openness” with “only a few colleagues” as for example Heinrich Scholz.137 All of the M¨ unster professors however rendered unconstrainedly and without exception their wholehearted loyalty and obedience to Adolf Hitler: I swear: I will be loyal and obedient to the Leader of the German People, Adolf Hitler, obey the laws and fulfill the duties of my office, so help me God.

The refusal to take the oath of 20 August 1934 would only have led to expulsion from one’s government position. Instead, the observers of this oath follow precisely the laws against Jewish and political undesirables or even disliked professors. In Deutsche Mathematik one reads: Logistical Logic and Foundations Research: The departmental representative of this field, for which M¨ unster is the first German university with its own instructional and research group, is Prof. Scholz. Prof. Scholz lectures 4 hours on logistical representation of the Foundations of Arithmetic. The lecture is accompanied by required recitation for Dedekind’s little book on Arithmetic. In the 2 hour Logistic Working Group with the assistant Hermes the central questions of the syntax of propositional and first order predicate calculus will be covered. In the additional two hour recitation with Mr. Hermes we will work through the propositional and predicate calculus in nonaxiomatic form. In the 14 day two hour per day Logistic-Physical Working Group Prof. Kratzer will present a logistical axiomatisation of classical mechanics. In June Dr. Ja´skowski–Chozyw (Poland) will speak here on his own work and on the system of the Warsaw logistician St. Le´sniewski, Dr. Gentzen–G¨ ottingen will speak on his consistency proof for elementary number theory.138

And that’s not all. Under the title “Research” is published: “A new completeness proof for the reduced Fregean axiom system for the propositional calculus. By Hans Hermes139 and Heinrich Scholz”, pp. 734-758. Why was the subject “logistic” so successful in M¨ unster? Every student that wanted to complete a teaching degree for high schools had to take an additional philosophy exam. The Office for High Schools recognised logistic as philosophy in the sense of the certification tests simply because of the prospective mathematicians.140 135 Adolf

Kratzer had instructed David Hilbert on the progress of relativity theory and journeyed with Heinrich Scholz to L  ukasiewicz in Poland. 136 1968, p. 268. 137 Reference by Petra Kess (1993, p. 148). 138 Vol. 1 (1936), p. 274. 139 Hans Hermes (*1912) studied mathematics and philosophy from 1931 to 1936 at the universities in Freiburg im Breisgau, Munich, and M¨ unster. He was promoted in M¨ unster in 1936, took his teaching degree in 1937, and became assistant to the Mathematical Seminar at Bonn in 1938, where he mainly worked on complex function theory. In 1941 he became a soldier. After the war’s end he returned to the Mathematical Seminar as an assistant, habilitating in 1947. In 1953 he became Professor of Mathematical Logic and Foundational Research at the University of M¨ unster. In 1966 he became Professor of Mathematical Logic and Foundations of Mathematics at Freiburg. Hermes became known on account of his work “Eine Axiomatisierung der allgemeinen Mechanik” [An axiomatisation of general mechanics] (Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, no. 3 (1938)); cf. in this regard pp. 120-122 in: Max Jammer, Der Begriff der Masse in der Physik, Wissenschaftliche Buchgesellschaft, Darmstadt, 1964. 140 Cf. Hermes, 1986, p. 43.

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Heinrich Scholz names the year 1936 as the year that the “M¨ unster School” was born.141 The actual and official date that “Mathematical Logic and Foundations of Mathematics” was institutionalised through the official recording of H. Scholz’s professorship by the Federal Ministry of Education was 21 May 1938, published in the records of the University of M¨ unster on 10 June 1938. The sort of upright man Scholz was could easily be seen in his personal files.142 In 1938 Heinrich Scholz submitted to the National and Prussian Minister of Science, Education and National Education, Bernhard Rust, his “Reflections on the new mathematical logic and foundational research”. He notes there that mathematical logic, despite Hilbert and his school, had fallen disproportionately behind. For review he included, with his reflections, the evaluation of the Polish logician Jan L  ukasiewicz. He had written in 1937 in his publication “W obronie logistyki” (In defense of logistic): Whenever I work even in the tiniest logistic problem, e.g. trying to find the shortest axiom of the implicational calculus, I always have the impression that I am confronted with a mighty construction, of indescribable complexity and immeasurable rigidity. I sense that structure as if it were a concrete, tangible object, made of the hardest of materials, a hundred times stronger than steel and concrete. I cannot change anything in it; by intense labour, though, I discover in it ever new details, and attain unshakable and eternal truths. Where and what is this ideal structure? A believing philosopher would say: it is in God and His thought.143

The chair of philosophy became in 1938 a chair of the philosophy of mathematics and natural science and a logistic seminar was founded. One month before this transformation the Polish logician Le´sniewski had visited M¨ unster for eleven days, and following all this on 5 December 1938 there was a membership (but in what?). However already on 17 April 1940 the question of the insufficient preparation of the students was raised, from which was concluded an “un-German position” on 19 August 1940. On 23 January 1936 Scholz received the printing cost increase for volumes 3, 4, 5 of the Forschungen zur Logistik. . . —actually it was called Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften. Neue Folge. Unter Mitwirkung von W. Ackermann, F. Bachmann, G. Gentzen, A. Kratzer, herausgegeben von Heinrich Scholz. Stuttgart: S. Hirzel 1937. It can be seen from the DFG records concerning Forschungen zur Logistik und Grundlegung der exakten Wissenschaften,144 that Ludwig Bieberbach had to a large degree promoted the publication of H. Scholz and his collaborators. The Forschungen had received for example RM 5000.-, in particular Heinrich Scholz as reimbursement for the Heine printing costs on 30 July 1938 for volume 4: Gentzen, mathematical foundations research from the German Research Council for example RM 5000.-. And all of this despite the fact that the paper had also appeared in the journal Deutsche Mathematik ! What was kept “secret” was the fact that at the same time the printing of Gentzen’s “New 141 Heinrich

Scholz, “Die mathematische Logik und die Geisteswissenschaften”, Studium Generale 11, no. 3 (1958). 142 “Heinrich Scholz” (Reichserzichungsministerium (REM) [National Ministry of Education] –Personnel file “Sch. 188” of the Berlin Documentation Centre). I haven’t looked into his personal files at the University of M¨ unster. 143 Reprinted in: J. Slupecki, Z zagadnien logiki i filozofii, Warsaw, 1961. 144 Bundesarchiv Koblenz R 73/11060.

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version of the consistency of pure number theory” in Deutsche Mathematik was financed, although this work also appeared in Forschungen.145 This was however clear to the Deutsche Forschungsgemeinschaft (DFG) [German Research Council], in particular to Mr. Griewank. Heinrich Scholz writes laconically: We want to serve the standing of the German spirit in the world through our good work and not wear ourselves down in our struggle for the subsistence that has been a given for our Polish and transatlantic friends.

And the president of the DFG, Mentzel, approved funding also on 5 September 1938: In the year 1938/39 several issues under the general title Research in Logic by Prof. Dr. Heinrich Scholz, M¨ unster/Westphalia were supposed to appear in the journal Deutsche Mathematik. For this purpose we need to grant special funds in the amount of RM 5,000.-.146

After publishing Gentzen’s work Bieberbach rejected the very beautiful and highly praised work “Semiotik” of Hans Hermes, the publication of which in the Forschungen would have been financed by the DFG.147 In short, Scholz and Bieberbach agreed with regard to matters of publication. Ackermann’s work on the consistency of elementary number theory had to be handed over to Mathematische Annalen by Scholz in 1939 because neither Bieberbach nor Scholz received funding from the DFG. In 1940 and 1941 Scholz received two printing grants from the DFG, granted for his Forschungen, but within the framework of the title of Deutsche Mathematik, so that Schr¨ oter’s “Kalk¨ ulbegriff ”, wherein Steck would be sharply criticised, could appear as number 6 of the Forschungen. In 1942 Ackermann’s System der Typenfreien Logik I appeared as number 7. As number 8 Schr¨ oter’s Axiomatisierung der Fregeschen Aussagenkalk¨ ule appeared in 1943 in an edition of 300 copies.148 Bieberbach had financed the publications of the M¨ unster School via the DFG and his cost centre, Deutsche Mathematik. Thereby Vahlen arranged that the physics budget would be burdened with these costs. Who could have imagined that? On 6 June 1936 H. Scholz proposed the “editing of the Frege Nachlass” and on 12 June asked for a stipend for Dr. Hermann Schweitzer. On 13 February 1939 he was denied a mimeograph machine and on 3 December 1941 was hit with a “thunderbolt” on account of “contact with Polish scientists.”149 Scholz requested the mimeograph machine in recognition of the mathematical logic and foundational research through the Polish foundational research, which was supposed to be a reference to the “prestige of the German intellect in the world.” Admittedly, the unster had just Philosophical and Natural Scientific Faculty of the University of M¨ awarded Jan L  ukasiewicz an honorary doctorate on his 60th birthday.150 H. Scholz regarded the Warsaw and Lemberg Schools and the United States with Harvard as 145 Edition

of 300 copies, payment files “Deutsche Mathematik”, Bundesarchiv Koblenz, R 73/15936. 146 Payment files “Deutsche Mathematik”, Bundesarchiv Koblenz, R 73/15936. 147 Edition of 300 copies, payment files “Deutsche Mathematik”, Bundesarchiv Koblenz, R 73/15936. 148 Invoice of the Heine Press in Gr¨ afenhainichen of 29 June 1943. (Bundesarchiv Koblenz R 73/15934.) 149 Cf. the relevant DFG files (Bundesarchiv Koblenz R 73/14484). 150 Official register of the Westf¨ alischen Wilhelms-Universit¨ at M¨ unster of 10 January 1939.

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worldwide leaders in the area of logic.151 He regarded Leibniz, Frege, and Hilbert as the intellectural predecessors of his own institute. He made this claim dramatically: We Germans now stand before a fateful question. Will German research succeed in joining in in the productive cooperation in the recent worldwide logistic research or not? If it succeeds, then something is won, which is worth the blood of the precious stone. If it does not succeed, then a possibility of effectiveness of the first order is lost to German research and an obligation, with which we Germans are burdened more heavily than any other great people, is shrugged off. . .

He referred to his group in M¨ unster, the recognition by Bertrand Russell, the examination of the Frege nachlass and the project of a Frege monograph (together with Friedrich Bachmann) and the “Parisian success” on which he had written.152 Here he explained the approach of scientific philosophy following the paradigm of mathematics and next to Leibniz, Kant, Frege, Russell and Carnap praised above all Reichenbach, Schlick, Tarski, Le´sniewski, Frau Lutman-Kokoszynska and Ajdukiewicz. Then he referred to his co-workers Albrecht Becker and F. Bachmann and his endeavours concerning the Frege nachlass.153 On 5 December 1942 he requested an allocation of paper for the work of Dr. Schr¨ oter154 on the logistic research in the setting of Deutsche Mathematik. This he again sent a reminder of on 6 January 1943. However, his “Hilbert Remembrance” project would already have merely been noted on 16 January 1942. In 1943 his chair in philosophy would be transformed into a “Chair for Mathematical Logic and Foundational Research”.155 156 Thereby, for the first time in 151 Cf.

the application for the support of his Forschungen zur Logik und Grundlegung der exakten Wissenschaften (Bundesarchiv Koblenz 73/11060). 152 Heinrich Scholz, “Der Pariser Kongress f¨ ur Philosophie” under the heading “Kultur der Gegenwart” of the cultural supplement of the K¨ olnische Zeitung, 29 September 1935 (no. 46) and 3 October 1935 (no. 47). 153 On this theme, see also Heinrich Scholz, “Die neue logistische Logik und Wissenschaftslehre”, Forschungen und Fortschritte 11, no. 13 (1935), pp. 169-171. 154 “Karl Schr¨ oter was my teacher; he’s been dead for approximately 10 years (22 August 1977). Till 1948 Schr¨ oter was in M¨ unster; he knew Gentzen well, but never spoke about him, at least concerning personal matters. In a certain manner both belonged to a logical movement. Schr¨ oter, however, was older, born 7 September 1905 in Wiesbaden.” Letter from Gerd Robbels to EMT of 3 July 1988. “After the 1941 promotion with the work ‘Ein allgemeiner Kalk¨ ulbegriff’ and the 1943 habilitation thesis on the axiomatisation of Frege’s propositional calculus. S. was lecturer from 1943 at the University of M¨ unster.” He was strongly influenced by H. Scholz. 155 Why is there no “Institute for Logistic”? Steck had attacked “logistic”. That seems to have had a “late effect”, in that H. Scholz let his seminar be renamed an “Institute for Mathematical Logic and Foundational Research”, in a timely manner just before “logistic” would, following the military linguistic usage, be used for industrial supply systems. (Prof. Dr. Gisbert Hasenjæger to me, 11 March 1997.) 156 Decisive for the actual conversion of Scholz’s institute with approval of the Reich’s education minister, i.e. Bernhard Rust, was most probably a positive report from Ludwig Bieberbach that till now, however, has not been found. On H. Scholz and his relation to the Polish scientists in their rˆ ole in his memorandum to Rust on the restructuring of the institute, cf. Volker Peckhaus, “Moral integrity during a difficult period: Beth and Scholz”, Philosophia Scientiæ 3, no. 4 (1998/99), pp. 151-173. But elsewhere too there are entirely serious changes. An example: On 1 April 1943, 20 months before the end of the “Third Reich”, in the midst of the confusion of a world war, the “Institute for the History of Natural Sciences” was founded in Frankfurt am Main. Almost a year later, in May 1944, it was completely destroyed by the hail of Allied bombs. The Institute would be founded by Willy Hartner, who had been a guest professor in Harvard in 1935. Hartner was a specialist on the astrolabe.

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Germany, there was a philosophical institute which philosophically supported mathematics and itself carried out mathematics in order to be able to create for itself a standard of exactness. The chair and the seminar would first be redesignated as a proper institute in the 1950s. The financing of the Hilbert Remembrance was later taken up by the DMV. Nevertheless the extraordinary Professor of Philosophy in Jena, Dr. Paul A. Linke— one of the few who had corresponded with Gottlob Frege—published an expert opinion on 25 February 1944: “The philosophical faculty completely endorses the projected memorandum,” namely that of all people Heinrich Scholz was the proper man for the re-occupation of the ordinary chair of philosophy (succeeding Bruno Bauch!) initially with a recommendation to step in for the Altona librarian Dr. phil. habil. Helmut Groos. Scholz stood at the top of the valuators before Th. Hæring, Herrigel, Glockner, Spranger, Max Wundt, all—according to today’s research literature—150% convinced National Socialists!—and others followed, and they praised one—I can’t stop being astonished—who was truly an “independent thinker”, “researcher”, and strict “Lutheran”, strong opponent of idealism and consistent determinist. Scholz was regarded by all as a deep, true and credible representative of logic. What Max Planck was for physics, Heinrich Scholz was for the field of mathematical logic! Scholz’s reports on the books of Groos, Der deutsche Idealismus und das Christentum (1927) and Die Konsequenten und Inkonsequenten des Determinismus (1931), reveal an idea of logic as an unyielding controlling authority, whose thought can only be sustained by one of Wittgensteinian dimensions. While Scholz was institutionalising his institute at the University in M¨ unster under the rule of National Socialist groups, he learned Polish to be able to follow the research of the Warsaw School. On page 73 of his Geschichte der Logik (1931) he had written: “And generally Poland has become a leading nation. . . of logistic research in the last decade.” In December 1938 he travelled to Warsaw in order to award L  ukasiewicz the degree and honorary doctorate from the University of M¨ unster. L  ukasiewicz celebrated his 60th birhday. “Clearly the recommendation from L  ukasiewicz played an important rˆole in the granting to Scholz of the special teaching post for logistical logic and foundational research” (Peter Schreiber).157 In 1939 Heinrich Scholz published in Organon 3 of the Mianowski Institute the 30-page essay, “Sprechen und Denken. Ein Bericht u ¨ber neue gemeinsame Ziele der polnischen und der deutschen Grundlagenforschung” [Language and thought. A report on new common goals of the Polish and German foundational research]. Behind the scenes Heinrich Scholz worked for the maintenance of international relations.158 He tried to save a logician and with that to preserve the honour of his institute. The fear of the consequences of the war, among others the murder of so many Polish mathematicians, for the mathematical communication, would be clear in the endeavours of the M¨ unster logician Scholz to make possible the resettlement of the Polish logician J. L  ukasiewicz, with whom he worked closely in M¨ unster. Scholz also engaged Bieberbach in this. 157 Cf.

¨ Peter Schreiber, “Uber Beziehungen zwischen Heinrich Scholz und polnischen Logikern”, in: Michæl Toepell (ed.), Mathematik im Wandel. Anregungen zu einem f¨ acher¨ ubergreifenden Mathematikunterricht. Band 1, Franzbecker Verlag, Hildesheim, 1998. 158 I have not attempted to find out if his actions cannot also be religiously motivated. The point here, I believe, is purely the rescue of a logician.

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On Bieberbach’s half-hearted mediation attempt one reads in the minutes of the general assembly of the Berlin Academy of 16 November 1939: Mr. Bieberbach reported that he was excited to move for support of the Polish scholar Prof. L  ukasiewicz. Mr. Konrad Meyer opposed any support of Polish scholars because it was exactly the circle of the Polish intelligentsia that had led the peoples’ resistance most fiercely. . . It was therefore decided merely to report to the Reichsminister of the state of affairs, however, without any recommenda ukasiewicz did finally get his freedom then this could not tion. . . If in the end L in any case be attributed to the efforts of the Berlin Academy, in which men like the co-author of the fascist “General Plan, East” Konrad Meyer would have a say.159

Heinrich Scholz praised further the “crystalline clarity” of the papers of J. L  ukasiewicz160 and the Warsaw School—for example in the Jahrbuch u ¨ber die Fortschritte der Mathematik (1941)161 —he particularly liked reviewing the writings of Soboci´ nski, Le´sniewski, Wladyslaw Hetpur and many others. We must not forget that the head of the planning division of the Reich security head office, Prof. Dr. Konrad Meyer, on Himmler’s order, put in place the “General Plan, East”, which took for granted the “final solution of the Jewish question” for the Germanisation policy in the east.162 That Bieberbach—whether bashful and cautious is not the question here—followed in general the wishes of Scholz and intervened before the SS on behalf of the German friendly L  ukasiewicz163 I do regard as noteworthy. Heinrich Scholz reported in a letter to Evert Willem Beth of 24 August 1946: I have not only saved Mr. and Mrs L  ukasiewicz, but, in a sort of underground manner, I have also been able to maintain the connection between Mr. Tarski in the USA and his wife and two children left behind in Warsaw until with my help Mrs. Tarski finally secured exit visas to the USA for herself and her 159 Reinhard Siegmund-Schultze, “Uber ¨ die Haltung deutscher Mathematiker zur faschistischen Expansions- und Okkupationspolitik in Europa”, in: Martina Tschirner, Heinz-Werner G¨ obel (eds.), Wissenschaft im Krieg– Krieg in der Wissenschaft, 2nd edition, Interdisziplin¨ are Arbeitsgruppe Friedens- und Abr¨ ustungsforschung an der Philipps-Universit¨ at Marburg, 1992; here, p. 193. 160 On L  ukasiewisz’s modes of thought in mathematical logic cf. Jan Wolenski, “On Tarski’s background”, in: Jaakko Hintikka, From Dedekind to G¨ odel, Reidel, Kluwer, 1995. 161 65 (1939), p. 23. 162 The rational planner Konrad Meyer became ordinary professor for state planning at Technical University Hannover in 1956, and member of the appropriate academy in 1957.— Earlier Konrad Meyer served the Reichsf¨ uhrer SS with the rank of SS regiment leader. From 1942 on he was planning representative for the settlement and land reform by the Reich manager for agrarian politics, by RMEuL, and by the Reich’s farmers’ leader and manager of settlement committee for the occupied eastern regions (cited from Buchheim, “Feststellung des Reichskommissars f¨ ur die Festigung des Deutschen Volkstums”, in: Gutachten des Instituts f¨ ur Zeitgeschichte, Munich, 1958; here, p. 274). On the mad plan of an ethnic cleansing, cf. Mechthild R¨ ossler and Sabine Schleiermacher (eds.), Der “Generalplan Ost”. Hauptlinien der nationalsozialistischen Planungsund Vernichtungspolitik, Akademie Verlag, Berlin, 1993; Bruno Wasser, Himmlers Raumplanung im Osten, Birkh¨ auser Verlag, Basel, 1993; G¨ otz Aly, “Endl¨ osung”, Fischer Verlag, Frankfurt am Main, 1995. For Konrad Meyer’s biography, see pp. 378 and 175 in: Notker Hammerstein, Die Deutsche Forschungsgemeinschaft in der Weimarer Republik und im Dritten Reich. Wissenschaftspolitik in Republik und Diktatur, C.H. Beck, Munich, 1999. 163 L  ukasiewicz had once received a sinecure through a formal position in M¨ uhlheim an der Ruhr, arranged through the Lord Mayor Hasenjæger (the father of the logician Gisbert Hasenjæger). As L  ukasiewicz later received a professorship in Dublin, he was reproached on account of his “collaboration”. (Letter from Gisbert Hasenjæger to me of 13 June 1997.)

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children by the most difficult indirect means.164 I have finally saved one of the best theological students of L  ukasiewicz, Mr. Salamucha, from the concentration camp, before the worst had happened. It is an unfortunate accident, which I will never forget, that this remarkable man was murdered in the battles surrounding Warsaw in August 1943.∗ (∗ Not by the Germans!). . . I will not relate here all of what I have risked. But I will be allowed to say that the Gestapo visited me three times and that our Minister let it be known to me after the release of Mr. S. from the concentration camp that if the incident repeated itself they would start disciplinary proceedings against me with the goal of relieving me of my position.165

This would be remembered in the 1960s: During the war he would rescue the Cracow Professor Salamucha from the concentration camp, for which he was threatened with removal from office if it were ever repeated. In July 1944 he succeeded in bringing Professor L  ukasiewicz and his wife to M¨ unster just before the Russian push into Warsaw.166

In November 1943 H. Scholz lectured in Z¨ urich and Basel on “Logik, Grammatik, Metaphysik”,167 , where he defended the concept of general validity and a welldefined truth—also in opposition to Nicolai Hartmann—within a formal language with the ideas of Tarski, Carnap, and Russell. But he was fighting on two fronts. He would not be denounced as a positivist and was open for the “Metaphysik als strenge Wissenschaft”168 [Metaphysics as a strict science]. He was certainly a metaphysician and for this reason is not opposed to a consequent logic. His battle with Steck in 1943 was about this. Only Steck would not have noticed how closely inwardly related Heinrich Scholz’s ideas were with those of the Lemberg-Warsaw School, especially in their approach to metaphysical questions.169 All resentment and feelings of revulsion with regard to Scholz’s German nationalistic point of view should on account of the facts be regarded skeptically in evaluating Scholz’s logic and politics.170 Scholz was always opposed to the Treaty of Versailles and the Weimar Republic. But from his Protestant theology he acquired 164 “I am perfectly aware that you did your best to help not only me but all your friends and colleagues in Poland. . . ; and this applied even to people whom you have never met in your life, although maintaining relations with some of them—because of their so-called racial origin— constituted a crime under the Nazi regime. . . there are few human traits which I hold in as great esteem as I do inner integrity and civil courage; that these virtues are essential elements of your character, your actions during the past few years have proved beyond doubt.” (Alfred Tarski, letter to H. Scholz from 21 October 1946. Scholz-Archive at the Institute for Mathematical Logic at M¨ unster University.) The friendship of H. Scholz and J. L  ukasiewicz is well documented in Hans-Chrisoph Schmidt Am Busch, Kai F. Wehmeier, “On the relations between Heinrich Scholz and Jan L  ukasiewicz,” History and Philosophy of Logic 28 (February 2007), pp. 67-81. 165 Arie L. Molendijk, Aus dem Dunklen ins Helle. Wissenschaft und Theologie im Denken von Heinrich Scholz, Rodopi, Amsterdam, 1991, p. 61. 166 Cf. the longer note at the end of the chapter. 167 Heinrich Scholz “Logik, Grammatik, Metaphysik”, Archiv f¨ ur Rechts- und Sozialphilosophie 36 (1943/44), pp. 393-433; new printing with minor changes in: Archiv f¨ ur Philosophie 1 (1947), pp. 39-80. Possibly he would have been invited on the invitation of Karl Barth. 168 In one breath “Carnap, Dubislav, Scholz, Burkamp” would be named by Gerhard Lehmann, Die deutsche Philosophie der Gegenwart, Kr¨ oner, Stuttgart, 1943. In the chapter “Rudolf Carnap” Heinrich Scholz is named as one who “energetically defended” himself against a necessary connection of logistic and positivism (p. 293). 169 On this unfortunately there is still no work. Cf. at least the contributions in Szaniawski (ed.), The Vienna Circle and the Lvov-Warsaw School, Kluwer, Dordrecht, 1989. 170 Heinrich Scholz is justly famous for having institutionalised logic with his institute outside mathematics and philosophy, and this he achieved under National Socialism. He had attracted

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—in contrast to his colleague Emmanuel Hirsch and many others—his strength to oppose the Nazis. 16. Bieberbach, Max Steck and Jænsch Bieberbach read Max Steck’s writings171 and discovered things that displeased him. Eventually he worked on a lecture or essay opposing Steck. It exists in three failed attempts in the partial Nachlass. The first text in the folder is named “Zur Ontologie der Mathematik” [On the ontology of mathematics] and consists of 27 pages written on used paper—e.g. on a letter from Georg Linck of 25 March 1944. He came out against the intrusion of “Dinglerism” into mathematics by Steck and wrote: This Dinglerism also attempts thus with Steck to make an entrance into mathematics. Only in order to counter (during the times to counter) it I hold going into Steck’s theses to be appropriate.

In the final sentence one reads: And we therefore maintain that the axiomatic method has taught us to recognise the true ideas and ideals, the mathematical ideas in their true form and shape. And that this succeeds is not least to the credit of the German researchers, above all Hilbert.

The second is called “Form, Inhalt, Gestalt und Sinn” [Form, content, Gestalt and meaning], and consists of 10 pages. The text is not complete or self-contained. There one reads, for example: Pure mathematics has already and more recently with particularly vocal efforts been subject to the accusations that it bogs down in formal inferences and that it lacks contentuality in its procedural methods. Thus writes the Munich docent Max Steck on page 4 in issue 5 (1941) of the journal Die Gestalt. . .

And there Steck referred to a passage in Bertrand Russell and recorded carefully “(translated by L. Bieberbach)”. It seems to me that this reference alone already suffices to show that Bieberbach felt himself attacked after the motto: Steck holds me to blame for the dissemination of these ideas; he kicked the dog but meant the master. A further indication of Bieberbach’s vanity is how carefully in the folder every mention of his name in newspapers and magazines has been collected. many students from Ernst Tugendhat to Karl Schr¨ oter to Hans Hermes. Some of these emancipated themselves and achieved their results first after the war, though this led to the emigrants having felt like “The Peripherals” (Ernst Gr¨ unfeld, Die Peripheren. Ein Kapitel Soziologie, Amsterdam, 1939), for they had received no offers of positions in Germany. The “state of the art” of H. Scholz in logic was provincial; one simply has to compare the publications in Germany and in the USA from 1945 to 1965. It has remained so. Not be underestimated against this is the mediatory function of Scholz and his school in M¨ unster, because even the conservative philosophy officials like Odo Marquard, Hermann L¨ ubbe, and Joachim Ritter learned at least modern logic from Leibniz via Bolzano and Frege to Brouwer, Hilbert and Weyl. 171 Max Steck found in his review of Bieberbach’s “V¨ olkische Verwurzelung der Wissenschaft” [Folkish roots of science] (1940) that the application of Jænsch’s type theory to the personalities of some great mathematicians has proven itself “extraordinarily valuable” (Geistige Arbeit 4, 20, February 1941, p. 4). Thereby Bieberbach saw the limits of the “methods of integration typology”: “Its penetration with the scientific results of race psychology is called for.” He saw this lecture of Bieberbach in connection with both of his others, “Pers¨ onlichkeitsstruktur und mathematisches Schaffen” [Personality structure and mathematical creation] (1934) and “Stilarten mathematischen Schaffens” [Styles of mathematical creation] (1934) and desired a larger work on the subject from the Berlin professor. But Bieberbach was in no way pleased with this review.

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187

One notes by inspection of this sketch of a rebuttal of Steck that Bieberbach failed in his retort. First he goes into the Gestalt volume, cites Speiser, van der Wærden and Kepler, cites and defends Hilbert, but stays at the same level of argumentation as Steck. Then Steck’s book appears, and Bieberbach probably saw as the best solution to appoint Heinrich Scholz as an expert, as a mathematician to reply to the attack. Steck was not always opposed to a discussion of Hilbert’s views. In a review of Heinrich D¨ orrie’s Triumpf der Mathematik,172 he wrote: The axiomatic research method in the sense of Hilbert has not in general come into its own. Especially because of its foundational significance for the space problem the author had to go into it somewhat.173

How can one approach the problem of an adequate presentation of Bieberbach in Nazi times and of the subject history, which is a history of its institutions, associations and unions? Not everything can be discussed as vanity, opportunism, ambition and whatever other individual vices there are. The philosopher Prof. Dr. Eberhard Simons suggested the concept of “institutional conspiracy” [Institutionsintrige], and in this connection stimulated a “conspiracy research”.174 The concept of a conspiracy and its necessity comes from the theatre and comedy, but there few men turn whole situations upside down. How did mathematics and its institutions present themselves? Which forms were of the frail and vulnerable sort, that a few succeeded in “forcing into line”, in which a few were involved? From which realities did the Nazis proceed, from which normalities, which would be declared for it? What kind of network of connections was there and how would it be used? These questions about processes, actors, developments, etc., are all for a long time unanswered. In his essay “Wege und Ziele der Psychologie in Deutschland”,175 Jænsch in no way mentioned or cited the efforts of Ludwig Bieberbach. On the contrary: Continuing intuitions, which the hereditary conditioning of all actual characteristics maintain absolutely, may prove to be mere hyperbola, and, like all exaggerations, signify more a hindrance than a promotion of the duration of the movement; in this case therefore above all, because each factually unfounded “too much” in the propaganda for the heritability viewpoint can discredit and maim the so necessary medical and cultural-political measures which are directed toward the healing of the damage caused by the environment.

Why did Jænsch use a word here that also marks a mathematical state of affairs? Why did he not use the Quintilian expression for exaggerated thought, the “hyperbole”, which recognises the steps of enlargement (amplificatio) and the gradual increase through comparison, etc., where thus the credibility of that expressed should retreat in favour of an evident insight of an impression? Was this a reference to Bieberbach? He expands on this on page 19: There each folkish cultural expression which is the prevalent basic form in the relevant people, or too—as in Germany, England, and France—of the fruitful tension-relations of two basic forms, so the type theory gives a key to the hand to the understanding of the appearances of the folk experience and at the same time an instruction to the fulfilment of values. The distant goal of this endeavour 172 F.

Hirt, Breslau, 1933. Arbeit, no. 9, 5 May 1934. 174 Interview, in: Dilemma—Zeitung des Instituts f¨ ur Soziologie, no. 9, 23 July 1996. 175 The American Journal of Psychology, 1, nos. 1-4 (1937), pp. 1-22. 173 Geistige

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is a rich concrete human culture in which all peoples bring their peculiarity into sharper expression and respect and supplement one another, thanks precisely to this peculiarity—in contrast to the impoverished “abstract” human culture which is exclusively founded upon what is the same in all men, i.e. above all the subordinate in them.

Jænsch did not busy himself with the set pieces from the race theorist G¨ unther as did Bieberbach, but rather distanced himself from it. Emphatically, misunderstanding must above all be opposed; the consideration of typological and racial typological viewpoints could lead to the result that suitable professional candidates would therefore be rejected because they do not meet with our leading typological and racial typological ideas.

Jænsch wanted that considered, but “without mutilating the particular standards of professional selection”.176 Jænsch is far removed from Bieberbach’s ideas, particularly the use of these ideas as the basis of the student boycott against Landau. 17. Bieberbach and Erich Jænsch The Deutsche Allgemeine Zeitung of 15 April 1934 wrote on Bieberbach’s speech: The mathematical style should be full of blood and be a proper object [of study] for the race- and type-theory of today. Its result is that the political rootedness gives thought the style, the practical requirement, the rejection of non-Aryan mathematics, which cannot be considered as German, whereby this rejection will also expressly be extended over the shapes of the past.

From there it is a short path, as the Bavarian culture minister writes: Reason—what belongs to it? Logic, calculation, speculation, banks, bourses, interest, dividends, capitalism, career, string-pulling, usury, marxism, bolshevism, rogues, and villains.177

Here the Jewish cult of the intellect and formalist logic would be played out against German substance-logic, which culminated in Himmler’s remarks that it would be good if the Germans were na¨ıve. Erich Jænsch developed in his “Grundformen menschlichen Seins. Mit Ber¨ ucksichtigung ihrer Beziehungen zu Biologie und Medizin, zu Kulturphilosophie und P¨ adagogik”178 [Basic forms of human existence. With consideration of their connections to biology and medicine, to cultural philosophy and pædagogy] a human typology just as Klages, Kretzschmer, etc., had. Why were typologies so interesting for this time? What did one want to say with integrative and disintegrative types?179 Jænsch came out against Heidegger’s view of Kant’s concept of imagination, because it opposed an anthropological position. Jænsch wanted to contribute absolutely “Zur Philosophie der Wahrnehmung und psychologischen Grundlegung der 176 P.

22. 118 in: Hans Schemm spricht. Seine Reden und sein Werk, G. Kahl-Furthmann, Bayreuth, 1935. 178 Otto Elsner, Berlin, 1929. 179 On the biography and works of the Nazi philosopher Jænsch, cf. Chapter 6.2 in Norbert Kapferer, Die Nazifizierung der Philosophie an der Universit¨ at Breslau 1933-1945, LIT-Verlag, M¨ unster, 2001. 177 P.

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189

Erkenntnislehre”180 [To the philosophy of perception and the psychological foundation of the theory of knowledge]. Thus in the same volume he proceeded against “the latent Cartesianism of modern science and its structural-typological foundation.” Here he criticised “pure thought” and the “mathematical” interpretation of those facts, thus a conception after the example of mathematics. Every defence of the “law licence of a priorism” is like French to him. A detailed review of the development of Jænsch’s philosophy up to its conversion into the philosophy of the Nazis would interest me very much. But where can one read these things? Mind you, in an “internal German” source one finds two agreeing larger references—again from nonexperts—in E.R. Jænsch and Fritz Althoff, “Mathematisches Denken und Seelenform. Vortr¨ age der P¨ adagogik und v¨ olkischen Neugestaltung des mathematischen Unterrichts”.181 The investigations are justified with a quotation from Felix Klein: Psychology stands first in the beginnings of the sort of investigations which I with many colleagues happily greet. Because we hope that in our science and its business many differences of opinion which now necessarily remain unresolved will disappear if we will first be more exactly informed on the psychological preconditions of mathematical thought and their individual differences.182

This work is also dedicated to him: “To the memory of Felix Klein as a German educator and early pioneer of the German-natured science.” Jænsch suggested in the foreword that the paper stood to his book Der Gegenstand as a special case to the general: Bieberbach’s expositions of the various style-forms of mathematical work can provide the student and future teacher with the immediate starting points.183

“Click your heels!” orders Jænsch, and I read further: As Bieberbach has already occasionally referred to our earlier typological works, so conversely his explanations of mathematical style contribute in this domain of thinking to the extension and deepening of our depiction of personality forms.184

But the relation of race typology and style-forms is not more exactly worked out. The investigations of Ludwig Bieberbach show that the results of our investigations, which were already carried through in the years 1930/31, also provide a thoroughly accurate picture of the great mathematicians, an agreement which must be all the more remarkable, as the results were found in completely mutual independence. We serve ourselves—exactly like Bieberbach—to the description of the various forms of the mathematical talent of the results of the typology of E.R. Jænsch, which is extrordinarily well-suited to capture the basic forms themselves of such special talents as those for mathematics even in their finer exceptions.185

180 In: Erich Jænsch and associates (eds.), Uber ¨ den Aufbau des Bewußtseins. Teil 1, A. Barth, Leipzig, 1930. 181 Supplement 2 to Zeitschrift f¨ ur angewandte Psychologie und Charakterkunde, edited by Otto Klemm and Philipp Lersch, 1939. 182 Felix Klein, Uber ¨ Arithmetisierung der Mathematik, 1895. 183 P. 25. 184 P. 67. 185 P. 72.

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Another example: Bernhard Bavink, Ergebnisse und Probleme der Naturwissenschaft.186 It was the most successful popular book on natural science under National Socialism. And it shows how uninfluenced by National Socialism and in contradiction to some of its Paladins of the “embattled science” one could write undisturbed solidly on science within a definite framework. Bavink referred to himself mostly as a thinker whose main work was a product of the days of the empire or the Weimar Republic. Relativity theory would be mentioned with praise, and it would be pointed out that every justification for “cheap propaganda” against it lacked; Dingler like Carnap was referred to with praise; Jænsch would be referred to. It would be established that the race concept had its justification; however everything of this sort was in such a state of flux that one couldn’t commit oneself. There is no talk in any case of Jews, Bieberbach will not be mentioned; opposed to that Klein and Hilbert are praised. Bavink is no opponent of National Socialism and declared support for the justification of race theory, therefore especially his book was an example of the nonideological sort and manner to drive science in National Socialist times. Max Planck grandly intervened in the discussions of the “embattled science” with lectures. In the lecture “Die Physik im Kampf um die Weltanschaung” [Physics in battle over the world view] of 6 March 1935 he took a mathematical example about brooding over meaning and nonsense from criteria external to the subject which would be brought in from outside. The lecture of 27 November 1936, “Vom Wesen der Willensfreiheit” [On the nature of free will], is entirely free of “racelike” requirements and gave folkish views a negative reply through each disregard; i.e. he judged these thoughts to be irrelevant. 18. Steck’s Attack on Hilbert Leads to Bieberbach’s Commissioning a Defence of Mathematical Logic by H. Scholz and Publishing It in Deutsche Mathematik The accusation of any relativism was dangerous because it had been written with the blessing of the “Official Party Examining Board for the Protection of NS-Literature”: Relativism is an attitude of mind especially clearly embodying the modern liberalistic-individualistic helplessness and lack of principle deliberately taken up in a calculating manner by Jews (especially section 605) out of disposition and to subversive ends.187

Anyone charged with this would be a lecturer no longer. Eduard May had written an award-winning book, Am Abgrund des Relativismus [At the abyss of relativism], awarded the prize of the Prussian Academy of Sciences in Berlin.188 After May thanked Nicolai Hartmann and Eduard Spranger (1941), he went flat out: No small thanks I owe the President of the Prussian Academy of Sciences, SS Oberf¨ uhrer Prof. Dr. Theodor Vahlen, who has not shied away from the trouble to once again read through my work in galleys. . . I must therefore emphasise with particular vigour, because, without the explanation which I constructed from the 186 8th edition, S. Hirzel, Leipzig, 1944 (first edition 1914). Dedicated to German science. 25,000 subscriptions, the editions of 1940 and 1941 were sold out: because of the paper shortage the printing only satisfied half the subscriptions. 187 Meyersche Lexikon (the so-called Nazi-Meyer ) (1937), vol. 9, section 290. 188 Dr. Georg L¨ uttke, Berlin, 1941.

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191

works and letters of Dingler, it would not have been possible for me to capture the relativism problem in its complete depth and to treat it with such beautiful success. So my thanks apply not last to the highly meritorious and genuine founder of the exact science, the great “master of the teaching of thinking”, as Philipp Lenard has with justice called him.189

The battle against relativism would be taken up on behalf of Dingler and his hyper-empirical, found in the pre-scientific experience of the a priori and willfully converted into a definite course of action. However, instead of pursuing forms of the a priori not founded in reality-connections, the experience of positive action would be operative and would penetrate, and in this way the volunteerist and activist priority of philosophy would suffice: only so can the collapse of science be avoided. The will is the absolute grounds for validity and it is the validating principle of logic! Thereby May came out against the theory of racial-folkish dependence of knowledge acquisition as it appeared and was represented to him by Krieck, B¨ ohm, and E.R. Jænsch. Instead inconsequential quotations and passages from letters from P. Lenard would be cited as counterarguments. Popper, whose ongoing relativistic erection of a pragmatism (pp. 86ff.) in the modern physics which Einstein “perceived as one it its best supports,” would be described as the bitterest enemy of Dingler (p. 276). It would therefore be criticised by May (pp. 110ff.). Only Dingler and Lenard are man enough to order a halt to this nonsense. The high point of the modern decline of the a priori however received its justification by “M. Schlick, H. Riechenbach, R. Carnap, O. Neurath, Ph. Frank, H. Hahn, and K. Popper” (p. 197). The fanatic battler against the a priori with “open confession to unrestricted reign of relativism” was R. Carnap: Take for example from Carnap’s main work the final conclusions, and one comes to an I that receives as a truly god- and reality-abandoning “solus ipse” [solipsism] out of a distant, foreign, to him never even analogously recognisable reality, individual sensory data, with which it drives its formula game with the aid of self-made tautological transformation rules, strictly considered therein, only each datum is also conceded a winning position. In place of a propositional structure situated on the truth-line over this and that part of reality there enters a multitude of mutually contradictory combining schemata, from which no single something out of reality is pronounced and which consequently are all theoretically equally justified.

But as Gentzen did not participate in this discussion because he was doing military service and therefore only worked pro forma at a university, the accusations of a representative of “German mathematics” caught Heinrich Scholz, the dead Hilbert and his students. Steck, May, Dingler, Th¨ uring and many others definitely demanded, clearly and expressly, their elimination in favour of a “German line”. This “German line” of mathematics (N. von Cues (Cusanus), Kepler, D¨ urer, Leibniz, Lambert, etc.) must again be taught in German universities, and “the formalists and logicists who deny this line and its achievements must fall.”190 There was at that time cause to ask anxiously on which fields and to which front the representatives of these schools of thought of mathematics should fall. Theodor Hæring—he also reported on the question of Bauch’s successor in Jena, as did Scholz, Max 189 G¨ ottingen, 190 Max

New Years 1940/41, Eduard May. Steck, Das Hautproblem der Mathematik, Georg L¨ uttke, Berlin, 1942, p. 210.

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Wundt and many others—welcomed Steck’s theses in a review.191 Only Edmund Hlawka and J. Jorgensen192 criticised Steck. Hans Hofer wrote: In Zentralblatt f¨ ur Mathematik, similarly to both of the last mentioned, Heinrich Scholz judged in his statement on Steck’s mathematical-historical work, “Unbekannte Briefe Freges u ¨ ber die Grundlagen der Geometrie und Antwortbrief Hilberts an Frege” [Unknown letters from Frege on the foundations of geometry and Hilbert’s answer to Frege]: “To be sure Hilbert was praised, but this praise would meanwhile be paralysed through a revised opinion of the editor, who makes exactly the same Hilbert answerable for the decadence of the mathematical spirit in Germany.” Steck has not at all noticed Hilbert’s main contribution, but rather has concealed through new “ontological” and “epistemological” formulations of questions. . . on which one can discuss for a long time to no end. On the actual correspondence between Frege and Hilbert and the location of the remaining unabridged originals unknown to him, the editor could have been able to completely inform him through a question to the referee” (Zentralblatt, vol. 36, 1942). And Scholz did not go on into the open and in many places intended antisemitism of Steck. Max Bense hid his book, Geist der Mathematik. Abschnitte aus der Philosophie der Arithmetik und Geometrie (Munich/Berlin, 1934), on the title page behind a quotation from L. Bieberbach: “As evidence for the folkish necessity of mathematics one refers mostly to the applications. It seems to me it suffices to refer to the fact that the folkish characterisation forcefully reveals itself in mathematical creation.” And one should not seek after the assumptions of the “revelations”. But Steck did not understand: he sensed “universalism” and criticised Bense: “Thereby he overvalues the accomplishments of this school (of logicism), which lie mainly in the logical and not in the mathematical regions, and Bense wishes to allow them” (Geist der Zeit 7, 5 May 1940). Bense however knows Bieberbach is behind him.

Scholz would be properly attacked by Steck and companions through their Hilbert criticism. Bieberbach understood immediately: Steck’s attacks on Hilbert gave cause in the end for the editor of Deutsche Mathematik, L. Bieberbach, to invite Heinrich Scholz—member of the M¨ unster School— to report on the “effective meaning of Hilbert’s foundational research” (Heinrich Scholz, “Was will die formalistische Grundlagenforschung?”, Deutsche Mathematik 7, pp. 206-248).193

Why, however, did Bieberbach not have a mathematician respond to Steck’s attack against the generally respected Hilbert? Why did he strengthen the position of Heinrich Scholz? Why did he have this appear as the official voice in Deutsche Mathematik ? Paul Bernays decisively came to the aid of Scholz in 1944 in the Journal of Symbolic Logic: Max Steck, in his book Das Hauptproblem der Mathematik (Berlin, 1942), attacked the formal methods of Hilbert’s axiomatics and metamathematics, characterizing them as a phenomenon of decadence. The author was asked to answer this attack by a report on the effective meaning of the Hilbertian method of 191 Zeitschrift

f¨ ur Deutsche Kulturphilosophie 9 (1943), pp. 66ff. professor of philosophy in Copenhagen, is regarded as a practitioner of that sort of scientific philosophising like that of the Vienna Circle represented in the journal Erkenntnis. Jorgensen was considered anti-Nazi and was interred a long time by the Germans. (Cf. Mogens Blegvad, “Vienna, Warsaw, Copenhagen”, in : Klemens Szaniawski (ed.), The Vienna Circle and the Lvov-Warsaw School, Kluwer, Dordrecht, 1989; here: p. 4.) 193 Cf. the longer note at the end of the chapter. 192 Jorgensen,

18. SCHOLZ DEFENDS LOGIC

foundational research, and has taken this as an opportunity to render prominent some main aspects of recent foundational investigations. Concerning axiomatics he emphasizes the requirement, in connection with the theory of models, that the primitive or indefined concepts of an axiom system under consideration—the example is used of Peano’s axioms for number theory— be replaced by corresponding variables, and also the requirement, which apppears in connection with questions of deductive independence, of making the rules of logical inference precise. Apropos of the first point Scholz brings out the inadequacy of the often used manner of characterizing formal axiomatics by speaking of “implicit definitions”. The second point is illustrated by the instance of the principle of duality in projective geometry. In fact, for the proof of this principle it is essential to see that the automorphism effected upon the axioms of projective geometry by the dual transformation of the primitive concepts extends to the consequences derived from the axioms by logical deduction, in virtue of the circumstance that the logical deduction makes no reference to other properties of the primitive concepts than those formulated by the axioms. However, in order to make this character of logical deduction really evident (not merely plausible) it is necessary to state explicitly the processes by which it is constituted, and this means formalizing the logical deduction. Generally, the need for strengthening the preciseness even beyond the degree which usually suffices in mathematics is noticed by Scholz as a characteristic feature of metamathematical inquiries. As an instance of what can be accomplished by such increased preciseness he mentions the result concerning the non-existence of certain decision procedures, obtained by rendering precise the notion of a decision procedure. Two examples of method are discussed in detail by Scholz. One of these is Dedekind’s logical definition of an enumerable class (“einfach unendliches System”) as given in 70 1/2 1(§§5-6), together with the passage therefrom to Peano’s axioms for number theory and to the ordering relation for numbers. The other is the characterization of the system of implication logic, on the one hand by its formal structure (the calculus being defined as a certain quadruplet of sets), on the other hand with respect to its interpretation. The way of doing the ideas of Tarski is in particular directed to the aim of avoiding references to processes (actions). Discussing the Hilbert problem proper in metamathematics, Scholz expresses a view concerning the failure of Frege’s system of classical logic similar to that expressed by Quine in his VII 100 (end of §VII). He says that by this failure our feeling for what is evident has been discredited. On account of this situation, he argues in accordance with Hilbert’s point of view, it would be very unsatisfactory simply to transfer our trust to a revised system of classical logic without something’s being done of a decisive nature by which, so far as it is possible, our renewed trust shall be justified. And he maintains Hilbert’s confidence that a proof of non-contradiction for a revised system of classical logic will succeed in such a way that the consistency of our trust in the revised logic will be implied by it. At the end of the paper, in addition, the Eudoxus theory of proportions is treated as the first known historical instance of a foundational inquiry which had already the tendency of abstraction that appears in the modern theories of abstract algebra and foundations.

What did Scholz himself write about this? Baffling, however, is the assessment of Hilbert, which makes the doyen of German mathematics answerable for a supposed decadence of the mathematical spirit in

193

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the German world. With reference to this judgement the managing editor of Deutsche Mathematik, Herr Bieberbach, has asked me to prepare a short report on the effective meaning of Hilbert’s foundational research. (pp. 207ff.)

And on the core of the Frege-Hilbert relation he wrote on pp. 242ff.: Frege had wanted to lay the basis of our belief on the truth of classical logic. . . It is, however, undeniable that this attempt has not succeeded. Nevertheless it is to be hoped that Frege’s attempt will still, after 40 years, enjoy a kind of resurrection, which satisfies the present day demands. . . Then, however, there remains to us still only the Hilbert Programme.

In Zentralblatt, W. Ackermann reviewed the above mentioned work of Scholz: The essay contains a detailed explanation of the methods and goals of the formalist foundational research, in response to a criticism which M. Steck (Das Hauptproblem der Mathematik, Berlin, 1942) had exercised against this. . . The opinion that with formalism mathematics degenerates into a meaningless game is to be sharply rejected, but rather the interpretation of the calculus should be recognised straightaway as an unconditional requirement of a complete formalised theory. . . It is to be stressed that, regarding the evidence-feeling [intuition] with which Max Steck opposes formalised foundational research, the passage from Frege to Hilbert, of truth to consistency, would have been produced directly through a foundational crisis given rise to by the evidence-feeling.194

And this means that the editor of the Frege-Hilbert correspondence could have no understanding of the matter treated. Scholz would have been applauded by the Nazis if he had published Frege’s diary195 —he was in possession of a copy of the compromised diary since 3 or 4 November 1938, a copy which had been placed at his disposal by Frege’s adopted son, Alfred Frege (1903-1945). But Scholz saw clearly that the publication of Frege’s diary would have been only a senseless, fruitless, even damaging act in this situation. On the contrary: In the propaganda newspaper Weltwacht der Deutschen 196 Scholz praised only the logical contents of Frege’s teachings. And at the end one reads: The path of the mind is the detour. Outstanding Polish researchers discovered Frege soon after the world war and on the basis that he had created, and in the deeply thought through style which it is worthy of, created this new form of philosophising. In Germany the M¨ unster School is till today the only recognised advanced school for this new form of philosophising. Of philosophising in the Leibnizian and Fregean sense. It has still not been carried through as much as it can be. Not by far. But we will not grow weary of securing for it its place in the sun, of which it has an undeniable right. And ever again we will say that the matter to which we deploy ourselves is not only a good thing, but at the same time it is a German thing, and at the not customary stage on which those who can judge go together with us into the entire world. They are correct to envy us our Frege; for in the same manner as Leibniz, he is a respected freeman of the German people, a freeman of the German mind, one of the few undisputed who with that which they have left behind for us belong at the same time to humanity. 194 Zentralblatt

28 (1943), pp. 103ff. Eckart Menzler-Trott, “Ich w¨ unsche die Wahrheit und nichts als die Wahrheit. . . Das politische Testament des deutschen Mathematikers und Logikers Gottlob Frege. Eine Lekt¨ ure seines Tagebuchs vom 10.3 bis 9.5.1924”, Forum 36, no. 432 (20 December 1989), pp. 68-79. 196 1941, January, no. 2, Cultural supplement: p. 4. 195 Cf.

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195

This paragraph is exciting because it shows how Scholz always attempted to stress the internationality of research and this yet with a nationalistic vocabulary that even used the then popular phrase of Bernhard von B¨ ulow—the “place in the sun.” In this essay Hilbert would at the same time be described as a “doyen of German mathematics.” Frege and Hilbert as forerunners of a German logic and mathematics, which because of that—and with the “M¨ unster School”—is leading internationally: that is the tenor of Scholz’s writings under National Socialism. He saw Steck’s strike against the whole of mathematical foundational research. One could not use a diary skirmish to go into action against Steck’s view that the correct, German, folkish mathematics had to be taught at the technical colleges, while the decadent, non-Aryan heresy of a mathematics made degenerate via faulty concepts and problems had taken its place at the universities. Scholz had, however, enough cover from Ludwig Bieberbach so that he did not require unintentional support from Frege. The logician and mathematician Karl Schr¨ oter, a co-worker of Scholz’s, gave a three-hour practice lecture in August 1943 before the Philosophical and Natural Scientific Faculty of the University of M¨ unster on “Der Nutzen der mathematischen Logik f¨ ur die Mathematik” [The usefulness of mathematical logic for mathematics]. Therein Scholz and his subject were plainly defended: The title of the essay can raise the impression as if in the following a justification of mathematical logic should be sought through the usefulness which mathematical logic possesses for another science, namely mathematics. Of this sort of justification there will in no way, however, be any talk. Because mathematical logic requires just as little any sort of justification through any sort of usefulness as any other abstract science, like theoretical physics or mathematics itself. (This conviction stands in opposition to that of M. Steck; cf. M. Steck, Das Hauptproblem der Mathematik, Munich, 1941 [correction: Berlin, 1942, although the foreword is dated from June 1941 from Munich-Solln—EMT].) With purely theoretical questions it is important only with regard to the hereby obtained contribution to knowledge. According to M. Steck mathematical logic offers no contribution of the sort, because it occupies itself admittedly only with research on abstract calculi. One can only come to a judgement of this kind if one willfully overlooks certain parts of the literature. Herr Steck’s “Hauptproblem” has been treated by several mathematical logicians, with final exactitude, for example by A. Tarski (“Der Wahrheitsbegriff in den formalisierten Sprachen”, Studia philosophica 1 (1936), pp. 261-245). H. Scholz, so very condescendingly condemned by M. Steck, has held a three-semester lecture series (“Formalisierte Sprachen mit einem widerspruchsfreien Wahrheits- und Allgemeing¨ ultigkeitsbegriff ”. Lectures Summer semester 1938, Winter semester 1938/39, Summer semester 1939, M¨ unster i.W.) on the same calculus interpretations. It must be said with emphasis that the initiative for occupation with these questions (cf. also K. Schr¨ oter, “Was ist eine mathematische Theorie?”, Jahresberichte der Deutschen Mathematiker-Vereinigung 53 (1943), pp. 69-82) does not come from Herr Steck.197

I know of no case where Steck is contradicted in so clear a fashion. Steck’s book Das Hauptproblem der Mathematik was reviewed by the respected Andreas Speiser.198 Speiser gave a written endorsement of Steck, that he had 197 Reprinted from the old matrices in: Archiv f¨ ur Mathematische Logik und Grundlagenforschung, September 1950, pp. 1-16; and in: Archiv f¨ ur Philosophie (4), no. 1 (1950) pp. 81-96. 198 In Jahrbuch u ¨ber die Fortschritte der Mathematik. Andreas Speiser (1885-1970) wrote among other things the first textbook in group theory to appear in Germany, “the most beautiful

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“encompassing subject knowledge of the area of mathematics” and could “present [his] thoughts in an impressive manner.” But Speiser also added: The author’s polemic against Hilbert and above all Scholz raises reservations. Each is yet in his late research still a true mathematician, and the problem of consistency can be as good an “idea” as the Fermat Problem. Scholz is factually in any case in agreement with the author because he wants not bare formulæ but rather metaphysics. But these objections do not keep me from acknowledging that the book fulfills a need of the times, and that one can only be pleased if its sort of thoughts acquire a larger public.

Wilhelm M¨ uller saw in Steck’s Hauptproblem a book in which perhaps for the first time on the mathematical side the formalist direction which the exponents of Hilbert, the logistic of the Scholz group, and the Vienna Jew Circle will be fundamentally attacked and battled as intellectualhistorical apparitions.199

M¨ uller succeeded in 1942 in bringing Steck temporarily from TH Munich to the Ludwig-Maximilian University Munich. There Perron, Carath´eodory and Tietze reported in a memorandum of 12 December 1942 that Hellmuth Kneser had rejected a call to the university on account of Dingler and Steck.200 They saw the faculty abandoned through the recruitment of such “pseudo-mathematicians” to the “ridicule” of the other colleges and wanted guarantees that such “nulls” would again be discharged and that sort of mistake would be avoided in the future. Once already Steck had been in discussion with the Ludwig-Maximilian University. On uller as successor to Constantin 27 January 1941 Steck was suggested by W. M¨ Carath´eodory and Th¨ uring, Dingler, and the Heidelberg astronomer Vogt schemingly supported him. But on the basis of an objection by W. Blaschke to the Berlin Ministry of Education, Steck would no longer be in consideration. Perhaps also from that comes a particular need for revenge on Steck’s part against mathematicians in office and title. Max Steck was also considered by people like Wilhelm Blaschke to be a “figure of the 5th order”.201 In a report by Beurlen, M¨ uller’s successor as dean, Perron, Carath´eodory and Tietze would be seen as standing “fundamentally on the soil of formalistic mathematics.”202 He said further: In a review I attempted to arrive at a somewhat more factual attitude of the mathematicians to the whole problem, for in fact the formalistic foundation of mathematics rested on a very narrow-minded world view attitude and next to that other possibilities of a foundation of mathematics must also be acknowledged as at least possible for discussion. This review came to nothing, for the attitude of the mathematicians was absolutely rigid. Discussions which I had in this connection with Prof. Blaschke–Freiburg and Prof. Bieberbach–Berlin confirmed that the special mathematical achievement of Steck is relatively slim.203 introduction into group theory” (van der Wærden). Speiser studied from 1904 in G¨ ottingen and was promoted by H. Minkowski. From 1932 until 1955 he occupied himself with the philosophy of mathematics. 199 Litten 2000, p. 135. 200 Ibid., p. 136. 201 According to Maria Georgiadou, Constantin Carath´ eodory. Mathematics and Politics in Turbulent Times, Springer-Verlag, Heidelberg, 2004, pp. 359 and 579. 202 Litten 2000, p. 139. 203 Litten 2000, p. 140.

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19. Interlude: May and Dingler Provide Arguments for Steck The entomologist Eduard May led the entomological institute of the Ahnenerbe Stiftung [Foundation of Ancestral Heritage] in the Dachau concentration camp and, under control of the Reichsf¨ uhrer SS, carried out “research” on the “naturalbiological” pest control, which meant nothing other than Heinrich Himmler’s “Pestproject”. He was closely tied to Hugo Dingler. After 1929 he further educated himself in mathematics, physics, and philosophy and this the more intensively as through the study of Einstein’s relativity theory and the uncertainty in the fundamental philosophical-natural scientific questions I gained the conviction that the neglected and the arbitrariness of every single abandoned middle realm between philosophy and natural science required a strict systematic exploration.

He published a few natural-philosophical discourses in the Zeitschrift f¨ ur die gesamte Naturwissenschaft, which with regard to reform and the founding of natural science on the spirit of Indo-Germandom followed the same tendencies as I.204

On the Zeitschrift f¨ ur die gesamte Naturwissenschaft as a “broadsheet against the growing positivism”, cf. the apologetic quotations of the publisher K.L. Wolf in the framework of the “denazification”.205 K.L. Wolf and Wilhelm Troll published the series of brochures Die Gestalt. Markus Vonderau distinguished and separated a “Gestaltkreis” [Gestalt circle] in Halle in Die Gestalt, in which Max Steck published several times, from the “folkish science” in the Zeitschrift f¨ ur die gesamte Naturwissenschaft. I would like to see this distinction (but it is no separation) worked out. Dingler would be described by May in his book Am Abgrund der Relativismus [At the abyss of relativism] of 1945 as the actual founder of the exact science. This book would receive a prize from the Prussian Academy of Sciences. The prize question read: “The inner grounds of philosophical relativism and the possibility of its conquest”; the prize judges were Eduard Spranger and Nicolai Hartmann. Therein May came out against Karl Popper’s demand to acknowledge scientific character only to findings falsifiable in principle (Logik der Forschung, 1935), and endeavoured to put down Popper’s conclusion that neither truth nor probability can be achieved through science as untenable. On the basis of his prize writing, Am Abgrund der Relativismus, on 13 March 1942, against the opposition of the mathematician Oskar Perron, May would be habilitated at the University of Munich in the Natural Scientific Faculty (Dean Wilhelm M¨ uller) for “Natural Philosophy and History of Natural Science”.206

Oskar Perron (1880-1975) wrote on 31 May 1942: After the Herr Dean had informed [us] in the faculty session on 29 April 1942 that the Reich Ministry desired the docenture of Herr Dr. May in our faculty, the question was understandably finished for me and I made no further opposition. . . Because Herr May had only an entirely amateurish knowledge of mathematics 204 Cited from Ute Deichmann, Biologen unter Hitler. Portrait einer Wissenschaft im NSStaat, 2nd edition, S. Fischer, Frankfurt am Main, 1995; here, p. 233. 205 Markus Vonderau, “Deutsche Chemie”. Der Versuch einer deutschartigen, ganzheitlich schauenden Naturwissenschaft w¨ ahrend der Zeit des Nationalsozialismus, Dissertaton, School of Pharmacy and Food Chemistry, Marburg, 1994. 206 Deichmann, op. cit., p. 234.

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and he likewise understood nothing of physics, as I’ve been told by a series of currently leading physicists in Germany.

Perron protested the bestowal of the lectureship as a licence for incompetence and cited Dingler: His special teaching post had so little to do with mathematics that the mathematicians were neither consulted as experts before the granting of the post nor did they learn about it after the granting other than through a notice in the newspaper. But in the index of lectures there appeared, in spite of the objection of the mathematicians, Herr Dingler’s lectures under the heading of Mathematics and they were also posted on the blackboard of the mathematical seminar— I know not on whose instructions, in any case without consultation of the board of directors. In the preceding semester Herr Dingler had held a lecture course on “Foundations of Geometry” in which, as some listeners remarked and related to me, various falsehoods on Hilbert’s axiomatics would be expressed, which in part are also in Dingler’s writings. After his habilitation, Herr Dingler had indeed written on mathematics, some of which was correct but hardly new, and also a great deal which one can only describe as pure nonsense; and—by the way—his physical theories also have nothing to do with real physics, as is revealed for example from a review of his latest opus in volume 5 of Deutsche Mathematik published by the Oberf¨ uhrer Vahlen and thus certainly no journal suspect of Jewish mentality.

W. M¨ uller turned this letter over to Dingler with the request for a prompt response. Dingler angrily replied on 11 June 1942: Already before 1942 Herr Perron was a driving element of that group of the faculty that, under the leadership of Sommerfeld (and in contrast to the still really significant minds of the previous generation of professors, Lindemann, Seeliger, Voss) had taken up the cause of the battle for the so-called modern physics and Einstein’s relativity theory. Since Sommerfeld’s departure, Herr Perron has obviously taken over leadership of this group. The opposition of this group against me stems from my critical position against the set theory of the Jew Georg Cantor207 (since 1911) and against the relativity theory of the Jew Einstein (likewise since 1911, but especially since 1919).208 This group has lamented since 1933 the removal of the Jewish mathematicians and physicists from German science, as these had been the main representatives of the scientific direction they represented.

Then Perron was characterised as an incompetent, embittered, petty grumbler, a one-time technician who had never achieved his goal to be a researcher and who embraced the most primitive intuitions on scientific foundational questions.209 I have been reproached for having painted pictures of Hugo Dingler or Oskar Becker and even Max Steck that are too simplistic, thus without the rich inner life, and not at all objective or true to life. And should not Hugo Dingler finally be incorporated, where he belongs, into the history of logic as actually done in Ivor Grattan-Guinness’s presentation (2000)?— In my opinion he has no business there. That’s right: these three are much worse, but I didn’t want to besmear the biography of Gentzen through the depictions of the misdeeds of these two friends 207 Cantor was not Jewish by religion. As of 1987, attempts to establish Jewish ancestry have turned up nothing. Cf. Walter Purkert and Hans Joachim Ilgauds, Georg Cantor, Birkh¨ auser Verlag, Basel, 1987. 208 The year Eddington verified Einstein’s theory of gravitation observationally. 209 Cf. the foreword to Perron’s Irrationalzahlen, 2nd edition, 1939.

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of Wilhelm M¨ uller any more than I already had.210 They agree completely with my comments on the pair. Interesting details are added by Georgi Schischkoff and Rudolf Tomascheck. Dingler, according to another reproach, could not have been an outsider to mathematics. That follows from the fact that Dingler was promoted in foundational research.— I can’t see that: Throughout his life Dingler would not be accepted by the mathematical community, respectively his peer group, would almost always be unfavourably reviewed and in fact excluded from any discussion and career. He was considered an incapable “last founder”, who never burdened himself with the necessary knowledge of detail.211 At the mention of his book (referred to by Perron as his “Opus”) Die Methode der Physik, Dingler mounted to top form: If he were honest, he would have to mention that in the same journal shortly before there had been an extraordinarily appreciative criticism from the first group on the same book (by Bruno Th¨ uring). The author of the second review, the only one referred to by Herr P., Professor Kratzer in M¨ unster in Wuppertal is a Sommerfeld student and a leading representative of the formalistic-logistic school of M¨ unster, which is a subsidiary of the notorious “Vienna Circle”, and which stands quite openly on the side of Einstein’s physics.

A digression on Th¨ uring: Whoever always uses the physicist Bruno Th¨ uring as a witness must explain why he would want to do so. Th¨ uring is of interest in that his compromised ideas still have an effect today in the world of science, if not directly in physics, via the Dingler detour and then via the detour of the theory of science uring issued a Persil-Schein for Dingler. of the “Erlangen School”. Incidentally, Th¨ Bruno Th¨ uring, occupant of the Chair in Astronomy in Vienna—also a Black Magician of Astrology of the Yellow Star—referred in a series of essays to the ruin of each theory of the universe—as is to be read in the title of one of his essays— helped himself to a noneuclidean geometry. That means Einstein’s relativity theory. Hugo Dingler, a Lysenko of physics in the Third Reich, had a good look at the destructive consequences of the quantum theory of Planck and Einstein.212

And the writings of Th¨ uring, who merrily continued publishing after the war, are still accessible to everyone in the Zeitschrift f¨ ur die gesamte Naturwissenschaft,213 so one can in any case read here “Albert Einstein’s Umsturzversuch der Physik und seine inneren M¨ oglichkeiten und Ursachen” [Albert Einstein’s attempt to overthrow physics and its inner possibilities and reasons]. Certainly whoever is as good as he at knowing the difference between Jews and German physicists would be an excellent witness of whether someone, especially if it concerns a close friend and work colleague, should be counted a member of Nazi science. And Dingler also pointed out how there came to be a double review: The managing editor, Prof. Bieberbach–Berlin, had first sent the book to Kratzer, whereupon protest from the opposition followed, which led to the first review. 210 One can look up the details on Dingler, May and Steck and the “Munich Circle” in the book of Freddy Litten (2000). 211 Cf. the longer note at the end of this chapter. 212 Imre T´ oth, “Wissenschaft und Wissenschaftler im postmodernen Zeitalter. Wahrheit, Wert, Freiheit und Kunst und Mathematik”, in: Hans Bungert (ed.), Wie sieht und erf¨ ahrt der Mensch seine Welt?, Buchverlag der Mittelbayerischen Zeitung, Regensburg, 1987. 213 Edited by E. Berdolt, F. Kubach, B. Th¨ uring, “Physik und Astronomie in j¨ udischen H¨ anden”, in: Forschungen zur Judenfrage, 1941.

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However at the urging of Kratzer and others, then the other counter-review would also be published.

In his statement of 30 August 1946—he attempted to regain his unpaid lectureship—May compared his difficulties with the habilitation with “the later habilitation of Herr Georgi Schischkoff” and bemoaned the lack of competent judges for science-historical and natural scientific-philosophical border areas. In his report Dirlmeier praised the discipline and purity of May’s thinking and pronounced himself in favour of the admission of May to proceed further with the habilitation: Indeed the basic direction of the path which Herr May follows is fixed through the critical realism of Nicolai Hartmann and the volunteerist impact of the philosophy of Hugo Dingler, as well as through certain elements of the philosophy of Dilthey and Driesch.

Peter Koch was more precise: On 4 October 1943 May received a commission for his research from the Reichsforschungsrat on “humanly harmful insects” with military job number of the Urgency Level SS. This job was connected with the biological warfare researches carried out by Kurt Blome. (p. 236)

May worked day by day in close contact with those who carried out human experiments in Dachau.214 E(duard) May (*14 June 1905-†10 July 1956) wrote in his vita on the occasion of his habilitation on 10 November 1941: The publication of this prize writing brought me into contact with the Zeitschrift f¨ ur die gesamte Naturwissenschaft (Organ of the Expert Group of Natural Science of the Reichsstudentenf¨ uhrung Ahnenberbe Stiftung Verlag, Berlin-Dahlem), which followed the same tendencies as I with regard to the reform and the laying of the foundations of natural science on the spirit of Indo-Germandom.

And he wrote on commission of Dr. Georg L¨ uttke Verlag an Einf¨ uhrung in die philosophischen und erkenntnistheoretischen Hauptprobleme unter besonderer Ber¨ ucksichtigung rassischer und v¨ olkischer Gesichstpunkte 215 [Introduction to the main philosophical and epistemological problems with special consideration of racial and folkish points of view]. On 30 August 1946, in his request to resume his earlier unpaid lectureship May indicated that he never was a member of the Nazi Party, possessed no rank in the army, and had no position in the economy. And then followed: As one generally known in expert and industrial circles as a specialist in applied entomology, I received in 1943 a research commission from the Reichsforschungsrat concerning insecticides. At the same time I would be appointed in the frame of the civil scientific war deployment (I was invalidated out of military service on account of double-sided chronic otitis) with the setting up and directing of an entomological laboratory for the police and the Waffen-SS. I carried out this work, as all my other official and private tasks, for a fee. . . In June 1944 the laboratory was put into service and in May 1945 I turned my copies of the results attained till then over to the American security police in Munich. In July 1945 once again two American military entomologists sought me out in my apartment and questioned me on details of the insecticide newly developed by me.

In short: May would be allowed by the military government to resume earning his bread and butter, as he was politically unobjectionable. A board of inquiry was 214 Cf. pp. 150 and 156 in Friedrich Hansen, Biologische Kriegsf¨ uhrung im Dritten Reich, Campus Verlag, Frankfurt am Main, 1993. 215 Habilitation file E H 2412, Archive of the Ludwig-Maximilian University, Munich.

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never considered in his case. On the basis of the submission of the same form by the University of Munich he would, however, be suspended, his bank account frozen. In the 1950s he would succeed Hans Leisegang at the Free University in Berlin. May’s student, Professor H. Stachowiak, unfortunately did not speak of May’s career in Berlin. In 1951 he [May] was named ordinary professor of philosophy at the Free University of Berlin. His co-worker Sch¨ utrumpf habilitated in K¨ oln and was named unscheduled professor. (p. 237)

In a toned down version of Poggendorf one reads: Eduard May (*14 June 1905 in Mainz, †10 July 1956 in Berlin-Kladow) became Dr. phil. nat. at the University of Frankfurt am Main. From 1942 to 1945 he was unpaid lecturer at the University of Munich. From 1951 he was extraordinary professor for History and Methodology of the Natural Sciences at the Free University of Berlin.216

Why wasn’t there at least a hint that May was the prize winner of the Academy competitions of Leipzig in 1936 and Berlin in 1941? May emphasised this in the blurbs of his 1949 books. Further, according to Vonderau: Possibly in his exposition Ernst Krieck aimed at, among other things, an essay in the Zeitschrift f¨ ur die gesamte Naturwissenschaft, in which May stressed the mechanical explanation of nature in its significance for physics: Eduard May, “Die arung und ihre Bedeutung f¨ ur die physikalisIdee der mechanischen Naturerkl¨ che Wissenschaft”, pp. 2-23 in: Zeitschrift f¨ ur die gesamte Naturwissenschaft 5 (1939).

But Krieck was, despite his biological world view, against a rejection of or even a hostility toward mechanical sciences.217 And Krieck battled against Dingler and May: Lenard had declared empiricism as German, constructive rationalism as Jewish. His students now preach, after they have raised the Dingler-I on the shield, a priori rationalism as German and empiricism as Jewish. The prize-winning May, however, tosses empiricism and relativism into the same pot. I am rendered so dumb by all of this, as if a mill-wheel were going around in my head. Only one thing is certain in this chaos: Newton is holy to the empiricists and rationalists, the relativists and absolutists, the Jews and anti-Semites: they are all his stragglers at the end of physics.218

Characteristic of thought about the “German physics” of Lenard and Stark around 1937 is the review of A. Becker (1936) by W. Hillers (1937) (p. 99): Spirit and content of the individual contributions are entirely different from each other, if however they have the same fundamental philosophy. Lenard and J. Stark expressly stress their conception that physics must remain a science of intuition, that mathematisation driven too far has led to fantasies—to relativity theory—and to illusion. In unsurpassably curt manner they also dismiss ruthlessly every German physicist—regardless of earlier contributions—who helps himself to nonintuitive mathematical research tools (relativity theory, matrix computation), as captivated by the Jewish mentality, and warn the public against them. 216 Vonderau,

“Deutsche Chemie”, op. cit., p. 183. Krieck, V¨ olkisch-politische Anthropologie. Erster Teil. Die Wirklichkeit. Weltanschauung und Wissenschaft. Band 1., Armanen Verlag, Leipzig, 1936, p. 22. 218 Krieck, p. 222 in Natur und Wissenschaft, Quelle und Mayer, Leipzig, 1942. Cf. in this regard p. 181 of Vonderau, “Deutsche Chemie”, op. cit. 217 Ernst

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The mechanical explanation of nature condemned by the “German chemists” as being un-German was also thoroughly discussed in the conforming press.219 Despite all that, anti-Semitism and National Socialism were the uniting basis of Krieck, Dingler, and May. Nowadays that becomes evermore difficult to understand. Dr. habil. Eduard May wrote further in 1949, when he came to a standstill in National Socialism. He proudly referred in his books Kleiner Grundriß der Naturphilosophie and Einf¨ uhrung in die Naturphilosophie 220 and to his publications under National Socialism and criticised mathematical foundational research, logistic, and relativity there, but now with great care and in a manner barely open to attack. And certainly so, that he and Dingler’s theories connected internationally. In 1956 Eduard May published the book Heilen und Denken 221 [Healing and thinking]. As one sees, the SS-sympathetic publishing house had not changed. The honorary professor Herbert Stachowiak wrote: With Hans Leisegang, the investigator of philosophical “thought-forms”, I joined in a respectful, admiring but also warm relation. After his sudden death Eduard May took his place. He too, who would become my philosophical teacher and friend, died much too early. With the deaths of these philosophers a philosophical thought-style of this particular kind was submerged for a long time, actually even till today.222

Now May didn’t “step” into anybody’s place, but rather he would be called upon— incidentally, against the impressive objection of Oskar Perron—and straightaway the historical writing of Stachowiak is an example of an uncritical, apologetic historical writing, which passes itself off as reminiscences and thereby conceals what is important.223 20. Steck and Scholz in Dispute Max Steck stood by Dingler’s side in his determined battle against Hilbert and his conception of mathematics. Hilbert had been short-circuited with an inferior 219 On this see E. May (1939) 23 (cf. also the remarks on p. 81 of Markus Vonderau, “Deutsche Chemie”). 220 Both: Westkultur Verlag Anton Hain, Meisenheim am Glan, 1949. 221 Hans Haferkamp (ed.), Arzt und Arznei 1. Mit einer medizinischen Einf¨ uhrung von Hans Freiherr von Kress, L¨ uttke, Berlin, 1956, p. 56. 222 Herbert Stachowiak, “Berlin 1945 bis 1951: R¨ uckblick auf eine unvergeßliche Zeit”, in Heinrich Begehr (ed.), Mathematik in Berlin. Geschichte und Dokumentation. Zweiter Halbband, Shaker Verlag, Aachen, 1998; here, p. 262. E. May was the first referee of Klaus Heinrich’s dissertation, “Versuch u ¨ ber das Fragen und die Frage”, with which he was promoted on 22 December 1953. 223 In the journal founded by May, Philosophia naturalis. Archiv f¨ ur Naturphilosophie und die philosophischen Grenzgebiete der exakten Wissenschaften und Wissenschaftsgeschichte, I have found in the 32 years up to now no article critical of E. May. I consider it scandalous that the journal did not expressly distance itself from May’s science-philosophy and history of the humandespising SS. Why does the publisher insist on E. May as the giver of tradition? Should the SS-Ahnenerbe be not defended as much as not celebrated? Why don’t the editors and editorial advisors expressly distance themselves from E. May? An editor wrote me on 2 January 1995, “Dear Mr. Menzler-Trott, in fact we have allowed the name of E. May to disappear from the title page of Philosophia nauralis. Over and above that in the latest volume of our journal a short explanation of the grounds that have brought us to this is to be published by the editors. The unanimous opinion of the editors is that the case is settled for us. . . ” The explanation reads: “From this volume on the name Eduard May as founder of the Philosophia nauralis will no longer appear on the title page. The publishing house and editors felt obliged to this measure after they learned that Eduard May had played a self-incriminating rˆ ole in the time of National Socialism.”

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concept-formalism, while the good values of contentual material, sensible meaning, and certainty of formulæ stood opposite on the side of Hugo Dingler. This support for Dingler224 would appropriately be reviewed by W. Ackermann.225 Steck had wanted to draw another companion into the fight: The formal, nominalistic thinking introduces a pattern of thought, which. . . has become a decadence of the mathematical spirit, especially in pure mathematics which will result in complete absurdity and genuine error and has already had an effect, as in all other mathematised sciences where this error is already entirely clearly recognised and has been brought to light by P. Lenard, J. Stark, H. Dingler, W. M¨ uller, B. Th¨ uring, and others.226

As a counterprogramme to “rational procedures”, which Steck had criticised in Hauptproblem, Steck wanted to make apparent the rˆ ole of the folkish spirit in the holistic way of looking at things: In the presentation of these complete works we also aspire to approach the potential of the mind in its idea-creating and -seeing activity and function only in connection with its folkish, racial, blood- and home-like ties. We also seek again in science the ideal German mind type as such. . . As we proceed from the general into its folkish categorisations, we are attached to the parts only insofar as we are able to see and recognise them as the carriers of meaning of the whole. Cosmopolitan-international views will thus be deliberately rejected. Apart from the fact that it ruins man in his creative individuality and stops his work, it yields, as the previous intellectual-historical development of all cultures has taught us, only narrow-mindedness and presents itself with its alleged “strict scientific objectivity” for the most part as camouflage for the drawing together of heterogeneous mentalities.227

Max Steck, one-time assistant to the anti-Semite and “Aryan” physicist Hugo Dingler, wrote in his pamphlet “Mathematik als Begriff und Gestalt” [Mathematics as concept and form], which he had “respectfully dedicated to Herr Privy Councillor Prof. Dr. P. Lenard on his 80th birthday on 7 June 1942”: Comparable to the armed struggle in which our nation stands, that will also bring and signify the liberation of the German intellect in science from western influence, and all the prior submissiveness and bending under the rule of English empiricism and sensualism and under western nominalism will be broken and will have been gotten rid of in the future German science.

And here a “Gestaltmathematik” [ideal type of mathematics228 ] should come to the rescue. He “shows” that neither Hilbert nor Brouwer can grasp the “essence of the mathematical” and continues: H. Scholz and the earlier so called “Vienna Circle”, which are tied to Frege with his “Begriffsschrift”,229 to Whitehead-Russell and their “logistic”, have in the end contributed only something purely logical to the epistemological problem area, and have even denied the Gestalt problem in mathematics if they wish 224 Max

Steck, “Mathematik als Problem des Formalismus und der Realisierung”, Zeitschrift f¨ ur die gesamte Naturwissenschaft 7, pp. 156-163. 225 Jahrbuch u ¨ber die Fortschritte in der Mathematik 67 (1941), p. 30. 226 Hauptproblem (1942), p. 214. 227 Max Steck (ed.), Johann Heinrich Lambert: Schriften zur Perspective, Georg L¨ uttke, Berlin, 1943, pp. xff. 228 “Gestalt” means form or shape, but under the Nazis it meant something like “ideal type”. 229 Frege’s “Begriffsschrift” was an early form of first-order logic in which logical formulæ were represented by graphs. The word itself would translate to “concept script”.—Trans.

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to derive everything from logic and logical grounds, or at least [they] haven’t seen it and have not recognised the essence of the mathematical. Moreover, logistic has decayed into a narrow-mindedness without equal, which can be interpreted neither as mathematical theory nor as logical discipline. Under a complete misunderstanding of a few fragmentary assertions by Leibniz, to which they always refer, they have “theologised” in an intangible middle ground between logic and metaphysics and achieved corresponding “results” thereby.

There he could not once again mock, “The mathematicians count him among the philosophers, and the philosophers count him among the mathematicians,” but instead here Steck only reproached university logic—he refused any meaning to an expression like “mathematical logic” (p. 8)—for charlatanism and unscientificness, and of implicitly being under the thumb of the Western enemy. Scholz answered immediately. He gave a lecture “Leibniz und die mathematische Grundlagenforschung”230 [Leibniz and mathematical foundational research] at the Kaiser Wilhelm Society for the Promotion of Science on 21 January 1942. It presented a completely different tradition of “German thought”. Scholz explicitly praised for example Hilbert, Max Planck, Leonardo, Leibniz, Mahnke, Schr¨ oder, Frege, and Boole. In a counter-rhetorical turn to Max Steck, he likewise found a “change of fortune”: We also always expect that the responsible publisher of our large reference books will take care that one will learn something about Frege from them. They are persistently quiet about him. Also in our histories of philosophy one will seek Frege just as futilely as with Leibniz, whose reputation he has established in so exemplary a fashion. The standard work for this Leibniz was written by a Frenchman: Louis Couturat, La Logique de Leibniz, Alcan, Paris, 1901. It is about time that we finally reflect on what we are indebted to this Leibniz for. That just makes me all the more determined; because we are once again and in the most serious sense presented with a fateful question. It is the question of the German spirit. Look out where you stand! But how this question will also be answered: each answer will be incomplete, which this Leibniz did not take into account.

And then Scholz enthused about the illuminating type of clarity and of the faithfulness in small things if it is directed to the large things. And of the founder of the German logical tradition a Frenchman wrote the standard work. And the question of the nature of the German spirit would be relieved of all responsibility with a line from a song, “Each look where he stands!” Steck231 criticised H. Scholz with a citation from Rickert and opposed him with the Leibniz interpretations of H.L. Matzat and G. Doetsch—who once favourably reviewed Dingler—that Scholz referred to Leibniz as a founder of exact science and logic. Against the “theses of logistic” he referred to his book Das Hauptproblem der Mathematik [The main problem of mathematics]. oter seconded Scholz explicitly with “Was ist eine mathematische TheKarl Schr¨ orie?”232 [What is a mathematical theory?]; expressly mentioned was Hilbert’s 80th birthday on 23 January 1942. The essay took Hilbert’s ideas as a point of departure for his own, and he cited Hermes, Scholz, Frege, Tarski, among others. 230 Jahresbericht 231 Max

der Deutschen Mathematiker-Vereinigung 52 (1942), pp. 217-244. Steck, “Die moderne Logik und ihre Stellung zu Leibniz”, Geistige Arbeit 11, nos. 10-

12, 1944. 232 Jahresbericht der Deutschen Mathematiker-Vereinigung 53 (1943), pp. 69-82; received on 15 February 1942.

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Steck would label Hilbert a formalist whose success in “foundations” was based upon a Jewish perspective of philosophy which had been laid down by Cohen, Vaihinger, Schlick et al. Formalism changes mathematics into a “system of science from pure tautologies.” In this Steck stuck closely to Tornier: It is namely the typical Jewish-liberalistic thesis that the criterion of right to existence of a mathematical theory is its “æsthetic beauty”, whereby a logically closed. . . construction from definitions will be meant.233

Formalism begets, ever more complex and varied, a subtlety which is accessible only to a circle of esoterics who ban the German mathematical idea and in the end work only with arbitrarily set names. Steck described formalism as mathematical futurism and cubism: The pinnacle of this absurd development is to be found in the books and works of E. Landau, A. Rosenthal, H. Minkowski, M. Dehn, I. Schur, S. Bochner and many other Jews in mathematics.234

The correct mathematics, however, proceeds from Euclid, where intuition plays a decisive rˆ ole: . . . the intuition, as the highest intellectual power of man, to seek in pure ideas of the mathematical, to examine and symbolise the ideal adequately.235

The consistency of an axiom system is to be given through geometry, because if the basic assertions of the axioms of a system were to be logically or intuitively compelling and illuminating, they would also be considered to be true. Steck himself indulged in a variation of constructivism. He wanted to build mathematics “stepwise synthetic-constructively completely from the ground up”— as in the science of Herr Dr. Dingler—and in this he was sympathetic “in that it demanded a reduction of all truth to the intuitively given,”236 to the idealistic significance of mathematical objects of intuitionism—which he, however, also polemically equated with the symptoms of decline of logistic if it appeared convenient to him to show himself original. “Steck attempted a synthesis. . . , which proceeded ‘idealistically’ in its main direction taking care to meet the demands of consistency of formalism.”237 Alongside formal axioms an Axiom der Gestalt [Axiom of form, or, Axiom of ideal type] must be given in order to arrive at a “morphology of the exact sciences.” Steck’s view of a “German mathematics” is best combined with Krieck’s view of an integrated, holistic science. In his essay “Zur Wissenschaftslehre der Mathematik und der exakten Wissenschaften” [On the theory of science of mathematics and the exact sciences], the National Socialist philosopher and pædagogue Ernst Krieck wrote how one must drive out all schematic abstractness and “objectivity”, so that mathematicians may once again take their place in the folkish reality of life: Scientific truths, thought-forms and methods can only be born in a definite folkish place and to a historical period. . . Even mathematics and the exact sciences will, whether they want to or not, be drawn into the revolutionary decision. This force will bring their downfall so far as the mathematician himself is not able through 233 E.

Tornier, Deutsche Mathematik 1, pp. 8ff. Hauptproblem, p. 184. 235 Ibid., p. 5. 236 “Mathematik als Begriff und Gestalt”. 237 Hans Hofer, p. 88. A side comment: The “Gestaltmathematik” of Hermann Friedmann, sketched in his book Wissenschaft und Symbol. Aufriss einer symbolnahen Wissenschaft (Biederstein Verlag, Munich, 1949), had nothing to do with this mathematics. 234 Das

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his work and knowledge to fall into line with the decision, in order to gain from it new power and vision, a new total-responsibility and total-task incorporated kind of problem-posing and answer-finding.

Of course, Steck’s theses “within mathematics” can be connected to those of Dingler. Dingler’s attempt at an empirical foundation of geometry from a priori valid norms of action would be accepted by the Nazi mathematician Friedrich Requard as welcome volunteerism: Rigorous, definite, creative mathematical work is not, as many believe, quiet contemplation, pure observation, submissive sinking into the empire of truth; rather. . . production of genuine deed-actions. The true concept of the actual is not that of an existence generally independent of consciousness, but rather only that of an active I in a repellent actual.238

And by this misunderstood Fichte -nonsense the “acting I” is socially conditioned for him239 and again fits marvellously into the conception of science of Krieck and Rust. Against the “Western imperialism” of a science-ideology of objectivity, freedom from assumptions, unconditionality, internationality, value-free-ness, neutrality, and detachment, the individual elements of nation, race and history would be led into the field. Truth should remain the way and formation law of science, but it should remain tied to a folkish life-order and find its soil in the world view, something which itself is not science. The complete baselessness and lack of direction of science, the liberal idea of humanity and objectivity, will finally be detached through a National Socialist science: science is to be conditioned by reality, thus received from folkish and racial origins. To be sure, the bond to background, location, denomination, profession and home, are valid, but the results retain their validity permanently in the cultural circles which stand in the same sense-direction of common life. The scientist comes from the population, works within the frame of a folkish science and returns his results back into the people as a whole. Because science is tied to concrete circumstances and is developed in them, this certainly does not mean that it is thus subjugated to these circumstances. But Steck had no success with his theses. His Wissenschaftliche Grundlagenforschung und die Gestaltkrise der exakten Wissenschaft 240 [Scientific foundational research and the Gestalt crisis of the exact science] would be briefly and devastatingly reviewed in Zentralblatt by the Dane J. Jorgensen: “The treatment is mainly of a programmatic character and contains no newly established results.”241 Thus mathematical proof is used as a form of rejection of ideological claims; it is the same with other tendencious writings. Max Dræger’s “Mathematik und Rasse”242 —he referred to the writings of F. Klein, Vahlen and Bieberbach—would be finished off by the Stuttgart mathematician Eugen L¨ offler: in Zentralblatt:243 In the closing section number theory is praised as an essential component of German mathematics and the overemphasis on applications is disapproved of as 238 Friedrich Requard, “Strenge Mathematik und Rasse”, Rasse. Monatszeitschrift der nordischen Bewegung IX (1942), p. 17. (The expression is obscure in the original German.) 239 “Rassenlehre”, the theory of race, would be scientifically recognised at the 2nd Congress for Anthropology and Ethnology held in Copenhagen from 1 to 6 August 1936. 240 Akademische Verlagsgesellschaft Becker & Erler, Leipzig, 1941. 241 Zentralblatt 25 (1942), p. 3. 242 Deutsche Mathematik 6 (1942), pp. 566-575. 243 Zentralblatt 27 (1943), p. 145.

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Americanism. . . This is not the place to take a critical position on these individual verdicts and interpretations.

Max Steck remained unteachable. Still in 1945 in the foreword to his edition of Proclus’s commentary244 he “justified” the publication for “inner reasons”: It deals with the whole of Greek mathematics as that mathesis universalis, which has worked an effect in the German intellectual space, not so much as rationalism, but rather in the decisive connections as idealism of the Germans as such, in science and art and in the shaping of the Reich. These theses will certainly arouse the opposition of the most modern mathematicians of formalism and logistic. They will also find little or no agreement among the philosophers of empiricism, sensualism and positivism, likewise among all those thinkers who believe that in the construction of the being one must call upon only rational methods, procedures and elements for support. The modern formalistic mathematics and its special form of logistic has till now interpreted Euclid and his work in a narrow-mindedness which straightaway submerged its genuinely deep impact on the formation of the intellectuality of the West and its culture, because they no longer observed it and (partly consciously, partly unconsciously) believed it was no longer active and could be disregarded. That is why, in the future, however, one will be able to avoid considering the Euclidean Elements against the background of the Platonic philosophy out of which it has grown and lay decisive value on it in mathematical foundational research.

But the long list of rational, mechanical, empiricist-sensualistic and positivist “pronouncements of the philosophers which [determined] the course of scientific and intellectual things—and in many cases the political developments,” with “their so numerous and today still functioning and office-holding representatives of materialistic attitudes of mind at the high schools of the Reich” all hated German idealism. Positivism, mechanism, materialism, empiricism, nominalism, “metaphysics-hostile formalism and logicism” are all hostile to German idealism. From the present difficult struggle over a new emergence of the West the German ideal mental type must lift itself purposefully and before all things enter into the consciousness of all those who follow us and come into the intellectual inheritance of the Reich and must carry on. May the Procleic work be a sure guide thereby, which preserves us from false paths, detours and dead ends, in which we are advised through the extended feeling of power of positivism and materialism also in science. . . The mind will and must triumph!

Max Steck wrote that and passed off a form of mathematics as a stay-the-course motto for a final victory for which at this point in time there was certainly no longer any prospect of success. And because it was recognisable that there would be no military victory, the thought-form of the scientist after 1919 appeared again: it concerned the West and it concerned the German mind and the German science. For the moment Max Steck confirmed the propaganda of his Hilbert criticism in his introduction till page 152 in the name of a “mathematical idealism”. Steck continued his theses in Grundgebiete der Mathematik 245 [Basic areas of mathematics]. In 1946 Steck saw himself in this book “again at a significant turning point of the spirit of the west” and actually “produced” in mathematics “the laws of the mind.” Mathematics is supposed to help find the intellectual means. 244 Proclus Diadochus. Kommentar zu Euklids Elementen, Deutsche Akademie der Naturforscher, Halle, 1945. How did he obtain paper for the printing of this in those days of severe quotas? 245 Max Steck, Grundgebiete der Mathematik, Winter Verlag, Heidelberg, 1946.

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In Chapter 1, “The significance of mathematics and its intellectual historical place among the sciences”, he was true to himself. He swore to the connection of mathematics and art, world view and basic foundations, and celebrated German idealism “in the long line from Nikolaus von Cues over Copernicus and Kepler to Leibniz, Lambert, Kant and Fichte.” In Chapter 2, “Fencing off, structure and method”, he separated arithmetic and geometry, space and number, intuitive and conceptual, and mathematics and logic. Logic is not characteristic for the mode of thought of mathematics: “Whoever allows the methodical of mathematics to be created and established in the logical shows thereby only that he has not penetrated the depths of the nature of this high science and climbed down” (p. 71). It is the old song: he battled against assertions which nobody mentioned; he battled against a bogey. Then, however, he praised the “proper proof theory of mathematics”: “One knows a little more about it as a methodical core of mathematics than by the teachings of the prior cases, mainly through the recent researches of axiomatics and the researches of logistic restricted to the logical part of mathematical proof ” (p. 75); i.e. Gentzen remained unmentioned this time. Afterwards with W. Troll, “the leading morphologist of biology,” and A. Speiser he would once again criticise “the logistic with its claims to universality of the broadest sort in the formulations of H. Scholz” on pp. 81 and 82, and still follow this with an attack on Schlick, Reichenbach and Carnap on account of their “fatal mixture of ideal- and real-validity of the mathematical” (p. 90). Theodor Vahlen is a “commendable old master of mathematics” (pp. 91, 92, 95), and Dingler would be lightly problematised on p. 92. He thus hadn’t changed much.246 In the postwar period Steck insisted on art history and celebrated nonstop the German spirit with Albrecht D¨ urer in various high-circulation books. That should be explained, especially against this background. Although the departure of Vahlen from the Reichsforschungsrat weakened Bieberuring and their gang would soon bach’s position, the theses of Steck, Dingler, Th¨ no longer be taken seriously by those responsible in the Nazi Party or by mathematicians, and they would thus be robbed of their danger. 21. Max Steck as Denouncing “Expert Witness” and Publicist After his habilitation in 1939 Max Steck received a salaried lectureship for general sciences in the domain of mathematics at the Technical University of Munich. Even before 1933 it was “Jew-free”. He taught nothing but geometry at Prof. Dr. Loebell’s chair for Descriptive Geometry. 246 Michæl

Toepell, Mathematiker und Mathematik an der Universit¨ at M¨ unchen. 500 Jahre Lehre und Forschung, Institut f¨ ur Geschichte der Naturwissenschaften, Munich. Therein one reads “Max Steck, *1907 in Basel, †1971?, 1935-1939 Assistant TH Munich, 1938 Habil., replaced Robert Schmidt 1941, 1952 Acad. for Applied Technology Nuremberg, 1957 Acad. for Structural Engineering Munich (Poggendorf) 7a, 504.” From 1941 to 1944 a special teaching post for geometry. Typical of a castrated historical description is the following evaluation: “In his monograph on Das Hauptproblem der Mathematik (Berlin 1942, 2nd edition, 1943), which is to be seen against a background of its time of origin, Steck pursued a bridge from intuition to formal mathematics and sought at the same time to establish a connection with philosophical foundations.” The impudent and disingenuous phrase “. . . which is to be seen against the background of its time of origin. . . ” is the unacceptable formulation, which everyone who must have feared for his physical intactness on account of Steck spat at a second time. Hugo Dingler would naturally still be mentioned in the same castrated bad manner.

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On 12 March 1944 he wrote a pompous letter to Rector Pistor.247 Together with Dingler’s “Memorandum” the letter is the most disgusting text which has ever been written by a mathematician. He, Max Steck, has been requested by the chancellery of the F¨ uhrer to actively participate in the official Party examining board, in particular to manage mathematical and natural scientific literature. . .

He continues: Further, I am likewise requested by the chancellery of the F¨ uhrer, [to give] for a start a situation report on the mathematics in the German high schools [i.e., colleges].

The chancellery of the F¨ uhrer wanted thereby to see how in 1933 the Jewish professors would be removed, which German talents to appoint in their place, and whether in general a change and a replacement of the strongly Jewish influenced science would be achieved.

He wanted evidence for the “machinations” of the “Jewish infiltrated cliques of mathematicians” and the “Jewish infiltrated Aryans” delivered for a definite goal: for their “wholehearted eradication.” He couldn’t ease off, and with the thwarting of his appointment for three years still rankling, he wrote further: I will work out an extensive monograph with all evidence and documents, a copy of which I am prepared to send you confidentially, highly respected Herr Rector, in case you wish and have an interest therein.248

In my opinion the seizing of the Jews as a preparation for murder would be carried out completely only in the discipline of music, in the lexicon of Jews in music published by Herbert Gerigk. According to the foreword the registry served “the quickest eradication of all incorrect remaining residue from our cultural and intellectual life. . . There the lexicon should be a secure guide.”249 Dingler’s and Steck’s desire for liquidation, however, found no resonance among professional mathematicians. But Dingler and Steck proceeded opportunistically, with the inner readiness of fellow travellers seeking advantage. Mind you the fascination of Dingler’s philosophy was held only within a definite philosophy of science in Germany. On 6 March 1943 he held a lecture in the Gestalt Colloquium in Halle. The goal was the elaboration of an all-encompassing nomography of the mathematical as a form-theory within a general morphology of the sciences: By this is meant a form-like description of those laws which all mathematical ideas without exception and in totality are subordinate to and obey.

And this attempt to establish a dictatorship in the world of mathematics was judged by him: May the writing thereof be seen as a contribution to a new idealistic definition of the position of mathematics and its thoughts for the present reconstruction of the concept of science in general and in particular be judged. The intellect orients itself again! 247 I thank Frau Fuchs of the university archives of the TU Munich for knowledge of this letter and the personal files. 248 I will surrender a copy of this obscene letter to anyone who would occupy himself with Max Steck or his doxography. 249 Cf. in this respect Eva Weissweiler, “Ausgemerzt!” Das Lexikon der Juden in der Musik und seine m¨ orderischen Folgen, Dittrich Verlag, K¨ oln, 1999.

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Steck thanked the president of the Leopoldina, Prof. Dr. Emil Abderhalden, that Steck had been allowed to publish his edition of the Euclid commentary of Proclus Diadochus in a special edition. Emil Abderhalden (1877-1950) was professor for physiology and physiological chemistry in Halle and became famous through a case of error or fraud. He wanted to show that the body builds certain protein-destroying enzymes as soon as foreign proteins appear. In the 4th edition of his book Die Abwehrfermente [The defence enzymes] (1914), Abderhalden referred to 451 works which describe the practical application of his methods. In 1914 the biochemist Leon Michælis published a critical work with the result that each “protection enzyme” was ineffective. No wonder, because this defence enzyme does not exist. Josef Mengele carried out “experiments” in concentration camps in order to determine with the “defence enzymes” the susceptibility of various “races” to infections in order to acquire therefrom a biochemical test for racial identification (“race-specific tuberculosis susceptibility”, “malaria susceptibility tests” by the Hamburg tropical medicine emeritus Claus Schilling in Dachau, theory of race-specificity of proteins by Verschuer, etc.).250 In the years 1945 to 1954 Abderhalden was the fourth most cited in Germany by the biochemists who did not emigrate. Was there contact with May’s biological and chemical warefare project? And no less he thanked the professors Dr. W. Troll and Dr. K.L. Wolf, who as editors of the series of writings “Die Gestalt” promoted my plan and have helped me to support it intellectually.

And again a nominalist-positivist science is criticised251 and a sideswipe is taken at “Heinrich Scholz and the modern logistic of the M¨ unster circle.” Carath´eodory attempted in April 1944 to block Steck from editing a Lambert edition through a letter to W. Blaschke. This failed because the Heidelberger Hellmuth Kneser wanted to prevent its becoming a purely Swiss edition and entrusted Steck with it, although Tietze was also opposed. Although Steck published a volume,252 the Swiss Andreas Speiser, a good friend of Steck, became the official editor of Lambert’s Opera mathematica, which appeared between 1946 and 1948.253 For his Lambert edition of the Optics Steck received in mid-1944 the Lambert Prize of the Schnyder von Wartensee Foundation in Z¨ urich with the sum of 700 francs. He was supported by Prof. Dr. Mentzel to carry out a collection of Lambert’s works, which was also supported by the Reichsforschungsrat. He wanted to use his trip to Switzerland to collect the prize to put into motion a German-Swiss cultural undertaking. In Z¨ urich Dr. A. Speiser and R. Fueter gave him the prize, and Dr. Wilhelm M¨ uller praised him in the V¨ olkische Beobachter on 14 June 1944. On 11 April 1945 the V¨ olkische Beobachter/Neueste Nachrichten reported that the lecturer Max Steck had become (on 3 April) an ordinary member of the Leopoldina. He cited this membership in letterheads and at all opportunities his entire life. He 250 The connections of Abderhalden, Verschuer and Mengele and their theories and experiments is described for example in Achim Trunk, Zweihundert Blutproben aus Auschwitz. Ein Forschungsvorhaben zwischen Anthropologie und Biochemie (1943-1945), Max Planck Gesellschaft, Berlin, 2003. 251 P. 136. 252 Max Steck (ed.), Johan Heinrich Lambert. Schriften zur Perspective, L¨ uttke, Berlin, 1943. 253 Cf. M. Georgiadou, pp. 398ff.

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had his portrait embossed on relief postcards and did everything that he could to compensate himself for not being a German professor. On 2 November 1945 Steck was temporarily kicked out of office on instructions of the military government. Die Neue Zeitung was indignant on 6 January 1947 over Steck’s book Grundgebiete der Mathematik [Basic areas of mathematics]. There, in a domain in which one would not have thought possible, was such hatred as in Nazi speeches. Paper was scarce and was surely not intended for this sort of stuff. The sale of the book would be stopped after protests. I appreciate that Steck wanted to improve his prospects of a professorship. The work—perhaps it concerned itself with a further elaboration of the above mentioned study guide on mathematics for the Oberkommando der Wehrmacht [Supreme Command of the Armed Forces]—was perhaps already typeset, and then there is the alternative: publication in altered form in a few years or immediately publishing and hoping that it would remain unnoticed by intellectuals or—in case of discovery—it could be passed off as idealism in the West. The possibility that Steck remained a keen National Socialist up to his Nuremberg activity at the Georg Simon Ohm Polytechnikum I’d rather not discuss in detail here, although naturally there are indications for it. Max Steck made multiple applications for reemployment, which would repeatedly be rejected. On 29 July 1948 Dean Gistl, for example, wrote to Rector F¨ oppl: From the foreword [Das Hauptproblem der Mathematik, i.e. “the main problem of mathematics”–EMT] I became acquainted for the first time with Steck’s relations to the leading Nazi circles; the original book reveals to the most painful astonishment of all who knew him previously not only an unrestrained anti-Semitic tendency, it gets down to a disrespectful, unjustified criticism of the most significant mathematicians, above all Hilbert. . . The attempt to induce the author to a revision remains essentially unsuccessful, as the second edition appearing a year later shows.

Steck, who in this new period described himself in letters as a student of Heinrich Liebmann, was assigned to the circle of the physicist Wilhelm M¨ uller. On 3 April 1954, Steck was in Nuremberg. The above mentioned Lagebericht zur Situation der Mathematik of Steck has till now not been found. The theme though is very similar to Dingler’s “Memorandum” of 1933. A conspiracy theory that Wilhelm M¨ uller, Eduard May, Hugo Dingler, Bruno Th¨ uring and others knew each other or made some sort of arrangement is not necessary. It is sufficient that there be a large commonality of thought- and argumentation-forms—perhaps independently of each other—but nonetheless common or similar goals would be followed and much more. But as little as there is a definite “alcoholic” type of man, so little is there “the Nazi” in mathematics and the natural sciences. Between 1934 and 1945 Max Steck had more than 20 publications to show, among which a contribution in P. Ernst (ed.), Mathematische Abhandlungen. Heinrich Liebmann zum 60. Geburtstag am 22. Oktober 1934.254 Thus his late statement that he was a student of H. Liebmann is to be taken seriously. The accusation of Loebell that the quality of his work was not sufficient for a high school [i.e., college] 254 “Dedicated by friends and students. With contributions by: F. Engel, S. Finsterwalder, G. Kowalewski, Max. M¨ uller, O. Perron, A. Rosenthal, S. Salkowski, W. Schaff, M. Steck, O. Volk”, published in the Sitzungsberichte der Heidelberger Akademie der Wissenschaften. Math.-nat. Klasse, Heidelberg, 1934.

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is in my opinion untenable. Steck’s bibliographical works on Euclid, Lambert, and Proclus are still today of great usefulness. Steck was a geometer255 —the only mathematician in the Dingler circle—and taught geometry as private docent at the Technical High School in Munich. He was obviously filled with indignation that the domain of mathematical logic would be internationally fashioned for a long time in the specialist journals and reference works and this domain could defend itself against the “incorporation” into the ring of “German sciences”. The “struggle over a German logic” would be carried out in the journals, and a desideratum is the survey of all the engagements in all relevant journals. Here as an example I cite the Geistige Arbeit: “Docent Dr. Steck, T.H. Munich” reviewed May’s book Am Abgrund des Relativismus [At the abyss of relativism] in the Geistige Arbeit 256 as “one of the most significant recent publications in philosophical and philosophico-natural-scientific literature of the present,” because it was about the central problem of science, that of the validity of scientific propositions. Against relativism the return of an “a priori ” would be called for and May was cited: The names M. Schlick, H. Reichenbach, R. Carnap, O. Neurath, Ph. Frank, H. Hahn and K. Popper signify. . . the climax, in which the complete a priori decline, or rather the fanatic struggle against the a priori, consistently joins forces with the open declaration of belief in the uncircumscribed reign of relativism.257

One need not substitute the word “race” for the a priori in order to know why Steck paid attention to the book and praised it with the words: “One will still consult this book, when the concoctions of the earlier ‘Vienna Circle’ of logistic and of relativity theories have for a long time gone to the happy hunting grounds of rubbish; one will today no longer be able to ‘hush it up’ and simply ‘bypass it’ with Jewish impudence. The truth must and will conquer. May has enlightened the path to this victory and seen the meaning of the scientific in general in a clarity and developed it as few thinkers before him.” Eduard May saw in Hugo Dingler’s final foundational claim a way out of the crisis of relativism. According to May, Dingler “is known to have succeeded” in framing “all fundamentals of the so-called ‘classical’ physics (to which also Aristotelian logic, Euclidean geometry, and arithmetic belong) as ideal requirements arising from the desire for unambiguity, with which a realisation in matters of natural knowledge of relativism will be radically overcome and the construction of a system guaranteed” (May 1941, p. 276). May undoubtedly knew this via some infallible criterion, but he is silent about it. The founder of “critical psychology”, Klaus Holzkamp, an icon of the Berlin student movement, dedicated his book Wissenschaft als Handlung. Versuch einer neuen Grundlegung der Wissenschaftslehre [Science as action. Attempt at a new grounding of the theory of science] (DeGruyter, Berlin, 1968) to Eduard May [“Eduard May zum Gedenken”] and also occupied himself in later publications with Hugo 255 Everyone knows the remarks of Aristophanes in the “Clouds” on geometers. But also in M¨ orike’s “Maler Nolten” one reads: “A delicate dandy came into the house. Geometer, or what he is, a widely travelled cousin from the neighbouring city.” (Albrecht Goes (ed.), Eduard M¨ orike. Werke in einem Band, Tempel Klassiker, p. 153.) The geometers, and therewith already in general the mathematician of any shade is meant, are mostly treated in æsthetic literature with coarse “humour”. Where does that come from? 256 Issue 1, p. 5, 5 January 1942. 257 P. 197.

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Dingler as an alternative to Karl Popper, on whose soil “the powers inclined to dissolution and decomposition could unfold without restraint” (E. May). But on 20 April 1942, on Hitler’s birthday, there appeared in the Geistige Arbeit a wonderful article by J¨ urgen von Kempski (Berlin), “Ernst Schr¨ oder—der Algebraiker der Logik” [Ernst Schr¨ oder—the algebraist of logic], in which one reads among other things: “It [Schr¨oder’s scientific Nachlass] has been examined there by Prof. Heinrich Scholz, who has created the centre for these researches in Germany in his logistic Seminar, and his collaborator Friedrich Bachmann, and awaits further working out.”258 On 20 May 1942 H. Scholz himself appeared in the Geistige Arbeit with his earlier aphoristically formed text “Leibnizsprachen”. Steck answered this in number 19 on 5 October 1942 with “Mathematik und Erkenntnis” [Mathematics and knowledge], where he denounced “formalism and logistic” anew: “There only remains left the idealistic formulation of the mathematical as total solution, which incidentally the intuitions of Brouwer and Weyl still don’t have.”259 Furthermore, the conception of Riemann’s intuition would be represented, as Max Kommerell saw it: “This view, as it in particular is set down in Riemann’s habilitation thesis, is unfortunately almost entirely wiped out through the usurping of Riemann’s thoughts on the part of the mathematical representatives of the ‘Vienna Circle’ and would be reinterpreted in favour of pure set theoretic-formalistic-logistical construction of mathematics and its applications in the frame of the so-called “relativity theories”, whose original roots Th. Vahlen has recently so tremendously exposed.” Steck referred to his book Paradoxien der relativen Mechanik [Paradoxes of relative mechanics], Leipzig, 1942. In the last two paragraphs Steck recommended that logistic—thus H. Scholz—meticulously consider whether it should not examine and correct its image of Leibniz. But Steck is obviously completely in the dark about his objection and its effect, and in October 1944, in the last issue of Geistige Arbeit, his “Die moderne Logistik und ihre Stellung zu Leibniz” [The modern logistic and its position on Leibniz] appeared,260 wherein he “answered” Scholz’s essay in a virulent manner and found against Scholz, Bachmann, Kratzer and Hermes that logistic referred wrongly to Leibniz. J¨ urgen von Kempski reviewed Alfred Tarski’s Einf¨ uhrung in die mathematische Logik und in die Methodologie der Mathematik 261 in the Geistige Arbeit 262 and recommended the book for beginners as especially useful as well as for philosophers “for thorough study” because it imparted the spirit of the discipline splendidly. Against nominalism, tautologism, schematism, and formalism Steck confessed to an archetypical-noetic manner of thought, where geometric ideas are never to be captured in abstract operations with formulæ. An adulteration of the correct mathematics had been brought into the world with Descartes. The future mathematics would be holistic, idealistic, and all-encompassingly German, but Steck could not formulate practical suggestions or positive goals. With his attacks against Hilbert as “decadent” he came into Ludwig Bieberbach’s territory, and Bieberbach allowed Heinrich Scholz a defence against the assault in Deutsche Mathematik. Scholz, like Steck, was also not a recognised expert mathematician, because neither possessed a chair in mathematics. The Deutsche Mathematik however had counted Scholz’s 258 P.

2. 2. 260 Pp. 55ff. 261 Julius Springer, Vienna, 1937. 262 Issue 12, p. 7, 20 June 1941. 259 P.

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institute as a mathematical one since 1936. Bieberbach is in any case to be thanked for having given the Dingler circle no opportunity to respond to Scholz’s views. He also did not allow any discussion of these things by Steck and his gang to take place in the Jahrbuch u ¨ber die Fortschritte der Mathematik. To what extent Max Steck was embroiled in the events around the “German uhrer, Bruno Th¨ uring, and Rudolf physicists” like Wilhelm M¨ uller, Wilhelm F¨ Tomascheck or whether he played a rˆole in the founding by Hans Frank in December (and closed in September 1942) of the “Institute for the Technology of the State” at T.H. Munich, I cannot explain in greater detail here.263 By 1940 mathematical foundational research, especially the formalism propagated by the Hilbert school, was almost without influence. At G¨ ottingen foundational research was no longer in existence. The M¨ unster Institute under Scholz worked the more industriously. Although a follower of Ludwig Bieberbach, Steck briefly inclined for tactical reasons to Weyl’s form of intuitionism. With the publication of parts of the correspondence of Frege and Hilbert regarding questions of axioms and implicit definitions, truth and existence, the furious judeophobe wanted to demonstrate that Frege had indeed misunderstood his adversary Hilbert but somewhere or other was still correct. So Steck built Hilbert up as the greatest logician of all time, the better to annihilate him later. Steck saw the folkish rooting of mathematics as endangered by Hilbert’s formalism. Max Steck’s “Ein unbekannter Brief von Frege” [An unknown letter from Frege] was reviewed by Ruth Moufang:264 The editor supplies the faithfully reproduced original letter with a foreword, in which he goes into the significance of the foundational research of Hilbert and 263 Cf. pp. 256ff. in: Ulrich Wengenroth (ed.), Die Technische Universit¨ at M¨ unchen. Ann¨ aherung an ihre Geschichte, Technische Hochschule M¨ unchen, Munich, 1993. 264 Ruth Moufang (1905-1977) would be denied a lectureship in 1937 on account of her gender. She was the daughter of the one-time director of the Staatlichen Porzellan Manufaktur Berlin, Dr. Nicola Moufang, who would be rabidly attacked by the Nazis. Cf. Gottfried Zarnow, Gefesselte Justiz. Politische Bilder aus Deutscher Gegenwart, Bd. 1, J.F. Lehmans Verlag, Munich, 1931, p. 95. Also Paul Bernays wrote a review in the Journal of Symbolic Logic 7 (1942), p. 92. Bernays went into Hilbert’s ideas. Ruth Moufang is not forgotten. She is mentioned for example in Wilhelm P.A. Klingenberg, Mathematik und Melancholie (Von Albrecht D¨ urer bis Robert Musil), Franz Steiner Verlag, Stuttgart, 1997:

My academic teacher, Friedrich Bachmann in Kiel, called my attention to a problem with which one of the female students of David Hilbert, Ruth Moufang, who taught in Frankfurt, had been occupied with for decades without finding the solution. The question arose from Hilbert’s investigations on the foundations of geometry from the beginning of the century. . . It concerned the clarification of the relation between two so-called closure theorems [“Schließungss¨ atze”. There apparently is no standard English translation of this term. I follow S.H. Gould’s translation (Fundamentals of Mathematics, II, Geometry, MIT Press, Cambridge (Mass.), 1974, p. 68) of Heinrich Behnke, Friedrich Bachmann, and Kuno Fladt (eds.), Grundz¨ uge der Mathematik. Band II. Geometrie. Teil A: Grundlagen der Geometrie. Elementargeometrie, Vandenhoeck & Ruprecht, G¨ ottingen, 1967, pp. 71ff. —Trans.] in a general affine plane, the figures (D) and (S). The solution by Klingenberg then follows. Ruth Moufang was promoted by Max Dehn, who was a student of Hilbert. On Ruth Moufang, cf. Renate Tobies (ed.) “Aller M¨ annerkultur zum Trotz”. Frauen in Mathematik und Naturwissenschaften, Campus, Frankfurt am Main, 1997; Gerhard Burde, Wolfgang Schwarz, J¨ urgen Wolfart, Max Dehn und das Mathematische Seminar, Institut f¨ ur Mathematik, Universit¨ at Frankfurt, 1984 and 2002.

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the currently still only historical value of Frege’s objections. Frege’s criticism directed itself generally against the character of Hilbert’s axiomatics and specifically against the proof of the independence of the axioms of Euclidean geometry.265

But what did Steck intend with this “presentation”? In the same volume266 Steck reviewed Hermann Weyl’s “The mathematical way of thinking”.267 After praising Andreas Speiser’s Die mathematische Denkweise [The mathematical way of thinking] it sets off again: Finally, the author describes as a characteristic trait of each mathematical way of thought the axiomatic method. Thereby the standpoint of formalism will first be explained. . . The weakness of the axiomatic method will be obvious through the investigations of K. G¨ odel.

One can scarcely believe that Steck desired the emigrants Weyl and G¨ odel as allies in battling against Hilbert. In his book Das Hauptproblem der Mathematik,268 published by the near-Nazi publishing house of Dr. Georg L¨ uttke, Scholz and Hilbert—like all representatives of modern foundational research—would be mocked. The attacks of the conceited geometer Steck were dangerous, because they aimed if anything at the abolition of the subject and the closing of the corresponding institutes. But for the official Nazi view, I cite a letter from W. Erxleben to the Nazi mathematician Fritz Kubach of 18 February 1943: To the Rosenberg office his [H. Scholz–EMT] work is too formal and without substance.269

Reich Student Leader Kubach had organised a boycott of lectures by “non-Aryan” mathematicians in 1935, and belonged to the circle of Eduard May, Th¨ uring, and Dingler. Kubach’s lack of interest saved Scholz. And therefore in 1943 he was rewarded with the conversion of his philosophical institute into one for mathematical logic and foundational research (official report by Bieberbach). Nevertheless H. Scholz wrote to Evert Willem Beth on 24 August 1946: I must not say to you what we had feared from certain places on account of this book [Max Steck’s Das Hauptproblem der Mathematik –EMT]. I must risk a countermove or I would be jointly responsible for it if our research died, because in those days I had to reckon with the possibility that the war would turn out such that the system would survive it somehow.270

It is necessary to think back about it, whether Scholz had to publish thus in the Reich. It is clear that without support and publications in near-Nazi journals he would have had much more difficulty with the furnishing and defence of his institute, collaborators and their works—yes the entire mathematical foundational work in Germany and the survival of these institutions.

265 Jahrbuch

u ¨ber die Fortschritte der Mathematik 66 (1940), p. 24. 1190. 267 Science 92, pp. 437-446. 268 Berlin, 1942; second edition, 1943. 269 The letter, according to G. Leaman 1993, is in the Institut f¨ ur Zeitgeschichte. 270 Arie L. Molendijk, Aus den Dunklen ins Helle. Wissenschaft und Theologie im Denken von Heinrich Scholz, Rodopi, Amsterdam, 1991, p. 60. 266 P.

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22. The Exception: The Dedicated National Socialist, Logician and Historian of Mathematics, Oskar Becker, Remains Neutral One who did not participate in this bitter fight was Oskar Becker. “Oskar Becker was of a physically extremely delicate, almost timid nature.”271 Oskar Becker (1889-1964) studied mathematics, psychology and philosophy in Oxford and Leipzig and was promoted in 1914 in mathematics. He habilitated in 1922 in Freiburg under Edmund Husserl writing on “Mathematical existence”. He was assistant to Martin Heidegger until 1927, when he became unscheduled professor. In 1931 he moved to Bonn, where he received emeritus status in 1955 and died on 13 November 1964. Karl L¨ owith described “Dr. B”—as Becker is there designated—in his memoirs272 as inspired by the National Socialist revolution. From his Freiburg period Becker was acquainted with the “Rassenseelenforscher” [racial psychology researcher] Ludwig Ferdinand Clauß and applied the latter’s theory to philosophy. Thus he placed opposite the “Near Eastern desert existenceinterpretation of the world” that of the “Nordic productive men”: The true, unspoilt Nordic researcher will never acknowledge that the magicbelieving world view of a Congo Negro in its kind could be as good as the results of his laborious observation of nature and conscientiously thought through conclusions. He knows much more: he alone sees nature as it is. . . The technology grounded on the nordic natural science has conquered the world, not the magical art of the primitive people.273

Consequently after the pogrom of 9 November 1938 Becker wrote to Otto Neugebauer that it appeared impossible after the most recent events that a German scientist could still further collaborate with a journal in which one of the editors was a Jew, and Neugebauer, because of this necessity, should ask Otto Toeplitz (18811940) to resign as associate editor of the Quellen und Studien zur Geschichte der Mathematik on account of the latter’s Jewish origin. Neugebauer responded with his own resignation and the remark: You write to me that you as a National Socialist appear to have a different point of view from mine, despite your deep personal respect for Mr. Toeplitz. I can only answer to that that I am not in lucky possession of any “world view” and therefore feel obliged to consider, in each individual case, what I should do, without being able to retreat to a predetermined dogma.274

Becker described himself in a letter to Neugebauer of 9 December 1938 as a National Socialist.275 Consistently, Oskar Becker remained professor in Germany. Even today it is thought highly of him that he publicly admitted to being a Nazi. 271 Hans Georg Gadamer, Philosophische Lehrjahre, Vittorio Klostermann, Frankfurt am Main, 1977, p. 174. On Oskar Becker, a teacher of J¨ urgen Habermas, Hans Sluga, Paul Lorenzen, Otto P¨ oggeler, Karl-Otto Apel, Kurt-Heinz Ilting, or Hermann Schmitz (who stood in close scientific contact with Paul Lorenzen and whose literature was often thoughtfully corrected by the latter), cf. Wolfram Hogrebe, Echo des Nichtwissens, Akademie Verlag, Berlin, 2006. The recent literature on O. Becker is also mentioned there. 272 Mein Leben in Deutschland vor und nach 1933, Metzler, Stuttgart, 1986. 273 Oskar, Becker, “Nordische Metaphysik”, Rasse. Monatszeitschrift der nordischen Bewegung 5 (1938). 274 Reinhard Siegmund-Schultze, Mathematiker auf der Flucht vor Hitler. Quellen und Studien zur Emigration einer Wissenschaft. Dokumente zur Geschichte der Mathematik. Band 10, Vieweg Verlag, Wiesbaden, 1998, p. 142. 275 Ibid., pp. 141ff. Cf. more generally, Hans Paul Hopfner, Die Universit¨ at Bonn in Dritten Reich. Akademische Biographien unter nationalsozialistischer Herrschaft, Bouvier, Bonn, 1999.

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It seems generally to have been a hallmark of the mathematician, that few— like even the modal logician Oskar Becker—described themselves whole-heartedly as “National Socialists”. Except for Weber, Teichm¨ uller, Tornier and perhaps in foggy moments Mohr or Nikuradse, none occur to me. “No one wants to be the only National Socialist” (Alexander Kluge). Oskar Becker at “heart” a modal logician and J. Goebbels in mind as guiding star of thought is for some historians of science today a still unachieved life goal. Those who were silent with regard to the Nazis and profited from them today have a higher reputation than the exiles and the murdered. This failure would be reckoned an achievement today, because if everyone had left, “we would have no intellectual elite, above all we would have stood defenceless against the colonisation after the war.” Such could also be argued against the “disgrace of denazification”. Better Konrad Meyer than Richard Courant—that remained the motto of the postwar period. Becker launched the theory of “para-existence” first in 1943. The accomplice of the Kapp-Putsch,276 he was usually excellent in the First World War—so much for his “timid” nature—and was a keen forerunner of the Nazis. His membership in the National Socialist Teacher Union and in the National Socialist Cultural Community made a membership in the Party unnecessary. His “Nordic Metaphysics” is not “anti-Semitic”, because all Jews at the university departed, Thomas Mann was stripped of his doctorate, and he could proceed from the assumption that, in principle, Germany was “Jew-free”. Not mentioning the “self-evident” can also be an indication for “successful” anti-Semitism. Thus he later came back to bare “synthesising ideas”, to which he still misused Nietzsche. Oskar Becker, whom a scholars’ calendar from 1940 correctly records as a “Rassenseelenkundler” like his colleague and likewise Husserl-assistant L. Clauß, was a leader of the Nazi race theory, who polemicised against the “Near eastern desert delivered people” and would with logical consistency become professor in Bonn again in 1951. Gadamer, whose single contribution to a view of National Socialism is also worth an examination—cf. the work of Teresa Orozco—remembered then and also later otherwise: Who went into the Party, in order to keep his position or to gain one, and then as teacher of philosophy studied rational philosophy, is to me ten times preferable to such people as (Oskar) Becker or (Hans) Freyer, who were not in the Party, but who spoke like the Nazis.277

Hans Heiber reports that Becker—a friend of Ludwig Ferdinand Clauß, author of Rasse und Seele. Eine Einf¨ uhrung in den Sinn der leiblichen Gestalt 278 [Race and soul. An introduction to the meaning of the physical form], likewise a student of Husserl but later oppressed on account of his Jewish wife—reacted spiritedly to a Mr. Mense with the habilitation thesis “Die metaphysischen Grundlagen der nationalsozialistischen Politik” [The metaphysical foundation of National Socialist politics]. Such a vivid description of the species-specific powers embodied in the F¨ uhrer would still be advisable; here from large points of view is an attempt at a 276 Wolfgang Kapp (1858-1922), co-founder in 1916 of the Fatherland Party, led an attempted coup in Berlin on 13 March 1920. The attempt at a nationalist revolution failed, and Kapp fled to Sweden.—Trans. 277 Gadamer 1990, cited from George Leaman 1993, p. 145. 278 122,000 of the edition, J.F. Lehmanns Verlag, Munich, 1943.

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deduction of the entire National Socialist body of thought from the latest metaphysical principles and basic positions, with which the author had done genuinely great service to the German state-philosophy. Oskar Becker is considered an established philosopher and logician. In Mathematische Existenz. Untersuchungen zur Logik und Ontologie mathematischer Ph¨ anomene 279 he determined the “life-meaning” of formalism and intuitionism. The opposition “intuitionism–formalism” is rooted in the basic philosophical opposition of the anthropological and the absolute conception of knowledge and finally of life itself. Formalism in mathematics goes together with absolutism, intuitionism with anthropologism. In the review of this work,280 H. Scholz compared the categorisation of Poincar´e of pragmatists (=intuitionists) and Cantorians (=formalists) “in the same rubrics”. And with the categories of intuitionism and formalism the History of Mathematics is bullied. However, what remains as a problem, in addition to those listed in the review by Scholz, is the connection and the direct classification of the diverse concepts of mathematical existence, intuitionistic Brouwerian choice sequences and historical time concepts, exact philosophy and mathematics, philosophy and metaphysics. Oskar Becker was clearly a dedicated Nazi, as follows from his letters to Neugebauer mentioned above. Why did he not denounce anybody in the domain of mathematical logic? Because this was like the University of Bonn, already “Jew-free”, and he, contrary to Steck, knew that one could construe no disastrous penetration of “Jewish abstraction” into logic and that a “German Mathematics” could not be successful? But one can positively record that Oskar Becker never participated in actions of “German Mathematics”. That was possible without risk. 23. Resistance as a Mathematician Was Possible under National Socialism Though the world expected otherwise from those who remained: In the summer of 1933 I was invited to a lecture at G¨ ottingen, the capital of mathematics. The professors had already packed their bags, and a student mob ruled the field. And then came the most painful disappointment of my life. What I had held as entirely impossible occurred. An entire row of outstanding scientists, not only opportunists who believed themselves now to be able to get rid of their competition, sounded Hitler’s horn and were not afraid to speak of the good Aryan and the bad “foreign” Jewish physics and mathematics, and therewith to make their German fatherland ridiculous in front of the whole world. With this I had much to dispute and my wife was often anxious that I would open my mouth. But I was skillful and never came into conflict with the Gestapo, only once with a wretched block leader, who wanted to forbid me the Frankfurt newspaper and whom I naturally, angrily told to get lost. The local experts were naturally on my side in the battle against the Nazis. There was no being forced into line. Orders about this would be circumvented or ignored. Never before would so many Jewish authors here be recommended in the lecture as after the ban. Therefore also after the final victory under the Americans, mathematics especially, here at the university as well as at the technical high schools, has remained intact whereas, for example, one battled to the utmost by us, but still forced on us by the Ministry as representative of theoretical physics [Wilhelm 279 Niemeyer, 280 Deutsche

Halle, 1927, pp. 184ff. Literaturzeitung 14 (1928), column 685.

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M¨ uller], who often sought to frighten us with his good relations to the F¨ uhrer headquarters, must have vanished immediately.281

24. Kurt Reidemeister’s Additional Contemplations on Politico-Scientific Power Play in “German Mathematics” What does the Marburg mathematician Reidemeister, whose wife, Elisabeth, would be the most important “witness” for Constance Reid’s books on Courant and Hilbert, write about “German Mathematics”? ¨ We take his book Uber Freiheit und Wahrheit 282 [On freedom and truth] as ¨ an example. In it he combined 3 newspaper contributions: I. “Uber Freiheit der Wissenschaft” [On the freedom of science], II. “Entartung und Norm der Wissenschaft” [Degeneration and norm of science], and III. “Zum Erlebnis von L¨ uge und Wahrheit” [On the experience of falsehood and truth]. He asks first for the determination of the relation of state and science, thus the responsibility of the civil servant, who would be appointed to look after science: what stands in his job description: “Are those appointed to care for science first of all scholars or civil servants performing tasks for the state?”, i.e. what is their responsibility? Can extra-scientific criteria be given for the assessment of science? Were the usefulness of knowledge, the realism of the science, the service for one from the state each to the set goal the proper task of the science, then it had experienced no better time as in the most recent past. . . Never had the utility even in the base sciences experienced such a complete recognition by the state as in the moment when it decided intentionally to intervene in the business of science. The objectivity of the research via this plan would be perhaps de facto, but in principle not touched.283

But each plan sets priorities, limits one, shortens, extends, modifies and is per se subject to “external goals”. There are indeed men who are not able to recognise the difference between state “censorship” and private “sponsorship”. One can normatively demand a lot, as Reidemeister does: “The institutions of scientific business must be competently managed. And the science, where it will help, must be able to speak freely and its assistance must afterwards be measurable, so far as it gains a hearing.”284 Freedom in an “inner public” of science might be necessary, but what does one do if it—as happened in the Third Reich—will simply no longer be guaranteed? One can cite a villain, who obviously saw the question of the “external goals” differently: The history of German science is most tightly connected with this its greatest academy. Since 1711 the name of no impactful scholar in Berlin is missing from it. Is it, however, also connected with German research? This question was raised in the session of 6 July 1939, held yearly as the “Leibniz session” in remembrance of the great founder, in the eyes of a wide, one must say, the German public, and, beyond that, of the world. The new president came forth and opened the new epoch in the history of the academy of sciences. He is Professor Dr. Vahlen, expert mathematician and until recently responsible ministerial director of the 281 Freddy Litten, “Oskar Perron–Ein Beispiel f¨ ur Zivilcourage im Dritten Reich”, Frankenthal einst und jetzt, no. 1/2 (1995), pp. 26-28. 282 Kurt Reidemeister, Uber ¨ Freiheit und Wahrheit, Carl Habel Verlagsbuchhandlung, Berlin, 1947. 283 Ibid., pp. 8ff. 284 Ibid., p. 11.

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Reich Science Ministry. He recently turned seventy years old and should now for the next five years be the president of the academy, in order to carry out its reconstruction. The crux of his opening address, which in tone and manner put forth the great and difficult tasks of the Third Reich, is contained in an idea: science must again become more research, and certainly in the sense of the founder Leibniz, who said, science must direct itself toward the general good. The path through the centuries shows compellingly what damage this turning away from true to life research has brought. So the life insurance industry that had been called to life around 1700 collapsed because it lacked the mathematical foundation. Leibniz created this and ever since the German insurance industry has thrived.285

Afterwards, one of the two secretaries, the mathematician Bieberbach, welcomed his colleague Georg Hamel. This “Professor for Applied Mathematics dedicated himself to the motion of water and succeeded thereby to a genuine research domain for the German reconstruction.” The Nazi tirades of Vahlen and his gang were strengthened through a like-minded press or members of the Reich’s Literature Chambre. Reidemeister specified that: . . . the currency of science in the general public of the German state in May 1933 would be destroyed, the management of the institutions would be adversely affected, but the inner public of science remained on the whole intact, and a struggle over it had developed, which with tenacity would be carried on into the final hours of the Hitler state.286

I would gladly have read exactly the description of the struggle in mathematics and management. Instead I read assurances: Just so decisive and general as the approval of this struggle over the objectivity in the business of science was and the judgement is of those who have attempted to introduce the power of the total state through application of world-view language to scientific questions as means of persuasion in place of reason. The representatives of “German Mathematics” and “German Physics” have as such enjoyed no respect. There is no doubt about it, that they have gone against the fundamental duty of the scholar: the fostering of the possibility of objective discussion, and that they will no longer be able to be counted as full citizens in the republic of scholars.287

And too Reidemeister finds it much more difficult to make clear how the politicoscientific power play in science management was: What happened in discussions of faculties and senates? Which spirit waved in the representative celebrations of the university? Were there also here general ideas and positions, over which one could speak in principle, over which we may speak without violating demands which ought to be silenced?

That the scientist is bound to the idea of objectivity and is free in it, comparable with the judge who is bound to the law and yet freely administers justice, Reidemeister held highly and in respect. But the reality looked otherwise. The “structure of National Socialist science” was seen by Reidemeister as “degeneration”: The degeneration consists here virtually only in uncritical, contradictory, and narrow-minded attempts to derive from a super-scientific meaning of science a 285 Hans

Hartmann, Forschung sprengt Deutschlands Ketten, Union Deutsche Verlagsgesellschaft, Stuttgart, 1941, pp. 195ff. 286 Op. cit., p. 12. 287 Ibid., p. 13.

25. LONGER NOTES

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“German mathematics” and a “German physics”. The two sciences have remained unshaken. “In fact”, Bieberbach said in an essay on the folkish rooting, “it will be difficult to give a correct mathematical proposition which not every mathematician recognises as correct.” The desperate courage of Steck, to dismiss the refutation of one of his established geometric propositions with the aid of algebra, [claiming] it rested on Jewish logic, remains a singular case.288 Still, “every National Socialist will be certain,” said Bieberbach in the above mentioned essay, “that we all, what we independently do, are dependent on and influenced by the gifts which our extraction laid down for us in the cradle, and certainly more so, the more we ourselves are in our achievements.” If we pursue mathematics independently, then we thus work only our talent for mathematics, and in order that we ourselves be whole thereby, we must determine the kind of our natural abilities and to pursue the, to us appropriate, German type of mathematics. To the solution of this task the “natural scientifically oriented German” study of race and psychology offer a hand. Thus Bieberbach. I call this treatment of the question of the meaning of mathematics degenerate because it does not take the generally binding correctness of mathematics as the basis of its sense, much more insofar as [under it] mathematics derives from requirements, from which the general obligation of its propositions definitely cannot be seen. The racial-psychological outcome of Bieberbach’s business is poor enough. Corresponding to the three I-types of Jænsch are three types of German mathematical talent. Only the S-type is un-German, and the fostering of German mathematics consists thus in nothing other than the eradication of and battle against S-typical traits and dispositions.289

Correct science is critical science which strives for the idea of objectivity. But: The objectivity of science is an idea and although science rests on rules, so we are still not in the position to paragraph those rules.290

Earlier one could ascertain what is not science if, for example, it would be argued that a factual criticism can be interpreted as a criticism of the state, if there are acts of violent destruction of the inner public, if the concept will become “ammunition”, which passes off an act of violence as scientific work. 25. Longer Notes 1

On the history of mathematics under the Nazis, cf., in addition to the sociologically oriented book by Sanford L. Segal (2003), above all the appropriate literature of Norbert Schappacher, Reinhard Siegmund-Schultze, and Herbert Mehrtens. The interested reader can find further references in: Norbert Schappacher and Martin Kneser, “Fachverband–Institut–Staat”, in Gerd Fischer et al. (eds.), Ein Jahrhundert Mathematik 1890-1990, Vieweg, Wiesbaden, 1990; Helmut Lindner, “ ‘Deutsche’ und ‘gegentypische’ Mathematik. Zur Begr¨ undung einer ‘arteigenen’ Mathematik im ‘Dritten Reich’ ”, in: Herbert Mehrtens and Steffen Richter (eds.), Naturwissenschaft, Technik und Ideologie, Suhrkamp, Frankfurt am Main, 1980; R. Siegmund-Schultze, Mathematische Berichterstattung in Hitlerdeutschland, Vandenhoeck & Ruprecht, G¨ ottingen, 1993. On G¨ ottingen cf. in all cases Norbert Schappacher, “Das mathematische Institut der Universit¨ at G¨ ottingen 1929-1950”, in: Heinrich Becker et al. (eds.), Die Universit¨ at G¨ ottingen unter dem Nationalsozialismus, K.G. Saur, Munich, 1987. The prehistory is solidly described by David E. Rowe, “ ‘Jewish mathematics’ at G¨ ottingen in the era of Felix Klein”, 288 Ibid.,

pp. 19ff. pp. 20ff. 290 Ibid., p. 27. 289 Ibid.,

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Isis 77 (1986), pp. 422-449. An outstanding reference is Herbert Mehrtens, Moderne– Sprache–Mathematik, Suhrkamp, Frankfurt am Main, 1990. As a former participant Saunders Mac Lane reports in “Mathematics in G¨ ottingen under the Nazis”, Notices of the American Mathematical Society 42, No. 10 (1995), pp. 1134-1138. On the blooming of mathematical logic in Germany in the 20s and 30s and after the sudden transition under National Socialism, cf. Christian Thiel, “Folgen der Emigration deutscher und osterreichischer Wissenschaftstheoretiker und Logiker zwischen 1933 und 1945”, Berichte ¨ zur Wissenschaftsgeschichte 7 (1984), pp. 227-256. Cf. in this respect Rolf Schaper, “Mathematiker im Exil” and Andreas Kamlah, “Die philosophiegeschichtliche Bedeutung des Exils nicht-marxistischer Philosophen zur Zeit des Drittens Reiches”, in: Edith B¨ ohne and Wolfgang Motzkau-Valeton (eds.), Die K¨ unste und die Wissenschaften im Exil 19331945, Verlag Lambert Schneider, Gerlangen, 1992. For more recent surveys on the natural sciences under National Socialism, I suggest Monika Renneberg and Mark Walker (eds.), Science, Technology and National Socialism, Cambridge University Press, Cambridge, 1994; or J. Olff-Nathan (ed.), La science sous le Troisi`eme Reich, Paris, 1993; Christoph Meinel, Peter Voswinckel (eds.), Medizin, Naturwissenschaft, Technik und Nationalsozialismus, GNT-Verlag, Stuttgart, 1994; and Helmuth Albrecht (ed.), Naturwissenschaft und Technik in der Geschichte, GNT-Verag, Stuttgart, 1993. The literature on “science in the Third Reich” (cf. the book of the same title by Peter Lundgreen, Suhrkamp, Frankfurt am Main, 1985) is endless and of most extremely variable quality. Unreliable, but encyclopædic is Helmut Heiber, Universit¨ at unterm Hakenkreuz, K.G. Saur, Munich, 1991. Also interesting is the “grey literature”, as for example Die braune Machtergreifung: Universit¨ at Frankfurt 1930-1945, ASTA der Goethe-Universit¨ at Frankfurt am Main, Frankfurt am Main, 1989. Classics in physics are the works of Paul Forman, Alan D. Beyerchen, Karl von Meyenn and Klaus Hentschel. For technology, consult K.H. Ludwig and H.L. Dienel, Ich diente nur der Technik. Sieben Karrieren zwischen 1940 und 1950, Nicolaische Verlagsbuchhandlung, Berlin, 1995; for rocketry: Michael J. Neufeld, The Rocket and the Reich. Peenem¨ unde and the Coming of the Ballistic Missile Era, Free Press-Simon & Schuster, ¨ New York, 1995. For biology, cf. Anne B¨ aumer, NS-Biologie, S. Hirzel Verlag, Stuttgart, 1990. The best source for the history of a technical university is Ulrich Kalkmann, Die technische Hochschule Aachen im Dritten Reich (1933-1945), Wissenschaftsverlag Mainz, Aachen, 2003. 11

On the circumstances concerning the confidence man Tirala, who would be supported by the publisher J.F. Lehmann, Julius Streicher, Bouhler, and Philipp Lenard, cf. Helmut Heiber, Universit¨ at unterm Hakenkreuz. Teil 1. Der Professor im Dritten Reich, K.G. Saur, Munich, 1991, pp. 445-460; Claudia Schorcht, “Gescheitert– der Versuch zur Etablierung nationalsozialistischer Philosophen an der Universit¨ at M¨ unchen”, pp. 291-327 in: Ilse Korotin (ed.), “Die besten Geister der Nation” Philosophie und Nationalsozialismus, Picus Verlag, Vienna, 1994. “Lothar Gottlieb Tirala was born on 17 October 1886 in Br¨ unn and already from 1927 was a member of the Nazi Party in Czechoslovakia and in 1928 Vertrauensarzt [untranslatable: A Vertrauensarzt is a physician who “examines patients signed off sick for a lengthy period by their private doctors” (Collins Dictionary. German-English. English-German; Unabridged, 4th ed., Collins, Glasgow, 1999)] of the Party. As of 1 November 1933 he was a Professor of Race Hygiene and Director of the Institute for Race Hygiene of the University of Munich, and from 1 March 1934 member of the German Nazi Party.” (Markus Vonderau, “ ‘Deutsche Chemie’. Der Versuch einer ahrend der Zeit deutschartigen, ganzheitlich-gestalthaft schauenden Naturwissenschaft w¨ des Nationalsozialismus”, Dissertation, Marburg, 1994, p. 168.) “The aptitude for science is to be found only in a few peoples; however, the gift of creative science is found only by the Aryans, in particular by the Nordic races. . . Only from a conception of the world can proper natural science develop. . . and that is the conception

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of the world of the Aryan race” (Tirala (1936), pp. 27ff.; cited on p. 169 in Markus Vonderau, “Deutsche Chemie”). And what distinguishes the Aryan scientist? 1. Abandoned devotion to nature 2. Pleasure in observation 3. Power of abstraction 4. Discernment of the important and unimportant, i.e. an eye for the essentials 5. Pleasure in the phenomenon 6. Pleasure in repetition 7. Modesty, not to step prematurely into the foreground with one’s ego 8. Power of concentration 9. A streak of asceticism or. . . solitariness, because without this trait one will be torn away from life and cannot do research 10. Joy in battle with the object, joy in the hunt (Tirala (1936), p. 32; cited on p. 169 in Markus Vonderau, “Deutsche Chemie”.) 36

Now Bieberbach also took up “supplements”. He wanted to publish W. Weber, Die Pellsche Gleichung, and Vahlen, Abstrakte Geometrie, which first appeared in 1905. Vahlen was to appear in an edition of 800 copies. By the end of 1941 there were still 704 copies in the warehouse; 59 had been sold. On 14 April 1941 Deutsche Mathematik would switch over from Fraktur to Antiqua. On 31 December 1941 the Heine Press reported the stock of Deutsche Mathematik. From 1936, 1,467 completely sewn copies each of issues 3 to 6 were lying around. Hirzel was more precise on 21 March 1942: Year 1936

1937 1938 1939

Issue Issue 1 (edition 3000) Issue 2 Issue 3 (edition 3000) Issues 4-6 Issue 1 Issues 2-6 Issues 1-6 Issues 1-2 (edition 700) Issue 3 Issue 4 Issues 5-6

Stock 2110 2125 1467 1467 each 667 467 each 167 each 167 each 157 166 167 each

Proposed for Pulping 1500 1500 1000 1000 each 200

If thus 2,110 copies from an original 3,000 were in stock in 1942, despite massive advertising and giveaways of the issues, then this cannot be celebrated as a “marketing success”, and even less as a propaganda offensive. Consequently, the first issue was pulped straight away. 53

Bieberbach’s anti-Semitism dates back at least to 1940. Cf. Ludwig Bieberbach, “Die Unternehmungen der Mathematisch-Naturwissenschaftlichen Klasse”, pp. 23-29, in: Wesen und Aufgaben der Akademie. Vier Vortr¨ age von Th. Vahlen, E. Heymann, L. Bieberbach und H. Grapow, Walter de Gruyter & Co., Berlin, 1940. Bieberbach reported that under the editorship of Professor Geppert the Jahrbuch u ¨ber die Fortschritte der Mathematik and the Zentralblatt der Mathematik combined and the work would be done by five mathematicians and a few office personnel. Also, connections existed to the Encyklop¨ adie der mathematischen Wissenschaften, an organ of the Association of German Academies. It begins to appear in a second edition: “Naturally in all times there are people for whom it is a thorn in the eye that for half a century German industry and German leadership managed in the Jahrbuch an indispensable tool for the whole of science, and brought to completion an otherwise never attained achievement. Recently this envy

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intensified under the leadership of some emigrants to a plan of a rival undertaking, which has begun to make its appearance with the support of Jewish money. That the German work is inadequate, the emigrants and Jews have not once been able to maintain. But that emigrants be at the head and Jewish money finance the undertaking explains sufficiently well in place of many words the motive of the re-establishment.” And he mentioned still the works on the history of the universe, systematic of biology, the atlas of German Lebensraum, the Leibniz edition and the new edition of the writings of Copernicus, where Fritz Kubach would be rewarded with the publication. A small side comment: The possibility was underestimated that vain, greedy, stuckup twits like Bieberbach could build up personal animosities—like his abhorrence of the personality and style of Landau—which only tangentially had to do with his Jewishness— into a struggle over world view. Bieberbach wanted to be a sort of mandarin in the Nazi Party; he wanted less actual power than magnificent representation and most preferred, I conjecture, every day to wander around the Mathematical Seminar with the chain of office of a president of the Kaiser-Wilhelm Society around his neck, to hold court and be in and out of the Ministry every day to dictate his directives to Minister Rust. But the would-be F¨ uhrer of the F¨ uhrer was badly misguided: in their vanity Heidegger, Vahlen, Bieberbach and many others would be seen by the Party’s own “golden pheasants” like Rosenberg and Himmler as rivalry, which was baseless, because in race questions members of the Ahnenerbe Stiftung and others knew their way around better than Bieberbach, Vahlen, Dingler, M¨ uller, or Stark. Bieberbach’s unintended consequences, like the classification of Hilbert, however, lost him good standing in mathematical society. And as he saw that thereby he no longer marched at the head of mathematical society, had to step down from offices, lost his reputation, Bieberbach quickly turned pensive and withdrew and atoned for his Judas betrayal of Hilbert with the assessment of Scholz’s Institute in M¨ unster. I am also firmly convinced that he wanted to rescue L  ukasiewicz and others, but was simply afraid of the SS-henchmen. Without backing by the mathematicians he could hardly open his mouth in the Academy: He was caught between two fronts. As diplomat and science organiser, Bieberbach was a washout. 63 Bieberbach may have criticised Hilbert strongly in 1925/26 because the latter with his formalism, falsely understood by Bieberbach, would have repressed intuition, but from 1941 he supported Hilbert via Scholz. Cf. pp. 205ff. in: Herbert Mehrtens, “Ludwig Bieberbach and ‘Deutsche Mathematik’ ”, in: Esther R. Phillips (ed.), Studies in the History of Mathematics, The Mathematical Association of America, 1987. On Bieberbach’s intellectual sphere cf. Herbert Mehrtens, “Anschauungswelt versus Papierwelt. Zur historischen Interpretation der Grundlagenkrise der Mathematik”, in: Hans Poser and Hans-Werner Sch¨ utt (eds.), Ontologie und Wissenschaft. Philosophische und wissenschaftshistorische Untersuchungen zur Frage der Objektkonstitution, Technische Universit¨ at Berlin, Berlin, 1984.— It is, however, completely false if one says, “Bieberbach’s new campaign from summer 1933 was thus also an attempt at revenge against ‘Hilbert’s Programme’ all along the line” (p. 23, Norbert Schappacher, “Auswirkungen des Nationalsozialismus auf die mathematische Forschung in Deutschland”, preparatory material to an inaugural address, Bonn, 8 June 1988, version II). If that were so, then he could have silenced the supporters of a Hilbert Programme ideologically in a single blow. On Bieberbach’s conduct, cf. Herbert Mehrtens, “ ‘Was verstehen Sie von deutscher Wissenschaft?’ Das ‘Dritte Reich’ und die politische Moral der Naturwissenschaften”, in: Christoph Hubig (eds.), Verantwortung in Wissenschaften und Technik, Technische Universit¨ at Berlin, Berlin, 1990. NB. If E.J. Gumbel criticises Bieberbach’s “pig-herd tone,” then it should not be forgotten that it is a question here of a return. The inventor of the “pig-herd tone” was the social democratic Professor Paul Lensch, who received this honorary title on the grounds of his lead articles in the Leipziger Volkszeitung. Hugo Stinnes held him to that

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in his national paper, the Deutsche Allgemeine Zeitung, p. 94, in: Paul Fechter, An der Wende der Zeit. Menschen und Begegnungen, C. Bertelsmann Verlag, G¨ utersloh, 1949.— Also David E. Rowe wrote in his review of Gerd Fischer’s Ein Jahrhundert Mathematik 1890-1990 (Mathematical Intelligencer 13, no. 4, pp. 70-74; here: p. 73): For example, they note Bieberbach’s rˆ ole in the earlier battle that ensued when Hilbert sought to oust Brouwer from the editorial board of Mathematische Annalen, an episode recently chronicled by Dirk van Dalen in the Mathematical Intelligencer (vol. 12, no. 4). The authors [Schappacher and Kneser—EMT] conclude quite correctly that this battle and the whole intuitionist-formalist debate that echoed throughout the Weimar period carried broader political implications, and these set the stage for much of what followed during the early Nazi years. I expect answers on why intuitionism was not at least publicly mentioned by a Nazi mathematician as official foundational research or—more importantly—wasn’t used in his mathematical writing. I expect a discussion of these “broader political implications”, which allegedly should have been the stage in National Socialism for the debate within mathematics to have been played. However, Bieberbach defended and permitted the defending of formalism. Why did the other Nazi mathematicians not join in in this discussion, rather were publicly content with the expositions of H. Scholz? That my father met with so much hostility after the war, lay among other things in the personal domains of Einstein and Perron. I have nowhere found a professionally grounded disparagement. My younger brother, then 5 years old, asked Einstein while he was once visiting us, why he stood with naked feet in his shoes. Foolishness and vanity are universal. (Ulrich Bieberbach, to the author from Oberaudorf on 27 November 1998.) And this vanity—on Ludwig Bieberbach’s side—is allegedly attested to through a letter from Albert Einstein to Max Born from the year 1919: “Mr. Bieberbach’s love and admiration for himself and his muse is most delightful.” (Cited from Norbert Schappacher, “The Nazi era: the Berlin way of politicizing mathematics”, in: H. G. W. Begehr et al. (eds.), Mathematics in Berlin, Birkh¨ auser, Basel, 1998; here, p. 131. No source is given there.) The text in the German original by Einstein goes further though, and I remember that Einstein wanted to express that Bieberbach, wholly like a classical scholar from bygone times, an “original”, lives only for his specialty interests. In the 1920s Bieberbach was definitely acquainted also with Hermann Weyl, and they corresponded (cf. the partial Nachlass of Ludwig Bieberbach). He did not appear then to have been an anti-Semite. The picture of Ludwig Bieberbach that appears to us in this collection is that of a vain geometer, who in the Weimar Republic stood loyal to the Republic, but who was vulnerable to the comments of Issai Schur, amongst other things. It seems to me that his active standing up for the Nazi Party as one of its spokesmen within the mathematical community was fueled by the opportunity to exercise revenge on all those who at one time—knowingly or unknowingly—had humiliated him or whose style, like Landau, was too drawn-out and slow. Also, the defence of the Hilbert Programme and especially its metamathematics against the assaults of the geometer Max Steck seem to feed on a perhaps personal aversion to a high school geometer who took it upon himself to play judge in a university field. His vanity perhaps also didn’t allow him to support talented students or even to have anyone around him who could lay claim to such honorary offices as he held. I emphasise that this can only be a motive strand in a many-stranded lynch rope, with which he thought to strangle others. In fact he gathered Nazi mathematicians around himself: Teichm¨ uller, Tornier, Weber or Geppert. And this perhaps because he sat in the neighbourhood of the ministry in Berlin and expected therefrom honour as the founder of a new tradition. What else, one could speculate on. There is no historical writing which looks at the earlier old battles fought in Bieberbach’s speaking up for National

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Socialism, understandably without any proof of this thesis, which through constant repetition is therefore not more plausible. Schappacher repeats (Norbert Schappacher, “The Nazi era: the Berlin way of politicizing mathematics”, in: H.G.W. Begehr et al. (eds.), Mathematics in Berlin, Birkh¨ auser, Basel, 1998; here, p. 131) that Bieberbach had stood on Brouwer’s side against Hilbert since the 1928 ICM Congress in Bologna. Schappacher even sees 3 fundamental conflicts in this controversy, of which one would be “formalism, which Bieberbach could share in, on Brouwer’s side, thanks to his predilection for the geometric approach over the algebraic one, and his emphasis on the rˆ ole of intuition in mathematics”. Only Bieberbach, who represented the line of Felix Klein, explicitly fought then—and this really matters—on the side of “formalism”, on the side of Hilbert, Scholz, Schr¨ oter, Ackermann, Gentzen, etc., just as Hasse had done as well. For the theses of Schappacher and Rowe there is no evidence. 70

Cf. the two biographical statements for the section “Nationalsozialistische Mathematik”, p. 1, in Leben im Dritten Reich, Bundeszentrale f¨ ur politische Bildung, 1989. It concerns Brennecke, Handbuch f¨ ur die Schulungsarbeit der HJ, p. 53, and “Mathematik im Dienste der nationalpolitischen Erziehung”, cited from Kurt Zentner, Illustrierte Geschichte des Widerstandes in Deutschland und Europa 1933-1945, 2nd edition, S¨ udwest Verlag, Munich, 1965; here, p. 348.— By way of contrast look at the year 1957 in the Federal Republic. What appeared? Robert Schmidt, Praktische Ballistik. Artilleristisches Rechnen, Einf¨ uhrung in die Ballistik und in die Lehre von der Treffwahrscheinlichkeit [Practical ballistics. Artillery computation. . . ]. This is supplement 1 of the Wehrtechnischen Monatshefte [Defence-technical Monthly]. New editions like Kutterer, Ballistik (Vieweg, 1957) or Athen, Ballistik (Quelle & Meyer, 1957) return to the application of probability theory to shooting from the year 1906 (Eberhard von Sabudski, Wahrscheinlichkeitsrechnung, Stuttgart) and above all Cranz, Lehrbuch der Ballistik, 4 volumes (Springer-Verlag, 1925). Artillery mechanics are contained in H¨ anert, Gesch¨ utz und Schuß (Springer, 1935).— However: the year 1957 was accompanied by books like Prof. Dr. Wilhelm Treue, Invasionen 1066-1944 ; Retired Major General Alfred Philippi, Das Pripjet-Problem [According to the Wikipedia (online), the Pripyat Marshes occupy 38,000 square miles of Belarus and divided the Eastern front into two separate theatres of operation during the Second World War. They were largely impassable to large German forces and provided a hideout for the Soviets. One German solution to the problem posed was to drain the marshes and colonise the area, but this proposal was nixed by Hitler, who feared it might result in a dustbowl.]; Dr. Wolfgang Marienfeld, Wissenschaft und Schlachtflottenbau [Science and battle fleet construction]; First Lieutenant Eike Middledorf, Handbuch der Taktik ; Retired Captain K.A. M¨ ugge, Fernmeldetechnik [Telecommunications engineering]; etc.—only in the same publishing house. What has happened? The ideology had hardly changed, but one differentiates science, ideology, and history in autonomous domains. Everything remains almost unnoticeably similar in the domains of science, ideology, applied mathematics (ballistics), but through the segregation everything becomes “purer”, more harmless, and every connection may and will be denied—that is the ideology up to the present day. From 1957 everything is “ideology-free”. Why do we regard applied mathematics as morally better since then? Because in the Federal Republic it was not in the service of any dictator. Of the corresponding applied mathematics in East Germany I have till now only heard eloquent silence. 76 A small aside: That matches the epistemological theory of Dr. Roland Freisler. To him it concerned a new knowledge through which the old juridical, bourgeois sham knowledge is distinguished: “A knowledge that depends on the capability to exploit it, independent of capability of judgement, independent of the clear view for life and its necessities, independent of all connections to all the peoples is available and yet cannot

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exist. A knowledge that therefore merely exists for itself is worthless. A knowledge that is without connection with discrimination, clear life view, conscious valuation of all things and powers in their relation to all the peoples is certainly not worthless, however is yet only of secondary significance.” Against this he set the new law: “Rules are certainly important, but rules alone create no new law and certainly no new life and more certainly no new life powers.” More important should be men who govern over the purity of the law, who must become the servants of law and fighters for the law. Camp commander Spieler [my father’s superior—EMT] expanded: “Understandably the camp leadership is clear about that, that one can neither learn nor teach National Socialism. One must focus oneself entirely on being a National Socialist, or one will never become such. But in many men who previously, owing to the company they keep or their prior way of life, have not become acquainted with National Socialism, or who have even disapprovingly stood opposed to it, National Socialism unconsciously lies dormant; it only requires an impulse to be awakened.” The world view instruction is only one means out of many. What else can one rely on? “Through the assembly of the leadership and this contact with other circles of people care is borne, that the old true SA-spirit, i.e. the National Socialistic world view, will be brought to the civil servant trainee and will remain in them.” Therefore the camps, out of necessity. “The division leaders occupied in camps are in part jurists, who as civil service trainees have already put the J¨ uterbog camp behind them and are former foot-soldiers, in part active SA-leaders. The Zugf¨ uhrers [chief guards] and uhrers, and instructors of the former Reich’s Gruppenf¨ uhrers are SA-, respectively SS-F¨ Committee for Cross-Country Sport.” The goal is the simulation of a “proper peoples’ society” under the absolute leadership principle, a depiction of the goal, a vision, like that National Socialism presented to the larger society outside in Berlin and elsewhere. “The leaders come from various professions. They recruit themselves from craftsmen, farmers, jurists, merchants, etc., and thereby give the trainees the opportunity to get to know other circles also. So snobbery cannot arise.” This intermixing together with the model-function of its Lager-, Zug- and Gruppenf¨ uhrers becomes continually emphatic. The camp appears to the outsider as a breeding ground of the permanent revolution in the sense of Ernst R¨ ohm and Gregor Strasser. 117

“Evidence is always a renunciation of actual grounds.” One doesn’t believe it, but Hugo Dingler trumpeted that toward the exiled Jew Paul Bernays in the year 1954. That was the practised argument against the axiomatic method with its arbitrary continuation of axioms and consequent antinomies and for the folkish interpretation of the “foundational crisis”. W. Stegm¨ uller opposed this in 1954 in his book Metaphysik, Wissenschaft, Skepsis (Vienna). Dingler, the opponent of noneuclidean geometry, wanted to construct science from the practical knowledge of everyday life. And Tarski, Behmann, and Arnold Schmidt listened motionless. Cf. Alfred Tarski, “Discussion of the address of Alfred Tarski”, Revue internationale de philosophie 8 (1954), pp. 15-21. The civilising of the Federal Republic always took place on the backs of the victims. NB : H. Behmann had given a clear deciagen zur Algebra der Logik sion procedure for the narrow function calculus in his “Beitr¨ und zum Entscheidungsproblem” (Mathematische Annalen 86 (1922), pp. 163-229) and thus got into an amusing priority battle with G¨ odel.— In general the old Nazi-satraps in mathematics were audacious: Fritz Requard published; Georg Hamel understandably participated in 1953 at the DMV conference; and the exiles threatened with death, like Paul Bernays and Alfred Tarski, had to be set at a table with the masterminds and carry on professional discussions. In the postwar period this happened constantly. Without this wonderful readiness of the emigrants, refugees, and exiles, Germany would possibly still be as provincial today as it was under the Nazis.— “We ask ourselves: ‘How should one behave toward previous friends? Greet them on the street? Offer one’s hand?’ How to behave 1) toward the Nazis? 2) toward the collaborators? Opportunists with sarcastic

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smiles. Each has an alibi; murmured: ‘At heart we condemned the Nazis.’ They hid their honour and their money. While we were robbed and hunted, wandered through four to five lands—for 12 years!” Alfred Kerr, PEN-Club, London, 1945. Cited from Alternative 52: Berlin 1967, therein, “Briefe aus dem Exil”. Perpetrators and followers however showed their unteachable political stupidity, their unyielding, aggravating and dangerous know-all attitude in that they did not wish to call emigrants to positions and chairs. 128 I do not go into the mathematical logic or philosophy of Heinrich Scholz because it is neither of mathematical nor philosophical relevance (cf. the entry by Eckehart K¨ ohler (pp. 324ff.) in volume 7 or Paul Edwards (ed.), Encyclopedia of Philosophy (Collier, London, 1967), which represents Scholz as a Platonist). For a contrary view cf. Heinrich Scholz: Logiker, Philosoph, Theologe, ed. Hans-Christoph Schmidt am Busch, Kai. F. Wehmeier, mentis-Verlag: Paderborn 2004. Nonetheless, because of his education, it has not remained without consequence. Therefore Scholz will be seen in this book merely from the point of view of an effective organiser and successful protector of logistic under National Socialism. Scholz was a politician who defended a scientific line. He gained acceptance for a mathematical orientation at a philosophically oriented faculty in a miserable time and thus proved a great ability. With him mathematical logic set itself apart from a philosophy which was occupied by National Socialism and would be fiercely contested. The coincidence of his “exact philosophy” as stubborn removal from ideology met his will and way, and is no escape. At first Scholz was a firm follower of the Fregean theory (cf. Hermes, 1986, p. 45). He gave this up himself in 1941 (Hermes, 1986, p. 49) and then went over to the Hilbert Programme, for which he himself however developed no affinity (Hermes, 1986, p. 49). He sat quasi-overall between two stools. On mathematical conception, only that which Hans Hermes reported: “In this type theory [of Bertrand Russell—EMT] Frege’s arithmetic constructions could be comprehended, though only if an axiom of infinity were adjoined, which was dispensable by Frege. . . In other words it is conceivable that one can already build up arithmetic in a—naturally type-free—subsystem, which is perhaps consisuchen tent” (Hermes, 1986, p. 48). Henrich Behmann showed in 1931 (“Zu den Widerspr¨ der Logik und der Mengenlehre”, Jahresbericht der Deutschen Mathematiker-Vereinigung 40, pp. 37-48) that one can avoid the Russell antinomy in a type-free logic. Scholz then prompted F. Bachmann to a simplification of Fregean logic (Friedrich Bachmann, Untersuchungen zur Grundlegung der Arithmetik mit besonderer Beziehung auf Dedekind, Frege und Russell, dissertation, M¨ unster, 1934). Hermes remarks, “One must say, however, that in the end it remains open whether the type-free logical principles applied are really consistent, for these will not be given detached from the individual constructions. It should be remarked that in 1941 Ackermann had attempted to systematise Behmann’s idea; he would be forced thereby to continuing encroachment in the certitude of classical logic” (Hermes, 1986, p. 45). Frege’s programme was done in; only Hilbert’s path remained open. Against that stood G¨ odel’s results, and so Scholz pinned his hopes on Gentzen and Ackermann with their consistency proofs. But for each (Hilbertian) formalism Scholz wanted to provide a precise semantics, with which the ontological dimension of the logic would become clear. The reservatio mentalis of Scholz against formalism as a Glasperlenspiel [glass bead game—a reference to a novel by Hermann Hesse in which the highest intellectual practice is a glass bead game devoid of significance—Trans.] hindered every approach to Hilbert and thereby too to Gentzen. By Gentzen, however, Scholz felt genuine efforts towards a mediation of all three forms of foundations known at the time.

130

Volume 1 of the Forschungen zur Logistik und zur Grundlegung der exakten Wissenschaften [Researches on logistic and the foundations of the exact sciences] appeared in 1934. Consisting of 78 typerwritten autographed pages, it was Friedrich Bachmann’s Untersuchungen zu Grundlegung der Arithmetik mit besonderer Beziehung auf Dedekind,

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Frege und Russell and was published on commission by Meiner in Leipzig. In 1937 there appeared the N.H.F. (Neue Heft Folge) [New series of volumes]. How “the Party” thought is easy to read in the entries in the 8th edition of Meyers Lexikon (Bibliographisches Institut, Leipzig, 1938). Hilbert was highly praised (cf. the entry “Hilbert” in column 1202 of volume 5): “In Grundlagen der Geometrie (1899, 1909, 3rd. edition) he expounded new trains of thought (Axioms of order, connection and continuity). The Grundz¨ uge der theoretische Logik (with W. Ackermann, 1928) treated matters of principles of mathematics, particularly the foundations of arithmetic.” As opposed to this, “Logistic” was condemned (volume 7, 1939): One reads, “in contrast to traditional (Aristotelian-scholastic) school-logic together with its psychological and epistemological-theoretical discourses with respect to the totality of the attempts, with the help of the thought process of mathematics, but not simply adopting mathematical theorems, to formulate anew the theorems of formal logic and to draw up and apply correspondingly new logical ‘calculation procedures’ (‘algorithms’). Today there are many incommensurable systems of logic. The most widespread today rests on the concept, that is to say, the operation of implication; it derives from G. Frege (*1848, †1925) and would particularly through Russel [sic] and Whitehead be systematically elaborated. The beginnings of L. are to be found already in the Middle Ages, among others by Nikolaus von Amiens and Raymundus Lullus, to the beginning of the modern era in the multiple beginnings of ‘geometric’ methods in philosophy by, among others, Descartes, Spinoza, Erhard Weigel. Leibniz grasped the problem of L. for the first time in fundamental clarity. The founder of modern L. is the American George Boole (*1815, †1864). The latest representatives, from time to time standing close philosophically to materialism, often embrace the most intolerant intellectualism, thus in particular the ‘Vienna Circle’.”— And in the literature Frege, Couturat, Whitehead and Russell, Carnap and the journal Erkenntnis are cited, but also Hilbert and Ackermann. One understands now quite easily why it was important for H. Scholz to save himself from the “good” tradition line, i.e. to secure himself—and for him this was Gentzen. 166

Herbert Luthe, “Die Religionsphilosophie von Heinrich Scholz”, Dissertation, Munich, 1961, p. 13. Cf. in this respect the detailed text and the documents in Peter Schreiber (1995). On the relations between Heinrich Scholz and the Polish logicians, see the bibliography. Consider also the literature in David Pearce and Jan Wolenski (eds.), Logischer Rationalismus. Philosophische Schriften der Lemberg-Warschauer Schule, Athen¨ aum Verlag, Frankfurt am Main, 1988. Also good is Peter Simons, Philosophy and Logic in Central Europe from Bolzano to Tarski, Kluwer, Dordrecht, 1992. On the rˆ ole of Jan Salamucha (1903-1944) in the circle of Cracow logicians as the “life and soul of the circle,” cf. Joseph M. Bochenski, “The Cracow Circle”, in: Klemens Szaniawski (ed.), The Vienna Circle and the Lvov-Warsaw School, Kluwer, Dordrecht, 1989. Salamucha is supposed to have been killed later during the Warsaw Uprising. It is possible that this was still a German deed.— On the later connection of logic and politics in the postwar period, I mention only: A leading light of West German logic in the Cold War period was the Polish Dominican father Joseph Maria Bochenski, whose history of “formal logic” is widely known. The neo-Thomist energetically carried on a lifelong battle against Marxism in the form of dialectic materialism. “As the process [against the German Communist Party in 1956— EMT] before the Federal Constitutional Court in Karlsruhe began to flounder, Bochenski was invited to give an expert opinion. He delivered, and that in the end was probably the decisive factor. The KPD in any case was banned” (Konrad, Adam, “Scholastik. Joseph Bochenski gestorben”, Frankfurter Allgemeine Zeitung, 11 February 1995). In any event Bochenski thought himself the originator. Cf. Niklaus Meienberg (Heimsuchungen. Ein ausschweifendes Lesebuch, Diogenes, Z¨ urich, 1986, pp. 103ff.), who describes Bochenski’s biography, especially his love of riding in a Jaguar E.— It is curious that his attitude toward religion would not itself be seen as a world view. How much different logicians

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of a common nationality can be also baffled Jan Kott (Leben auf Raten. Versuch einer Autobiographie, Alexander Verlag, Berlin, 1993). On the logician, artist and writer Leon Chwistek (1884-1944), he reported that he could possibly have been poisoned in Moscow on political grounds as a Polish patriot (p. 69), while the logician and resistance fighter Boleslaw Soboci´ nski (1906-1980) was denounced as “a determined anti-Semite” Hanna Krahelska and Professor Handelsmann “because of his connections with communists”, whereupon both would be abducted and killed. The “multiplicity of reality” (as in a book title by Leon Chwistek) could not be dealt with by mathematical logic so much in those days. In a commentary on number 23 of the list of the one hundred non-fiction volumes of the century composed by the publisher Random House, one reads, “No one will probably read this through from front to back”; Russell himself admitted, “I once knew 6 people who had read the latter part of the book. . . Three of them were Poles and were, I believe, murdered by Hitler. The other three were Texans” (S¨ uddeutsche Zeitung, Munich, 12 June 1999). 193 Hans Karl Hofer, Deutsche Mathematik. Versuch einer Begriffsbestimmung, dissertation, Vienna, 1987, pp. 94ff. In 1998 Bieberbach’s son Ulrich turned part of Bieberbach’s Nachlass over to Prof. Dr. Menso Folkerts to work on. From this the following are significant for my theses: 1. File box, labelled “Fachaufs¨ atze” on the back, containing: 2 letters from H. Kneser to Bieberbach and 1 letter from Bieberbach to Kneser (for clarification of the relationship Bieberbach-Kneser-Gentzen, also because of the collaboration on Deutsche Mathematik ). 2. Unnumbered box labelled “Mathem. Manuscripte. Tschebyscheff ” on the back: folder with the manuscript “Zur Ontologie” of mathematics (with criticism of Dingler and Steck) and “Form, Inhalt, Gestalt und Sinn” because of the relation to Dingler and Steck in contrast to H. Scholz, Schr¨ oter, Gentzen, etc. 3. Box 3, folder 1: Newspaper clipping O. Perron and Eva Manger as well as “further clippings, works and manuscripts of Bieberbach and others on these themes.” By “these themes” one means the concept of German mathematics, the opinion of Oskar Perron and Eva Manger thereon, and similar things. 4. Box 6, folder on Arithmetic: manuscript “Der Zahlbegriff ” and revision of the manuscript. Here perhaps Bieberbach’s opinions on Frege, on logic, or similar things are dealt with. The relation of Scholz to Bieberbach may be deduced indirectly from two letters. Scholz wrote from M¨ unster on 28 November 1947 to L. Bieberach: “That I finally answer is caused by an external impetus. Tomorrow my Dr. Schr¨ oter travels for a couple of days to Berlin on account of an invitation by the mathematical faculty to hold a lecture on the consistency of number theory. I have asked him that he, if it is possible, might call on you in my name, that he give you my greetings and ask about your present existence. That he cannot however see clearly whether his time will be sufficient for a visit in Dahlem, I send this sheet separately, just in case; for you should know that I have not forgotten the good you have done for us, and that I have sincerely felt deeply for you in these difficult things which are happening to you. . . I will now be relieved through Dr. Schr¨ oter, who had habilitated himself during the war, and I have got the two best young mathematicians as collaborators for our good cause. In September Herr Bernays from Z¨ urich was here for 14 days: the first case of this sort in M¨ unster. We have thoroughly discussed with him the results which we have worked out, and we have been able to establish that we have not been overtaken. They will appear in the journal of our transatlantic friends, who have kept faith with us in exemplary fashion. This also holds of our remaining friends in the larger world, including the few good Polish friends who were not murdered by the Germans. None of them has still forgotten where I was during the war and to the bitter end was to be found. On this point I have not erred. Now I have said the most necessary of that which can interest you. And now I say to you once more that I will not forget you.”

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And Bieberbach answered from Gelfertstrasse in Dahlem on 7 December 1947. He praised Scholz’s independent mind, held at the time to be dishonest and insincere, which however did not prevent him from arranging relations and winning what they could for science: “So I have held it too in the interest of others with the German mathematics. The elements hostile to science progressively took it over for their own ends. Therefore I have used that little reputation that I had known to acquire for myself in the leading circles. Many have fed off that who today have ‘forgotten’ me. That doesn’t disturb me.”— He lamented his fate with the sentimentality of the time and continued, “. . . I don’t complain about this. For by what I learn from you, Germans have treated your Polish friends still much more badly. I have endeavoured at that time to interest the Academy at your wish in your friends, your M¨ unster freemen. Encouraged by Vahlen—you will know—how miserably one bid and let him die—I put that forward with the suggestion, to whom one has turned into a freeman, by him one must also stand. An Academy member, whose name I prefer not to betray today, then cut me short, that Vahlen be declared president immediately, under the circumstances one could not pursue the matter. . . ” (I thank Ulrich Bieberbach, Oberaudorf, for the look into the correspondence.) 211

J¨ urgen Mittelstraß, who lectured on the pre-theoretic and “practical foundations of science” as a programme of a science before some Leninists likewise saw it as such and construed a connection between “social system and knowledge system” and saw an anthropological foundation with technical constructibility and changeable stages of knowledge, emerged as a critic of Popper like all latter-day Dinglers of Paul Lorenzen’s “Erlangen School”. In his book, Die M¨ oglichkeit von Wissenschaft (Suhrkamp, Frankfurt, 1974), next to the usual curiosities of the “constructive theory of science” is also the polemical essay “Wider den Dingler-Komplex” [Against the Dingler complex] (pp. 84-105). There, contrary to Wolfgang Stegm¨ uller—“Dinglerism”, as it is only given in Dingler’s writings— is sharply separated from the constructive theory of science. Also, however, Mittelstraß wants no “erudite Dingler-Interpretation” (why not?), but rather to clarify “Dingler’s intention.” What “Dingler completely clearly” was, Paul Lorenzen amplified and “made precise.” Dingler’s conception of a “recurring science [based] on normed actions” would be reformulated “as reconstruction of pre-scientific action-intentions.” Thereby Dingler’s suggestions proved themselves “worthy of revision in detail as in considerable scope.” So one wished to withdraw oneself from Dingler’s conception of a formation of thought and action grounded on will—that was in fact a Nazi concept of science, which would have nothing against a principle of methodical and pragmatic order, on the contrary!—in order to be more faithful to his intentions, whatever that might mean in a democratic post-war Germany, where Paul Lorenzen worked on an “ortholanguage” which should have nothing to do with Goebbels’s linguistic rule. The relation between identity and reality is always in favour of—as by Hegel—the ideality as demand determines: The Gedankenexperiment “is first considered merely a ‘challenge’ or a ‘procedure’, in order to predetermine in reality only the possibility by existing form in this” (Hugo Dingler, Das Experiment. Sein Wesen und seine Geschichte, Ernst Reinhard, Munich, 1928, p. 58). Dingler had shown that will and world view were the dialogic deciders of a winning game, also therein, whether one was Jewish or practised Jewish physics. Dingler could verify this methodically and stepwise constructively. And certainly therefore there is no mention of all this by Mittelstraß. And thus today Mittelstraß is still a celebrated, serious scientist, because he had ignored all that, saw it as unimportant and inconsequential, castrated Dingler and his political past first from Roman Herzog (student of the Nazi jurist Theodor Maunz who was promoted by the dreadful jurist Filbinger), and silenced the necessary “impartiality”. When Mittelstraß surfaced in old chancellor Dr. Helmut Kohl’s Technology and Science Museum (Bonn) in the catalogue and was represented in commissions and committees, one no longer wondered at it. Just as the SPD under Helmut Schmidt

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declared Karl Popper as house philosopher, the neo-Dinglerists worked to bind the destiny of the CDU to that of the Fascist house philosopher Dingler. It did not succeed.

https://doi.org/10.1090/hmath/033/05

CHAPTER 5

Recovery and Docent Position 1942 to 1944 1. Final Discharge from the German Army The mathematicians Herbert Seifert (1907–1996) and William Threlfall (1883– 1949) visited Gentzen on Easter Sunday, 5 April 1942, and entered into their diary: “14:00 to 14:30 by bycicle to the aircraft-sanatorium in Mescherode to visit Dr. Gentzen, who is mentally ill there. One recognizes nothing of that. His doctor is Kentenich. Gentzen is to be discharged as unfit for duty.—Kneser drafted” (Tagebuch von William Threlfall and Herbert Seifert, vol. XVII, p. 734, ed. Philipp Ullmann, 2006, not yet published). Gentzen informed the curator of G¨ ottingen University by postcard that he had been discharged from the army as of 14 June 1942: “Since I’m unable to exercise my profession due to my illness, I intend to take a longer vacation.” But the Director of the Mathematical Institute, Kaluza, wrote to the curator1 on 4 July 1942: “Dr. Gentzen has resumed his position as assistant at the Mathematical Institute as of 15 June 1942.” The curator sent a request to the Ministry on 8 July 1942: whether in the immediate future the discharge would be followed by recall, after Dr. Gentzen was discharged from the army on 8 June 1942. If he is not recalled then the question is whether the assistant is suited as a candidate for the position of university teacher. Deadline: 2 weeks.

On 17 July 1942 Kaluza informed the curator sincerely, that the continued activity of Dr. habil. Gerhard Gentzen is urgently needed also after the conclusion of the second year of service as assistant at the Mathematical Institute. Dr. G. Gentzen’s suitability for the position of university teacher is clearly demonstrated in that Mr. Gentzen possesses the rank of Dr. habil.

The rector and Dozentenschaft had nothing against an extension of the position. On 19 November 1942 he was declared officially to be “w.u.” [wehrunf¨ ahig = unfit for duty] by the district military command (“nervous breakdown as a result of stressful duty as radio operator with an airborne watch regiment”). On 2 December 1942 the curator entered into his personal file: I have not pronounced the discharge by recall of Assistant Dr. habil. Gerhard Gentzen in accordance with §5 of the federal regulations regarding assistants. Resubmission on 1 January 1944.

After a short recovery in a sanatorium his condition improved quickly and he began his logical research again—commuting between Putbus, Liegnitz and Sigmaringen: The awakening from mental slackness and dazedness changed inappreciably, mind you, for weeks. [23 April 1942 to Hellmuth Kneser] 1 I.e.

the chief financial officer of the university. 233

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2. Hans Rohrbach Commandeers Gerhard Gentzen to Prague through the Osenberg Initiative At the end of 1942 he was promoted by Professor Hans Rohrbach to the position of docent in Mathematics. Rohrbach knew that he would not be comfortable in the army: When I heard how little in his way he was able to conform to the army drills, it was clear to me that I should try to help him get out. Otherwise he would fall apart, or rather be prime for a mental hospital.2

Hans Rohrbach related relative to his motivation: I put him in this position because he had been drafted into the German army and felt by no means good there. I had been with Gentzen in G¨ ottingen before my appointment to the German University in Prague. [Hans Rohrbach became Senior Assistant at the Institute on 1 April 1938—EMT.] As a result we knew one another well and I could understand that he wanted to get away from the army. The Osenberg Act had as its charge and authority to release professionals from military duty for work essential to the war effort. That is why I put in an application to have Gentzen granted this release. I would keep him occupied in Prague with mathematical work essential to the war effort. It was in this way that Gentzen was released from military service and I was obligated, according to the Osenberg Act, to report on Gentzen’s work.3 2 Letter

from H. Rohrbach to Dipl.-Ing. G. Gentzen of 21 February 1986. to the author from H. Rohrbach of 8 September 1988. Germany, with its mystic emphasis on “blood and soil”, did not see much military use for science and scientists. With the exception of research conducted for G¨ oring’s Luftwaffe, there was no mobilisation of science, scientists, or research until late in the war. G¨ oring took over the Reich’s Research Council, and following his decree of 29 June 1943, the Planning Office was created as an autonomous organisation of the Council by Albert Speer. Werner Osenberg, a Professor of Mechanical Engineering in Hannover and a good Party member, was placed in charge. Like many such political appointees, he was not particularly talented in science or management, but he had a passion for organising information and compiled a list of some 15,000 German scientists and technicians: 10,000 of these were too important for war work to be drafted, and some 5,000 were already in the military and were to be brought home. He managed to get Hitler to sign a decree in December 1943 to have these 5,000 released from the armed forces. This decree, dubbed the Osenberg Action, was the basis for his popularity among German scientists, despite mixed estimates of its degree of success. (Samuel Goudsmit (Alsos, 1947) says about 2,500 out of the 5,000 were sent home; Alan Beyerchen (Scientists Under Hitler, Yale, New Haven, 1979) estimates 4,000 out of 6,000.) Osenberg described himself thus: 3 Letter

He [i.e., Osenberg himself—EMT] founded, as ever more able-bodied professors and specialists of the scientific institutes were being inducted, in the year 1943 the Planning Office of the Reich’s Research Council, after he had earlier convinced important authorities that the German researcher, whether old or young, had to work at his research station rather than to waste his powers ineffectually in subordinate military positions. He succeeded not only to preserve some 10,000 scientists through indispensibility exemptions for research, but also beyond that to withdraw some 5,000 particularly talented and requested scientists from the front, who would in addition be distributed to the scientific institutes. However successful or unsuccessful the Osenberg Action may have been, Osenberg’s card file proved useful to the Americans after the war’s end, first to the Alsos mission anxiously looking for any German scientists who might have been part of an atomic bomb project, and then by Operation Paperclip, the American military’s secret plan to recruit German scientists and technicians for American industry. Gentzen was not among those singled out for recruitment. A card bearing his name exists in the American collection of Osenberg’s files held by the Library of

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But Gentzen had already been discharged as unfit for duty from the army. And Rohrbach had gotten him the SS job and had only to “report” to Osenberg. Should Gentzen have been afraid of being sent to the front? Hans Rohrbach (1903-1993) was a great cryptologist.4 His work, he wrote in the FIAT-Report, was exactly the right kind of assignment for Gerhard Gentzen. Because of a breakdown in communication the G¨ ottingen faculty was not informed about the position in Prague as lecturer, and Gentzen continued to receive his salary from there. Gentzen was initially supposed to teach “Beginning Mathematics”.5 From Liegnitz he requested of the curator of the University of G¨ ottingen on 1 January 1943 the granting of teaching competence in mathematics and appointment as docent. I asked the Faculty of Natural Sciences of the German Charles University for release from the trial lecture.6

In G¨ ottingen the curator wrote to the Director of the Institute of Mathematics on 30 January 1943: On 31 March 1943 the Assistant Dr. habil. Gerhard Gentzen will have served in that position 4 years. I ask for word as to whether the discharge through recall will take place at the specified point in time.

But Kaluza wrote on 2 February 1943 to the curator of the University of G¨ ottingen that they “request the extension of the appointment of Dr. habil. Gerhard Gentzen as Assistant in the Mathematical Institute beyond 31 March 1943.” There cannot be the least doubt about the scientific qualification of Mr. Gentzen. His previous published researches are also known outside of G¨ ottingen and are highly regarded. The Institute puts a great deal of value on retaining Mr. Gentzen as an Assistant as long as possible.

The rector supported this and the Dozentenschaft had no objection. Gentzen’s dismissal was not announced, and he received 454.81 RM in monthly salary. Gentzen visited his mother and sister in Liegnitz and from there sent a mixed lot of his papers to Sigmaringen, where his mother’s sisters lived. Was this foresight?7 In the Fall of 1943 he was appointed by Prof. Dr. Hans Rohrbach, Non-resident Cryptologist and Director of the Mathematical Seminar of the German Charles University, to be unsalaried lecturer in Prague. For this position Gentzen wrote a curriculum vitae: Transcript Curriculum vitae. G¨ ottingen, 25 August 1943. Congress, but it says only “SS-Forschungsauftrag” [SS-research contract]; he does not appear in Osenberg’s cards stored in the Federal Archives in Berlin. It is not clear if the American holding is the original or a sanitised copy. (The American military and the State Department did not agree on the desirability of importing former Nazis into the United States, so the military altered a few records.) 4 Cf. F.L. Bauer, Entzifferte Geheimnisse. Methoden und Maximen der Kryptologie, 2nd expanded edition, Springer-Verlag, Berlin, 1997. 5 I especially thank Prof. Dr. Miroslav Kunstat of the Archives of the Charles University and the mathematician Dr. Milan Vlach for their aid and searches in the Prague archives. 6 I heartily thank Dr. Ulrich Hunger of the G¨ ottingen University Archives for a copy of this statement. 7 These papers would probably have been turned over to Prof. Dr. Christian Thiel by Frau Waltraut Student in 1988 for decipherment, as they were written entirely in shorthand.

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I was born on 24 November 1909 in Greifswald as the son of the attorney Hans Gentzen and his wife Melanie Gentzen n´ee Bilharz. I attended the humanistic gymnasium in Stralsund from 1920 to 1928 and there completed final examinations at Easter 1928. Finally I studied mathematics and physics at the Universities of Greifswald, Munich, Berlin and G¨ ottingen. In G¨ ottingen I took the degree of Dr. phil. in 1933 and in the same year passed the state examination for my teaching certificate in mathematics and physics as majors and applied mathematics (astronomy) as minor. From 1934 to 1935 I continued my scientific work in the area of mathematical logic and foundational research, partly supported by a research stipend of the German Research Society. On 1 November 1935 I received the appointment as unscheduled Assistant at the Mathematical Institute of the University of G¨ ottingen, which was converted on 1 April 1939 into a regular Assistant position which I still occupy today. From 1935 to 1938 I have held individual lectures in my area at the Universities of Greifswald, G¨ ottingen, Leipzig, M¨ unster and T¨ ubingen, and then at the meeting of the German Mathematical Society in 1937 as well as as a member of the German delegation to the International Congress of Philosophy in Paris in 1937. Since November 1933 I have been a member of the SA, since 1 May 1937 of the NSDAP, since 1 January 1939 of the NSD Dozentenbund.8 From 28 September 1939 to 8 June 1942 I was a soldier in the Air Force and from 28 September 1939 to 21 January 1942 on duty in the home war zone. I was dismissed and discharged from military service due to nervous exhaustion. My health has since then improved to the degree that I am again capable of scientific work and teaching. In December 1940 I earned the degree of Dr. phil.-habil. at the University of G¨ ottingen. signed G. Gentzen

3. Gentzen’s Teaching Position in Prague: “Kepler’s Laws of Planetary Motion” On 19 and 20 May 1943 in two hour-long sessions he gave his lecture “Kepler’s Laws of Planetary Motion” in a slow but clear manner. Already in his youth he had read on the subject.9 Moreover, a theme like “Kepler” was already considered in Prague as “urdeutsch”.10 Only a title like “Kepler” was therefore already uncontroversial. The lecturer could be certain that he would not be exposed to inconvenient questions about the subject of his lecture and that he would not have a disagreeable seminar attendance. There are two expert opinions on his Trial Lecture in the Habilitation Acts of the Bundesarchiv. The first reads: 8 According

to the Berlin Document Center (BDC) (in Gentzen’s own hand), but in the files the date is given as 1 January 1941. 9 Cf. the chapters “Johannes Kepler” and “Isaac Newton” (pp. 164-182) in Max Valier, Der Sterne Bahn und Wesen. Gemeinverst¨ andliche Einf¨ uhrung in die Himmelskunde, R. Voigtl¨ ander Verlag, Leipzig, 1924. (Recall mention of this work in Chapter 1.) 10 Cf. for example Max Steck, Uber ¨ das Wesen des Mathematischen und die mathematische Erkenntnis bei Kepler, Halle, 1941; A. Speiser, Mathematische Denkweise, Z¨ urich, 1932; F. Kubach, “Johannes Kepler als Mathematiker”, Ver¨ offentlichungen der Sternwarte zu Heidelberg 11 (1935); E.A. Weiss, “Kepler”, Deutsche Mathematik 5 (1940); Max Kaspar, Kopernikus und Kepler, Munich-Berlin, 1943; Franz Hammer, Joh. Kepler, Stuttgart, 1943—the “German” literature is legion.

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Expert opinion on the trial lecture of Dr. habil. Gerhard Gentzen on 19 May 1943. Dr. Gentzen chose as the topic of Trial Lecture Kepler’s Laws of Planetary Motion. After a short historical introduction he treated in two lectures the Ptolemaic Epicycle Theory,11 Kepler’s Three Laws,12 the Newtonian Theorem on Surfaces and the Derivation of the Laws of Planetary Motion from the General Law of Gravitation. The execution of the lecturer showed that he is capable of treating within the context of a lecture a scientific topic with clarity before a larger audience. Glaser,13 Prague, on 14 June 1943.

And the second: The Mathematical Institute of the German Charles University Prof. Dr. H. Rohrbach Prague, 1 July 1943 Expert Opinion Dr. habil. Gerhard Gentzen held his public Trial Lecture before the Natural Science Faculty of the German Charles University of Prague on the topic: Kepler’s Laws of Planetary Motion He began his presentation with a reference to the local connections of his topic, since Kepler had worked many years in Prague and the necessary material for the development of the laws of the planets was present in the observations of Tycho de Brahe. His historical overview of the efforts of the human spirit since antiquity to describe the movements of the planets made its connection here. He closed part 1 of the lecture with an interesting demonstration that the theory of epicycles of Ptolemy is mathematically equivalent to Kepler’s First Law. Part 2 of the lecture was dedicated to the theoretical derivation of Kepler’s Laws from Newton’s general law of gravitation.14 Dr. Gentzen showed with this conception and with the execution of his Trial Lecture a considerable didactic aptitude. His presentations were clear and understandable. The link with historical events and positioning in a broader context showed that Dr. Gentzen possesses the necessary view to frame a lecture interestingly and vivaciously. This judgement is not affected by the current slowness while lecturing of Dr. Gentzen caused by nervous disease. In conclusion it should be noted that Dr. Gentzen is very capable of executing a docent position.

Because of the conditions of the National Habilitation Regulation of 17 February 1943 the request from the director of the German Charles University that Dr. phil. habil. Gerhard Gentzen be granted authority to teach was sent to the Federal Minister. The annotation to the public trial lecture was: Dr. Gentzen has a clear command of his lecture. For a mathematician whose scientific works lie within the highest level of abstraction (Hilbert School), the presentation was remarkably understandable and clear. Gentzen also appears capable of holding lectures of benefit both to advanced students in mathematics, 11 Apparent retrograde or stationary motion of planets was explained by the planets travelling in small circles, called epicycles, which themselves travelled in larger circles around the Earth. 12 Kepler’s Laws assert: 1) a planet travels in an ellipse around the sun in such a way that the sun is at one of the two foci of the ellipse; 2) the line drawn between the sun and the planet sweeps out equal areas in equal time intervals; and 3) the square of the time it takes a planet to orbit the sun is proportional to the cube of its average distance from the sun. It is not known how far Gentzen if possible offered a new insight, as did R. Feynman, who formulated Kepler’s First Law purely in terms of plane geometry and thereby obtained a proof different from Newton’s. 13 Walter Glaser, extraordinary professor of Theoretical Physics. 14 The law asserts: The gravitational force between two bodies is directly proportional to the product of their masses and inversely proportional to the square of their distance r. From this derives the oft-used expression “1/r 2 Law”.

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as well as to beginners and students that only use mathematics as an auxiliary science.15

His first certificate of recognition as docent of the Berlin Federal Ministry for Science and Education is dated 5 October 1943: In the name of the Fuehrer I appoint in cooperation with the Federal Agent in Bohemia and Moravia Dr. phil. habil. Gerhard Gentzen as docent I fulfill this charter in the expectation that the named true to his oath of office fulfills scientifically the duties of his position and justifies the trust that is shown him through this appointment. At the same time he can be sure of the particular protection of the Fuehrer. Berlin, 5 October 1943 The Federal Minister of Science, Schooling and National Education By Order, signed Groh

He was made aware that he acquires “no right to and no responsibility for the granting of a salary or appointment to a regular chair.” 4. The First Courses in November 1943 On 2 November 1943 the dean retroactively requested permission for the announced lectures and exercises by Docent Dr. Gerhard Gentzen for Winter 1943/44: Repetition and Supplement to School Mon., Wed., Thurs. 11-12 Mathematics Exercises in School Mathematics Tues. 4-5 With the teaching was associated a “Research contract of the SS”16 —arranged by Prof. Dr. Rohrbach and the Director of the “Mathematics Working Group of the National Research Council”, Wilhelm S¨ uss—for calculations on the V-2m,17 whose results go first to Peenem¨ unde, then directly to Mittelbau Dora near Nordhausen. On 29 June 1988 Professor Rohrbach wrote to Mrs. Waltraut Student: Your brother had accepted responsibility for carrying out mathematical calculations for the technicians of the production group of the V-Weapons in Peenem¨ unde. They concerned, as far as I know, primarily ballistics problems, i.e., solution of differential equations. For this he was assigned a group of young girls—they were school students of the senior class in Prague—who were to perform statistical and other purely computational problems for the theoretical work of Gerhard Gentzen as a contribution to the war effort.18

Professor Rohrbach wrote to me on February 5, 1990: As far as I know the tasks that Gentzen was supposed to do for Peenem¨ unde were statistical. In addition he had a large group of high school girls that were supposed to help him with them. 15 Habilitation 16 Card

files of the Bundesarchiv. from the Osenberg Index in the Bundesarchiv in Koblenz. It thus deals not only with

“priorities”. 17 To enhance the propaganda value of German rocket weapons, Goebbels christened them Vergeltungswaffen [retaliation weapons, generally called V-weapons in English], changing the designations of rockets. The Fi-103 became the V-1 and the A-4 became the V-2. 18 The letter is in the possession of Frau Waltraut Student.

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Gentzen became head of a computating group. Gentzen approached his assignment with dedication but without any particular conviction: his SS research assignment seemed to him to guarantee that for the duration of the war he would no longer be used in an active military role. In the meantime there is considerable doubt about what exactly Gentzen was calculating.19 Mehrtens20 writes: Rohrbach was also professor at Prague, from 1941, while working part-time for the AA. In Prague he had a small group doing mathematical work for various industrial firms and also for the HVP.21

There is no indication—as with all references to this computing group—that calculation work was actually done for the Army Experimental Station at Peenem¨ unde (HVP). And what were the “various industrial firms”? The possibility that Gentzen’s Calculation Group—that got its assignments from Rohrbach—was to have worked for the flight research station Hermann Goering, as Mehrtens22 insinuates— has never been corroborated. The Army Experimental Station at Peenem¨ unde was an army institution over which the National Research Council had no control. Prof. Osenberg had a bad reputation with us. With typical Nazi arrogance he wanted the HAP-technicians at his disposal. On my recollection he demanded a card index from HAP of all qualified personnel employed there. He would later be given to understand that the RFR [Reichsforschungrat– EMT] would not be the client of the HAP and that he (Osenberg) possessed absolutely no authority over 19 Did Gentzen know at all “definitely for what the works should serve?. . . The prototype of the V2/A4 was already launched on 3 October 1942. The ensuing questions concerned not the theory, but rather the changeover from prototype to mass production.” Also the idea that Gentzen carried out calculations aimed at preventing “aerial disintegration” (a problem that had occurred with the V-1) seems implausible. “The following explanation seems easier to me: In Peenem¨ unde fundamental questions and various rocket studies and projects would be worked on. . . The contact via Osenberg would probably have led to the development group, but probably not to the production group of the V2 in Mittelwerk. The contact between the development group and the production group succeeded ‘institutionally’ through engineers expressly appointed for such.” (Dr. Gunther Engler) “I do not know what Rohrbach’s group in Prague calculated. My interest in Rohrbach was always directed at his activity in the Foreign Office. I was not at all aware that during this time he also occupied his Chair in Prague. Unfortunately, Lammel, who likewise was then in Prague, has also died. It will not be easy to resolve the open questions. In my estimation of Rohrbach’s character, the activity of the computing group could have been a Potemkin village for the rescue of human lives. But that is naturally a pure speculation supported by nothing. I have treasured Rohrbach’s human traits greatly, even if his missionary concerns got on my nerves.” (F.L. Bauer on 31 January 1999 to the author.)— Perhaps the computing group worked on tasks for the cryptological requests of the Foreign Office? In the political archives of the Foreign Office one finds no hint of Gentzen. For Prof. Dr. Rohrbach, on the other hand, there is a personal file. From it we learn that since 10 May 1940, thus at a time when he was still senior assistant in G¨ ottingen, he was temporarily employed as a scientific labourer in the Foreign Office and was assigned to the Personnel and Management Department, Project Z, Cyphers and Communications. Otherwise we find in the file no reference to other people. In Rohrbach’s personal file Ernst Mohr is mentioned only once as a lecturer in Prague, and certainly in connection with Rohrbach’s replacement during his activity in the Foreign Office in Berlin. By the way, this file, in accordance with the Federal Archives Act for the use by a third party, is still closed. This German law—we expressly have no “Freedom of Information Act”—is not there to protect the culprit, but rather is a health precaution for researchers, whereby they will not overwork. 20 1996, p. 112. 21 “AA” is an acronym for “Ausw¨ artiges Amt”, or “Foreign Office” (cf. footnote 19). “HVP” is an acronym for “Heeresversuchsanstalt Peenem¨ unde”, the “Army Experimental Station at Peenem¨ unde”. It was later called the “Heeresanstalt Peenem¨ unde”, or “HAP”. 22 1996, p. 117.

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the qualified personnel employed there. On the contrary, Prof. Osenberg was completely incapable of supplying the lacking qualified personnel of the HAP.23

He considered it certainly possible that Rohrbach’s computing group worked for the SS and, to be sure, in its “Research Centre” in Pilsen (Skoda Works): It is surely possible that through Skoda the SS being “up in the air”, i.e. without knowledge of the actual boundary conditions, assigned tasks on the A4/V2 ballistics.

The HAP in any case maintained its own computing groups for the carrying out of the extensive numerical computations for ballistic and navigational problems. There the printout of computation results on “wallpaper-length” paper strips resulted in the “War-Aid Girls” [Kriegshilfsm¨ adchen] being called simply “Wallpaper unde Archive, where over 2000 scientific Women” [Tapetenfrauen]. In the Peenem¨ reports are gathered, Gentzen is just as little mentioned as in Dr.-Ing. R. Strobel’s publication “Ballistik ferngelenkter Geschosse”24 [Ballistics of remote-controlled projectiles]. Strobel was the laboratory director for ballistics in Peenem¨ unde. The work for a manufacturing group and not for a development group would clarify some things. It explains why the files of the development group surrounding Wernher von Braun shipped to America contain no archival material on Gentzen’s computing group. But Professor Hans Rohrbach insists on this type of problem for Gentzen: By these means [Osenberg Action– EMT] I could, through a corresponding proposal, bring Gentzen to my mathematical institute in Prague, where I had already begun to carry out computational works for the V-weapon manufacture in Peenem¨ unde. I could release Gentzen for this and transfer to him the direction of the workgroup, which I had already formed. At the time these were young girls from the upper classes of the higher schools, who with certain mathematicalstatistical computing works, to which they would be trained, must carry out assistance work, also a sort of military action. Gentzen performed his task well and worked out good results for Peenem¨ unde; the problems would be posed from there. This shows that he was also capable at other mathematical works.25

This was confirmed in the memories of one of his former co-workers in a similar way. Dr. Franz Krammer26 stated in a letter: In the frame of his research mission Prof. Rohrbach had, probably in collaboration with the lecturer Gentzen, furnished a branch of the Institute. Under Gentzen’s direction, mathematical tasks, mostly for the Reich’s Research Council Prof. S¨ uss, were worked on. Gentzen and I prepared the computational instructions so that they could be carried out by 4 to 6 girls at the calculators (antiquated monsters to you). (The girls, who had to break off their studies, were happy that they didn’t have to work in a munitions factory, where the women

23 Letter

to the author from Dr.-Ing. H.R. Reisig of 14 March 1997. ¨ Molitz-Strobel, Aussere Ballistik, Springer Verlag, Berlin, 1963. 25 Letter from Hans Rohrbach to Dipl.-Ing. G. Gentzen of 21 February 1986. 26 Franz Krammer (*30 March 1915 in Udoli u Kaplic) joined the Nazi Party in 1938 and was promoted in 1942 at the German University in Prague with the work “Absch¨ atzung mit Hilfe konvexer Funktionen” [Estimation with the aid of convex functions]. After the war, imprisonment, forced labour, and expulsion, he went into the school service. He lives today in Munich. 24 In:

4. THE FIRST COURSES IN NOVEMBER 1943

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lost their hair.27 From April ’45 all the girls would gradually be given leave to go home.)28

The question of how the communication between Gentzen and Osenberg functioned was answered by Rohrbach: Gentzen’s assigned tasks came via the Osenberg Office in the Hartz from Peenem¨ unde; his results returned via the Osenberg Office, generally by mail, in March 1944 through my personal messenger service because the post office was already no longer functioning fully.29

In 1943 Gentzen became acquainted with Ernst Mohr.30 Although both had been promoted by Weyl, they hadn’t known each other previously. Mohr remembered little of Gentzen: 1) He gave an impressive and masterly lecture on the foundations of mathematics; 2) he admired and respected Bertrand Russell above all! 3) it pleased him greatly when he could still acquire a copy of the second edition of Perron’s Irrationalzahlen in 1943 in a Prague bookstore.31

In his second edition of Irrationalzahlen, Perron mocks Bieberbach: 27 Women working in munitions factories mostly had yellow-red discoloured hair. By the drawing off of the highly toxic liquid dinitrobenzol, a cheap variant of the explosive TNT, it came into contact with skin and clothing. But also from the decomposition of TNT, a mixture of toluol, sulfuric acid and nitric acid, there developed a chemical zoo of dangerous substances like aromatic amines and dinitrotoluol. However, not only poisoning but also fires and explosions made this “workplace” not very attractive. 28 Letter from Dr. Franz Krammer to Dipl.-Ing. G. Gentzen of 12 February 1986. Numerical computations for the solutions of differential equations would be broken down into individual computation steps by qualified mathematicians, which then could be carried out for ever new constants by less qualified computing groups. Which calculator was used by this computing office with 8 girls I cannot say. A survey of the calculating machines available and customary in 1943 and their principles of use and modes of functioning can easily be found in the little book of Prof. Dr. Friedrich Adolf Willers, Mathematische Instrumente, Verlag von R. Oldenbourg, Munich and Berlin, 1943. Alfred Frege (n´ee Fuchs), the adoptive son of Gottlob Frege, worked for the firm Brunsviga (Grimme, Natalis & Co.) from 1935 to 1937 as a factory assistant supervising the production of calculating machines. 29 Letter from Prof. Dr. H. Rohrbach to me of 9 August 1988. ALSOS found Werner Osenberg and his files, including the ones concerning the Reichforschungsrat RFR, in Nordheim, near Harz in Thuringia. 30 An article by Hans Ebert brings up Ernst Mohr. Mohr was supposed to be a member of Gentzen’s computing group. According to Ebert, he must also have concerned himself with textbooks for schools and high schools in the future Reich’s commissionership of the Caucasus (p. 227). But Ebert says:

In March 1943 at the request of Johann Nikuradse of the R¨ ustungskommando [weapons command] of the Reich’s Minister for Arming and Munition, Mohr would be declared a “key employee” for the “execution of important military work.” In this capacity collaboration was guaranteed. I don’t believe that Mohr wrote textbooks and take that as a cover name for this computing group. In any event it is reported that the SS Raw Materials Office could procure fully automatic calculating machines, even hand calculators, in mid-September 1944. These should have served to furnish a mathematical division in the Sachsenhausen concentration camp. Perhaps Gentzen’s computing group tested the plausibility of the results of these prisoner groups? (p. 234) All this, however, is endless speculation that leads to questions. See: Hans Ebert, “ ‘H¨ aftlingswissenschaftler’ im Einsatz f¨ ur die SS 1944/45”, pp. 219-242, in: Heinrich Begehr (ed.), Mathematik in Berlin. Geschichte und Dokumentation. Zweiter Halbband, Shaker Verlag, Aachen, 1998. 31 Letter to me from Professor Dr. Ernst Mohr (1910-1989), student of Hermann Weyl, of 19 August 1988.

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Only by L. Bieberbach in 1924 was fault found in the German literary newspapers where Dedekind’s theory was presented; it would have been more natural to choose the theory of Cantor and M´eray, the purity of method combines with a sense for elegance. One may not turn a deaf ear to an exhortation from so competent a side, especially when its weight is significantly increased by famous new research on the J- and S-types of mathematicians. . . Third, I believe that it is always permitted a German, who has a choice between a German creation and an in-itself beautiful and valuable creation of foreign origin, to follow the voice of his heart and prefer the German, because it is German.32

On 28 August 1943 Gentzen received a certificate from the medical officer in G¨ ottingen. He weighed 61 kg (134 21 lbs) at a height of 170 cm (5 feet, 7 inches) and had a “proper” nervous system and was also otherwise fully healthy. Rohrbach requested on 13 November 1943 that “a salaried lectureship be transferred to Herr Dr. Gentzen,” because he got his income as a scheduled scientific assistant through ottingen: the University of G¨ His scientific accomplishments must be described as outstanding and are generally respected in expert circles. Dr. Gentzen has for several years already been associate editor of the scientific journal Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften. It is to be expected that in the foreseeable future he will receive a call to a permanent professorship. I consider it just and necessary to give his scientific accomplishments and his employment by the military the proper respect by transfer to a salaried lectureship.33

The dean, rector and curator approved this request without reservation. On 24 January 1944 the Reich’s minister for science, education and adult education of the German ministry for Bohemia and Moravia approved him in his position as salaried lecturer with effect from 1 April 1944. He received RM 511.02 per month, leaving him RM 395.12 to live on after taxes. His share in lesson fees remained untouched thereby: “You are removed from your assistantship at the end of March 1944.” The curator in G¨ ottingen stopped paying his salary on 31 March 1944. On 2 December, Gentzen wrote Uncle Willi and Aunt Iga: I now govern an entire “computing office” of 8 young girls, mostly students who were not permitted to continue their studies and were employed by us. Thus you can imagine that I scarcely have a calm moment. “Incidentally” I must also hold my lectures. However, there are not many students; one must be happy if half a dozen come together. Under these circumstances I apply there less effort in the preparation of the lectures. . . Is Dr. Herbst also in Putbus? For me he has been the closest friend of all my teachers. . . 34

Uncle Willi was an earlier school friend of his father from their shared gymnasial days in Stralsund. His daughter was Hertha Michælis. Looking back, Prof. Rohrbach spelled out the situation: 32 In

his bibliography, Perron cites: L. Bieberbach, “Pers¨ onlichkeitsstruktur und mathematisches Schaffen”, Forschungen und Fortschritte 10 (1934); G.H. Hardy, “The J-type and the S-type among mathematicians”, Nature 134 (1934). 33 Bundesarchiv Koblenz R31/379 folio 1.— Der Kurator der deutschen wissenschaftlichen Hochschulen in Prag, Habilitationsakte Dozent Dr. Gerhard Gentzen; available in part under Sen. Prot. Nr. 876/1943 in the Archives of the Charles University, Prague. The “Habilitationsakte” is split into two parts: 1) Gentzen’s personal file from G¨ ottingen to 26 January 1944; 2) Gentzen’s personal file from Prague. 34 The letter is in the possession of Frau Waltraut Student.

5. THE LAST KNOWN SCIENTIFIC LETTER OF GERHARD GENTZEN

243

Also, the work in Prague satisfied him, made him friends, and with a clear conscience, having also given his best therefor, he allowed himself, as I would after be told, without contradiction, without emotion to be taken away to his end. I could imagine that there he had already finished his life, i.e. his mathematical work.35

Hans Rohrbach worked in the area of Algebra and Number Theory and had no mathematical points of contact with Gentzen.36 They rarely saw each other in Prague. For “Uncle Willi” Gentzen also looked for mathematical books. These include Grunert, Loxodromische Trigonometrie (1849), Das Verebnen der Kugeloberfl¨ ache [Flattening of the surface of the sphere] (1892), and any useful literature on the analytic geometry of the sphere. However, he found nothing more on the subject. In August 1944 Gentzen spent 14 days vacationing with Uncle Willi and Aunt Iga in Putbus. 5. The Last Known Scientific Letter of Gerhard Gentzen Gerhard Gentzen wrote to Paul Lorenzen, who had taught at the Weserm¨ unde maritime school since 1942: G¨ ottingen, 12 November 1944 Very respected Herr Lorenzen! I have looked through your attempt at a consistency proof, not in detail, for which I lack the time. However I say this much: The consistency of Number Theory cannot be proven so simply. Permit me to give you some well-intentioned advice: Do not start with so difficult a problem, but rather begin with, say, the propositional calculus, as I have done; this is very suitable practice in selfthinking and gaining familiarity with logical concepts. The path into predicate logic is truly stonier! A pair of pretty problems is for example: a proof of the completeness of my “structure”-inference rules, or: to give a handy decision procedure for the intuitionistic propositional logic. With the best greetings Your G. Gentzen37

During the war years Gentzen wrote simultaneously two things: his book Die mathematische Grundlagenforschung [The mathematical foundational research] and on the problem of the consistency of analysis. This is shown by his stenographic sketches which have survived. Pages AL 137-138 contain a decision procedure for the intuitionistic propositional calculus. Gentzen’s “handy decision procedure” for 35 Letter

to Dipl.-Ing. G. Gentzen of 21 February 1986. also published books like: Naturwissenschaft, Weltbild, Glaube, 13th ed., R. Brockhaus, Wuppertal; or Sch¨ opfung—Mythos oder Wahrheit, R. Brockhaus, Wuppertal, 1990. 37 Shortly before his death, Prof. Dr. Paul Lorenzen wrote to me: “Around 1937 Gentzen lectured on his consistency proof to the Mathematical Society. From this I knew the problem of tertium non datur for theories with variables over infinite domains (arithmetic and analysis). My own consistency proof for analysis (axiomatised as ‘ramified type logic’) I had sent in outline to Gentzen in 1944. And received the enclosed letter as answer. My work was published in 1951 in the Journal of Symbolic Logic (‘Algebraische und logistische Untersuchungen u ¨ ber freie Verb¨ ande’ [Algebraic and logistic investigations on free lattices]). Contrary to Gentzen’s opinion, consistency proofs do allow themselves to be ‘so simply’ proven. (You can now find out about the subject in my Lehrbuch der konstruktiven Wissenschaftstheorie (Bibliographisches Institut, Mannheim, 1987).)” 36 Rohrbach

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this calculus is a semantic procedure to decide the intuitionistic correctness of formulæ.38 Gentzen’s suggestion to Lorenzen that the completeness of the structural rules would be a nice problem should mean developing a sequent calculus using no structural rules.39 6. Gerhard Gentzen in 1944: Teaching Functions, Computing Office, and War On 20 January 1944, the curator of the Georg August University (G¨ ottingen) wrote to the curator in Prague, who had requested Gentzen’s personal files: I beg to know when Dr. Gentzen has assumed his activity as lecturer in the natural scientific faculty of the German Charles University in Prague, as I have been informed neither of his appointment as unsalaried lecturer in Prague nor of his leaving G¨ ottingen and thereby his activity as scientific assistant at the Mathematical Institute of the University of G¨ ottingen, for which he still regularly receives his salary from me. Thus I ask Dr. Gentzen be advised with regards to his official duties to me as his superior and to arrange the rest.

The personal files were enclosed. The curator informed the Reich Minister via the State Minister of Bohemia and Moravia and everything went as it should. In the summer semester of 1944 (1 April to 31 July) Gentzen offered in the Personal- and Lecture-Schedule40 of the German Charles University: Analytic Geometry I 4 hours, p Mon.-Thurs. 11-12 Exercises in Analytic Geometry I 2 hours, p Wed. 16-18 However, Dr. Krammer wrote: At my time (from 1 June 1944) Gentzen held no lectures. (Hitler: In this time, when the young men fulfill their duty for Germany, the girls should also not study, rather make their direct contributions to the final victory.)41

However, Gentzen held seminars (privatissime et gratis). The illness made itself apparent with the ongoing war: Mathematical Institute of the German Charles University, Prof. Dr. H. Rohrbach Prague, 29 July 1944 To the Curator of the German Scientific High Schools of Prague I request to allow Docent Dr. Gerhard Gentzen of the Mathematical Institute of the German Charles University to take a longer convalescent leave. Dr. Gentzen 38 For

A&B and A ∨ B the correctness reduces to that of the components: in the first case both A and B must be correct, and in the second at least one of them. The interesting case is the implication A → B. It is correct if B is correct assuming A is correct. Stated differently: The correctness of B follows from the correctness of A and then the implication is also correct. The distinction from the classical concept of correctness is this, that one does not consider the possibility that the antecedent of the implication is not correct. That the definition of intuitionistic correctness is not trivial can be seen when the antecedent consists of iterated implications. An example is Peirce’s Law: ((A → B) → A) → A. The implication is correct if A is correct given that (A → B) → A is correct. Let us consider if this follows. Now (A → B) → A is correct if the correctness of A follows from that of A → B. This holds if B is correct whenever A is correct. From none of these conditions does it follow that A is correct, and thus Peirce’s Law is not intuitionistically correct. 39 Cf. Chapter 4 of Sara Negri and Jan von Plato, 2001. 40 Lecture Schedule, p. 59. The “p” in the second column stands for “privat” and announced that the students would not have to pay lecture fees. As Gentzen was now salaried, he did not need to charge them. 41 Letter from Krammer to Dipl.-Ing. G. Gentzen of 12 February 1986.

6. GERHARD GENTZEN IN 1944

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needs the leave for the restoration of his capacity for work. For more detailed explanation I cite the following: Dr. Gentzen had to a large extent overexerted himself through his difficult and subtle mathematical investigations when the war broke out and he was drafted as a radio operator for the Luftwaffe. He was three years from 1939 to 1942 in military service through which particularly stressful service as radio operator he suffered from nervous exhaustion, on the basis of which he would finally be discharged from the military as permanently unfit for duty. He had to avoid mental activity for a full year and is only now in reduced measure capable of intellectual work. Since the 1943 winter semester he has held lectures in Prague. He can however carry out this lecture activity during the semester only if he imposes on himself a complete rest in the lecture-free period. I am strongly relying on his support in the mathematical instruction for the coming semester. I ask therefore that he be granted the implementation of a convalescent leave of 8 weeks for the restoration of his capacity for work. The replacement of Docent Dr. Gentzen in the research tasks which the Mathematical Institute has carried out will be taken over by Dr. Franz Krammer and the secondary school teacher Walter Tietze, who have been released from the army for these tasks and will shortly arrive in Prague. Their address reads: Prague II, Weinberggasse 7. Heil Hitler! The Director of the Mathematical Institute of the Charles University Rohrbach

A month later the relevant certificate reached the curator from G¨ ottingen: University Nerve-Clinic Polyclinical Office Hours 11-1 a.m. Telephone no. 3648 G¨ ottingen, 22 August 1944 Medical Certificate Docent Dr. Gerhard Gentzen of Prague, born 24 November ’09, is already known to me from his time in G¨ ottingen. He suffers from periodic conditions of depression with headache, apathy, lack of concentration, stomachache, and other nervous manifestations, which arise in these depressed states and have already often made him incapable of work for a long time. With his kind of work and his not very hardy physical constitution, complete relaxation and quiet, preferably under a physician’s care, is required from time to time. I consider it necessary, if an acute condition is not imminent, that Dr. Gentzen be granted the 8 week convalescent leave requested by him, in order that a recurrence of nervous and depressive disturbances which would certainly be accompanied by an unfitness for work as previously experienced be prevented in time. Prof. Dr. Ewald, Director of the Clinic

In the winter semester 1944/45 (1 November 1944 to 28 February 1945), Gentzen offered:42 Differential and Integral 4 hours Mon., Tues., Thurs., Fri. 11-12 Calculus I Exercises in Differential 2 hours, p Wed. 14-16 and Integral Calculus Analytic Geometry II 2 hours, p Wed., Sat. 11-12 42 Lecture

Schedule, p. 67.

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7. Hans Rohrbach’s Report on the Conditions in the Mathematical Institute in Prague On 3 October 1944 Rohrbach informed the curator: Prague, 3 Oct 1944 German Charles University To the Curator of the German Scientific High Schools Prague Report on the Development of the Mathematical Institute of the German Charles University I take the move of the Mathematical Institute into the now completed new rooms as an opportunity to give a short report on the development of the Institute in this last year. Until the end of 1943 no such research tasks in the Institute would have been carried out. The two established professors were directly appointed to war work and additionally continued their instructional activity. The assistant was in the army, and further scientists were not available. However, since the beginning of my presence here, my endeavour has always been to have the Institute itself involved in war-related research. For this I needed more space and more collaborators. My requests for the enlargement of the Institute having been granted in satisfactory manner by the curator, the work for the setting up began a year ago. In the meantime I had acquired Dr. Gentzen as lecturer for the Institute and trained Miss Garkisch as a computer. In the hope of an early finish of the preparations for the new rooms, I began at the start of 1944 to accept the first tasks of war-related works to be performed at the Mathematical Institute. As computers, besides Miss Garkisch, now and again students were employed; the direction of the research work was taken over by Dr. Gentzen, as I myself was tied up with my activity in Berlin and can only be in Prague in reduced extent. At the same time I requested through the Planning Office of the Reichsforschungsrat the return of mathematicians from the army. The first of these arrived in July of this year and could be installed in further research tasks. At the beginning of August the new rooms were first temporarily moved into. The procurement of the necessary fittings naturally caused difficulties and was delayed for some time. In any event it may be said that the Institute already carries out essential work on research tasks. At present the following scientists are busy at the Institute: Docent Dr. Gentzen, Dr. Franz Krammer, secondary school teacher Walter Tietze, Dr. Paul Armsen. In addition there are the following helpers: the students Helmut Wolf, Wolfgang Fleischmann, Maria Burian, the technical assistant Edith Garkisch. Additional collaborators and helpers are requested. The work of the Institute is coming along as planned, in accordance with the demands of total war. In summary I express the conviction that it was the correct decision of the curator to commission the preparations for the Mathematical Institute. I hope that the development of the Institute depicted and, through the new study requirements, the heightened significance of mathematics have shown the necessity of providing the Mathematical Institute of the German Charles University full support also for the remaining preparations and acquisitions still to come. The Director of the Mathematical Institute of the German Charles University Rohrbach

8. WHY DID GENTZEN BANISH ANY THOUGHT OF FLIGHT?

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On 17 December 1944 Gentzen dedicated the “Zusammenfassung von mehreren vollst¨ andigen Induktionen zu einer einzigen” [Combination of several complete inductions into a single one] to Heinrich Scholz on the occasion of his 60th birthday.43 8. Why Did Gentzen Banish Any Thought of Flight? From fear of being shot—on-the-spot executions and hangings by the SS for the smallest “defeatist remarks” were the order of the day—or as sacrifice to be sent to the front by the “Heldenklau” [hero-claw] General (later: Field Marshall) Sch¨ orner (“He who is caught facing west will be shot!”)—also from a misunderstood sense of duty and the expectation that as an “innocent” mathematician nothing would happen to him—Gentzen remained until the “Revolution” of the Czechs in Prague and went to the weekly exercises of the German territorial army. He would hear nothing of the pleas from Dr. Franz Krammer and Max Pinl to leave Prague.44 Perhaps Gentzen remained in Prague also because, after all the insanity, he could go forward with building a small institute. Where should he, who daily completed mathematical lines of thought, flee to? Did he, in the face of a collapse which he as an “intellectual worker” however was certain to survive, feel responsible for caring for the Institute and its library? This friendly, reserved, intellectually oriented man was not an example of the Nazi ideal of hard and fast decisiveness. The decision to stay or run receded and attack was to him possibly simply too arduous and laborious, unnecessary or irrelevant. Pinl did not belong to the Mathematical Institute, but had a separate workroom on the same second floor. As I suggested the Mathematical Institute consisting of only us two move to my small hometown, Gentzen referred me to the leader of the Prague office of the Reichsforschungsrat, Prof. Gudden (1892-1945): “If you believe that you have no more tasks to fulfill in Prague, then I must transfer you to the military!”45

Although the Prague mathematician Premysl Vihan has unearthed a Dr. med. Gentzen in Prague from an index of members of the German territorial army in the possession of the Prague Police Directory, all entries point to Gerhard Gentzen. Therein one discovers the routine handling of the militarily unfit private first class Gentzen from simple Sturmmann to SA-Rottenf¨ uhrer and his ensuing employment by the German territorial army (Platoon III – 3rd Group, membership card 19). This may have been the “official” charge filed against Gentzen by the Czech authorities and police—Gentzen, by the way, was never informed either publicly or in front of witnesses, but he would be arrested simply because one wanted to shoot “German rats”. Mind you there is neither in the personal file nor in the party files in the Berlin Document Center any reference to his position as a “Rottenf¨ uhrer”.46 The collaboration with high school teacher Walter Tietze, Dr. Franz Krammer (who was assistant to Rohrbach), Ernst Lammel, Paul Armsen (who was assistant 43 It

appeared posthumously in Archiv f¨ ur Mathematische Logik und Grundlagenforschung, vol. 5, no. 1 (1954), pp. 81-83. 44 “Against the advice of the author of this report he would not willingly leave the position in Prague at Easter 1945,” p. 173, in: Max Pinl, “Kollegen in einer dunklen Zeit”, Jahresberichte der deutschen Mathematiker-Vereinigung 75 (1974), pp. 166-208. 45 Letter from Dr. Franz Krammer to Dipl.-Ing. G. Gentzen of 12 February 1986. 46 Based on the source this fact seems to me unquestionable. I thank Premysl Vihan (Prague), who made available a copy of the paper from the territorial army book in which the entry on Gentzen is plain to read.

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to Rohrbach and helped out with the lectures and seminars, but was not a do¨ cent),47 Docent Karl Maruhn48 (he had just published “Uber eine Klasse ebener Wirbelbewegungen in einer ideellen incompressiblen Fl¨ ussigkeit”49 ), and others, and the contacts with Gerhard Kowalewski and Ernst Mohr were restricted to the necessary. As he thought of music as martyrdom, he was left with theatre, crime novels, coffee and card games as entertainment. Gentzen’s mother and sister fled the Red Army in 1945, leaving behind all papers, furniture, and the rest of their belongings. They travelled to Sigmaringen, where they would be well received by the sisters. On 12 February 1945 (2 days before Dresden) Prague received its first aerial assault. A stray 10kg bomb detonated on the meadow next to the buildings and ripped half of a side wall out of the Zoological Institute (below us on the first floor). Gentzen’s room was a few metres farther away from the point of impact. . . Naturally we followed—completely passively—the military and political situation, but said not a word on the matter. Together we went to the exercises of the German territorial army, but were never quartered in weekly rotation as were other Germans. . . In the display of personal concerns we were both largely reserved. . . At the end of April I went home to retrieve my bicycle. G. did not

47 According to the research of Maria Georgiadou (Constantin Carath´ eodory. Mathematics and Politics in Turbulent Times, Springer-Verlag, Heidelberg, 2004, p. 574), Paul Armsen was a “repatriated Balt from Estonia” (Feigle to S¨ uss on 19 July 1943, Freiburg University Archives, Nachlass Wilhelm S¨ uss 89/51) working at the Institute for National Psychology of the Heydrich Foundation, named after Reinhard Heydrich, one of the worst characters of the SS. He got his ¨ doctorate from Carath´eodory with the thesis “Uber die Strahlenbrechung an einer einfachen Sammellinse” (Munich, 1942, 49 typewritten pages, reviewed by Carath´eodory in Zentralblatt f¨ ur Mathematik 28, pp. 269-270). Using Armsen’s work, Carath´eodory published his article “Die Fehler h¨ oherer Ordnung der optischen Instrumente” [The higher order errors of optical instruments], Bayer Akad. d. Wiss. M¨ unchen, math.-nat. Abt. (1943), pp. 199-216, and reprinted in the second volume of his collected works. 48 Karl Maruhn was born on 5 December 1904 in Chemnitz. He studied mathematics and physics in Leipzig and T¨ ubingen. In 1930 he was promoted in Leipzig with the work “Ein Beitrag zur mathematischen Theorie der Gestalt der Himmelsk¨ orper” [A contribution to the mathematical theory of the formation of the celestial bodies]. After the teaching examination in 1931 he was a teacher at various schools in Leipzig until 1935. In 1939 he was drafted into the Deutschen Versuchsanstalt der Luftfahrt [German experimental station for flight] in Berlin-Adlershorst and “occupied himself with rigorous solutions for the surrounding currents of ellipsoids, and thereby had obtained beautiful, also numerically valuable results” (p. 119, in: J. Weissiger, “Erinnerungen ¨ an meine Zeit in der DVL, 1937-1945”, Jahrbuch Uberblicke Mathematik 1985, Bibliographisches Institut, Mannheim, pp. 105-129). At the same time he held lectures at the Technical High School in Berlin-Charlottenburg after habilitating there in 1937. From 1944 to 1945 he held the vacant chair for applied mathematics at the German University in Prague. After the war Maruhn would be lecturer and ordinary professor in Jena. From 1945 to 1949 he was managing director and teacher at the Mathematical Institute in Jena with potential theory as area of expertise. In 1949 he received a chair at the Technical High School in Dresden and became director of the Institute for Pure Mathematics. Together with H. Grell and W. Rinow he was answerable to the Deutschen Verlag der Wissenschaften [German publishing house for the sciences] for the publication of the “Hochschulb¨ ucher f¨ ur Mathematik” [High school (i.e., college) books for mathematics]. In 1959 he then followed a call to Giessen, where he taught until he became emeritus in 1973. He died on 8 February 1976 (all details from the biographical sketch in Mitteilungen aus dem Mathem. Seminar Gießen, Heft 123 “Dem Andenken an Karl Maruhn gewidmet”, Selbstverlag des Mathematischen Instituts, Giessen, 1977. I thank Frau H. Bertram, administrative assistant at the mathematical seminar in Giessen, for this reference). 49 Jahresberichte der deutschen Mathematiker-Vereinigung 45 (1935), pp. 194-201.

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feel free to accept my invitation to come along. On 30 April I cycled from there the 180 km back to Prague in a heavy snowstorm.50

The computers for Gentzen’s calculating machines would be sent home in succession from April 1945. Dr. Franz Krammer could put a bed in an empty assistant’s room, where they would be looked after by the Institute nurse, particularly with breakfast. In March I departed from him and brought a completed work to the Hartz, where Osenberg had his quarters,

wrote Prof. Dr. Hans Rohrbach, who in 1944 through skillful diplomacy helped to rescue the mathematician Ernst Mohr, who had been sentenced to death by the Peoples’ Court.51 Prof. Rohrbach came to Prague as Director of the Mathematical Institute only on Friday and Saturday from his Berlin post. He held a 2-hour lecture and carried out the accumulated Institute business. (Dr. Franz Krammer)

Hans Rohrbach described it more dramatically: I had concluded my activity there for the winter semester 1944/45 (end February) and was in Berlin, where during the war I was busy full time at the Foreign Office.52 I looked after the service in Prague only two days in the week during the semester, travelling there at night in a sleeping coach, two nights later back in a sleeping coach, thus working 4 days a week in Berlin, 2 days in Prague. During the breaks I was busy only in Berlin. I had the intention to return to Prague for the summer semester 1945 (end of April). I had as always left behind in the Institute everything personal, in particular books, lecture manuscripts, etc. But at the beginning of April 1945 I was already in a lay-by of the Foreign Office in Thuringia.53

And Dr. Franz Krammer added: 50 Letter

from Dr. Franz Krammer to Dipl.-Ing. G. Gentzen of 12 February 1986. wrote “that Mohr performed computations in connection with the development of V-weapons in Peenem¨ unde for a group in Prague founded by him and led by Gerhard Gentzen, and that he delivered to Mohr the material to be worked on in Pl¨ otzensee.” (Freddy Litten, “Ernst Mohr– das Schicksal eines Mathematikers”, Jahresberichte der deutschen MathematikerVereinigung 98 (1996), pp. 192-212; here, p. 204.) I have the impression that Rohrbach used this computing group and the same purpose to rescue men quite often. Did the group actually have a different purpose? Would it actually have counted? 52 I don’t know exactly how much H. Hasse or Hans Rohrbach, thus the Naval and Foreign Office, or Hasenjæger (answerable to the security of ENIGMA in OKW/Chi [OKW = Oberkommando der Wehrmacht, or Supreme Command of the Military. The OKW/Chi was the cipher office of the OKW]), or whoever else busy in the deciphering divisions (Georg Hamel, Helmut Grunsky, etc.) had an understanding of the recursion theories of Turing, Welchman, von Neumann and others. Why wasn’t Gentzen assigned to this task?— Perhaps it is good that Gentzen and Germany were spared that. “That you probably also know F. Bauer’s book, Entzifferte Geheimnisse [Enciphered secrets] (Springer, Heidelberg, 1995), in which I would be “consoled” on p. 343, you probably understand if I reflect thereon, what Gentzen, if he, instead of yours truly, had been brought in by Heinrich Scholz would probably have arrived at regarding the Enigma weaknesses.— To the question ‘Did ULTRA decide the war?’, one already read before: No, but it could have lasted about half a year longer—And where then the two atom bombs would have been introduced, one can have no doubts about that” (Gisbert Hasenjæger to me on 11 March 1997). Cf. on this theme the excellent book: F.L. Bauer, Entzifferete Geheimnisse. Methoden und Maximen der Kryptologie, 2nd expanded edition, Springer-Verlag, Berlin, 1997. 53 Letter to me from Hans Rohrbach of 11 December 1987. 51 Rohrbach

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In the remaining days Docent Gentzen took over direction of the reduced Institute.54

There are those who maintain that Ernst Mohr—again according to the single information source Prof. Dr. Rohrbach—must have undertaken computations in the concentration camp for Gentzen’s work group. I have never been able to confirm this. The hero-claw General Sch¨ orner managed to make Bohemia a last military and political bulwark. Although the Americans were a long time in Munich, the Soviets stood in Berlin and Vienna, for 3 days after Hitler’s suicide the swastika flag fluttered at halfmast in Bohemia. Every “defeatist” remark could mean onthe-spot execution. In my south Bohemian hometown still on the second of May 3 men were publicly hung, resp. shot, by the SS. In Prague one attempted to force an “and-yet voice”. So on 20 April we too “heiled” inaudibly to Hitler’s birthday in the Lucerna (with the largest hall). (Dr. Franz Krammer)

Ferdinand Sch¨ orner (1892-1973) was a submissive military officer who wanted to bring the German army in his sphere of influence unconditionally into line with the wishes of the Nazi Party.55 Already at the end of 1943 he had announced: An officer who teaches a National Socialist motto and deviates from it in practice has forfeited his right in my corps just as the one who today still sees in National Socialism an imposed form of the spiritual-psychological attitude.56

On 17 April 1945 Dr. Karl Maruhn collected a “salary declaration” for Gerhard Gentzen. Therein one reads: All public cashiers and payment offices of the Reich, the lands and municipalities are requested, by way of assisting the administration, from 1 June 1945 to pay the above named civil servant 395.12 RM tax-free every other month upon presentation of this certificate and a photo ID.

Gentzen signed the certificate on 20 April 1945, and this is the last sheet in his personal file.

54 Letter

of Dr. Franz Krammer to Dipl.-Ing. G. Gentzen of 12 February 1986. November 1943 Sch¨ orner became Chief of Staff of the National Socialist Command in OKH (Obercommando des Heeres [Supreme command of the army]). As ruthless and cruel executor of all orders to hold out, he would gladly be deployed for the “stabilisation” of the endangered sections of the Front. Since 25 January 1945 Sch¨ orner was commander in chief of the military group from the centre of Lusatia into Czechoslovakia. On 1 March 1945 he became general field marshall and on 30 April he was the last commander in chief of the German army. Mr. J¨ org Harms of N¨ otsch, Austria, informs me: Sch¨ orner demanded of us “on 10 May 1945 (two days after the official capitulation) not to lay down our arms; he would lead us ‘home’ into the Reich. Afterwards, he put on lederhosen and flew away in a Fieseler Storch [The “Stork”, produced by the Fieseler Company, was a small liaison airplane that lasted well after the war] into the still enemy-free Alps, where he would be snatched by the GIs, handed over to the Russians, held the quarter measure (25 years), residing in 1948 with two orderlies in a villa in Moscow, there sketched plans for Stalin for deployment against the Yanks, would be ‘freed’ by Adenauer in 1955, held in the Federal Republic for another four years for ‘homicide’. . . From my birth year 1922 over fifty percent were sacrificed on the altar of the fatherland, the blood-smeared slaughterhouse of the General.” From 1957 until 1960, Sch¨ orner found himself in prison in Bavaria and lived out his last years comfortably in Munich. 56 Manfred Messerschmidt, Die Wehrmacht im NS-Staat, Hamburg, 1969, p. 445. 55 In

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Now I must reproach myself that I had brought him to Prague at all! Together with Dr. Maruhn, who wanted to go from Prague to his family in Thuringia, Gentzen made plans to journey back together, over Thuringia to G¨ ottingen. They had already acquired maps.57

But Gentzen remained in Prague.

57 Hans Rohrbach on 3 January 1946 to Heinrich Scholz. The original letter is in the Institute for Mathematical Logic and Foundational Research in M¨ unster. I thank Prof. J. Diller for a copy.

https://doi.org/10.1090/hmath/033/06

CHAPTER 6

Arrest, Imprisonment, Death and Nachlass Here the need for knowledge of the circumstances of the death of Gentzen will be satisfied. One may find the various eyewitness reports of Gentzen’s death tiring because they repeat some of the facts.1 They don’t completely agree and bring in new details and demonstrate thereby ever anew doubt about the credibility of the accounts and memories. 1. The Last Days of Freedom in the Private Sector Gerhard Gentzen had a short but close friendship with Paul Armsen (1906-?), who was promoted by Constantin Carath´eodory in 1943 and who worked mathematically together with Hans Rohrbach. From February 1945 he limited his social dealings only to the Armsen couple. In 1944 Hans Rohrbach had had him installed as assistant and then as special lecturer. The third in the alliance, Paul Armsen’s bride, was Frau Hella, an interpreter. The tightness of the triple alliance was explained by Armsen by the fact that in Prague Gentzen had no mathematical equal at the time.2 Paul Armsen married Hella Armsen in February 1945, and Gentzen served as witness to the ceremony. An enterprising photographer offered his services and thus the last two photographs of Gentzen came into being. On G. as an internationally famous mathematician I know that since 1939 he carried around invitations to congresses in New York and Leningrad.3

Assuming they existed, I wonder if he carried these invitations around as a sort of “life insurance”. Gentzen was a very modest, aloof, one could almost say unworldly man, who lived only for his science and only allowed himself to take his mind off it with difficulty. For my wife’s sake he accepted under pressure an invitation to a concert, which we caught on too late must have been an ordeal for him. He explained to us after the concert that for him music was a most extremely unpleasant noise. In the last weeks in Prague G. visited us more frequently. The talks at the time naturally turned to when and how we would leave Prague. On 3 May ’45 he appeared with 2 bottles of wine, the remnant of a package received from his uncle. On 5 May the Czechs began their revolution.4 I experienced this beginning in a lecture of 1 The

publication of Premysl Vihan, “The last months of Gerhard Gentzen in Prague”, pp. 6-9 in Collegium Logicum. Annals of the Kurt G¨ odel Society, Vol. 1 (Springer-Verlag, Vienna, 1995) has fewer sources of information. 2 Letter from Dr. Paul Armsen to Dipl.-Ing. Gerhard Gentzen of 4 August 1986; Dr. Armsen went first with his wife to Erlangen, then to South Africa. 3 Ibid. 4 On the German occupation and the Prague popular uprising of 5 and 7 May 1945 see for ¨ example Helma Kuden (ed.), Die faschistische Okkupationspolitik in Ostereich und der Tschechoslowakei (1938-1945), Pahl-Rugenstein, Bonn, 1988. For the history and political conditions 253

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G. on “Introduction to Formal Logic”. He accompanied me still as far as the tram and took his leave with greetings to my wife and the advice not to speak on the way. On the 6th and 7th we spoke by telephone and on the 8th of May we were locked up by the Czechs.5

Franz Krammer saw it similarly. Rohrbach had rescued Krammer “in a for me wondrous action” (F. Krammer) from the disintegrating eastern front: On 30 May 1944 Prof. Rohrbach introduced me to the lecturer Gentzen. . . His black hair was always parted neatly; he was always very well shaved and his clothing carefully looked after. His finely structured form with the lively black eyes revealed an in no way na¨ıve but predominantly intellectually stamped personality. In spite of the gulf between his towering intellectual power and a nobody like me, Gentzen was always a warm and ready-to-help superior. In the evening and Sundays everyone went his own way. Gentzen spoke however of his dissertation, for which only with difficulty a proper judge could be found. . . G. used this calculus frequently for logical clarification—also for banal questions, whether he should go to the cinema. Thereby on notes he wrote tautologies, etc. The secretary probably observed this. . . Gentzen showed me a lovely invitation to deliver a lecture to the Institute for Advanced Studies in Princeton. The outbreak of the war hindered all lectures.6

Krammer reports: At the end of April Gentzen had already sent all of our computers7 to their relatives. I myself went home on 28 April. As Dr. Gentzen could not be persuaded to come (Gudden “branded” all members of the faculty who deserted Prague, and Dr. G. was always an idealist, unworldly as probably most mathematicians are), I traveled back to Prague again on 1 May with the hope that we would struggle through as I know Czech to some degree.8 Docent G. still held a lecture on Saturday, at which Herr Prof. Lammel was also present. (I have already spoken with Herr Prof. L. in Weilheim; he also told me of you.) Afterwards we went to the “Vegetarna” for lunch. In entering the restaurant we already heard tumult on the streets. Docent Maruhn and Dr. Armsen went home with their “secretary” (the name of the student doesn’t come to me);9 Docent G. and I went to the Institute, from which the Zwickelflagge 10 was already blowing against us (5 May).We did not leave the Institute again until our

of the Charles University of Prague, cf. Gerd Simon, Wissenschaftspolitik im Nationalsozialismus und die Universit¨ at Prag. Dokumente eingeleitet und herausgegeben von Gerd Simon. T¨ ubingen: GIFT-Verlag, 2004. 5 Armsen, op. cit. 6 Dr. Franz Krammer to Dipl.-Ing. G. Gentzen in a letter of 12 February 1986.— “In connection with my doctorate I would at the instigation of my promoter, Professor Hans Rohrbach. . . after 4 years as a soldier on the crumbling eastern front be pulled out (with reserve status) and called up as a “scientific assistant” in the frame of a research commission to the Mathematical Institute of the German Charles University in Prague. On 1 June ’44 I was introduced to Herr Docent Gentzen.” (Dr. Franz Krammer to EMT in a letter of 11 March 1988.) 7 That is, the women who performed computations by hand during the war. 8 Dr. Franz Krammer in a letter from Weilheim to Max Pinl in Oberahmede on 23 November 1946. (Nachlass Bernays: Hs. 975-1666.) In correspondence, “every letter writer from the USZone in 1946 counted as self-evident” (F. Krammer, 11 March 1988) his letters would be read, if not copied. The text on this matter is missing. 9 Edith Garkisch, later of the University Mainz—EMT. 10 A literal translation is “gusset flag”, which means nothing. The term was merely a derogatory name for the Czech flag, which had been outlawed under the occupation.

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arrest midday on 7 May. There were partisan patrols now and then, but they only asked about weapons; otherwise we were hopeful throughout.11

Dr. Karl Maruhn went to his apartment and would be arrested a few days later. Eventually, he found employment at the Mathematical Institute of the University in Jena. Gentzen was arrested in Prague on 7 May 1945. 2. The Arrest of Gerhard Gentzen and the Awful Imprisonment On 5 May 1945 about noon we went to lunch in a vegetarian restaurant about 400 metres distant. On the ground floor we met with Prof. Lammel, who came from a lecture on experimental physics. Prof. Gudden, in clear knowledge of the situation, held a formal lecture in which he gave a summary of the planned material for the semester. In the “Veget´ arna” we met, as usual, the secretary as well as Dr. Armsen (later Uni Erlangen). Around 1:30 p.m. the proprietor of the restaurant asked us in a distraught way to leave the establishment through the back door. As we saw a tumultuous mob in the lane, the secretary, who as a native of Prague spoke Czech with no accent and was also decked out in the Prague fashion, took over leadership and we ambled in the direction of the Institute. After a few metres we saw that people on the sidewalks were spitting onto the street. There lay a German-uniformed policeman, whose neck had been ripped open by a broken signpost. We then came to a wide arterial road, where initially German soldiers thundered towards the West in jeeps. Shortly there were only Czechs with the Zwickelflagge (forbidden under the protectorate) in the jeeps. The secretary, brightly chatting in Czech, then brought us across the street where we parted. Gentzen and I strolled to the Institute buildings where the Zwickelflagge was already hanging. Gentzen calmed down the excited porter, who said that the Czech engineers from the Technical Physics Institute (1st floor)12 could not prevent it. We gathered from “Mathematics” our sturdiest shoes and clothing and returned to the cellar, where we found the rest of the personnel who lived in the building. Electricity, water, telephone and radio functioned without disturbance. On 7 May Czechs from our University Quarter (medicine, natural sciences) came with armbands and rifles and said they had to bring us into “protective custody”. Already waiting on the street were Deputy Rector Denk13 (plant physiology), the highest official of the German Charles University after the suicide of the Rector Albrecht, Prof. Lorbeer14 (genetics), Dr. Lothing (scientific assistant like me) and the head gardener Horner from the Botanical Institute’s garden.15 Now, however, I’d like to inform you briefly about Gerhard Gentzen, with whom I was together almost until his death. Gentzen was arrested, like almost all Germans in Prague, on 7 May 1945 in the Mathematical Institute of the Prague university after which, with the rest of the members of the Institute present on the 5th, he was interned. Under the howling of the revolutionary mobs he would first be taken to police headquarters, as all of us, he was told it was only for a short verification of identity.16 11 Franz

Krammer to Pinl, op. cit. floor in American numbering. 13 Professor Dr. Viktor Denk, Director of the Institute for Plant Physiology. 14 Unscheduled Professor Dr. Gerhard Lorbeer, holder of the chair for genetics. 15 Krammer to Gentzen, op. cit. 16 Dr. Fritz Kraus of Friedberg/Hessen to Paul Bernays of 12 April 1948 (date of the postmark of the envelope, cancelled in Munich). 12 2nd

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From the police headquarters, where we would be correctly treated as prisoners, but where I saw the first act of bestiality in the courtyard through a cell window, we were delivered in a closed prison car to the district prison of the Neustadt city hall, not far from our Institute; here we would be prepared according to prison routine. (Valuables, watches, documents, writing paper, pencils, pocket knives, and suspenders would be placed in bags with our names.) In the cell we found, among others, Prof. Watzka, the dean of the medical faculty, and Dr. med. Dr. rer. nat. sub auspiciis Kraus,17 who often called on the Mathematical Institute. On 8 May (German capitulation) the Czech intelligence groups, who hated all Germans with the brutality of the Nazis, together with the mob lusting after atrocities, had conclusively seized power. Ever more Germans, who under dreadful circumstances or with refined tricks would be “snatched”, would be tossed into our cell. Together with 8 Czech “collaborators” there were up to 54 men in the cell.18 On 7 May we were taken by a large patrol. At the gate Prof. Denk and Prof. Lorbeer, Dr. Lothing, and head gardener Horner were already waiting. With the greatest hopes we were escorted over barricades in the direction of the police station in Bartholom¨ ausgasse. To the inflammatory cries in the streets, “Zastrelte ty krysy!” [Shoot the rats—EMT] the escort soothingly answered “To jsou jen botanici!” [They are only botanists—EMT]. At the police station our entry was filmed twice;19 in the building only our knives and matches were taken away and our names noted. Then we were brought into a cell (noon); around 4 in the afternoon particularly many Germans were brought into the courtyard, men and women beaten, some struck dead and shot. I myself saw this from the cell window. No place on the courtyard pavement was free of blood. On the next day the six of us were taken by auto the short distance to Okresni on Karlsplatz. There everything was taken from us, especially all papers. Rudely, but with no blows, we were brought to a 20-man cell where we remained for 3 months, often up to 64 men in the cell. The items taken from us we put into a little sack with the name of the owner. No one saw any of it again. I drew attention to the attendant that a world famous scholar with an invitation to an international mathematical congress in Princeton and with letters from American and Russian scholars could be found among us. Docent Gentzen had these letters on him. The reply: “Nedekejte si ostudu” [Don’t disgrace yourself—EMT]. The man merely shrugged his shoulders. On the next day, Dr. Kraus came into our cell. He was from a column on the march which was assured free exit by the “Narodni rada”, the Czechoslovakian and International Red Cross; the breaking of this assurance would be just one of many. With his promotion watch with Dr. Benes’s name engraved he tried in the following period at least to be brought before a higher 17 “Sub auspiciis” means “under the eyes of the head of state”. The promotee receives after the lecture on his subject area, to which also scientists from abroad travelled, a golden watch with the date engraved, in this case: “ ‘Dr. rer. nat. sub auspiciis. . . Kraus,. . . 1935. . . Dr. Edvard Benes’. The inscription ‘sub auspiciis’ would be allocated only twice per year from the (after Berlin and before Munich) second largest German university in Prague during the first CzechSlovak Republic.” (Dr. Franz Krammer in a letter to EMT of 11 March 1988.) “My husband, Dr. habil. Dr. med. Fritz Kraus (†7 August 1980). . . had received from there a golden watch from the President of the Czechoslovakian Republic with an inscription from him and a golden ring of the University. . . ” (Mrs. Olga Kraus (Augsburg) in a letter to the author of 7 August 1988.) The physician Dr. Fritz Kraus was born on 28 April 1903 in T¨ oplitz-Sch¨ onau (Teplice-Sanov), ¨ promoted on 19 April 1937 formally with the work “Uber konvexe Matrixfunktionen” [On convex matrix functions]. 18 Krammer to Gentzen, op. cit. 19 The entry into the prison was filmed by “Cechoslovensky Tyden u Filmu”.

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official, but did not succeed; he came out of the prison later as a camp doctor and may perhaps have reported to you of his further fate; he is now a while in Germany.20 Agonising as were the terribly nonhygienic relations in the fully crammed cell was the hunger. Devilish was the “protein ban”. Following the “Law of the Minimum” the body processes from each of 3 basic nutritional groups: carbohydrates, fat, protein, only so much as the minimum of one of the 3 basic nutrients is met. In the absence of protein the body takes it out of its own tissues. We became skin-covered skeletons. By further protein deficit either the heart or the sphincter is attacked. In each case death through circulatory failure is automatic. From the medical experts in our cell we were also informed that through too little fat the intestinal flora21 atrophy. Sudden fat influx immediately covers in the stunted remainders. In many cases it would drastically be demonstrated to us that death occurs in less than 24 hours. After a week in which we—without a clock—vegetated there, the call came into the cell: Who wanted to come “clear barricades” should come out bare chested and without shoes. (By earlier groups the very young Czech guards did not risk intervening against people from the slavering pack of hounds who robbed skeletons of their shoes and shirts.) On the last (4th) day there was only the still heaped up paving stones to load onto a heavy goods vehicle. Suddenly a woman in the encircling mob began to clamour, so shrill that I could understand nothing; she pointed in the direction of Gentzen, who with his finely shaped hands struggled to loosen a stone from the braces. Perhaps she believed she recognised a Nazi tormentor in the man with the tangled black hair and full beard(!). She then threw a paving stone without hitting Gentzen (as I believed). As I returned from the truck I saw that G. was being brought away by a tidy-looking passer-by. A short time later the whole group returned to the prison. In the cell I saw then that through the stone throw the small and ring finger of his right hand was almost crushed. The marvelous team of physicians in our cell succeeded with the healing in about a week, though both fingers remained stiff.22

3. Gentzen’s Physical Death Because of this he could no longer work. In the beginning I could bring him some bread from my workplace, but later we were most strictly inspected. The conditions in the cell were dreadful. In the 20 man cell there were sometimes 64 men; despite the 2-storeyed plank beds, mostly we could not stretch out at night. There were 1 or 2 buckets of water daily for the whole of us. Consequently we were soon completely bug-infested and lice-ridden. The worst, however, was the constant hunger. I would never have believed that hunger could be so agonising. Dr. G., who unlike the others had not the insignificant improvements through work, succumbed first to the hunger in the cell, after many had already died in other cells, on the 4th of August. In that typhus fever had broken out, we were cut off from work. With so many lice I find it incomprehensible today that fewer than 10 percent fell victim to the typhus. He was the first death in our cell, and so we were struck particularly hard by the death of this—I may use this word well here—noble man. Profs. Lorbeer and Denk spoke words of farewell. I myself suffered an uncontrollable fit of crying and would be first brought “to reason” 20 Krammer

to Pinl, op. cit. are the microorganisms that inhabit the intestinal tract and are vital to good health. 22 Krammer to Gentzen, op. cit. 21 These

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by being slapped by the guard. Dr. Gentzen died not of typhus but in fact of starvation. Later in many cases all piety before death was lost, warm items of clothing and shoes would be pulled off the deceased. The dead would be loaded, sometimes wrapped in a piece of paper, onto a lorry and driven away, where to I could not find out.23 From the police [station] he came to the prison at Karlsplatz, where, on the night from the 7th to 8th of May, we were together in a cell intended for 20 people but which already then had a gang of 36 (later over 60). I’d like particularly to emphasise that of none of us and naturally too not of Gentzen was any fact known other than the one, that we were German. No one concerned himself about whether anyone had been a party member or not; in Prague practically every German would be interned, one of the very few who was spared from that was Prof. Kowalewski, who lives now in Gr¨ afelfing by Munich, Akilindastraße 37. The circumstances in the cell were awful. 3 weeks we came not once into the courtyard for exercise, which is prescribed for prisoners. Daily rations were: 120 grams of bread, morning and evening, 1/4 litre coffee, at noon a very thin water soup without content. People went downhill rapidly. There was no soap and no washcloths; in exchange, however, lice and bugs in overwhelming number. Infections were the order of the day. Clothing was first washed after 7 weeks, shaving at first only every 14 days, later every 8 to 10 days. Later we had to go to work; I myself once paved one of the main streets of Prague with Gentzen. That was always very dangerous, because of the crowds who stood around, mainly the women; many were struck, often even struck down. We also presented a wretched picture; without shoes and socks, dressed only in trousers, we would be driven to work through Prague by heavily armed guards or partisans. On one such occasion a fanatic Czech woman threw a paving stone at Gentzen, which cut through two tendons in both last fingers of his right hand. For a long time he could no longer go to work and this again had catastrophic consequences, for occasionally by the work—by far not always—one obtained a piece of bread.24 After that the atmosphere outside the prison changed in a life-saving manner: We no longer saw slavering mobs; again we were taken off in our shoes and clothing by “mature men” who only roared their “shake those bones, march, march” in sight of the prison, but—cautiously at the boundaries—appeared to be somewhat blind. In the first days already I could slip a letter for my sister to an old woman who put the correct postage on it and mailed it off; the letter arrived. Gentzen also told me the address of an old professor of mathematics at the Czech University, which had been closed after the Heydrich assassination; I had already noted in the Institute time that between the two groups of mathematicians there was a warm relationship based on mutual high esteem. I also smuggled out a letter with the name of G. Gentzen and the cell number. The old man had no chance against the brutal machinery of destruction. I described to Gentzen again and again the life-saving turn in the work detail; too deeply shocked, bodily and spiritually weakened, he could not decide to join. Because I knew that the 250 grams of bran bread that would be brought to the cell every morning as our daily ration had a lot of protein, I took only a chunk of my ration and turned the larger remainder over to G.25 As the hunger tormented Gentzen so much, he attempted once more to go to work. That time I was in the same party. We had to beat rugs at the supreme 23 Krammer

to Pinl, op. cit. to Bernays, op. cit. 25 Krammer to Gentzen, op. cit. 24 Kraus

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administrative court without break from 9 to 1 o’clock, in the blazing sun, kneeling on the ground. As the sun moved farther along and we came into the shadow, the rugs would be pulled again into the sun, so that we again came into the sun. At the same time the guards, armed with submachine guns beat time on our backs. In carrying back the heavy rugs, Gentzen collapsed and was therefore terrifically beaten with rubber truncheons. I thought then it was the end of him! After that he no longer went out to work. He remained permanently in the cell and I see him still lying on his wooden plank (we had neither sacks of straw nor blankets) and thinking the entire day on the problems occupying him. He expressed to me once that he was really entirely contented, because he now finally had the time to think about the proof of the consistency of analysis. He also had the firm conviction that he would succeed in carrying it through. But he occupied himself with other questions, like that of an artificial language, and so on. Every so often he gave a little lecture in the cell; we did that frequently to take our minds off things. We had also other university professors in the cell, Prof. Denk (plant physiologist), Prof. Lorbeer (geneticist), these too were starved and at length died. Gentzen was firmly convinced we would soon be returned to freedom, the more so, as by the interrogations which were held with us, the complete harmlessness of the prisoners was determined. We would always be assured that the formalities of the release could only last a few days. Gentzen hoped to return to G¨ ottingen and there to be able to devote himself entirely to his studies in mathematical logic, foundational research, etc. He dreamed of an institute for these goals, perhaps together with H. Scholz. He asked me, if I should be released earlier than he, to report to his friends and teachers on his fate.26 A few days ago I received my first direct letter from Prague, from which I take for you the following lines (from an in the meantime released prisoner whose name I ask to be allowed to conceal)27 : “Unfortunately I must inform you that Mr. Gentzen died in the Charles Prison, exactly how is unknown to me (I was already away). I ask you to notify Professor Scholz, as he asked me to report to him on his fate. Mr. Gentzen had big plans for the future, worked a lot in his cell and was, as he informed me, really convinced he was to find the proof for the consistency of analysis, or, rather, was already on the path to it. Besides that he had the aim of establishing a standard language [Einheitssprache] and in this way to contribute to international understanding. He had the idea to build together with Prof. Scholz an Institute for Logistic and to consult Weyl concerning means.” So much from my source. He writes besides that he was together with Gentzen and some other young mathematicians of my Institute, as well as two to five botanists for 10 weeks in a cell of the prison on Karlsplatz, till he was taken out on account of intestinal illness.28

Dr. habil. Fritz Kraus entered a “communist work camp” as camp physician on 10 July. In the first days of August on my return from the work detail I found Gentzen already in agony; I bedded his head in my lap; thus he died, almost unconscious and peacefully after a short time.29 26 Kraus

to Bernays, op. cit. wrote to Prof. Dr. Thiel on 18 Novermber 1983: “I no longer have any reservations about naming the source who sent me the news of Gentzen’s death. It is probably identical too with that cited by Herr Szabo. It was Dr. med. Dr. rer. Nat. habil. Fritz Kraus, born 1903 in Toeplitz-Sch¨ onau.” (Information from Prof. Dr. Thiel to me on 28 December 2001.) 28 Letter from Hans Rohrbach in G¨ ottingen to Heinrich Scholz of 3 January 1946. 29 Krammer to Gentzen, op. cit. 27 Rohrbach

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Of the gardener Horner named in this report I know only that one morning, certainly if anything for roll call, Gentzen got down from his plank on the “first floor” (the planks were one above another),30 but was too weak to climb back up again. Horner offered him his “ground floor” bed, but Gentzen could not once raise his leg onto the plank. Shortly after he passed away. The evening before he had still held another lecture.31 In accordance with the familiar routine: The cell boss banged in a fixed way on the cell door; shortly after a guard came with the prisoners’ physician and an odd-jobs-man with a stretcher. The physician ascertained “death through circulatory failure.” As the boss and handyman carried the dead person out, I spun around and cried: So great a man and scientist to be murdered (zavrazdit) in this fashion, that will someday avenge itself. In Czech the passive is frequently expressed through the reflexive. The guard understood: That was avenged and gave me a powerful slap; as he saw my face, he turned away, shrugging his shoulders. I have multiply confirmed information about what happened to the dead: Without any registration the naked corpses were toppled into pits which would be covered up without markings when they were full.32

Since then Franz Krammer has written some affirmative assurances that the lecturer Gerhard Gentzen died in his arms in cell 34 on the ground floor of the district prison on Karlsplatz, Prague, on 4 August 1945 at 6:30 p.m. The prison physician of the “Pankras” determined the cause of death to be “circulatory failure”. On 6 May 1946 Franz Krammer wrote the following description to Melanie Gentzen, which the family found much comfort in: Another danger, the prison boredom, which leads to despair and self-abandonment, we bridged over via lectures from our work areas. Unforgettable to us all is the lecture which your son held on a Sunday afternoon on the infinite extension of the universe. On the next Sunday his soul had already entered into the infinity of the starry sky.33

4. Is Gentzen’s Death Understandable? I have often tried to make Gentzen’s death understandable to me, whilst one holds up to me the events in the erstwhile “Protectorate of Bohemia and Moravia”: the humiliation of the Czech’s through the Munich Accord of 1938, the powerless rage of the unarmed Czechs at the German invasion of 15 March 1939, the German ideas for a “repopulation “ of Bohemia and Moravia, the murder of two-thirds of the Jews living in Czechoslovakia, the snuffing out of the intellectual and economic elite, the razing of Lidice and murder of men and abduction of women to Ravensbr¨ uck after the assassination of Heydrich, reprisals also against nonparticipants, red posters with the names of victims on the advertising pillars and the deportation of teachers of higher education to concentration camps. The understandable Czech desire for expulsion of the Germans after 1942 is a natural consequence—this was prominent also in the Benes-Decree, till today a firm component of the Czech constitution— 30 Gentzen

had an upper berth. to Bernays, op. cit. 32 Krammer to Gentzen, op. cit. 33 The original letter is to be found in the Gentzen family archives and a copy by Dipl.-Ing. G. Gentzen. 31 Kraus

5. RUMOURS

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which after 1946 would also be continued officially by the Czech legislative organs, and the Allies also gave their approval. I have nothing to add, but wish to know how this obvious, general rage which expressed itself in murder and plundering, the expulsion of innocent women and children, also in the “Br¨ unner Death March”—the really guilty Nazis, the evil men and their collaborators had long since disappeared—could concretely be directed at Gentzen. I think that the causality of such things must be proven—and certainly not in abstract historical terms, but rather in the verification of concrete, nameable culprits and victims on both sides, before one is lost in endless speculation. The woman who threw the stone at Gentzen, which crushed his fingers, the watchmen who beat Gentzen and let him beat rugs in the heat—were they all guided by the general rage which is supposed to have kindled itself in them? That is too abstractly thought for me, and I suspect that without having these things to analyse correctly, we can merely charge it to account or at least solicit understanding. And “to understand” does not mean “to condone”. On the psychology of the “Revolution” that broke out on 5 May 1945, one finds much illumination in the novel of Josef Skvorecky, Feiglinge [Cowards] (Eichborn Verlag, Frankfurt am Main, 1993, original Czech edition 1988 in Prague). He shows for example on page 50 that the Benes ideas of unrestrained revenge and a total expulsion of the Germans was popular among the Czechs. 5. Rumours Before his death he still declared that he would succeed completely to secure the consistency of the smallest infinite.34

Max Pinl expanded this citation: To the end he hoped, after the successful proof of the consistency of number theory, also to be able to prove the consistency of analysis and dreamed of the founding of an Institute for Mathematical Logic and Foundational Research.35

Without citing his sources, Manfred E. Szabo stated this in sharper form in the Dictionary of Scientific Biography: Only a few days before his death, Gentzen had in fact announced the feasibility of a consistency proof for classical analysis as a whole.

Thus would the unpublished nachlass be mythically super-elevated. It is, however, not at all certain that Gentzen left behind anything exact on a consistency proof for analysis either orally or in writing. Possibly he was merely knocking an idea for the solution around in his head. Could he have been in the cell too long to be capable of such a proof? The source of Kowalewski and Szabo appears to be the letter from Dr. habil. Dr. Fritz Kraus to Bernays of 12 April 1948. But, in his letter to Melanie Gentzen of 22 May 1946, Krammer reported from small mathematical discussions which Gentzen held on the subject that he could barely do any scientific work, but wanted to inform Rohrbach on that. He mentioned the use of a strictly forbidden pencil stub with which they made notes on scraps of paper until the controls became tighter and more brutal, and everything 34 Gerhard

Kowalewski, Bestand und Wandel. Meine Lebenserinnerungen zugleich ein Beitrag zur neueren Gechichte der Mathematik, R. Oldenbourg, Munich, 1950, p. 108. 35 Max Pinl, “Kollegen in einer dunklen Zeit”, Jahresberichte der Deutschen MathematikerVereinigung 75 (1974), pp. 166-208; here, p. 133. How did Pinl know this? He was incarcerated together with Gentzen in prison, but had been released before Gentzen’s death.

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was destroyed at the wishes of the “Cell Commandant” Prof. Dr. Denk, as he would be liable for it.36 But of a consistency proof or any larger work in general on which Gentzen could have worked in the cell, he knew nothing whatsoever.37 Instead he reports that after his death and the parting words of Viktor Denk all the cellmates prayed an “Our Father”.

I can however imagine that from this mix of information, speculation and feelings the impression could easily arise that something wonderful, significant, important, urgent and essential for mathematical foundational research would have been left behind by Gentzen. And the idea of a “black bag” which—as by a magician—hid mysteries is something irresistible for curious and credulous people who wish to make such discoveries. But what if the bag had only excerpts? What if only papers for his seminar were collected therein or things in which he would have to work in again as a lecturer? It is pointless to speculate on this. Kurt Sch¨ utte reported that Gentzen was killed in prison after “both American and also Russian mathematicians [had] endeavoured in vain for his release.”38 He had heard this from Paul Bernays, but this fact is as little verified as the lecture invitations.39 Georg Kreisel wrote: I never met Gentzen, but I have talked to his fellow logicians Bernays and Sch¨ utte, who were unsympathetic to his politics, and his friend Witt, who was not. From all I heard I get the impression that Gentzen lived within his moral and emotional means and never harmed a fly.40

Kreisel cites two things: According to Bernays, Gentzen is supposed to have dismissed his reports on rioting against Jewish colleagues with the words, “The government wouldn’t allow such a thing.”41 The second bit of news coming from Kreisel is about Ernst Witt:

36 The

original letter is to be found in the family archive of Waltraut Student. from Dipl.-Ing. G. Gentzen of 12 February 1986. That should mean nothing, because Dr. Krammer had been working, thus was away the entire day. But wouldn’t it have gotten about in the cell? 38 Kurt Sch¨ utte and Helmut Schwichtenberg, “Mathematische Logik”, in: Gerd Fischer et al. (eds.), Ein Jahrhundert Mathematik 1890-1990. Festschrift zum Jubil¨ aum der DMV, Vieweg Verlag, Wiesbaden, 1990; here, p. 725. 39 “My assertion that both American and Russian mathematicians had worked for the release of Gentzen, I certainly cannot verify. I had only heard statements of that sort from Bernays and Arnold Schmidt only a few years after the end of the war.” (Kurt Sch¨ utte to the author in a letter of 30 November 1990.) I consider it possible that Gentzen had contact with A.A. Markov and A.N. Kolmogorov. The documents of A.A. Markov are to be found in stock 193 of the Leningrad division of the Archives of the Academy of Sciences of the USSR. The documents of A.N. Kolmogorov are administered by Professor Dr. Albert Nikolaevich Shiryaev of the V.A. Steklov Mathematical Institute, Scientific Commission A.N. Kolmogorov, Uliza Vavilova 42, 117966 GSP-I Moscow V333. I could not see any documents of either. 40 Georg Kreisel, “Review of M.E. Szabo, The Collected Papers of Gerhard Gentzen”, Journal of Philosophy 68 (1971), pp. 238-265; here, p. 256. Kurt Sch¨ utte saw Gentzen just one time, and Sch¨ utte listened to him. Neither spoke of politics. 41 Letter from Kreisel in Oxford to me of 4 May 1988. There is no other verification for this alleged uttering of Gentzen. Nothing similar is in Bernays’s Nachlass. 37 Information

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Also the mathematician Witt (in Hamburg) has told me himself with great enthusiasm of the war games in the National Socialist Student League, which was the cause of so much fun for both him and also Gentzen.42

Witt also liked to report openly about his past at the Institute in Princeton.43 Gentzen’s conduct with respect to Bernays would be dismissed by those born later or those who did not know him either as “pure hypocrisy” or “unworldliness”. I believe that with Gentzen a similar reaction could develop out of anxiety, hidden foreboding and panic-stricken fear, that it could be the case as others have long conjectured it. But today who is willing to testify to it all, to interpret from the few life dates and eyewitness accounts, or to clarify it definitely?44 In every case it is agreed that Gentzen certainly was in the Party and the SA, but made no use of that. He had never harmed a fly. That we know. That is verifiable. That Gentzen had gone on a hunger strike on account of the unreasonable treatment as Professor Rohrbach relates with the statement of Dr. Fritz Kraus to Gentzen’s mother, I could not verify. And whether Gentzen had attempted to get something into the Swiss newspapers on the situation of German mathematics or had tried to help Heinrich Scholz and his Swiss relations to bring L  ukasiewicz to the West must remain as rumours so long as there is no evidence for them. We want to consider here only those occurrences which have at least two independent witnesses. 6. Attempts to Rescue the Nachlass Rohrbach wrote on 3 January 1946 to Heinrich Scholz: It must be established whether Gentzen had still made written notes. Perhaps it is possible for you through your Polish friends to gain influence on the Czech mathematicians, that they search for anything left behind by Gentzen.45

Heinrich Scholz answered: I have, to be sure, always hoped for such a turn of events, because this unusually solitary man had since 1939 felt curiously sheltered in our company and especially in the company of my little lady. But that he would put the idea of this collaboration into action in such a definite fashion, this I have first learned through you. His death left its mark on me, which will not vanish. And now to his nachlass. . . I ask you that you place the G¨ ottingen papers into our hands. We would preserve them properly and work on them. . . Much more difficult will 42 Letter to me from Kreisel in Oxford of 6 June 1988. Later in Princeton Witt is supposed to have shown around his driver’s license, which was authorised with a swastika. This display is supposed to have led to misunderstandings. 43 Ina Kersten, who looks after Witt’s nachlass, found nothing therein pertaining to Gentzen. Ernst Witt never mentioned Gentzen’s name to her. “Witt joined the Nazi Party and the SA in Spring 1933, but his enthusiasm for this was quickly left behind, and he refused to join the NS Student League and later the Dozentenbund. . . Later he generally spoke openly about his party affiliation.” (Letter to me of 4 February 1997.) Frau Prof. Dr. Ina Kersten and Frau Witt are preparing a detailed curriculum vitae with numerous documents. Cf. Ina Kersten, “Ernst Witt”, Jahresberichte der Deutschen Mathematiker-Vereinigung 95 (1993), pp. 166-180. In 1941 Ernst Witt was called up for military service, came back to Germany in 1942 for health reasons and worked until the end of the war in decipherment service. Did he encounter Hans Rohrbach there? Who in the foreign office verifies that? 44 Ernst Witt, Hans Hermes, and many others had nothing to say to me about Gentzen, or were silent. 45 The original letter lies in the Institute for Mathematical Logic and Foundational Research in M¨ unster. I thank J. Diller for a copy.

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be the rescue of the Prague papers. I myself see no way at the present, as my Polish friend finds himself, in a for us inaccessible way, in France for an indefinite period.

On 17 June Heinrich Scholz reported to Gentzen’s mother that during Pentecost he had met Max Pinl, a lecturer in mathematics at the university in K¨ oln who had “existed in the same prison space until 14 days before his death” with her son: He told me that your son up to his separation from him had been of an exceptional mental activity and occupied himself without break with the consistency of analysis.46

After his studies at the University in Prague, Ladislav Svante Rieger (19161963) worked as a statistical computer in the national bank and from 1943 on in a construction office in the aircraft industry. In 1945 he became assistant at the Technical High School in Prague. He is supposed to have fought for Gentzen’s cause. Here too there are no documents or witnesses known to me. On 19 September 1947 Scholz wrote to Mostowski that Bernays had visited M¨ unster.47 It can be considered certain that Bernays likewise attempted to rescue Gentzen’s papers. Max Pinl wrote to the head of the government planning and building control office, K¨ uhnel, in M¨ unster in an undated letter: With much effort I have been able to establish that Mr. Gentzen had left behind a bag with valuable manuscripts in the then mathematical department, and that Prof. Dr. Hlavaty, Praha XIII, Mexicka 2 CSR, is supposedly in a situation to say or to establish what has become of this bag.48

Prof. Dr. H. Scholz wrote to Prof. Dr. Hlavaty on 23 December 1947: Through Docent Dr. Pinl, who is personally familiar to you and to whom I owe also your name and address,

and requested from him clarification concerning Gentzen’s nachlass. Professor Dr. Vaclav Hlavaty answered on 9 Jan 1948 from “Seminar pro geometria” in Praha II, U. Karlova 3: Very respected Herr Colleague, Immediately after receiving your letter of 23/XII I searched everywhere to find a trace of the manuscripts of Mr. Gentzen. Unfortunately I must inform you that my efforts were without success. If these manuscripts were in the former German math department, there is practically no longer any hope at all of finding them, because we have taken over the empty rooms. . . The news of the demise of Mr. Gentzen I learned of in the USA and must admit that it hurt me very much even though I did not know him personally. You will understand if I inform you that during the occupation 70 Czech high school teachers and some of my very good friends from the German university were either shot or died in concentration camps. . . Please give my greetings to my best friend Pinl. He was the only one of the German colleagueswho, during the occupation, found the way to us. . . I am sorry 46 Cf.

footnote 45. ¨ Schreiber, “Uber Beziehungen zwischen Heinrich Scholz und polnischen Logikern”, pp. 343-359, in: Michael Toepell (ed.), Mathematik im Wandel, Franzbecker Verlag, Hildesheim, 1988. 48 See footnote 45. 47 Peter

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that I could not give you the desired information although I had the best intentions. With particularly deep respect, Your very humble servant, V. Hlavaty49

On 9 March 1948 Arnold Schmidt wrote to Melanie Gentzen in Rottweil that he had discovered in an unobserved cabinet of the G¨ ottingen Mathematical Institute a series of stenographic notes of Gentzen himself, some mementos, more stenographic manuscripts of already published works, and a cardboard box with offprints from colleagues; and he requested the last as his entire library had been lost. In addition, he had an obituary to write—probably at the prompting of Rohrbach. The rest I cannot verify and am informed by accounts of Frau Waltraut Student: Hellmuth Kneser and Hans Rohrbach saw each other in September 1946 at the mathematicians’ conference in T¨ ubingen. Kneser had received from Arnold Schmidt stenographic notes of Gentzen’s, which he now returned. Both got down to the decipherment.50 Paul Lorenzen of the Mathematical Institute of the University of Bonn was appointed by Wilhelm S¨ uss—why exactly him?—to draw up a report for the Americans on the situation of mathematical foundational research from 1939 to 1946. In 1948 there appeared Naturforschung und Medizin in Deutschland 1939-1946. F¨ ur Deutschland bestimmte Ausgabe der FIAT Review of German Science. Band 1: Reine Mathematik, Teil 1. Herausgegeben von Wilhelm S¨ uss. Mathematisches Forschungsinstitut Oberwolfach/Baden 51 [Natural science and medicine in Germany 1939-1946. Edition intended for Germany of the FIAT Review of German Science. vol. 1: pure mathematics, part 1. Edited by Wilhelm S¨ uss of the Mathematical Research Institute Oberwolfach/Baden]. Helmut Hasse and Hellmuth Kneser are also among the authors. Paul Lorenzen reported on “Grundlagen der Mathematik” [Foundations of mathematics] (pp. 11-22). Oskar Becker is mentioned in an excessive manner, then “Klassische Logistik” [Classical logistic] in the spirit of Frege is reviewed along with the results of Tarski, Schr¨ oter, Hermes and Scholz. In “C. Finite Logistik” [C. Finite logistic], one reads: In his final treatment [“Beweisbarkeit und Unbeweisbarkeit von Anfangsf¨ allen der transfiniten Induktion”—EMT], Gentzen sharpened the method of his consistency proofs of arithmetic, in order to show directly the underivability of transfinite induction up to ε0 in arithmetic (which follows from the consistency proof only with the aid of G¨ odel’s Incompleteness Theorem). Transfinite induction up to any ordinal permits a natural formalisation within arithmetic. For the formalisation, the author adjoins to the natural numbers the ordinal numbers up to ε0 as terms. Further, a class variable E is added.

49 Ibid.

50 A new and readable copy of this decipherment was carried out by Prof. Dr. Manfred E. Szabo. On 12 January 1990 Hans Rohrbach wrote to me that there was only one steno-manuscript of Gentzen provided by A. Schmidt. “It would be deciphered at the instigation of Prof. H. Kneser in T¨ ubingen.” The widow of A. Schmidt told me that she had burnt the nachlass of him and that no paper of Gentzen survived. 51 Dieterich’sche Verlagsbuchhandlung (W. Klemm), Wiesbaden, 1948.

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There follow 17 further technical lines before Lorenzen presents in detail W. Ackermann’s “Widerspruchsfreiheit der Zahlentheorie” of 1940 and his “System der typenfreie Logik” of 1941.52 7. The Deciphering of the Stenographic Notes More than 40 years later, Hans Rohrbach, who had presented something on theology on the radio, received news from Waltraut Student, who had heard his broadcast, that Gentzen had sent a bundle of papers from Prague to Sigmaringen. These were either from 1943 in Liegnitz or the summer of 1944, when Gentzen is supposed to have worked on a generalisation of his consistency proof. The work is written on loose sheets, Flugwacht note paper, and such like. This news was further related to Christian Thiel. Frau Student personally handed two folders, one blue and one violet, over to Christian Thiel on 18 May 1984 during a visit to Rottweil so that Thiel and Szabo could look them over and determine their publishability. Because of the difficulty of transcription, Szabo soon withdrew from the joint work. (Report from Prof. Dr. Thiel of 28 December 2001.) Thiel has been laboriously deciphering the bundle since 1984 and will continue to do so.53 The bundle, including the main part of the fragment of the proof theory book, consists of two parts: AL reads “Aussagenlogik” [propositional logic], BG “Buch u ¨ ber Grundlagenforschung” [Book on foundational research], and these parts are—above all the BG-parts and the pieces of paper—mostly still concept notes for the planned book. The AL-part contains the beginnings of an attempt to find a semantics comparable to the classical for the positive (= negation-free) propositional logic; the syntactic side has been solved since Hilbert-Bernays and Gentzen.54

52 Paul Lorenzen reserves the last three pages for himself. His review does not give a good account of the actual situation, but it probably improved Lorenzen’s place in the new postwar mathematical world. 53 Chr. Thiel, “Research on the history of logic in Erlangen”, in: Ignacio Angelelli and Mar´ ıa Cerezo (eds.), Studies in the History of Logic. Proceedings of the 3rd Symposium on the History of Logic, DeGruyter, Berlin, 1996. 54 Christain Thiel on 16 April 1988. Gentzen had actually sketched a “handy decision procedure”. It is a semantic method, which is simpler than the so-called Kripke models.

https://doi.org/10.1090/hmath/033/07

Conclusion 1. Misapprehensions about the Life of Gerhard Gentzen Even today there remain peculiar ideas about Gentzen’s life. That Gentzen didn’t become a professor under the Nazis was not due to the Nazis nor even the opposition of the Nazis to logic. It was simply a result of the war that Hitler had set in motion. But this shouldn’t distract us from our topic. His illness after the beginning of the war not only made possible his discharge from military service but also gave him the opportunity to take up a docent position at the University of Prague.1

Seen against the background of comparable fates this sounds ideal. And the logician Wilhelm Essler continues: During the Nazi rule the mathematical logicians like David Hilbert, Wilhelm Ackermann and Gerhard Gentzen could pursue their research relatively undisturbed.2

I have been glad to relate just what this “relatively undisturbed” really looked like and, in so doing, to correct outmoded points of view. However, in comparison to Leopold L¨ owenheim and Kurt Grelling, Gentzen could actually research almost undisturbed. 2. Logic and Politics According to Essler it was supposed to have been such that the Nazis left the mathematical logicians in peace, because they had not drawn any political conclusion from mathematical logic. And one cannot come to any political conclusions from mathematical logic just as little as one could construct a logic out of the “crystal clear”, “compelling”, “plausible”, “mathematically clear” statements of Carl Schmitt. The Nazis knew this also. Only the private people like Dingler, Steck and May wanted to get dangerous and spoke of a race-based foundation of mathematics. It is a question of distinguishing the task of mathematical logic from that of everyday logic or philosophical logic and not burdening it with unrealistic expectations. Mathematical logic should be promoted as a means of reaching “better” socio-political goals and is just as sensible to learn as Latin in order to think more “logically”.3 1 Max Pinl, “Kollegen in einer dunklen Zeit”, Jahresberichte der deutschen MathematikerVereinigung 75 (1974), pp. 166-208. 2 W.K. Essler, “Die Entwicklung der Logik in der Bundesrepublik nach dem 2. Weltkrieg”, Ruch Filozoficzny 33, Nos. 3-4 (1974), pp. 255-264. 3 On the subject that the Latin language be learned to master logic because it suits more than any other (language) the logic of judgement, Hartmut von Hentig and Otto Seel expressed irritation already in the 1960’s (cf. Otto Seel, “Die lateinische Sprache”, pp. 416-585, in: Seel, R¨ omertum

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There are personal historical explanations for exaggerated expectations and hopes of individual professional logicians for the area of logic. Alfred Tarski (1901-1983) fled from Poland to the United States, forced to leave his wife and children behind. In September 1940 he wrote and sent to Harvard University his new preface to his Introduction to Logic and to the Methodology of Deductive Sciences (1941): The course of historical events has assembled in this country (the United States) the most eminent representatives of contemporary logic, and has thus created here especially favorable conditions for the development of logical thought. These favorable conditions can, of course, be easily overbalanced by other and more powerful factors. It is obvious that the future of logic, as well as of all theoretical science, depends essentially upon normalizing the political and social relations of mankind, and thus upon a factor which is beyond the control of professional scholars. I have no illusions that the development of logical thought, in particular, will have a very essential effect upon the process of the normalization of human relationships, but I do believe that the wider diffusion of the knowledge of logic may contribute positively to the acceleration of this process. For, on the one hand, by making the meaning of concepts precise and uniform in its own field and by stressing the necessity of such precision and uniformization in any other domain, logic leads to the possibility of better understanding among those who have the will for it. And, on the other hand, by perfecting and sharpening the tools of thought, it makes men more critical—and thus makes less likely their being misled by all the pseudo-reasonings to which they are in various parts of the world incessantly exposed today.4 und Latinit¨ at, Klett, Stuttgart, 1964). But that Latin compensates with Kantian depth, lacony, lapidarity and heavy pregnance for what it lacks in dazzle, elegance and formal method was already clear to many before, even if students like me learned Latin sooner as a quibbling, unfriendly, pseudogeometric school grammar, which could very well have been the fault of the teachers and their misunderstood pedagogy. Or did they mean by logic the cleverness, rigor, coldness, discipline, clarity, economy of thought, unshakeability, reverence for imperial representation that was taught in the German fatherland’s Latin philosophy since 1850? 4 The relationship between logic and mathematics was already sufficiently complicated with Hilbert. “It’s just one more unfounded prejudice to only regard mathematics as a means of developing logical thought. Despite the fact the criticisms of the logical thinking by mathematicians are often just as frequent as those from others—even Euclid’s books are not completely devoid of them—logical thinking occurs in the sense meant here just as automatically as does the use of one’s mother tongue and I’m not sure whether because of this mathematics lessons would be needed. No, as the mathematician Voss once put it, the strength of mathematics as a means of education leans primarily in an ethical direction and towards developing a free, creative understanding. In areas of history, specifically through the study of foreign languages, knowledge is developed. But such knowledge is not cognition. However it is facilitated by mathematics. He who understands the proof of a theorem has gained the conviction of having grasped a truth on the basis of one’s own work. This not only awakens more secure consciousness, that one can find truths through thought, but also self-confidence in one’s own mind, the power of critical judgement that distinguishes the truly educated from those who are captives of simple belief in authority. Self-confidence in one’s own ability, critical regard, the energy in overcoming difficulties that at first appear impossible to overcome, persistently toward the goal of a focused will, these are ethical powers and qualities, for arousing which there is no better means than working with mathematics. . . A very great such thinker, Georg Cantor, the founder of set theory, who was perhaps the most original mathematician who has lived, found the pregnant expression: The essence of mathematics lies in its freedom.” (Page 3 in: David Hilbert, “Wissen und mathematisches Denken. Vorlesung von Prof. D. Hilbert”, W.S. 1922/23; worked out by W. Ackermann and published by C.-F. B¨ odigheimer, Mathematical Institute of the University of G¨ ottingen, 1988). However: “What exactly does the success of mathematics rest upon? The answer sounds like: The essence of the mathematical method lies in the consistent formal development of the procedures that formal thought possesses, hence logical thought generally. Thought has at its disposal an unlimited supply of formal relations

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Even for men of good will none of these hopes were fullfilled. For Tarski this statement was also an appeal to the humanity of the Germans in Nazi Poland to allow his family to live. We will not however achieve a better life through logic. 3. Upshot I wasn’t able to say what Gerhard Gentzen’s goal in life was. One can gather a few facts toward doing so: We can surely read out of his affection and love for Hertha Michælis the need to love and be loved, i.e., entirely private happiness. While solving problems in mathematical logic it was important for him to be successful within mathematics, even more so if this succeeded professionally, and he was always pleased when this happened. Gentzen wrote at the latest in 1936 a letter on March 3 to Paul Bernays that he had a program. And this program was decisive for the post-G¨ odel period of mathematical logic. There were three essential levels in Gentzen’s ideas: • His conviction that the consistency of arithmetic could be proven con¨ das Verh¨ altnis zwischen structively can be seen from his paper “‘Uber intuitionistischer und klassischer Arithmetik”. • He proved this consistency in 1935 in his “Widerspruchsfreiheit der Zahlentheorie”. • And he proved, in his 1939 habilitation thesis, “Beweisbarkeit und Unbeweisbarkeit von Anfangsf¨ allen der transfiniten Induktion in der reinen Zahlentheorie”, in a direct way the unprovability of ε0 -induction with Peano arithmetic. And he achieved all of this as a consequence of the 1932/33 calculus constructed for this purpose. and it is a question of finding systems of formal relations that can be applied to the previously discovered relations in reality. . . In this way mathematics has two tasks: On the one hand it is a question of developing systems of relations and investigating their logical consequences, just as it is done in pure mathematics. This is the progressive task of mathematics. On the other hand it is a question of providing a more firm structure and the simplest possible foundation for theories based on experience. For this it is necessary to clearly work out the prerequisites and to everywhere distinguish between assumption and logical consequence. . . This is the regressive task of mathematics.” Unfortunately one comes within the realm of calculation with infinitesimals, where there rules an exact framework of concepts, to questionable results; however there are far more questionable ones in set theory. “Here can be seen particularly clearly how strongly the popular opinion of mathematical reasoning diverges from the real circumstances: The methods of mathematical reasoning are so little obvious that its exact foundation still remains as a difficult task. There doesn’t seem to be any prospect of deriving mathematical reasoning completely from the usual logic. One will more likely have to produce a system of axioms and rules, from which on the one hand all of mathematics can be derived, without doing unnecessary violence to standard mathematical reasoning, and from which on the other hand there is agreement that they never lead to contradictions. Attempts to do just this have already been made in mathematical logic: still the greatest part of that effort remains.” (Pp. 22ff. in David Hilbert, “Natur und mathematisches Erkennen. Vorlesung von Professor Hilbert. Ausgearbeitet von P. Bernays”, Fall Semester 1919, published by J¨ org Br¨ udern, Mathematical Institute of the University of G¨ ottingen, 1989.)— Those who oppose all connection between logic and politics often use of all things the history of logic as their ideological playing field. Thus for many years it was thought that logic without question worked against totalitarian ideologies and social systems. Thus one preferred to meet with logicians from the USSR and East Germany, Peru, and other liberal states.

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CONCLUSION

Unlucky in the sense of his professional career, he didn’t have a constant scientific or university life. The trappings of life were not important for him. He preferred to let himself be promoted, referred to and mentioned by others. Were it not for the war, he would by all accounts surely have reached a full professor’s position. It would have been a successful life. His was, in the sense of everyday life, almost a childish attitude, with games all the way up to the last days: Ping Pong, proving equivalences, playing cards as a hobby, like a “normal youth”. As far as things that had nothing to do with mathematics went, he was spiritually unambitious, without music, fine literature or philosophy. Materially he was used to spartan conditions and could live completely ascetically. These protestant personal conditions produced a great deal of “culture”, i.e., education. (“He who has not eaten his bread with tears . . . ”—Goethe). He responded to necessary scholarship per expectation, i.e., with consideration for mother and sister, with the highest level of performance. Seen in his rˆ ole as top pupil in his class and the way he assumed it was natural and accepted by his classmates. Through the end of his life he had a small circle of acquaintances and friends around him. There was no place in his world for happiness and expectations of “Better”: the infinity of the heavens, wherein Giordano Bruno and moreover a member of his own generation, Wernher von Braun similarly, promised innovation, expansion of world view, conquest of the unknown much more than metaphysical consolation. He didn’t seem to be receptive to religious things. He liked his work and his family haven. I know nothing of his social or political opinions, but they didn’t scare either Bernays or Cavaill`es away. He dealt with political situations as he could without enthusiasm. External catastrophes and crises, above all of course the war, made him so nervous that he became unfocussed and sick to the point of depression. There was no question of pleasures or vacations. On the contrary, it was all about mathematics, creation and convalescence. He was quiet, friendly, helpful and amiable. He had good manners, but was by no means eloquent or meant for parlour gatherings. But he could be condescending as soon as he had the feeling that someone had not thought through a problem sufficiently. For this reason he accepted editing work. One never heard any talk of his participating in associations. In mathematics he was respectful of authority, accepted advice but on the other hand tested his results and methods against others’ understanding; in this he was no egoist. He let his writings stand for themselves, as well as preprints and invited lectures, but little is known of other written contact or disputes. As an assistant at the university without habilitation this would not have been prudent. The “scientifically oriented person” is not vain, but rather modest, though self-aware. He knows of what he is capable. In this sense he can be stubborn and introverted if he is cobbling out a mathematical formula. His quiet thought without pen— to grasp it, think it through, and retain it—produced a lack of awareness of the present, in other words a little absent-mindedness. In his mathematical work he was free of any idealogical agenda, modestly preferring what for him was mathematically feasible to the discussion of “greater”

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foundational ideas. Gentzen appeared to be the ideal scientist, as Max Weber has portrayed him in his work Wissenschaft als Beruf. He was not conceited relative to position or education and interacted with those below him and colleagues with expected subservience. He could easily be sociable with up to three other people. There was never talk of tensions, but also never of resistance, protest or obstructiveness. His external life, lived in inhospitable times, appeared boring, but was just as much simple, plain and straight as virtuous. Does a thankful life exist?5 Life becomes exciting however if one takes up and investigates with the scientific methods the accessible, measured and adequate concepts and questions of this life, thus life wisdom coupled with analysis and rationale. But this is a completely different book. Here, where my book stops, it becomes once again exciting. However, this is a different story that I will tell another time.

5 Why is there frequently the biographer in the foreground? To make apparent that every biography is a mild form of fiction that is based on lack of experience in view of one who had not lived it because it was another’s life. I can only portray the life of Gentzen from a factual point of view because I’m a different person than he was. He was for me a pleasant fellow and I like him, but that an ingenious researcher and a dim biographer found each other was just as blind and accidental as fate can be. So the biographer has to explain where he’s coming from and what he wants so that it is clear from the kind of storytelling what could have been otherwise. Frege demanded that one inserted a “probably” or “possibly” if one wasn’t sure about the facts and the claimed statements. He was right: Never try to give the impression a biography makes a claim to truth! Try to write a Life of Gentzen; since Vasari’s The Lives of the Artists, I regard this as nonsense (cf. Ernst Kris and Otto Kurz, Die Legende vom K¨ unstler. Ein geschichtlicher Versuch, Suhrkamp Verlag, Frankfurt am Main, 1980). Sigmund Freud wrote on 31 May 1936 to Arnold Zweig: “He who becomes a biographer commits himself to lies, to secrecy, distortion, euphemism and even to the worship of his own lack of understanding; for there is no biographical truth and, even if there were, it would be useless.” Accordingly I believe that the distortion of a biography through the biographer through the disclosure of his interests becomes clearly recognizable, which is why there is no place for a castrated “objectivity”. On the other hand I think that the biographical truth is very much a help in understanding mathematical facts. For example, biography portrays the factual discussion well and chronologically correctly, and here what is needed is a truly factually embarrassing exactness without interpretation.

https://doi.org/10.1090/hmath/033/08

Tables of the Life of Gerhard Gentzen Chronology Childhood and Youth 1909 Gerhard Gentzen born on 24 November 1909 in Greifswald. 1910 Baptism on 28 March 1910. 1914 Death of Adele Bilharz, his maternal grandmother, on 14 October 1914. 1916 Enters the Volksschule Bergen at Easter after a year of private instruction by his mother, Melanie Gentzen. 1918 Enters the Septima of the Bergen Realschule at Easter. 1919 Death of his father, Hans Gentzen, in Bergen on 11 March 1919. Death of his grandfather, Wilhelm Gentzen, in Stralsund on 2 December 1919. 1920 Resettlement in Stralsund by his grandmother, Alwine. 1921 Beginning of his verifiable occupation with astronomy and mathematics. 1925 22 pages on “The position of Mars in the solar system and its moons”; poem “On the starry sky” in 9 stanzas. 1926 Gerhard builds a radio receiver from the ground up by himself. 1927 Death of grandmother, Alwine Gentzen, in Stralsund on 2 December 1927. 1928 Abitur with distinction on 29 February 1928. Confirmation on 5 April 1928. His Confirmation motto is: “Be happy in hope, patient in sorrow, constant at prayer.” Romans 12:12. Studies 1928 Beginning of study of mathematics and physics at the University of Greifswald. Financial support of his studies through a stipend from the Studienstiftung des Deutschen Reiches. Friendship with Lothar Collatz. 1929 Continuation of studies in G¨ ottingen on the advice of Prof. Dr. Hellmuth Kneser. Letter of reference to Richard Courant from H. Kneser. 1930 Continuation of studies in Munich under Carath´eodory (function theory), Perron (algebra) and Tietze (topology). Study of Hilbert-Ackermann, Theoretische Logik. 1930/31 Continuation of studies for a semester in Berlin. 273

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TABLES OF THE LIFE OF GERHARD GENTZEN

1931 Continuation of studies in G¨ ottingen. Friendship with Saunders Mac Lane. On advice of Prof. Dr. Paul Bernays, occupies himself with ideas of Paul Hertz. Richard Courant describes Gentzen as a “scientifically oriented man” in the recommendation for a Promotion-Stipend from the Studienstiftung des Deutschen Reiches. 1932 First publication: ¨ 1.) “Uber die Existenz unabh¨ angiger Axiomensysteme zu unendlichen Satzsystemen”, Mathematische Annalen 107 (1932), pp. 329-350. Received on 6 February 1932. Programmatic idea: Concentration on the nature of mathematical inference using only mathematical means and excluding all that is philosophical. ¨ 1933 Gentzen withdraws his manuscript “Uber das Verh¨ altnis zwischen intuitionistischer und klassischer Arithmetik” of 15 March 1933 after he learns that Kurt G¨ odel had obtained the same result shortly before. Dissertation 1933 Promotion with the work “Untersuchungen u ¨ber das logische Schliessen” on 12 July 1933 (day of the oral exam). Submission of the dissertation for publication in Mathematischen Zeitschrift on 21 July 1933. Dissertation published separately 1934. Official publication: 2.) “Untersuchungen u ¨ber das logische Schliessen”, Mathematische Zeitschrift 39 (1934-1935), no. 2, pp. 176-210, and no. 3, pp. 405-431. Entry into the Sturm-Abteilung (SA) on 5 November 1933 at the advice of others. Gentzen passes his state exam on 16 November 1933 with the work “Elektronenbahnen in axialsymmetrischen Feldern unter Anwendung auf kosmische Probleme” and receives the mark “Good”. Research Stipend 1934 Research stipend for “Beweis der Widerspruchsfreiheit der Analysis” from ur die Deutsche Wissenschaft for the period from 1 the Notgemeinschaft f¨ April to 30 September 1934. Official awarding of the doctoral degree 2 October 1934. 1935 Gentzen spied on by the Dozentenschaft because of contacts with Jews. Stipend-Assistant of the Deutschen Forschungsgemeinschaft from 1 July to 31 October 1935. Submission of “Widerspruchsfreiheit der reinen Zahlentheorie” to Mathematischen Annalen on 11 August 1935. On criticism from Prof. Dr. Paul Bernays, begins revision of the work. Unscheduled Assistant by David Hilbert 1935 Unscheduled assistantship (succeeding Arnold Schmidt) by David Hilbert until 1 November 1935. First verifiable meeting with the logician Jean Cavaill`es in December.

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Withdrawal from publication of “Widerspruchsfreiheit der reinen Zahlentheorie” and revision of parts of the proof on 11 December 1936. 1936 Entry into the NS-Lehrerbund on 1 March 1936 with membership number 335,247. Publications: 3.) “Die Widerspruchsfreiheit der Stufenlogik”, Mathematische Zeitschrift 41 (1936), pp. 357-366. 4.) “Die Widerspruchsfreiheit der reinen Zahlentheorie”, Mathematische Annalen 112 (1936), pp. 493-565. 5.) “Der Unendlichkeitsbegriff in der Mathematik. Vortrag, gehalten in M¨ unster am 27. Juni 1936 am Institut von Heinrich Scholz”, in: Semester-Berichte M¨ unster, Winter Semester 1936/37, pp. 65-80. 1937 Entry into the Nazi Party 1 May 1937 with membership number 4,237,555. Gentzen becomes an “Associate” in the publication of Heinrich Scholz’s series Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften. Neue Folge, the first volume of which appeared in 1937. Gentzen named a member of the German contingent to the Parisian Descartes-Congress in August 1937. Publications: Self-reports on publications 3. and 4. in Deutsche Mathematik 2 (1937), p. 130. 6.) “Unendlichkeitsbegriff und Widerspruchsfreiheit der Mathematik”, in: Actualit´es scientifiques et industrielles, no. 535, pp. 201-205, Paris: Hermann, 1937; and in: IXe Congr`es International de Philosophie, VI. logique et math´ematiques, Paris, 1937. Lecture at a meeting of the Deutschen Mathematiker Vereinigung in Bad Kreuznach on 21 September, “Die gegenw¨ artige Lage in der mathematischen Grundlagenforschung”. Extension of the period of service of the unscheduled assistant Gentzen for another year in effect from 1 October by H. Hasse. 1938 Publications: 7.) “Die gegenw¨ artige Lage in der mathematischen Grundlagenforschung”, Deutsche Mathematik 3 (1938), pp. 255-268 (First printing). 8.) “Neue Fassung des Widerspruchsfreiheitsbeweises f¨ ur die reine Zahlentheorie”, Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, vol. 4 (1938), pp. 19-44, Leipzig: Hirzel. 7. also appeared in Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, vol. 4 (1938), financed indirectly by Ludwig Bieberbach. The second edition of Hilbert-Ackermann, Grundz¨ uge der theoretischen Logik, appears with an acknowledgement of Gentzen in the preface. Extension of the assistantship for another year in effect from 1 October 1938. Invitation (together with W. Ackermann) for 18 November 1938 to a seminar in Switzerland from F. Gonseth; probably G. Gentzen and

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TABLES OF THE LIFE OF GERHARD GENTZEN

W. Ackermann did not attend on account of currency difficulties and general restrictions on foreign travel under National Socialism. 1939 Gerhard’s mother moves to live with his sister in Liegnitz. Gentzen critiques Hugo Dingler at the request of Heinrich Scholz (14 March 1939). Conversion of the unscheduled assistantship into a scheduled one on 1 April 1939. Last extant letter to Paul Bernays on 16 June 1939. Call-up to military service on 28 September 1939: radio operator for Flugwachkommando in Brunswick on the “homefront”. Meeting with Kurt G¨ odel in G¨ ottingen on 15 December 1939. 1941 Gentzen joins the NSD-Dozentenbund (1 January 1941 in the self-entry ottingen records the date is 1 January 1939). in the Prague file; in the G¨ Habilitation 1940 Scientific talk for his habilitation thesis submitted in 1939, “Beweisbarkeit und Unbeweisbarkeit von Anfangsf¨ allen der transfiniten Induktion in der reinen Zahlentheorie” in G¨ ottingen on 12 December 1940. Soldier in military service on the homefront (1939-1942). 1942 Admission to the hospital for a “state of nervous exhaustion” on 21 January 1942. Discharge from military service without reserve status (intact) as a war veteran on 8 June 1942. Resumption of his assistantship in G¨ ottingen (on paper) on 15 June 1942. Official demobilisation as “unsuitable for military service” from Wehrbereichs-Kommando on 19 November 1942. Requested by Prof. Dr. Hans Rohrbach for a lectureship in Prague on the Osenberg-Aktion in December 1942. Docent in Prague 1943 Gentzen requests from G¨ ottingen on 30 January 1943 the authority to teach mathematics and to be appointed docent. He requests till the completion of his probationary lecture to be assigned to the University of Prague. The registrar at G¨ottingen asks in turn on 31 March 1943 if Gentzen should be dismissed through compulsory cancellation, as he had been assistant already for 4 years. Publications: 9.) “Beweisbarkeit und Unbeweisbarkeit von Anfangsf¨ allen der transfiniten Induktion in der reinen Zahlentheorie”, Mathematische Annalen 119 (1943), pp. 140-161, received on 9 July 1942. Probationary lecture: “Die keplerschen Gesetze der Planetenbewegung” on 19 and 20 May 1943. Appointed docent without salary on 5 October 1943. Establishment of the “Rechenb¨ uro” [computation office] (precise task of the bureau is unclear and the exact dates uncertain). First lectures and academic practice in November 1943.

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1944 Approval of salary on 1 April 1944. Cessation of lectures (dating uncertain). Petition for an extended leave (July 1944). 14-day vacation in Puthus (R¨ ugen) in August 1944. Letter to Paul Lorenzen of 12 September 1944, Gentzen’s last known scientific letter. Rohrbach’s “Bericht u ¨ber die Entwicklung des mathematischen Instituts der Deutschen Karls-Universit¨ at” on 3 October 1944. Dedication of “Zusammenfassung von mehreren vollst¨ andigen Induktionen zu einer einzigen” to Heinrich Scholz on the occasion of his 60th birthday (published posthumously) on 17 December 1944. 1945 Detonation of a 10 kg bomb next to Gentzen’s housing on 12 February 1945. Gentzen’s Death 1945 Gentzen is a witness at the wedding of Paul and Hella Armsen. Hans Rohrbach hands over a mathematical calculation from Gentzen’s computing office at Osenberg in March 1945. Gentzen pays no attention to Pinl and Krammer, Easter 1945, and remains in Prague. Maruhn picks up Gentzen’s salary card for the “Reich” in April 1945. Last entry in Gentzen’s personal file on 20 April 1945. Toast to the wedding pair Armsen on 3 May 1945. Arrest of Gerhard Gentzen on 7 May 1945. Martyrdom. Gerhard Gentzen dies on 4 August 1945 at 6:30 p.m. at age 35 in the arms of Dr. Franz Krammer. Efforts to Secure the Nachlass 1946 Futile attempt to rescue the nachlass of Gerhard Gentzen. 1948 Deciphering of some stenographic notes by Hellmuth Kneser and Hans Rohrbach ( = H. Urban). 1984 Frau Student, Gentzen’s sister, turns over two folders, one blue and one violet, thus the major part of the fragment of the “Proof Theory Book” on 18 May 1984 during a personal visit from Prof. Dr. Thiel. He and Prof. Dr. M. Szabo attempt to transcribe and prepare the work for publication. Because of difficulties in transcription from the shorthand, Prof. Szabo shortly withdraws from the joint work. The transcription by Prof. Dr. Chr. Thiel continues till today (2006). Posthumous Publications 1954 10.) “Zusammenfassung von mehreren vollst¨ andigen Induktionen zu einer einzigen”, Archiv f¨ ur Mathematische Logik und Grundlagenforschung, in: Archiv f¨ ur Philosophie (ed. by J¨ urgen von Kempski) 5 (1954), no. 1, pp. 81-83. ¨ 1974 11.) “Uber das Verh¨ altnis zwischen intuitionistischer und klassischer Arithmetik”, received on 15 March 1933, withdrawn and published posthumously by Paul Bernays in: Arch. Math. Logik 16 (1974), pp. 119-132.

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12.) The original version of “Widerspruchsfreiheitsbeweis f¨ ur die klassische Zahlentheorie”, received on 11 August 1935 and then withdrawn and revised, published in: Arch. Math. Logik 16 (1974), pp. 97-118. Contemporary Assessments of Gentzen Page 16 17 29 31

31

38 42 43 44 51 51 54,56 55 58 60

60 62 63 63 64

66

Type of Assessment Evaluation for Studienstiftung des Deutschen Reiches Diploma Evaluation for the promotion-stipend of the Studienstiftung ¨ Review of “Uber die Existenz unabh¨ angiger Axiomensysteme zu unendlichen Satzsystemen” in Zbl. 5 (1933), pp. 338ff. ¨ Review of “Uber die Existenz unabh¨ angiger Axiomensysteme zu unendlichen Satzsystemen” in JFM 58 (1932) Correspondence with Heyting Evaluation of the dissertation “Untersuchungen u ¨ber das logische Schliessen” Review of “Untersuchungen u ¨ber das logische Schliessen” in JFM 60 (1) (1934), pp. 20ff. Review of “Untersuchungen u ¨ber das logische Schliessen” in Zbl. 10 (4) (1935), pp. 145ff. Award of the Doctoral Degree State Exam for higher teaching post Correspondence with Paul Bernays Correspondence with Hellmuth Kneser Correspondence between Bernays and Weyl on Gentzen Bernays discusses Gentzen’s results in Princeton, lays down his reflections in Logical Calculus and discusses this with H. Weyl, K. G¨ odel and J. von Neumann Correspondence with Helmut Hasse Request to the dean by H. Hasse for Gentzen as Hilbert’s assistant Review of “Die Widerspruchsfreiheit der Stufenlogik” in JFM 62 (2) (1936), pp. 43ff. Review of “Die Widerspruchsfreiheit der Stufenlogik” in Zbl. 15, no. 5 (1936), p. 193 Review of “Die Widerspruchsfreiheit der Stufenlogik” in Journal of Symbolic Logic 1 (1936), p. 119 Review of “Die Widerspruchsfreiheit der reinen Zahlentheorie” in JFM 62 (1) (1936), pp. 44ff.

Author

R. Courant A. Schmidt

Th. Skolem

H. Weyl W. Ackermann A. Schmidt

F. Bachmann A. Schmidt Alonzo Church

F. Bachmann

CONTEMPORARY ASSESSMENTS OF GENTZEN

Page 67

68 70

72 76 78 78

79 79

82 84

86

87

88

89 91

98 98 99 100

Type of Assessment Review of “Die Widerspruchsfreiheit der reinen Zahlentheorie” in Zbl. 14 (9) (1936), pp. 386ff. Mention in review of a paper by Bernays Review of “Die Widerspruchsfreiheit der reinen Zahlentheorie” in Journal of Symbolic Logic 1 (1936), p. 75 Correspondence between Bernays and Ackermann on Gentzen Mention in Einf¨ uhrung in das mathematischen Denken, Gerold & Co., Vienna, 1936, pp. 80ff. Review of “Der Unendlichkeitsbegriff in der Mathematik” in JFM 62 (1936), pp. 43ff. Review of “Die Unendlichkeitsbegriff in der Mathematik” in Journal of Symbolic Logic 2 (1937), p. 95 Correspondence with H. Kneser Mention in the report, “Denken und Erkenntnis des Abendlandes. Der Pariser DescartesKongress” in K¨ olnische Zeitung on 5 September 1937 Cavaill`es and Lautman on Gentzen Gentzen appears as “Associate” on the title page of Scholz’s Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften Review of “Die gegenw¨artige Lage in der mathematischen Grundlagenforschung” in Zbl. 19 (1938), p. 97 Review of “Die gegenw¨artige Lage in der mathematischen Grundlagenforschung” in Zbl. 19 (1938), p. 241 Review of “Neue Fassung des Widerspruchsfreiheitsbeweises f¨ ur die reine Zahlentheorie” in Zbl. 19 (1939) Bernays on Gentzen in his Swiss lecture Bernays, Church and W. Ackermann on Gentzen on the occasion of a Swiss conference in 1939 Evaluation of the extension of the unscheduled assistantship, 1938 “Problem child” letter to the REM of 14 October 1938 Bernays on Gentzen in the second volume of Hilbert-Bernays Grundlagen der Mathematik Bernays and Blumenthal on Gentzen in Hilbert’s collected works, 1935.

Author A. Schmidt

A. Schmidt Paul Bernays

F. Waismann W. Ackermann Barkley Rosser

Heinrich Scholz

Heinrich Scholz

H. Curry

A. Schmidt

A. Schmidt

H. Hasse W. S¨ uss P. Bernays

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Page 122

122

122

127

129 130 133

135 136 138 138 145ff.

233 237 237 238 241 242 244ff. 246 253ff. 254ff. 255ff.

Type of Assessment Review of “Die gegenw¨artige Lage in der mathematischen Grundlagenforschung” in JFM 64 (1) (1939), p. 26 Review of “Neue Fassung des Widerspruchsfreiheitsbeweises f¨ ur die reine Zahlentheorie” in JFM 64 (2) (1939), pp. 26ff. Review of “Die gegenw¨artige Lage in der mathematischen Grundlagenforschung. Neue Fassung des Widerspruchsfreiheitsbeweises f¨ ur die reine Zahlentheorie” G¨ odel invites Gentzen via Hasse to his lecture on 15 December 1939 on the consistency of Cantor’s continuum hypothesis Report on Gentzen’s habilitation thesis commissioned for H. Scholz Supplement to Ackermann’s evaluation Reference to Gentzen’s work without expressly naming him in an anthology prepared on the occasion of Hitler’s birthday Review of the habilitation thesis in Zbl. 28 (1943), p. 102 Review of the habilitation thesis in Journal of Symbolic Logic 9 (1944), pp. 70-72 Mention of Gentzen by L. Kalm´ ar Mention of Gentzen in Hellmuth Kneser’s “Notio et Notatio” Mention of Gentzen by Max Steck and Friedrich Requard in connection with a “racial foundation” of mathematics G¨ ottingen would like to retain Gentzen Evaluation of Gentzen’s probationary lecture Evaluation of Gentzen’s probationary lecture Remark on Gentzen’s computing group Remarks on Gentzen by Ernst Mohr H. Rohrbach proposes a lectureship with salary for Gentzen Request for leave of absence and certificate for this Evaluation on the situation at the Mathematical Institute in Prague Remarks on Gentzen Remarks on Gentzen Reports on Gentzen’s Death

Author W. Ackermann

W. Ackermann

Max Black; J. Barkley Rosser

K. G¨ odel

W. Ackermann H. Scholz H. Hasse

W. Ackermann P. Bernays L. Kalm´ ar Hellmuth Kneser M. Steck; F. Requard Th. Kaluza Glaser H. Rohrbach H. Rohrbach Ernst Mohr H. Rohrbach H. Rohrbach; Dr. Ewald H. Rohrbach Paul Armsen Franz Krammer F. Krammer; F. Kraus

PUBLICATIONS OF GENTZEN

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Publications of Gentzen ¨ 1932 1.) “Uber die Existenz unabh¨ angiger Axiomensysteme zu unendlichen Satzsystemen”, Mathematische Annalen 107 (1932), pp. 329-350. Received on 6 February 1932. ¨ 1933 “Uber das Verh¨ altnis zwischen intuitionistischer und klassischer Arithmetik”, received on 15 March 1933 by Mathematischen Annalen, withdrawn and posthumously published by Paul Bernays in: Arch. Math. Logik 16 (1974), pp. 119-132. 1933 2.) “Untersuchungen u ¨ber das logische Schliessen”, Mathematische Zeitschrift 39 (2) (1934-1935), pp. 176-210; and (3), pp. 405-431. 1935 Submission of “Widerspruchsfreiheit der reinen Zahlentheorie” to Mathematische Annalen on 11 August 1935. Withdrawn on criticism from Prof. Dr. Paul Bernays. Revision of the work begun. The original would be published posthumously by Paul Bernays in 1974. 1936 3.) “Die Widerspruchsfreiheit der Stufenlogik”, Mathematische Zeitschrift 41 (1936), pp. 357-366. 4.) “Die Widerspruchsfreiheit der reinen Zahlentheorie”, Mathematische Annalen 112 (1936), pp. 493-565. 5.) “Der Unendlichkeitsbegriff in der Mathematik. Vortrag, gehalten in M¨ unster am 27. Juni 1936 am Institut von Heinrich Scholz”, in: Semester-Berichte M¨ unster, WS 1936/37, pp. 65-80. 1937 Self-report on publication 3. and 4. in: Deutsche Mathematik 2 (1937), p. 130. 6.) “Unendlichkeitsbegriff und Widerspruchsfreiheit der Mathematik”, Actualit´es scientifiques et industrielles, no. 535, pp. 201-205, Paris: Hermann, 1937; and in: IXe Congr`es International de Philosophie, VI. logique et math´ematiques, Paris, 1937. 1938 7.) “Die gegenw¨ artige Lage in der mathematischen Grundlagenforschung”, Deutsche Mathematik 3 (1938), pp. 255-268. (First printing) 8.) “Neue Fassung des Widerspruchsfreiheitsbeweises f¨ ur die reine Zahlentheorie”, Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, vol. 4 (1938), pp. 19-44, Leipzig: Hirzel. 7. also appeared in Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften 4 (1938), indirectly funded by Ludwig Bieberbach. 1943 9.) “Beweisbarkeit und Unbeweisbarkeit von Anfangsf¨ allen der transfiniten Induktion in der reinen Zahlentheorie”, Mathematische Annalen 119 (1943), pp. 140-161, received on 9 July 1942. 1944 Dedication of “Zusammenfassung von mehreren vollst¨ andigen Induktionen zu einer einzigen” to Heinrich Scholz on the occasion of his 60th birthday (appeared posthumously) on 17 December 1944. Posthumous Publications 1955 10.) “Zusammenfassung von mehreren vollst¨ andigen Induktionen zu einer einzigen”, Archiv f¨ ur Mathematische Logik und Grundlagenforschung, in:

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Archiv f¨ ur Philosophie (ed. by J¨ urgen von Kempski) 5 (1) (1954), pp. 8183. ¨ 1974 11.) “Uber das Verh¨ altnis zwischen intuitionistischer und klassischer Arithmetik”, Arch. Math. Logik 16 (1974), pp. 119-132. 12.) The original version of “Widerspruchsfreiheitsbeweis f¨ ur die klassische Zahlentheorie”, Arch. Math. Logik 16 (1974), pp. 97-118. Reviews and Related Material by Gentzen Page 60 76 85 93

Author Malcev, A. Post, Emil L. Ackermann, W. Hilbert-Bernays

94 Rosser, Barkley 96 Hilbert-Ackermann 96 96 97 118 119 120 121 121

122 125 128 128 128 243

Orrin, Frank Jr. Schmidt, Arnold Pepis, J´ ozef Ono, Katudi Dingler, Hugo Hilbert-Bernays Rosser, Barkley a) Huntington, E.V. b) Kalm´ ar, L. c) Greenwood, Th. d) Birkhoff, George D. e) do Amaral, Ingacio M.A. Padoa, A. Bense, Max Moisil, Gr. C. Novikoff, P. Menger, Karl Lorenzen, Paul

Journal Zbl. 14 (9)(1935), p. 385 Zbl. 15 (5) (1936), p. 193 Zbl. 16 (5) (1937), p. 194 Proofreading and suggestions for improvement Zbl. 17 (6) (1938), p. 242 Corrections to Grundz¨ uge der theoretischen Logik, 2nd edition Zbl. 18 (8) (1938), p. 337 Zbl. 18 (8) (1938), p. 337 Zbl. 18 (9) (1938), p. 385 Zbl. 19 (6) (1939), p. 242 Remarks to Heinrich Scholz Zbl. 20 (5) (1939), pp. 193ff. Zbl. 20 (5) (1939), p. 194 Zbl. 20 (5) (1939), pp. 193ff.

JFM 64 (2) (1939), p. 923 Zbl. 21 (1939), p. 97 Zbl. 21 (7) (1939), p. 290 Zbl. 21 (7) (1939), p. 290 Zbl. 21 (7) (1939), p. 290 Letter

https://doi.org/10.1090/hmath/033/09

APPENDIX A

Gentzen and Geometry ´ ski C. Smoryn When Alfons Bilharz received a letter from his then 13-year-old grandson announcing a new geometric theorem, he had no way of gauging the originality of either the result or its proof. It is not a result found in every geometry text and is not later appealed to as a lemma in the standard higher mathematics courses. It appears at first sight to be an obscure geometrical exercise of comparable interest to Napoleon’s closely related theorem. Gentzen’s result, however, has a history. Recall Gentzen’s result. We are given a triangle ABC on the sides of which we erect exterior equilateral triangles. From each vertex of the original triangle, we draw the line segment connecting this vertex to the far vertex of the equilateral triangle opposite. Such a segment he called a “pereunt”. Gentzen’s Theorem states: Theorem 1. The pereunts of a triangle meet in a common point, the angles they form at that point all measure 60 ◦ , and the pereunts have a common length. The result was not new. Indeed, the common point has three names: the Fermat point, the Torricelli point, and Steiner’s point. In the case in which all three angles of the triangle are less than 120◦ , the point is the solution to a minimisation problem posed by Pierre de Fermat as a challenge to those who disliked his technique of solving max-min problems. Galileo’s student Evangelista Torricelli solved the problem by showing the point in question to be the intersection of the three circles circumscribing the equilateral triangles. Another of their contemporaries, Bonaventura Cavalieri, discovered a variant of the 60◦ angles. He showed: if the angles of ABC are all less than 120◦ , then the Fermat point—F for short—is the unique point P in the interior of ABC for which the angles ∠AP B, ∠AP C, and ∠BP C are all 120◦ . A century later, Thomas Simpson made the connection with the pereunts, now commonly called Simpson’s lines. The point, however, seems not to have been credited to him. It has, however, been credited to the 19th century Swiss mathematician Jakob Steiner, who, I have on good authority (Harold W. Kuhn, “ ‘Steiner’s’ problem revisited”, G.B. Dantzig and B.C. Eaves, eds., Studies in Optimization, MAA, 1974), contributed nothing new but who was championed by Richard Courant and Herbert Robbins in their popular volume What Is Mathematics? (Oxford, 1941). What Gentzen missed were the minimisation problem and the unicity result for the 120◦ angles. He also did not prove Napoleon’s Theorem, which will be stated later. He did prove F to be the point of intersection of the three circumscribing circles. All in all, in the matter of discovery, he did remarkably well, better than one could expect of today’s university undergraduates. That he missed the unicity result and the optimisation problem, which will be discussed later, should be expected: 283

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Gentzen was a young boy taking a geometry course, considering problems in a geometric vein and applying geometric methods to their solutions. So many characterisations of F suggest a multiplicity of proofs. One textbook I’ve seen begins with Cavalieri’s 120◦ property and is completely geometrical. Paul J. Nahin’s popularisation When Least Is Best presents a geometric proof due to the German math historian J.E. Hofmann that first solves the minimisation problem. While awaiting delivery of a copy of Gentzen’s letter, I worked out a simple computational proof based on the circumscribing circles. Gentzen’s proof is geometric and elementary, and it deals directly with the triangles and pereunts using no remarkable extra properties of the point F. Moreover, Gentzen’s proof works uniformly in all cases and makes no restriction on the sizes of the angles of ABC. Figures 1 to 3 illustrate the three possible configurations that arise according to whether all the angles are less than 120◦ , one equals 120◦ , and one is greater than 120◦ . B A  A   B  C  C       F   AJ A B
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 Figure 3 C Gentzen’s proof that the pereunts have a common point of intersection begins with a lemma about the 120◦ case. Lemma 1. If ∠C in a triangle ABC is 120 ◦ , then the pereunt CC  divides the angle into two equal 60 ◦ angles.

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At first sight it looks like he is going to prove his theorem in the special case of a 120◦ angle, and, indeed, he pretty much does this. But he never mentions this and uses the lemma as a tool in handling the general case. Here is the proof. Refer to Figure 2. We have AC = AB  AC  = AB ∠CAC  = ∠B  AB, the first two assertions holding true because their given line segments are sides of an equilateral triangle and the last because the angles are ∠CAB + 60 ◦ . It follows that the triangles CAC  and B  AB are congruent. Thus the angles ∠C  CA and ∠BB  A are equal. But ∠BB  A is 60◦ , whence so is ∠C  CA. The symmetric argument tells us that ∠C  CB is also 60◦ , and the lemma is proven.  As already stated, Gentzen does not mention it yet, but the congruence also yields the equality of the pereunts CC  and B  B. A symmetric argument yields this between CC  and A A. All that remains to prove the theorem in the special case is to extend CC  beyond C (= F ) and observe that it halves the angle ∠A CB  . Following the proof of the lemma, Gentzen states and proves the general result, referring to Figure 1. He never refers to Figure 3, the case in which ∠C is greater than 120◦ and F will lie outside the triangle. Whether this is an oversight or a following of the time-honoured geometric tradition of only presenting one diagrammatic case, I cannot say. I can state only that the proof works regardless of which of the Figures 1, 2, and 3 one is looking at, and this is not the case with all proofs of the theorem. Proof of the theorem. Of course, Figures 1 and 2 presuppose the pereunts to intersect in a single point, and this cannot be used to establish that they do so. However, each pair of pereunts will intersect in a single point, and one can erase temporarily the third pereunt from the picture. We don’t need to do this right away, as Gentzen begins by repeating the proof of the lemma, which made no mention of the intersection of the pereunts: AC = AB  , AC  = AB, ∠CAC  = ∠B  AB,

sides of an equilateral triangle sides of an equilateral triangle = ∠CAB + 60◦ .

Thus, triangles CAC  and B  AB are congruent and C  C = BB  . Similarly, C  C = AA . Thus, all three pereunts have the same length. The next step is to calculate the angles between two pereunts at their point of intersection. Let F be the point of intersection of the pereunts AA and BB  . By the congruence just cited, ∠ABB  (= ∠ABF ) is the same as ∠AC  C. Similarly, ∠A AB (= ∠F AB) is the same as ∠CC  B. But ∠AC  C + ∠CC  B = 60 ◦ , whence ∠ABF + ∠F AB = 60 ◦ , whence ∠AF B = 120 ◦ . Now, whether or not F lies on CC  , the segment F C  is a pereunt of the triangle AFB, whence the lemma applies: ∠AF C  and ∠BF C  are both 60◦ angles.

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If we let G be the point of intersection of the pereunts BB  and CC  , we can repeat the argument to conclude ∠A GB and ∠A GC are both 60◦ . To see that F and G coincide, consider Figure 4 below: B HH

A    # H F  # H HH ##    GH  HH A H  HB C Figure 4 Here I have placed G to the right of F simply because it must fit somewhere into the picture; one could as well have placed it to the left. I have not drawn in C because its exact position is irrelevant. The angle ∠AF B being 120◦ and AF A being a straight line, it follows that ∠A F B is 60◦ . Again, since BGF B  is a straight line and ∠A GB is 60◦ , it follows that ∠F GA is 120◦ . As the angles of triangle F GA must sum to 180◦ , it follows that ∠GA F is 0◦ ; i.e. G lies on AA , whence it coincides with F (both lying on the sides AA and BB  , they are the intersection of these lines). This pretty much finishes the proof. For the sake of completeness, we should repeat the application of the lemma to conclude that the last two angles ∠AF B  and ∠CF B  are 60◦ .  Possibly the most impressive thing about the proof is Gentzen’s ability to consider a pereunt of a different triangle (AF B ) to prove something about those of the given triangle (ABC ). This is further illustrated in his first corollary. Corollary 1. The Fermat point of the triangle A B  C  obtained by connecting the far vertices of the superimposed equilateral triangles coincides with that of ABC. Proof. The proof is rather clever. Let F be the Fermat point of ABC and let A B  C  be the equilateral triangle drawn on side A B  of A B  C  . This equilateral triangle is also drawn on a side of A B  F, whence the line F C  is one of this latter triangle’s pereunts. By the lemma, it subdivides ∠A F B  into two equal 60◦ angles. By the theorem, the pereunt CC  of the original triangle does the same. Hence F C  and CC  are collinear. Hence F, C, C  , C  lie on a straight line, and, in particular, F lies on the pereunt C  C  of A B  C  . Repeating the argument for the other two pereunts shows F to lie on all three, whence it is the Fermat point of A B  C  .  Insofar as the points A , B  , C  are not connected in any of Figures 1-3, it is to Gentzen’s credit that he considered the triangle at all. Nonetheless, what struck me was that he did not consider the triangle A B  C, which jumps out at you if you stare at Figure 3 long enough. It seems obvious that it shares its Fermat point with ABC. It does in this case (i.e., ∠ACB > 120◦ ), as is easily shown: Corollary 2. If ∠ACB > 60◦ , the Fermat point of triangle A B  C coincides with that of triangle ABC. Proof. If ∠ACB > 60◦ , then triangle A B  C shares AB  C and A BC as external equilateral triangles (if the angle is 60◦ , A CB  is a straight line; and if the angle is less than 60◦ , the triangle overlaps these two equilateral triangles). Thus, B  B

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and A A are two of its pereunts. The intersection of these two lines is the Fermat point of both triangles ABC and A B  C.  If the angle in question is less than 60◦ , the triangles have distinct Fermat points, but pursuing them takes us a bit far afield. Gentzen drew a final conclusion: Corollary 3. The Fermat point of a triangle is the intersection of the circles circumscribing the superimposed equilateral triangles. Proof. This is more an observation than a proof. Gentzen knew the characterisation of quadrilaterals that can be inscribed in circles as those in which a pair of opposite angles adds up to 180◦ . Assume none of the angles of ABC is 120◦ . Each of the quadrilaterals with vertices F and those of one of the equilateral triangles (e.g., ACF B  in Figure 3 ) have an interior angle of 120◦ at F (by the theorem) and an opposite angle of 60◦ (an angle of an equilateral triangle). Hence the quadrilateral is inscribed in a circle. Since the three vertices other than F are vertices of an equilateral triangle, the circle is that triangle’s circumscribing circle. F, lying on all three such circles, is their point of intersection. In the case of a 120◦ angle at C (Figure 2 ), F coincides with a vertex of two of the equilateral triangles and hence lies on their circumscribing circles. One only needs to apply the relevant portion of the above argument to the quadrilateral ACBC  to put it on the third.  Corollary 3 is the least impressive part of Gentzen’s letter in that it was clearly not a case of a conjectured discovery for which he had to find a proof, but a quick application of a result he had learned in class. Moreover, in referring only to Figure 1, his proof was, strictly speaking, incomplete. Gentzen finished his letter with the remark that they had studied geometry in class up to the Pythagorean Theorem, for which he had discovered his own proof. It is one of the standard simple proofs and need not be commented on here. That said, there may be something in this mention of the Pythagorean Theorem to motivate Gentzen’s discovery of Theorem 1. The familiar graphical representation of the Pythagorean Theorem depicts a right triangle with exterior squares drawn on its sides. This might suggest the analogy of a triangle with other regular polygons, e.g. equilateral triangles, so added. The famous Euclidean proof begins by drawing a perpendicular line from the far side of the square on the hypotenuse to the vertex of the right angle. With the equilateral triangles replacing the squares, there is no opposite side, but there is an opposite vertex. There being no preferred vertex in the original triangle, one might draw all the pereunts. The least amount of care in one’s drawing would necessarily result in the pereunts intersecting in a common point. And this observation would become a conjecture. This ends our exposition of Gentzen’s proof. I should like in the remainder to discuss what Gentzen didn’t prove. The proof just given applies to yield the uniqueness part of Cavalieri’s result: Corollary 4. Let each of the angles of triangle ABC be less than 120 ◦ . Then F is the unique point P inside ABC for which the three angles ∠AP B, ∠BP C, and ∠CP A are 120 ◦ . Proof. That F has this property is established by the theorem. If P is any other interior point making 120◦ with each pair of vertices, then the argument used in

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proving Corollary 3 makes P the point of intersection of the circumscribing circles of the superimposed equilateral triangles, i.e. P = F.  Thus, we see that the proof was well within the young Gentzen’s reach and we may surmise that he missed the result because his pereunts extended through F and he was concentrating on the 60◦ angles. Another result he missed, but was so close to, was Napoleon’s Theorem, actually discovered by Napoleon Bonaparte: Corollary 5. The triangle with vertices given by the centres of the three superimposed equilateral triangles is itself equilateral. Proof. Let P, Q, R be the centres of triangles ABC  , AB  C and A BC, respectively, and consider the line connecting P and Q. The picture looks something like Figure 5: F PPP   PP Q P  P  P PP   PP P A Figure 5 Now F P = AP and QF = QA by Corollary 3, whence the triangles QF P and QAP are congruent. Thus angles ∠QP F and ∠QP A are equal; i.e. ∠QP F is half of ∠AP F . Looking at P R we similarly conclude ∠F P R is half of ∠AP B. Thus ∠QP R = ∠QP F +∠F P R is half of ∠AP B = ∠AP F +∠F P R. But the angle from the centre of an equilateral triangle to two of the triangle’s vertices is 120◦ , whence ∠QP R is 60◦ . Repeating the above for each vertex shows triangle QP R to be equiangular, whence equilateral.  The proof does not use anything beyond the reasoning Gentzen used in his letter to his grandfather, and we can conclude it was well within his reach. As he was a young boy who loved adventure stories, writing them as well as reading them, I should think that had he been aware of the result and its imperial authorship, he would not only have derived the result but would have excitedly included it in his exposition to his grandfather. Hence I think we can safely conclude that he just didn’t think of the triangle P QR. The biggest thing Gentzen missed of course was Fermat’s optimisation problem. This is only natural. Gentzen was studying geometry in school, and other than knowing that the shortest distance between two points is a straight line, optimisation was a wholly different problem area. It simply would not have occurred to him to ask such a question. Or if he had, what experience had he had in proving such a result? Getting the full result would be asking too much of a budding genius. That said, Gentzen’s work does have some relevance. First, let me state the optimisation result precisely. Theorem 2. Let all the angles of the triangle ABC be less than 120 ◦ . Then the Fermat point of ABC is the point P that minimises the sum P A + P B + P C of the distances of P to the three vertices.

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J.E. Hofmann gave a simple geometric proof that doesn’t go beyond the methods we have been using. But the proof is inspired, and even someone familiar with the result ought not to be expected to find it. Proof. Let A, B, C be given, let AB  C be the equilateral triangle constructed on side AC, and let P be any point. Find P  by rotating the line AP 60◦ counterclockwise around the point A (cf. Figure 6 ). C  l B B B hhh P  h B ll h S S B l S l BP  l S  l S  ll S A B Figure 6 Now AB  = AC, ∠B  AC = 60 ◦ = ∠P  AP, and AP  = AP, whence triangles B  P  A and CP A are congruent, and we can conclude B  P  = CP, i.e. B  P  = P C. On the other hand, AP  = AP and ∠P  AP = 60 ◦ , whence triangle P  AP is equilateral (congruent by side-angle-side to an equilateral triangle) and P  P = P A. Putting things together, the overall length of the polygonal path B  P  P B is BP  + P P + P B = P C + P A + P B = P A + P B + P C. But B  P  P B ≥ B  B, whence P A + P B + P C ≥ B  B. The Fermat point, however, lies on B  B. Moreover, for P = F the angle ∠B  F  A = ∠CF A = 120 ◦ , and we have just seen that ∠F F  A = 60 ◦ , whence ∠B  F  F = 180 ◦ and F  also lies on B  B. Thus, for P = F, the polygonal line is straight and F A + F B + F C = B  B.  If one of the angles, say, that at C, is 120◦ , then C lies on the line B  B and the same proof applies. If the angle at C is greater than 120◦ , it can be shown that C still minimises the sum. As for F , if the angle at C of triangle ABC is greater than 120 ◦ , then F is the point P that minimises the sum P A + P B  + P C, F is the point P that minimises the sum P A + P B  + P C  . Moreover, if one considers the equilateral triangles constructed on A B  C and applies Corollary 1 to them, one sees that F is the point P that minimises the sum P A + P B + P C  . A small remark about this last observation: For P on the segment CC  , we have P C  = CC  − P C. So for such P, F minimises P A + P B + CC  − P C, and since CC  is constant, F minimises P A + P B − P C on the line segment CC  . In What Is Mathematics?

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Courant and Robbins claim that F minimises this sum for all points. This is demonstrably not the case and is an error that could not be made in today’s world of 3D graphics software programs. The one case I did examine suggests that for the angle at C greater than 120◦ , the expression P A + P B − P C assumes a local maximum at C and local minima at A, B, with the absolute minimum occurring at the vertex farthest from C.

https://doi.org/10.1090/hmath/033/10

APPENDIX B

Hilbert’s Programme ´ ski C. Smoryn Elsewhere among these appendices Jan von Plato will discuss Gentzen’s logical works and their place in modern logic. In the present appendix,1 I wish to present the background to Gentzen’s work, in particular the problem it aimed to solve and the programme it grew out of. The problem was the consistency of mathematics, and the programme designed to prove this consistency is known as Hilbert’s Programme after its founder David Hilbert. While much of this book deals with mathematics and politics, Hilbert’s programme deals with mathematics and psychology. Thus we must first give some background to Hilbert’s Programme, i.e. a background to the background to Gentzen’s work. 1. Constructive Prologue There are several historical threads leading up to Hilbert’s programme. The most dramatic of these are Leopold Kronecker’s open opposition to Cantorian set theory and the strong negative reaction to Hilbert’s success in solving the main problems in invariant theory. Less dramatic, but equally important, are Hilbert’s underlying philosophical assumptions expressed so forcefully in his confirmation of faith—his address to the International Congress of Mathematics held in Paris in 1900—and Hilbert’s work in the foundations of geometry in the 1890s. Kronecker’s opposition to Georg Cantor is legendary, and like all good legends it contains some good fiction as well as some fact. The biggest fiction is that Kronecker’s persecution drove Cantor mad, whereas Cantor was already mentally unstable. Nonetheless, Kronecker had strong views and wasn’t afraid to express them or to act upon them. When in 1882, for example, Ferdinand von Lindemann, Hilbert’s formal advisor, proved the transcendence of π, Kronecker asked, “Of what value is your beautiful work on the transcendental number when in reality there are no irrational numbers?” His oft-quoted, “God gave us the integers; all the rest is man’s work,” sums up his philosophy nicely. The extent to which he was willing to act upon his beliefs is illustrated by his attempt to convince Eduard Heine to withdraw his paper “On trigonometric series” when the latter was already correcting publisher’s proofs. With respect to publishing in Acta Mathematica, he wrote to Sonya Kovalevskaya: 1 Most of this appendix was written for my as yet unpublished second volume of the twovolume set Logical Number Theory. A version of it has already appeared in the Dutch journal CWI Quarterly, volume 1, number 4 (1988), pp. 3-59, and I repeat it here with the kind permission of the editor.

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Since the point is only the victory of truth, I think our friend Mittag-Leffler will gladly accept my considerations instead of Cantor’s blind assertions to the opposite.

Cantor believed the delay in publication of his famous 1878 paper on dimension could be squarely laid on Kronecker’s doorstep. Whether or not Kronecker was able to block any set theoretic publications, he lectured against it, in one seminar even calling Cantor a “Perverter of Youth”, and even threatened to publish a critique exposing the emptiness of the field. Kronecker was a petty and, I dare say, evil man. When the Swedish king sponsored a mathematical competition and G¨osta Mittag-Leffler passed Kronecker by in choosing judges, the latter reneged on his promise to recommend Mrs. Mittag-Leffler to a famous German physician. He also threatened to write to the king to tell on Mittag-Leffler. In a letter to Kovalevskaya, Karl Weierstrass summed up Kronecker’s megalomania with the simple remark that his claim to be the only expert and the key figure in algebra has become a welladvanced disease.

Algebra was not the only subject in which Kronecker felt himself expert enough to direct its progress. Indeed Weierstrass and his program of arithmetising analysis was a favourite target of Kronecker’s. And I have already mentioned his attempt to persuade Heine not to publish the type of mathematics he did not approve of. Reports on Kronecker are not uniformly negative. Indeed, he was helpful to Cantor early on, before Cantor, in his view, went off the deep end. Hilbert, who abhored Kronecker’s perceived abuse of power, had great respect for Kronecker as a mathematician. In fact, Hilbert built his own mathematical reputation in the 1880s in invariant theory partially upon Kroneckerian foundations. David Hilbert was born in 1862 in K¨ onigsberg, Prussia (now: Kaliningrad, a Russian enclave in Lithuania). It was under Lindemann’s direction at the local university that Hilbert would earn his doctorate in 1885 and where he would habilitate the following year. Both theses concerned invariant theory, a then hot topic, to which he devoted the next several years. The main problem in invariant theory was something we would now call a finite basis theorem. This had been worked on for years, the main result in this area being due to Paul Gordan, who was called the King of the Invariants. In 1890, Hilbert published a solution to Gordan’s problem, a solution that had unexpected consequences. Gordan’s work, as well as that of the other invariant theorists of the day, was concrete and constructive. Given an invariant problem, they actually constructed the finite basis; Hilbert’s proof was highly abstract and nonconstructive: he showed the basis existed, but not how to find it. In his historical work Vorlesungen u ¨ber die Entwicklung der Mathematik im 19. Jahrhundert [Lectures on the development of mathematics in the 19th century] (Springer-Verlag, Berlin, 2 volumes, 1926/27), Felix Klein reported Gordan’s reaction: “That is not mathematics; that is theology.” Hilbert’s advisor Lindemann labelled the proof unheimlich [“eerie” or “sinister”]. I do not know about Lindemann, but Klein reported Gordan’s change of heart: “I have become convinced that even theology has its uses.” Indeed, he later simplified Hilbert’s proof. But the damage had already been done. Klein does not report the occasions on which Gordan made his comments. But correspondence on the subject and its effect on Hilbert does exist. Hilbert had submitted the paper to Klein, one of the chief editors of Mathematische Annalen,

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and Klein in turn had forwarded the paper to Gordan for comment. Gordan was the leading authority in the field and an associate editor, thus the obvious source for an informed opinion. In a letter of 24 February 1890 from Gordan to Klein, Gordan remarked: You wanted my impression of Hilbert’s work. I’m sorry to say to you that I am very unsatisfied with it. The things are very important and also correct, thus that is not what my criticism addresses.

His criticism was that Hilbert’s proof did not deliver on the promise to produce the forms in question. Hilbert’s proof was fine for a first discovery, but insufficient for publication in an important journal like the Mathematische Annalen. He peppered his letter with a couple of choice comments on how Hilbert had “scorned” the idea of laying out his thoughts following formal rules and that Hilbert thought it sufficed that no one contradicted his proofs. He also reminded Klein that he had told the latter earlier he didn’t accept Hilbert’s methods of reasoning. Gordan’s letter would have been harmless had Klein not forwarded it to Hilbert’s friend Adolf Hurwitz, who passed it on to Hilbert. On 3 March, Hilbert wrote a very defensive letter to Klein saying that he had found Gordan’s letter “unusually harsh.” He pointed out that the proof in question was a year and a half old, that he had discussed it with numerous mathematicians either orally or in correspondence, and had taken into account when writing it up for publication the points they had difficulty with. He referred to Gordan’s “reproaches” [Vorw¨ urfe] and finished up remarking that, barring definite and irrefutable objections to his mode of deduction, he had no intention of either rewriting or withdrawing the paper. Hilbert was clearly hurting—so much so in fact that Klein’s letter to Hilbert of the 14th of the next month may best be described as (belatedly) comforting: Gordan had been visiting for 8 days and was completely converted; in fact, Hilbert couldn’t wish for a better outcome. He added a conciliatory postscript stating that Gordan’s remark on the unsuitability for publication of Hilbert’s paper in the Annalen had no actual significance, but was rather a general formulation for possible consideration for future works. Klein’s letter may have eased the pain, but Hilbert would be wary of constructivists thereafter, as if fearing the spirit of the recently deceased Kronecker would take hold, Mabuse-like, and destroy mathematics. 2. Problems in Paris In August 1900, Hilbert addressed the International Congress of Mathematicians in Paris. It was the last year of the 19th century, and Hilbert thought to inaugurate the coming century with a glimpse of the future. To this end he compiled a list of 23 problems2 covering a broad spectrum of mathematics. Many of the problems did indeed occupy the minds of 20th century mathematicians. More pertinent here, however, are not the problems themselves but the remarks prefatory to them. Hilbert began his discussion of problems by focussing on their importance for mathematics—and mathematicians. Hilbert cited several such problems, noting also that, “So long as a branch of science offers an abundance of problems, so long is it alive; a lack of problems foreshadows extinction or the cessation of independent development.” 2 A 24th has recently been discovered. It will be discussed elsewhere in this volume. In the lecture itself he cut the list down due to time constraints.

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Having explained that mathematics needs problems, Hilbert then addressed the question of where the problems come from. Initially, they are empirical, arising from geometry, physics, etc. Eventually, the problems arise from mathematics itself. Similarly, logic, which is one level further removed from the empirical world than mathematics, draws its problems from mathematics, computer science, philosophy, and ultimately from logic itself. This drawing on themselves is a not uncontroversial aspect of mathematics and logic. For Hilbert, however, this was not a matter of controversy, but a re-affirmation of the fruitfulness for mathematics of its problems. The third point Hilbert addressed is this: what constitutes a solution? I should say first of all this: that it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied by the statement of the problem and which must always be exactly formulated. This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigour in reasoning.

In his second problem he complements this: When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of the science. . . no statement within the realm of the science whose foundations we are testing is held to be correct unless it can be derived from those axioms by means of a finite number of logical steps.

This language is partially repeated in the statement of the tenth problem: To devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers.

The desiderata of the preface and of the second problem fit together like hand and glove. The prefatory requirement is, as Hilbert says, merely that of rigour; the second is the establishment of the relevant notion of rigour. Except for the tenth problem, a problem for Hilbert was (almost) a statement, the truth or falsity of which was to be ferreted out and proven rigorously. The statement concerns a science, e.g. geometry or algebra, and the science must be isolated and completely described axiomatically. From this, every problem of the science can be solved: either the statement in question or its negation will have a proof from the axioms. . . This, of course, is exactly what G¨odel proved not to be the case, but that fact gets us ahead of our story. Hilbert’s fourth point in his discussion of problems was a short enumeration of the difficulties that can arise in trying to find the solution, i.e., in trying to find the proof or disproof desired. It could be the case, for example, that the statement of the given problem is wrong (e.g., the hypotheses are insufficient, as when one neglects to require the continuity of the derivative. . . ). Despite difficulties, Hilbert believed a solution always to be possible. This was the fifth and final point of his general discussion— his expression of faith: Take any definite unsolved problem, such as the question as to the irrationality of the Euler-Mascheroni constant C, or the existence of an infinite number of prime numbers of the form 2n + 1. However unapproachable these problems may seem to us and however helpless we stand before them, we have, nevertheless, the firm conviction that their solution must follow by a finite number of purely logical processes. Is this axiom of the solvability of every problem a peculiarity characteristic of mathematical thought alone, or is it possibly a general law inherent in the nature

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of the mind, that all questions which it asks must be answerable?. . . The conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus.

Hilbert’s rejection of the ignorabimus was a reaction to the more negative beliefs of Emil DuBois-Reymond and his catchwords “ignoramus” [We don’t know] and “ignorabimus” [We will never know]. DuBois-Reymond, elder brother of Paul DuBois-Reymond (the inventor of the diagonal argument later used so fruitfully by Georg Cantor), studied electrophysiology and was also a populariser of science. DuBois-Reymond believed that there were insoluble problems in science. A lecture ¨ on this he gave in 1872 was entitled “Uber die Grenzen des Naturerkennens” [On the limitations of natural philosophy]. In an 1880 lecture, “Die sieben Weltr¨ atsel” [The seven world puzzles], he cited 7 questions to which we can only answer “ignoramus” or “ignorabimus”: i. the essence of force and matter ii. the origin of movement iii. the origin of life iv. the teleology of nature v. the origin of sense perception vi. the origin of thought vii. free will DuBois-Reymond’s views were widely discussed and attained a degree of popularity. Hilbert’s optimism ran counter to DuBois-Reymond’s pessimism. His address on problems was given at the turn of the century, a particularly optimistic time, and one may be tempted to attribute Hilbert’s faith to the mood of the day. However, the mood changed. There was soon to be the First World War, Heisenberg’s Uncertainty Principle, and, in the 1920s, a battle between Hilbert and L.E.J. Brouwer over just such an issue. Yet, on 8 September 1930, an optimistic David Hilbert addressed a radio audience on the occasion of his retirement in his hometown of K¨ onigsberg. Once again he banished the ignorabimus from mathematics: Instead of the foolish Ignorabimus, we call on the contrary our own motto: We must know, we will know.

The couplet, “We must know, we will know”, in its German original, adorns his tombstone. Both DuBois-Reymond and Hilbert were well aware of the difficulties inherent in the sciences. DuBois-Reymond declared some to be insurmountable; Hilbert said we just have to be patient. Perhaps the best indication of Hilbert’s assessment of the situation is given by an anecdote told by Carl Ludwig Siegel, a 20th century number theorist sometimes referred to in jest as the last of the great 19th century mathematicians. At one lecture in 1920, Hilbert was discussing problems in mathematics. He told the students, who numbered Siegel among them, that much recent work on the Riemann zeta function led him to hope that he’d yet see a proof of the Riemann Hypothesis. The Fermat problem was more difficult, but the youngest in √ the audience might live to see its solution. However, the transcendence of 2 2 was so difficult a problem that no one in the audience would live to see its solution. Within a few years, Siegel had proven this transcendence! (In fact, Siegel was not

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the first to obtain this result. Today, the Fermat problem has been solved, and the Riemann Hypothesis has been overwhelmingly corroborated numerically, but it remains an open problem.) What this anecdote tells us is that Hilbert’s optimism was not a Utopian fantasy. If he did misjudge on specifics, he was nevertheless realistic in recognising the great difficulty of mathematics. But was his appraisal of the nonimpossibility correct? Might there be an ignorabimus in mathematics? With our modern knowledge of algorithmic unsolvability, an affirmative answer seems most likely. The evidence available to Hilbert in 1900 seemed to support his rejection of the ignorabimus. Our earlier statement extracted from Hilbert’s second problem calls for a resoundingly strong rejection of the ignorabimus, namely the following: General Programme. Give a complete axiomatic description of mathematics from which any mathematical assertion can be decided by finitely many logical steps. This General Programme is not the programme usually referred to as Hilbert’s Programme, but it is good to keep it in the back of one’s mind in the following. 3. Hilbert and Geometry History—even chronology—tends to have an annoyingly nonlinear character. Another, apparently independent, prologue to the Hilbert-Brouwer dispute was Hilbert’s work on the foundations of geometry. The earliest recorded comment of his on the topic was an aphoristic remark he made concerning a lecture on geometry he had just attended: “It must be possible to replace in all geometric statements the words point, line, plane by table, chair, mug.” This remark, encapsulating, as Hermann Weyl put it, “the axiomatic standpoint in a nutshell,” was made in 1891. In the next decade, Hilbert was to think deeply on matters both geometrical and foundational, lecturing several times on geometry and finally, in 1899, publishing his lectures on Euclidean geometry, the Grundlagen der Geometrie. A first approximation to an understanding of the foundational importance of Hilbert’s book on the foundations of geometry is to be had by comparing Hilbert’s work to Euclid’s Elements. Although Euclid’s Elements had been regarded as the epitome of rigour for two millenia, the fact is that Euclid’s treatment is very far from being rigorous; there are gaps in Euclid’s reasoning one could drive a truck through. It seems Euclid did not replace (the Greek equivalents of) “point”, “line”, and “plane” by “table”, “chair”, and “mug” (nor, as Hilbert also suggested, by “love”, “law”, and “chimney-sweep”); Euclid allowed himself to use evident properties of points, lines, and planes not explicitly assumed in his axioms. Hilbert’s treatment of geometry may be viewed as a stopping of Euclid’s gaps. Hilbert gave a complete axiomatisation of geometry and used only the axioms cited in his further development of the subject. His treatment was much more rigorous. Describing Hilbert’s Grundlagen der Geometrie as a rigorisation of Euclid’s Elements or as the first purely deductive treatment of geometry, is, however correct, a superficial description from the foundational point of view. As Hermann Weyl put it, “It is one thing to build up geometry on sure foundations, another to inquire into the logical structure of the edifice thus erected. If I am not mistaken, Hilbert is the first who moves freely on this higher ‘metageometric’ level.” The Grundlagen is a milestone in the history of the axiomatic method. It marks a break with the old axiomatic viewpoint and a full acceptance of the newer, not previously recognised, one.

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In the traditional Aristotelian approach to axiomatisation, the axioms are evident truths about the structure being axiomatised, and truth is borne from the axioms to the theorems via the proofs. In the 18th and 19th centuries, this view was slowly changing. In 1733, Gerolamo Saccheri attempted to derive Euclid’s parallel postulate from the remaining ones by assuming the negation of the postulate and deriving a contradiction therefrom. Mistakenly, he thought he had succeeded. What he had really done, and what was consciously done in the following century by various mathematicians (including Ferdinand Karl Schweikert, Carl Friedrich Gauss, Janos Bolyai, Nicolai Lobachevskii, and Bernhard Riemann), was to develop axiomatically a consistent non-Euclidean geometry. By the mid-19th century, with three major geometries and only one physical space to be described by them, it was clear that the geometries couldn’t all have true axioms. Whereas initially most mathematicians ignored the non-Euclidean geometries, those familiar with these new geometries recognised their coherence. While some asked which of the three gave a correct description of space, no one denied the significance of any of the geometries. This became especially true in the latter half of the 19th century, when models of the non-Euclidean geometries were found. A minor episode of a similar nature occurred in England in the early 19th century in response to questions of the validity of the use of negative numbers. In the face of such doubts, William Frend attempted an algebra of nonnegative quantities. Such an algebra is an awkward reversion to the practice of the 16th century by which, for example, the equations x3 = ax + b and x3 + ax = b are of completely different character (as a, b must be nonnegative) and had to be solved separately. In response to this, George Peacock proposed his “symbolical algebra”. In it, he acknowledged that he had no idea what “negative quantities” were and proposed an algebra not dealing with quantities at all. His was a purely symbolic system of algebraic manipulation based on simple rules (like commutativity, associativity, etc.). Further British developments—Hamilton’s invention of quaternions with their noncommutative multiplication, Cayley’s numbers with their commutative but not associative multiplication, etc.—revealed the extreme axiomatic freedom mathematics could offer. In short, there was a shift from the Aristotelian conception of axiomatics by which axioms were true about a given structure to the modern mathematical conception by which a consistent set of axioms determines the subject matter under study. Hilbert’s Grundlagen der Geometrie is the first major work written in this completely modern spirit; it barely casts a nostalgic look at the old practice. Central to the book is not so much an axiomatic development of geometry but a study of the axiomatics of geometry. The axioms are divided into various groups, and the relevance of the various axiom groups for certain important results is examined. Further, the axioms are proven independently, and the consistency of the system as a whole is established by modelling the geometric axioms in the arithmetic of the reals. When the definitive account of the Hilbert-Brouwer dispute is finally written, it will devote an entire chapter to the Grundlagen der Geometrie and Hilbert’s resulting correspondence on this book with the philosopher Gottlob Frege, for it is in this correspondence that Hilbert, as far as his lack of patience with philosophy allowed him, explained his views on axiomatics and what he intended to accomplish. Given

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the limitations of space, I shall merely offer two quotes from Hilbert’s responses to Frege. These quotes do not encompass the full range of Hilbert’s thoughts on the matter and are not truly representative of his views; they do, however, illustrate how modern his outlook was. The first is, more or less, his dismissal of Frege and ends their correspondence on the subject: My opinion is simply this, that a concept can only be logically fixed through its relations to other concepts. These relations, formulated in precise statements, I call axioms and I add, that the axioms (possibly including the giving of names to the concepts) are the definitions of the concepts. I have not thought out this view to pass the time, but rather I saw myself pressed to it through the demand of rigour in logical inference and in the logical construction of a theory. I have come to the conviction that in mathematics and the natural sciences one can handle more subtle things with security only in such a way; otherwise one merely turns in circles.

So, rigour requires the use of the axiomatic method, but what check is there on the axiom systems? Hilbert had answered this question some months earlier: If the arbitrarily given axioms do not contradict one another with all their consequences, then they are true and the things defined by the axioms exist. This is for me the criterion of truth and existence.

In his own axiomatic work, Hilbert allowed himself some liberties in choosing the axioms—particularly in relaxing the Aristotelian requirement of the evident truth of the axioms—but he was far from arbitrary in such choice of axioms. He had, if not criteria, guidelines in the selection of axioms. Completeness and simplicity were two desiderata he cited in the introduction to the Grundlagen; consistency was, of course, another. None of these desiderata were entirely unproblematic. Hilbert’s knowledge of logic, indeed logic itself, was then too vague for us to determine exactly what Hilbert meant at the time by (axiomatic) completeness. Hilbert’s own axiomatisations are complete in that they are categorical: any two models are isomorphic. In the last section I cited his remark on completeness from the second of his problems in his 1900 Paris lecture calling for the setting up of an “exact and complete description” of whatever science one happens to be studying, be it geometry or arithmetic or physics. The General Programme I ended the section with took the “exact and complete description” to be complete enough to decide the truth or falsity of every statement. Semantically, such completeness follows from categoricity; syntactically, such completeness—derivation from the axioms by finitely many logical steps—fails, as would eventually be shown by G¨ odel. Ultimately, Hilbert would definitely mean completeness in this syntactic sense: Every sentence or its negation would be derivable from the axioms in finitely many steps. Possibly the only one who understood the distinction between syntax and semantics at the turn of the century was Gottlob Frege; the equivalence—for first-order logic—would first be posed by Hilbert as a problem in the late 1920s and solved by G¨ odel in 1929. Hilbert’s axiomatisation of geometry in the Grundlagen and his axiomatisation of arithmetic published in 1900 were not first-order. Indeed, each had a second-order Archimedean axiom and both (the axiomatisation of arithmetic and the axiomatisation of geometry given in later editions of the Grundlagen) had a “completeness axiom” asserting that the structure under consideration was maximal with respect to the remaining axioms; i.e. no proper extension of the structure

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could satisfy the remaining axioms. This “completeness axiom” is not even properly a second-order axiom, but goes beyond that logic. What Hilbert meant by the simplicity of an axiomatisation is unclear and not relevant to our discussion. I remark only that both in the Grundlagen and in his paper on axioms for real arithmetic, he paid close attention to the dependence and independence of the axioms given, but that, given the elegance of his work, “simplicity” must have had an aesthetic component as well. Although the meaning of consistency was clear—a theory is consistent if it doesn’t derive two contrary assertions (equivalently, if it leaves some assertion unproven)—the necessity of and means for proving consistency were unclear. The attitude of most mathematicians, even today, is that since the axioms are true and derivations preserve truth, consistency is obvious. Frege had said as much in his correspondence with Hilbert. He had not experienced, as Hilbert had, the opposition to the results of mathematical proof nor (apparently) the strong rejection by Kronecker of the infinite in mathematics—which rejection would preclude any proof of the consistency of theories of the infinite by reference to models. The problem, as Frege asked Hilbert, was: is there any way of proving consistency other than by exhibiting a model? Earlier mathematicians had proven the consistency of non-Euclidean geometry by modelling it in Euclidean geometry, and Hilbert had reduced the consistency of the latter to that of his axioms for the arithmetic of the reals. In the second of his 23 problems at Paris, Hilbert reiterated this and added that a “direct proof” of the consistency of this arithmetic was now needed. The only comment he offered on how such a proof could be given was so vague (“I am convinced that it must be possible to find a direct proof for the compatibility of the arithmetical axioms, by means of a careful study and suitable modification of the known methods of reasoning in the theory of irrational numbers”) that it is impossible to decide whether or not his later attempts conform with this remark. His earliest consistency proofs follow a different line. Before Hilbert published his first direct consistency proofs, Bertrand Russell discovered his famous paradox of the set of all sets which are not elements of themselves. If R = {x : x ∈ / x}, then R ∈ R iff R ∈ / R. Russell communicated his result to Frege, in whose axiomatic system the paradox was derivable. Frege went ahead with the publication of the second volume of his book on the inconsistent system, but added an appendix including Russell’s paradox and acknowledging there to be serious flaws in his work. News of Russell’s paradox created quite a stir. Philosophers uncovered newer paradoxes, and even an occasional mathematician took them seriously. Richard Dedekind, whose Was sind und was sollen die Zahlen? used the set of all sets, delayed publication of the third edition of his book which was due to come out in 1903 until 1911 because he was concerned about the foundations of his treatment. Hilbert was largely unperturbed. Writing to Frege in 1903 in thanks for a copy of Frege’s book, he noted that his student Ernst Zermelo had shown Hilbert the Russell paradox some 3 or 4 years earlier and he himself had known other, more convincing, ones for 4 to 5 years. Indeed, we now know that Georg Cantor had written

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to him as early as 1897 to explain that there is no set of all cardinal numbers—a fact Hilbert cited in his 1900 paper on axiomatising arithmetic and in his 1900 problems address. The paradoxes convinced Hilbert that, just as the paradoxes in calculus had engendered a need for rigour in that field, one needed rigour in all foundational work. As Hilbert had said in his problems paper, “the requirement of logical deduction by means of a finite number of processes is simply the requirement of rigour in reasoning. Indeed the requirement of rigour, which has become proverbial in mathematics, corresponds to a universal philosophical necessity of our understanding.” The paradoxes also convinced Hilbert that, contrary to Frege’s programme and what would become Russell’s programme of reducing mathematics to logic, logic alone was insufficient for the foundations of mathematics. 4. First Steps In August 1904, at another International Congress of Mathematicians, this one in Heidelberg, Hilbert offered his first examples of consistency proofs. Because logic already uses arithmetic notions, Hilbert insisted on simultaneously developing logic and arithmetic. The fragments of mathematics shown to be consistent are also logically fragmentary and the results are quite unimpressive. They do, however, illustrate both his approach and its weakness, and it will be instructive to consider an example. We begin by specifying a (non-first-order) language as follows. First, there is one constant 1 and three function symbols: I (for an infinite set), S (for successor),  and S (an accompanying operation); there are no variables. Terms are built up in the usual way. The only relation symbol is that for equality, and equations between terms constitute the atomic formulæ. Equations can be negated. The axioms and rules of inference are as follows: for any terms t, u and any formulæ ϕ, Axioms. t=t S(It) = I(S  t) ¬S(It) = I1 Rules. t = u ϕ(t) S(It) = S(Iu) , . ϕ(u) It = Iu Theorem 1. The system described is consistent. Proof. Define an equation t = u to be homogeneous if t and u contain the same number of symbols (not counting parentheses, which are used only for readability’s sake). A simple induction on the number of inferences applied shows any derivable equation to be homogeneous. (Such also shows that the rule of substitution of equals need be applied only to equations in deriving equations.) It follows that S(It) = I1 is not derivable, and the system is consistent.  The range of the function I, i.e. the set of terms I(1), I(S  1), I(S  S  1), . . ., is infinite in a weak sense: The addition of any equation of distinct such terms to the system results in an inconsistent system. Had the system included some logic, this would have meant the derivability of the inequations; as formulated, however, the system cannot prove even ¬I(S  1) = I(S  S  1). Nonetheless, Hilbert maintained that Theorem 1 proved the existence of an infinite set and that this method showed the incorrectness of the widely held view that the elements of a set must have their

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existence prior to the existence of the set itself. Through the axiomatic method, both the set (here represented by I) and its elements (here represented by the terms I(t)) come into existence simultaneously with the proof of consistency. The system of Theorem 1 was intended only as an example, and we should thus be careful in criticising the proof for what it doesn’t accomplish. That the system does not accommodate simple propositional logic (i.e. logic without quantifiers) is not a critique but a desideratum for extending the result. Similarly, that it does not include any arithmetic beyond the trivial successor function ought not to be put forth as an objection; more arithmetic can be added to the system with the extension then to be proven consistent. Hilbert viewed mathematics and foundations as an open-ended enterprise. One would start with something like the infinite set I and its proof of consistency and add additional structure and axioms, proving the consistency of such as one went along. Indeed, in his paper of 1904, Hilbert attempted an example of the extension of the system by adjoining an abstract set and some rules of inference, but this adjunction is (at least to me) not very convincing. One real criticism—valid for practically all of Hilbert’s contemporaries as well— is that Hilbert was fundamentally confused about the nature of logic. He included the notion of set in logic, held second-order axiomatisations like Dedekind’s as paradigms, and spoke of logical derivations proceeding in finitely many steps. Yet the system he treated was virtually logic-free and when he did mention adding some logic to the system, it was purely propositional logic. At this point in time, only Frege had any experience in working within a formal logical system—and his was inconsistent. At the end of the decade, Bertrand Russell and Alfred North Whitehead would be working on Principia Mathematica, a monumental case study of the formalisation of mathematics in logical systems. But it really wouldn’t be until the 1920s that the nature of the quantifier and the distinction between firstorder logic on the one hand and second-order logic and set theory on the other would be understood. A good part of this understanding would come from Hilbert and his disciples. Having said all of this, it must yet be stated that Hilbert had achieved something—he had shown how the consistency of a “theory” could be recognised without the construction of a model—and it must yet be stated that there was one bit of criticism that was levelled and that Hilbert could not at the time answer. This was Henri Poincar´e’s observation that the proof of Theorem 1 used induction: Hilbert’s stated goal of providing a “rigorous and completely satisfying foundation for the notion of number” rested on the notion of number itself. For Poincar´e, induction was a special mathematical intuition which did not need any justification—a fortunate thing since any justification, like Hilbert’s, required induction and was therefore circular. It would be over a decade and a half before Hilbert offered a formal response to Poincar´e’s critique, and by then Poincar´e would be a side issue, for, by then, Hilbert would already be locked in battle with Brouwer. 5. Enter Brouwer The contrasts between Hilbert and Luitzen Egbertus Jan Brouwer were many. Hilbert had little patience with philosophy, his own philosophy of mathematics being perhaps best described as na¨ıve optimism—a faith in the mathematician’s ability to solve any problem he might set for himself. His interest in the foundations of mathematics was largely a concern for rigour—establishing the rules of the game

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as it were; his proposal of consistency as the criterion of existence was ostensibly a libertarian suggestion intended to allow maximum freedom to the mathematician. As he was eventually to say, his foundational goal was to abolish foundational questions once and for all, i.e. to allow mathematicians to get back to business-as-usual. Brouwer, on the other hand, was as much a philosopher as a mathematician and was deeply interested in the foundations of mathematics for its own sake. Moreover, unlike Hilbert, who took the central problem of foundations to be the clarification and justification of (then) current practice, Brouwer took the epistemological problem to be central. According to Brouwer we have one fundamental mathematical intuition and mathematics is the working out of the consequences of this intuition. Those parts of mathematics that cannot be based on this fundamental intuition must be rejected. I must be careful here in referring to Brouwer’s rejection of parts of mathematics. Brouwer was not the cardboard replica of Kronecker that Hilbert was to caricature him as. When in 1912 Brouwer became Extraordinary Professor in Amsterdam with the recommendations of Hilbert and Felix Klein in G¨ ottingen and Poincar´e and Emile Borel in Paris, he chose as the topic of his inaugural address the two philosophies, intuitionism and formalism, represented by him and Hilbert, respectively. A quote from the English translation of the following year will illustrate the dispassionate style of the prose and the level of objectivity Brouwer could bring to bear on such issues: On what grounds the conviction of the unassailable exactness of mathematical laws is based has for centuries been an object of philosophical research, and two points of view may here be distinguished, intuitionism (largely French) and formalism (largely German). In many respects these two viewpoints have become more and more definitely opposed to each other, but during recent years they have reached agreement as to this, that the exact validity of mathematical laws as laws of nature is out of the question. The question where mathematical exactness does exist is answered by the two sides; the intuitionist says: in the human intellect, the formalist says: on paper.

Another quote: In the domain of finite sets in which the formalist axioms have an interpretation perfectly clear to the intuitionists, unreservedly agreed to by them, the two tendencies differ solely in their method, not in their results; this becomes quite different however in the domain of infinite or transfinite sets.

There are points of agreement: The notion of a “ ‘denumerably infinite ordinal number’ . . . has a clear and well-defined meaning for both formalist and intuitionist.” However, the theory of higher cardinal numbers has no meaning for the intuitionist and “illustrates so clearly the impassable chasm which separates the two sides.” Following some details on this separation, he concludes with “Thus my exposition of the fundamental issue, which divides the mathematical world. There are eminent scholars on both sides and the chance of reaching an agreement within a finite period is practically excluded.” Like Kronecker, Brouwer was led to constructivity and the rejection as meaningless of certain parts of mathematics. Whereas Kronecker’s rejection had been externally political, we find Brouwer simply stating his views and acknowledging, with typically Dutch tolerance, that there are other views. Brouwer’s writings would not become polemical until it was time to respond to Hilbert’s polemics.

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Brouwer formed his philosophical views in his youth and never changed them. They were pessimistic to say the least and included a strong disapproval of “extrovert science” (read: technology), the goal of which was to conquer nature. His dissertation, begun at the age of 23 in 1904, was largely an attempt to reconcile ole of mathematics as the chief tool of extrohis interest in mathematics with the rˆ vert science. The excessive philosophical content—and its excessively pessimistic nature—alarmed his advisor D.J. Korteweg (now remembered primarily for the Korteweg-de Vries equation in partial differential equations), and a small battle over the contents of the thesis ensued. When Brouwer finally published his thesis in 1907, the offending material was gone. Brouwer’s connection with Hilbert may be said to have begun with this thesis. Not only do some of the problems tackled come directly from Hilbert’s problem list, but—more pertinent to our purposes—the thesis was on the foundations of mathematics, and Brouwer offered critiques of Hilbert’s work in the area, both the Grundlagen der Geometrie and Hilbert’s 1904 paper on consistency proofs. I will not repeat these criticisms here as they are not central to our story. For one thing, Brouwer criticised aspects of Hilbert’s 1904 paper that I have not covered because the paper led in a direction Hilbert was never again to follow. One point perhaps worth mentioning is that the style of the criticism is that of a self-confident young man: Brouwer states simply where and how Hilbert is confused or outright wrong. For example, he says, “Making a mathematical study of linguistic symbols. . . can teach us nothing about mathematics,” and, “the most uncompromising conclusion of the methods we attack, which illustrates most lucidly their inadequacy, has been drawn by Hilbert.” The matter-of-fact style applied to matters of opinion was inappropriate, perhaps mildly offensive, but hardly hostile. The following year (1908) Brouwer published two articles that must be cited here. The first of these, entitled “Die moeglichen Mæchtigkeiten” [The possible powers], cited our one mathematical ur -intuition and its application to the development of the denumerable infinite both in the form of the order type of the natural numbers (iterate “next”) and that of the order type of the rational numbers (iterate “between”). He pointed out that such iteration allows only the construction of countable sets. As there occur higher powers in mathematics, there is a problem of reconciling this difference. According to Brouwer, there are two ways in which higher powers occur—in diagonalising and in constructing the continuum. In the first case, one really has a method and not a set. For example, constructing newer and larger countable ordinals from any sequence of countable ordinals does not show the uncountability of the set of countable ordinals but exhibits a method of construction of larger ordinals and shows, incidentally, that the set of such ordinals does not exist. As for the continuum, i.e. the real number line, we can compare it to the collection of all paths through the infinite binary tree. This tree really contains only denumerably many nodes, not uncountably many paths. While individual paths may exist, their entire collection does not, only the countable tree exists. The “collection” of paths ought to be thought of not as a set which is an object, but as a matrix in which to place objects. The continuum is similarly a matrix and not a set. Coming in 1908, Brouwer’s interpretation was not so revolutionary as it might now seem. Poincar´e similarly rejected the uncountable; Zermelo’s axioms for set

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theory were just appearing that year; and Russell and Whitehead’s Principia Mathematica, in which logic was a theory of types rather than sets, was yet to appear. Sets had not yet disengaged themselves from properties, and the paradoxes of set theory had not yet been resolved. In short, the situation was completely unsettled and many solutions were being proposed. Brouwer’s solution to the problem of higher powers was as plausible as anyone else’s. It does illustrate, however, the direction his research was taking and the extent to which his philosophy would lead him: he was willing to jettison the majority of Cantor’s theory of cardinal numbers. The other paper of 1908 that I wish to mention is more relevant to our discussion and illustrates more dramatically the depth of Brouwer’s thought and of his commitment to his philosophy. Entitled “De onbetrouwbaarheid der logische principes” [The unreliability of the logical principles], it is a paper worth reading. If in the former paper Brouwer was willing to jettison uncountable sets, he was now questioning the laws of logic. Centuries earlier, Galileo had already shown that not all properties of finite cardinals carry over to the infinite case. Brouwer now suggested that among the lost properties one might find some logical laws—in particular, the Law of the Excluded Middle, or tertii exclusi (or tertium non datur ): ϕ ∨ ¬ϕ. According to Brouwer, to assert ϕ ∨ ¬ϕ, i.e. that ϕ is either true or false, one must either have a construction accomplishing the task demanded by ϕ or a construction that stops the process of performing this task. It was by no means clear to Brouwer that this could always be done: the question of the validity of the principium tertii exclusi is equivalent to the question whether unsolvable mathematical problems can exist. There is not a shred of proof for the conviction, which has sometimes been put forward, that there exist no unsolvable mathematical problems.

The point here is that Brouwer placed mathematics in the intellect: to be true or false, a mathematical assertion had to be known to be true or known to be false. With infinite sets (as well as infinite methods and infinite matrices) we cannot necessarily perform the infinitely many verificational tasks necessary in finitely many steps; the principle of mathematical induction would occasionally allow this for denumerable collections, but would be of no use elsewhere. Brouwer never was to assert that the Law of the Excluded Middle was false in the sense that ¬(ϕ ∨ ¬ϕ) was true, because this was contradictory. In other words, he accepted ¬¬(ϕ∨¬ϕ)— whence the consistency of the Law of the Excluded Middle. But he distinguished between “provable truths” and “provable non-contradictories”. Eventually, as he refined his intuitionistic development, he would assert things like ¬∀F (∃x(F x = 0) ∨ ¬∃x(F x = 0)), which appears absurd at first glance. On a constructive-intuitionistic reading, however, it asserts the unassailable proposition that it is impossible to construct a uniform algorithm which, given a function F , will tell us whether or not F has a zero. The questioning of the validity of the Law of the Excluded Middle and the introduction of the dichotomy between truth and consistency stand in clear opposition to Hilbert’s views, but—despite all my advance remarks—neither they nor the remarks of Brouwer’s dissertation ought to be taken as the opening shots in the battle to come. Hilbert was the leading mathematician of the day, and he had spoken on

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the foundations of mathematics. Ergo, Hilbert was the yardstick by which all serious foundational remarks would be measured. Anyone putting forward original views on the subject would have to discuss them in light of Hilbert’s views. [Besides, Brouwer published his dissertation and this paper in Dutch; for effect he would have to have published in French or German.] The next few years saw two major developments in our story: Brouwer’s growing friendship with Hilbert and the spread of Brouwer’s fame. Let us begin with the second of these. Under some coaxing from Korteweg, Brouwer shifted much of his energy from foundations to mathematics itself. He did this with such success that by the time of his inauguration as extraordinarius in 1912, he was one of the leading mathematicians of the day. This rise is reflected in the recognition accorded him: 1907— 1909— 1912— 1913— 1915—

Ph.D. Privaatdocent (unpaid lecturer) at the University of Amsterdam Extraordinary Professor; member of the Royal Academy Ordinary Professor Associate Editor of Mathematische Annalen.

Two episodes of this development will give us a hint as to Brouwer’s personality. The first of these also introduces a minor character who was again to be a minor character in the later fight—Otto Blumenthal, the managing editor of the Annalen. Mathematische Annalen was at the time one of the leading mathematical journals, and Brouwer published several of his important early papers in topology in it. One of these was his proof of the invariance of dimension. When Brouwer submitted this paper to the Annalen, Blumenthal was visiting Paris and the subject came up in a conversation with Henri Lebesgue, who remarked to Blumenthal that for some time he had had a number of proofs of the result. Blumenthal did not understand Brouwer’s proof, nor did he understand the proof Lebesgue sketched for him; but such was his faith in Lebesgue that when Brouwer’s paper appeared it was immediately followed by Lebesgue’s simpler proof in the form of an extract from a letter from Lebesgue to Blumenthal. Lebesgue’s proof was simply wrong. Blumenthal had treated Brouwer unfairly, and Brouwer’s priority on this obviously major result was needlessly in question— and all because neither Lebesgue nor Blumenthal had a sufficient grasp of the complexity of the problem. Brouwer’s reaction betrayed no appreciation of the irony of the situation. He immediately submitted another paper to the Annalen, the tone of which Blumenthal found “unfriendly and unpleasant.” The paper went unpublished. However, Brouwer did manage to get a few digs against Lebesgue into print that year (1911): In one paper he remarked in a footnote that Lebesgue’s proof was incorrect, and in a second paper he omitted the proof of an important lemma, noting that Lebesgue was going to publish such knowing full well the result lay beyond Lebesgue’s reach. This latter was not mere impishness, but a tactic aimed at forcing Lebesgue’s public acknowledgement of mathematical impotence. The priority battle continued; as late as 1924 Brouwer was attacking Lebesgue in print. [The whole fight was pointless: Priority was predictably assigned along national lines, the French lining up behind Lebesgue and the Germans behind Brouwer. Now that both parties are dead, everyone agrees that Brouwer had priority and that Lebesgue’s proof was incorrect but that it had the germ of an important idea.] The obvious causes of an excessive and inappropriate response like Brouwer’s— youth, financial insecurity, ambition—were all there. But they do not fully explain

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his actions, for he was perennially embroiled in battles of this kind. The facts are that Brouwer was a highly emotional man who (i) had a rigid conception of right and wrong, and (ii) never completely matured emotionally. In theory Brouwer hated all people; in practice he enjoyed the company of others and formed a number of friendships. However, when one of his friends or acquaintances behaved in a manner consistent with his general theory of human behaviour, Brouwer seems to have been taken completely by surprise and to have reacted emotionally and strongly. A second anecdote of this period illustrates Brouwer’s immaturity and childish egocentrism. In 1913, Brouwer vacillated between keeping his extraordinary professorship in Amsterdam and answering the call to the more desirable ordinary professorship—but in the more provincial Groningen. Brouwer stuck with Amsterdam. Subsequently, in one of the rare altruistic gestures of mathematics, Korteweg vacated his chair and took the lesser position of extraordinarius so that Brouwer could be elevated to ordinarius in Amsterdam. [There is an apocryphal story that Isaac Barrow did the same for Newton, but his motives were different.] Brouwer soon thoughtlessly complained to Korteweg of all the work involved! As I remarked earlier, Brouwer’s growing acquaintance with Hilbert was another major development of the years 1909-1919 preceding the battle. They met in the summer of 1909. Writing to a friend in November, Brouwer described the experience thus: This summer the first mathematicus of the world was in Scheveningen. Through my work I was already in touch with him, [but] now I have over and over again walked with him, and talked as a young apostle with a prophet. He was 46 [actually: 47], but young in spirit and body, swam powerfully and climbed with the greatest pleasure over walls and fences of barbed wire. It was a beautiful new shaft of light through my life.

What did the young apostle and his prophet talk about? The mathematics in the ensuing correspondence—at least that which has been published—is topology. Later, Brouwer would maintain—in a footnote—that they discussed the foundations of mathematics. The only other comment on these discussions I know of is in a letter from Brouwer to Hilbert of 28 October 1909: “If I come into the dunes, I think always on our beautiful excursions, and for me such instructive and stimulating conversations.” It is evident from the above remarks that Brouwer had a very high opinion of Hilbert. Hilbert may have been something of a father-figure to Brouwer: When, in 1913, Brouwer was a young extraordinarius trying to decide whether or not to let Groningen spirit him out of Amsterdam with its offer of an ordinarius, he wrote to Hilbert for advice. Hilbert could not possibly have had as high an opinion of Brouwer as the latter had of him. He was, after all, the “first mathematicus of the world” and was intelligent enough to realise it. But he could and did appreciate talent. As I’ve already mentioned, in 1912 Hilbert was one of those recommending Brouwer’s promotion to the position of extraordinary professor. Moreover, between 1909 and 1919 Brouwer visited G¨ ottingen frequently and made numerous acquaintances there; it was, so to speak, his second scientific home. And, in 1919, on the eve of their battle, Hilbert tried to make G¨ ottingen Brouwer’s first home through the offer of a professorship— not the kind of offer lightly or easily made.

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Whether their relationship was purely professional or a genuine friendship is hard to say. Their first—and evidently prolonged—meeting was at the seaside resort of Scheveningen in the Netherlands and it was a meeting of families, not just men. Brouwer’s letters of the period seem friendly enough—the one cited above continues to say how nice the weather by the sea had been, to express concern about Hilbert’s news that Mrs. Hilbert’s knee had not recovered, and to introduce some comments Brouwer’s wife was to add. This certainly suggests friendship, but in those pre-telephone days letter writing was more of an art than it is today, and I am hesitant to read too much into such remarks. I note, for example, that scarcely a century earlier Thomas Carlyle and Johann Wolfgang von Goethe had a similar correspondence—right down to the womens’ appending their own comments to the letters—and yet the Carlyles and Goethes never met. Indeed, while Carlyle was expressing to Goethe his undying friendship and deep devotion, he was also writing a friend for an opinion of Goethe’s mental state. On the other hand, one of Brouwer’s psychological make-up is more prone to exaggeration than dissimulation, and I am inclined to believe Brouwer’s feelings, however overexpressed, to be genuine enough. Hilbert’s feelings remain a mystery to me. I have as yet said nothing about Hilbert’s personality. This remains something of a mystery to me. Constance Reid’s biography gives one the impression of a generally nice man, as does Blumenthal’s hagiography published in Hilbert’s collected works. Blumenthal had been Hilbert’s first doctoral student, receiving his degree in 1898, and he paints a picture at odds with the image of the later Hilbert we will encounter. Our Hilbert was in his 60s. It was the decade in which he simply killed the career of his student Wilhelm Ackermann because the latter married too early for Hilbert’s taste. Indeed, it has been reported that students of this later generation today sit up straight at the mention of his name. The image of the dictatorial German professor may thus be the one we should keep before us. And the poor old Hilbert haunted by the spectre of Kronecker, as I suggested some pages back? There may be some of that too. Barring genuine historical research, I can only suggest these possibilities and caution the reader that Hilbert’s assault on Brouwer will not reveal him in the best possible light. But enough about personalities! Let us get back to our story. From the time of his thesis until that of his inauguration in 1912, Brouwer devoted most of his energy to topology. With the security of his job, Brouwer returned to his foundational interests, but not completely: He continued his topological studies into the mid-1920s and even afterwards kept in touch through his young assistants and international visitors. However, he did return to his foundational work, and in 1918 and 1919 he published, in two parts, an important paper, “Begr¨ undung der Mengenlehre unabh¨ angig vom logischen Satz vom ausgeschlossenen Dritten” [Founding of set theory independently of the logical principle of the excluded middle]. As Brouwer was to say in an accompanying paper—a sort of commentary on the present one— his previous writings on intuitionistic mathematics had merely been fragmentary and his non-philosophical, mathematical work had—although he had strived for results which could be established intuitionistically—been proven in the classical manner. In the present paper, he actually gave an intuitionistic development that went some distance. This success was to be his undoing.

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6. Back to Hilbert Hilbert during this period was primarily occupied with mathematical physics, but in 1917 he did three things that concern us here. In the spring he hired Paul Bernays as his assistant, in the fall he gave a lecture in Z¨ urich entitled “Axiomatisches Denken” [Axiomatic thinking], and in the winter semester (1917/1918) he gave a course on “Principles of mathematics and logic” in G¨ ottingen. Bernays had impressed Hilbert with his knowledge of philosophy and would become the latter’s primary logical co-worker in the years to come. Bernays was an extremely modest man and kept mostly in the background; he will be underrepresented in the sequel. Of the three occurrences of that year, the one I wish to discuss is the Z¨ urich lecture. This lecture, presented on 11 September 1917 and published the following year in Mathematische Annalen, is a transitional one lying somewhere between his attempt of 1904 and his future programme to be undertaken with the modest Bernays. Mathematically, it is the least of Hilbert’s papers on foundations: it attempts nothing and announces no attempts. It is primarily a paean to the axiomatic method and sings of the victories the method had had in mathematics and physics. Towards the end, however, there are some points worth mentioning. With respect to inconsistency, Hilbert cited Cantor’s paradox of the set of all sets (which couldn’t possibly be smaller than its power set) and noted that, because of it, respected mathematicians like Kronecker and Poincar´e denied the legitimacy of set theory—and this despite the fact that set theory was “one of the most fruitful and most powerful knowledge-branches of all of mathematics.” Fortunately, said Hilbert, Zermelo had saved the day by axiomatising set theory. [Tiny commentary: (i) Hilbert did not cite Russell’s paradox—the one usually cited—because he had long known Cantor’s earlier one; (ii) Kronecker objected to set theory long before the paradoxes were known; (iii) Hilbert did not mention Brouwer; and (iv) Zermelo’s axioms seem not to have been motivated by the paradoxes but by the desire to clarify his proof of the Well-Ordering Theorem. It would be the later explanation of the cumulative hierarchy of sets as an intuitive model of Zermelo’s axioms that would rescue set theory.] As for proving consistency, Hilbert cited a few examples of consistency proofs by interpretation—e.g. his proof of the consistency of geometry by interpretation of his axioms in a theory of the arithmetic of the reals and the consistency of the latter theory by interpretation in the (second-order) theory of the arithmetic of the integers. For the arithmetic of the integers and set theory, however, the only possibility left along these lines was to reduce these subjects to logic. Such a reduction, begun by Frege, had been, Hilbert said, accomplished by Russell, whose axiomatisation of logic could be viewed as the high point [Kr¨ onung] of work on axiomatics. Four years later Hilbert would definitely be of the opinion that the work of Russell (and Whitehead) was insufficient and one needed a better consistency proof. At this time, however, he had a few other things to say. Related to the consistency problem were 5 other epistemological questions of a “specifically mathematical colour”: i. the problem of the solvability in principle of each mathematical question, ii. the problem of the additional [nachtr¨ aglichen] controllability of the results of a mathematical investigation, iii. the question of a criterion of simplicity of mathematical proofs,

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iv. the question of the relation between content and formalism in mathematics and logic, and v. the problem of the decidability of a mathematical question through a finite number of operations. Of these, the only one he elaborated on was the fifth, which he remarked was the best known and most frequently discussed. The first example he cited was his own solution to Gordan’s invariant problem. His first solution solved the problem but didn’t allow the effective calculation of the invariants; for this one needed his second solution. Finally, just before a short summary of his views, Hilbert noted that problems like this fifth one of the decidability via finitely many operations of mathematical questions seemed to require a new field of research, one in which proofs themselves were the objects of investigation. He added that the execution of this programme was yet an unaccomplished task. He did not add that he would soon be working on this task, but, given his hiring of Bernays, it can be assumed he was inclining toward, if not yet definitely preparing for, such a task. [Indeed, as I have already mentioned, Hilbert gave a course on “Principles of mathematics and logic” in G¨ ottingen during the ensuing winter semester.] I should say a few words about Hilbert’s questions i-iv, particularly the ones the meanings of which are not obvious, i.e. i, ii, and iv. Question i will be discussed shortly with Brouwer’s reaction to Hilbert’s paper, and question iv will be clarified in discussing Hilbert’s further papers. This leaves us with question ii and the meaning of the “controllability of the results of a mathematical investigation.” Hilbert was to repeat this phrase and even give something of an example of this controllability. Basically, he was referring to the practice of accountants of controlling, or partially checking, their calculations. Indeed, according to Samuel Johnson’s dictionary (admittedly not the most up-to-date source, but I am so taken with this explanation that I have not subjected it to the control of a more modern etymologist), the word “control” (in German: Kontrolle) derives from the word “counterol”, indicating just this practice of accountants (sitting at their counting boards with their copper counters) of checking their results. With pencil and paper computation, one can cite the technique of casting out nines as a control for large sums. Hilbert appears to be calling for something similar for all mathematical investigations, not just arithmetic calculations. I remarked earlier that Brouwer published a separate commentary on his 1918/1919 paper on intuitionistic set theory. In it he began by remarking that since 1907 he had in several publications defended two theses: (i) that the chief setexistence axiom of set theory (the Comprehension Axiom) was—even in Zermelo’s weaker form—inadmissible, and (ii) that Hilbert’s Axiom of the Solvability of Every Problem was equivalent to the Law of the Excluded Middle and was impermissible as a means of proof. In a footnote, he commented on Hilbert’s 1917 paper: This proposition, an axiom to Hilbert in 1900, was now—as question i—considered by Hilbert to be an open problem. To Brouwer, who saw the question as an open problem in 1908, the principle was now simply false (albeit not in the strong sense of being contradictory). Moreover, to Brouwer, Hilbert’s questions i and v were the same—solvability (in principle or in practice) meant solvability by a finite number of operations. In particular, he did not accept Hilbert’s assessment of what Hilbert’s first solution to the invariant problem had and had not accomplished. Brouwer

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said simply that proving a set (here: the set of invariants) not to be infinite was not the same as proving the set to be finite. In effect, Brouwer said that Hilbert’s first proof did not establish the existence of a finite number of invariants, just as Gordan had hinted years earlier. Although it had been written in German, Brouwer’s 1918/1919 paper appeared in a Dutch journal. The commentary was published in both a Dutch and a German journal, the latter being the Jahresbericht der Deutschen Mathematiker-Vereinigung [“Annual report of the German mathematicians-union”, hereafter simply Jahresbericht]. This was the main organ of the German Mathematical Society, and every German mathematician would see it. In September of 1920 Brouwer lectured at a meeting of natural scientists in Bad Nauheim, in Germany, on the question of the intuitionistic existence of decimal expansions of all real numbers. He published the paper with its negative answer to this question the following year in Mathematische Annalen. The single direct reference Hilbert would make to one of Brouwer’s papers would be to this one; everything else Hilbert would say about Brouwer would be about a Brouwer filtered through Hermann Weyl. 7. Weyl Stirs Things Up It is now time for Weyl to enter our story. Weyl, who had got his degree under Hilbert, was one of the leading young mathematicians of the day. Like Hilbert he worked in mathematical physics as well as in mathematics, and like Brouwer he was something of a philosopher as well. As a philosopher, he took the paradoxes more seriously than the average mathematician. In his view, paradoxes in the more remote regions of mathematics were symptomatic of a deeper problem in ordinary mathematics. His diagnosis of the problem agreed with Russell’s and his solution was in line with Poincar´e’s: the paradoxes arose from circular reasoning, particularly in using impredicative definitions— definitions in which the object being defined is defined in terms of some collection in which the object is included. This naturally leads to vicious circles. For example, the definition of the set R of Russell’s paradox, R = {x : x ∈ / x}, is impredicative in that R itself is an element of the possible range of its values. That this is bad manifests itself when one asks if R ∈ R. To determine the truth of this statement, one must check if R ∈ / R, which of course presupposes one to have checked if R ∈ R, thus taking one full circle. Like Poincar´e, Weyl proposed to base analysis on the natural numbers and predicatively defined sets thereof. His notion of predicativity was, however, extremely narrow and his reconstruction of analysis did not go as far as he would have liked. In 1918 Weyl published a short monograph, Das Kontinuum, containing his attempted reconstruction of analysis. In the following year he published a note entitled “Der Circulus vitiosus in der heutigen Begr¨ undung der Analysis” [The circulus vitiosus in the modern foundation of analysis]: Modern analysis was based on vicious circles and was in danger of collapsing under the weight of paradox. As late as 1930 he was to say, “It is true that so far no actual contradictions in analysis proper have resulted; we do not completely understand this fact at present.” Of greatest immediate importance for us, however, were his lectures in 1920 in the mathematics colloquium in Z¨ urich. Published the following year under

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¨ the title “Uber die neue Grundlagenkrise der Mathematik” [On the new crisis in the foundations of mathematics], these lectures caused quite a stir and roused Hilbert to action. Z¨ urich, where Weyl was a professor (ordinarius) at the time, was not Weyl’s only pulpit. On 11 May 1920 he had, for example, lectured on “Das Kontinuum” in Hilbert’s G¨ ottingen, and on 28-30 July he gave a series of lectures on his new foundation of analysis in Hamburg. His Z¨ urich lectures and the paper to come out of them had, however, a bigger effect. Everything about the paper was provocative. Weyl himself would later refer to the style as “bombastic”, reflecting the excitement of the postwar years. With less caution than Brouwer’s “I believe”, that one cannot apply the Law of the Excluded Middle, Weyl at one point says, “But this standpoint is absurd and untenable.” Moreover, at one point—after dramatically announcing he was abandoning his own foundational programme and joining Brouwer—he sings the praises of Brouwer to an embarrassing excess: “. . . Brouwer—that is the revolution!” and “Brouwer is the one to thank for the new solution to the problem of the continuum. . . Whether I have the right to describe Brouwer’s theory. . . is to me, mind you, doubtful.” It was not style alone that was provocative about Weyl’s lectures. There was also the content: Dedekind’s definition of the set of natural numbers is circular; there are circular definitions in analysis. In his analysis the convergence of Cauchy sequences still holds, as does the Intermediate Value Theorem, but the existence of suprema of bounded sets fails, and one cannot hope to save—one of Hilbert’s results—Dirichlet’s Principle. In accepting Brouwer’s continuum one had to give up points; Brouwer’s continuum was not a collection of points, but a “medium of free becoming”—the exact meaning of which (fortunately!) need not concern us here. Weyl gave the first clear and accessible (though not entirely faithful) account of Brouwer’s ideas. Brouwer’s big 1918/1919 paper had been primarily straight mathematics, and his commentary on it was somewhat sparse. Weyl, by then an experienced expositor, offered a more expansive explanation. Moreover, he phrased things more vividly. Discussing the problem with classical logic, he described the use of quantifiers as follows: An existential assertion—e.g. “there is an even number”—is not at all a judgment in the proper sense, which asserts a fact.

Rather, it was a “judgment abstract” to be compared with a slip of paper offering a treasure but not telling where the treasure was to be found. Continuing, The general “Every number has property E”—e.g. “for each number m, m + 1 = 1 + m”—is equally less a genuine judgment; rather [it is] a general instruction [Anweisung] on judgment.

Thus, Weyl more or less considers, ∃ : an I.O.U. ∀ : a payment slip. “In this light mathematics appears as a dreadful [ungeheuer] paper-economy.” For good measure he adds, “It is not the existence theorem that is valuable, but rather the construction carried out in the proof. Mathematics is, as Brouwer occasionally said, more of a doing than a teaching” [“mehr ein Tun dann eine Lehre”—“teaching” here in the sense of a doctrine].

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Weyl’s lectures were widely discussed. In G¨ottingen, Richard Courant joined Bernays on 1 and 8 February 1921 in reporting “On the new arithmetic theories of Weyl and Brouwer” in the G¨ ottingen mathematical colloquium. Also that year, the Jahresbericht reports lectures “On the Brouwer-Weyl number concept” in Breslau (now: Wroclaw) by L. Koschmieder and “On Weyl’s researches in the foundations of mathematics” in Frankfurt by E. Hellinger. 8. Hilbert Responds Hilbert had been thinking of returning to the problem of foundations since 1917 when he had hired Bernays as his assistant. Nonetheless, for the next three years he continued to work mainly in mathematical physics. Weyl’s public defection changed that: in the winter of 1920/1921, Hilbert once again turned his attention to foundations. Indeed, it seems he first announced his views only two weeks after Bernays and Courant lectured on Weyl’s lectures when, on 21 and 22 February 1921, he lectured on “A new laying of foundations for the number concept” in the G¨ ottingen mathematics colloquium. In the spring he lectured in Copenhagen and on 25-27 July, after an advance notice in the Jahresbericht, he lectured in Hamburg. These Hamburg lectures drew “numerous listeners—including also famous mathematicians” according to Kurt Reidemeister’s report in the Jahresbericht, the first, but inadequate, description of Hilbert’s new programme to appear in print. In September (in particular, on 23 September) of that year Bernays lectured at a meeting of the DMV (the German Mathematicians-Union mentioned above) in Jena “On Hilbert’s thoughts on laying the foundations of arithmetic”. The year 1922 was not so full of public action for Hilbert’s programme. The texts of his and Bernays’ lectures were published, and on 22 September Hilbert gave another lecture, the text of which was published in Mathematische Annalen the following year, at a meeting of the DMV in Leipzig. There were also some personal matters that, however, are best postponed until after discussing Hilbert’s new programme and the papers on it. Hilbert’s programme was, of course, to prove the consistency of arithmetic (with arithmetic taken in a broad sense to include analysis and set theory). What was new were the ground rules and method, as well as perhaps a sense of urgency. What was still missing was a convincing reason for the emphasis on consistency. Obviously, an inconsistent theory has no informational content, but why should consistency be enough? Bernays, who was more philosophically inclined than Hilbert, went further than Hilbert had previously gone in explaining this importance. By this I do not mean that Bernays directly addressed the issue, but that here and there in his paper one can find statements bearing on the problem: Whereas logic has to do with contentual universality [inhaltliche Allgemeinsten], (pure) mathematics is the general study of formal relations and properties;

and: What matters for the question of pure mathematics is only whether the usual, axiomatically characterised mathematical continuum is possible in itself; that is, it is a consistent creation [Gebilde].

To Hilbert, whose fundamental views on this subject may well have come from geometry, where the success of non-Euclidean geometry exemplifies the merely formal nature of mathematics and the sufficiency of consistency, this was second nature. Neither he nor Bernays realised at the time that, to Brouwer, consistency was a

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red herring: Brouwer was after truth, not mere consistency. He would hardly have been impressed by Bernays’ further remark that And what we have gathered from the investigations of Weyl and Brouwer is the conclusion that a proof of consistency via a replacement of existence axioms by construction postulates is not possible.

Brouwer could easily have objected that his goal was not to establish the consistency of a false theory, but to learn the truth. It never seems to have occurred to Hilbert, and did not occur to Bernays at the time, that the very goal of the programme—once achieved—would not be convincing and was itself in need of explanation. Perhaps, had Hilbert not been so impatient with philosophers, he would have been led to this realisation through his earlier correspondence with Frege. Eventually Hilbert would—accidentally?—hit upon such an explanation and the myth of Hilbert’s Programme (with a capital “P”) would be generated. But we will get to this later. Hilbert’s first paper, “Neubegr¨ undung der Mathematik, erste Mitteilung” (usually referred to simply as Hilbert’s First Hamburg Lecture), may be considered the first shot fired in the battle between Hilbert and Brouwer. Indeed, Hilbert fired a whole salvo of polemics—but polemics that appear to be aimed more at Weyl than at Brouwer: His criticisms refer directly to Weyl’s Z¨ urich lectures, not Brouwer’s writings; and whenever he refers to Brouwer and Weyl, he mentions Weyl’s name first. Was Weyl in the audience in Hamburg? A few quotes will illustrate the tone and the extent to which the battle between Hilbert and Brouwer, before Brouwer was even engaged in it, was already becoming one between Hilbert and the dead Kronecker: Weyl and Brouwer are searching for the solution to the problem—in my opinion— down the wrong path.

In fact: What Weyl and Brouwer are doing comes in principle to this, that they are wandering down the former paths of Kronecker: They seek to found mathematics in such a way that they throw overboard everything that has an uncomfortable [unbequem] appearance to them and erect a Verbotsdiktatur a ´ la Kronecker.

Against Weyl alone we read: The circulus vitiosus is artificially dragged into analysis by Weyl;

and if Weyl notices an “inner groundlessness of the foundations on which rests the construction of our Empire” and worries about “the menacing dissolution of the State of Analysis,” then he’s imagining things [literally: seeing ghosts].

Hilbert had his own political metaphors to offer, and offer them he did—in one of the most famous quotations from the entire battle: I believe that, as little as Kronecker in his day succeeded in banishing the irrational numbers. . . so little will Weyl and Brouwer succeed today; no, Brouwer is not, as Weyl believes, the Revolution, but rather only the repetition of an attempted putsch with old means, which, in its day more energetically undertaken, miscarried completely and now especially with the State so well-prepared and strengthened by Frege, Dedekind and Cantor, is condemned at the outset to failure.

In describing one of Hilbert’s later lectures, Weyl said that there were “anger and determination in Hilbert’s voice.” Can we hear that in the present paper? If

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so, there is also some humour in what we read: It cannot be with mere vicarious viciousness that we enjoy the “Revolution/putsch” comparison. Moreover, Hilbert’s good humour is further revealed when he gets down to introducing his theory. In stating that he is going to start with concrete symbols, he biblically intones, “in the beginning—so it goes here—is the sign.” In any event, any anger directed at Weyl was short-lived. Hilbert gave the lecture in 1921; in 1922 he tried to spirit Weyl out of Z¨ urich to join him in G¨ ottingen. Weyl declined, but in future years Hilbert would try again and eventually succeed in luring Weyl home geographically, if not philosophically. Although Hilbert’s indulgence in polemics is certainly a sufficient reason to mention his First Hamburg Lecture, it is not the main reason for doing so. The main point of interest in this lecture is, of course, the announcement of his new programme for defeating Weyl and Brouwer. As in 1904, the key to this programme was to be a consistency proof for arithmetic in its broadest sense (i.e. including set theory). There was, however, a major difference between his 1904 attempt and his new one. His approach in 1904 was to build up mathematics and logic a bit at a time, at each stage using what was currently available to prove the consistency of what was to be added next. His development at the time never reached a stage where the mathematics proven consistent was powerful enough to do anything, and he did not make clear what was to be allowed in the initial stages. His new attempt avoided these difficulties. To begin with, Hilbert distinguished between two kinds of mathematics—actual [eigentlich] mathematics (or: mathematics proper) and metamathematics. Actual mathematics is what mathematicians do—analysis, set theory, etc. Actual mathematics is abstract, infinitary, and has no empirical meaning; the sole check one has on its validity is its non-contradictoriness. Metamathematics, on the other hand, is intuitive and contentual mathematics; it is the direct combinatorial study of signs and their combinations. As we have already seen, Hilbert felt that the need for rigour imposed the axiomatic method on actual mathematics. He now went one step further and called for its formalisation: Actual mathematics was to be thought of as or be replaced by a formal system with precisely defined axioms and rules of inference, and the consistency of this formal system was to be proven metamathematically. The distinction between the formal mathematics—actual mathematics—and the contentual metamathematics was made by Brouwer in his dissertation in 1907. Hilbert very probably picked it up from Brouwer during their discussion in the dunes of Scheveningen in 1909; indeed, Brouwer would maintain this in years to come. In any event, the separation suited Hilbert’s purposes well. In the first place, metamathematical reasoning was acceptable to Weyl and Brouwer, and they would have to accept the consistency proof—though, as I said earlier, they need not accept the main consequence of consistency Hilbert envisioned—the correctness, in some sense, of the consistent theory. Moreover, by distinguishing between metamathematics and mathematics proper, Hilbert hoped to sidestep Poincar´e’s earlier charge of circularity—that Hilbert had been using induction to justify induction. According to Hilbert, metamathematical induction was of a simpler character than the full principle of induction. Whether one agrees with this is largely a matter of perspective, of whether one considers the properties to which induction is applied as central (in which case one will agree with Hilbert) or one considers induction as

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a property of natural numbers rather than of the formulæ inducted on (in which case one will still agree with Poincar´e). Hilbert named his programme Beweistheorie [proof theory] and, as in 1904, gave a sample metamathematical consistency proof of a formal system. Again as in 1904, the result was not very impressive and it need not concern us here. He would do better the following year in his Leipzig lecture. The Leipzig lecture of September 1922 was published in 1923 in Mathematische Annalen under the title “Die logischen Grundlagen der Mathematik” [The logical foundations of mathematics] and shows some real progress, both expository and mathematical. The tone is much milder, with only an occasional touch of the polemical—as in the opening remark where he describes the goal of his new programme to be “to banish definitively from the world the general doubt of the security of mathematical inference.” Moreover, he came very close to determining exactly what constituted his metamathematics. In doing so, he more or less reversed himself: in Hamburg he had said that the metamathematics, being intuitive and contentual, was not axiomatic: it had no axioms and could not be inconsistent. In Leipzig, he now offered a formal system of metamathematical arithmetic (i.e. the arithmetic portion of his metamathematics) and sketched a proof of the consistency of a fragment thereof. Neither the formalisation of nor the proof of the consistency of his metamathematics was necessary for the working out of his programme, but they illustrated most simply how he thought the programme could be carried out. Indeed, in the ensuing years his student Wilhelm Ackermann would almost succeed in carrying through Hilbert’s programme by following the lines of Hilbert’s sketch. It will be instructive for us to consider Hilbert’s formalisation of metamathematical arithmetic, his formalisation of “transfinite” arithmetic, and his sketches of consistency proofs for fragments thereof. First, however, a few words of explanation. Hilbert introduced a new word into his proof-theoretic vocabulary—“finit”, a Latin form of “finite” to stand alongside the usual German “endlich”. English equivalents abound—“finitist”, “finitistic”, and “finitary” are generally used. The use of the new word is conceptual as well as emphatic: One can imagine a finite set of infinite objects, e.g. {Z, Q, R}, but one would not call such a thing finitistic. In commenting on his metamathematical arithmetic, Hilbert said: The derivable formulas, which are obtained on this standpoint, all have the character of the finitary, i.e. the thoughts, whose images they are, can also be obtained without any axioms contentually and immediately through consideration of finite universes.

In such finite universes, there is no questioning of the applicability of traditional Aristotelian logic; in particular, the use of the Law of the Excluded Middle is unexceptionable. However, Hilbert went on to say that when we thoughtlessly apply procedures that are reliable in the finite case to the infinite case, we make mistakes left and right. This happens in analysis when we ignore convergence criteria in dealing with infinite sums and products. In logic it happens when one starts using quantifiers ∃ and ∀, which merely serve to abbreviate infinite logical sums and products, respectively. A universal arithmetic assertion ∀vϕv was to Hilbert an infinite conjunction ϕ(0) ∧ ϕ(1) ∧ . . . . Its negation had no precise content or meaning. Similarly, an existential assertion ∃vϕv was an infinite disjunction ϕ(0) ∨ ϕ(1) ∨ . . ., incapable of being negated. Exactly what Hilbert meant here is unclear, and the clarity would

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not improve in future presentations of his programme. In 1925, he would maintain that universal assertions ∀vϕv, with ϕ quantifier-free, were finitistic because they were mere infinite conjunctions, but that existential assertions were not finitistic because they were only partial judgments, omitting crucial information. He might have been expected to label the existential ones finitistic because their truths can be established by appeal to “finite universes”, while the truth of universal assertions cannot. Hilbert may have preferred the universal ones because, as we shall shortly see, they were subject to some control. More probably, he questioned existential assertions because he had to establish their validity (in some sense) because it was his existential theorems that had drawn fire from the critics—a fact he cited in this paper. Ignoring the question of the exact borderline between finitistic and infinitistic assertions, one can say that, according to Hilbert, quantifiers introduce an infinitistic element into logic. Quantificationally complex formulæ are meaningless and Aristotelian logic—in particular, the Law of the Excluded Middle—does not apply to them. Nonetheless, Hilbert would add such formulæ to his formal system of finitary arithmetic and apply the Law of the Excluded Middle to them. He compared this to practice in analysis and geometry: In my proof theory [we] adjoin to the finite axioms the transfinite axiom and formulae, just as one [introduces] imaginary elements to the reals in the theory of complex numbers and ideal objects in geometry. And the motive for doing this and the success of the procedure is in my proof theory the same as there: namely, the addition of the transfinite axiom achieves in a sense the simplification and rounding off of the theory.

[“Transfinite axiom” will be explained shortly.] The use of ideal elements in algebra and geometry can, in principle, be dispensed with. Any theorem, for example, about the real numbers proven via a detour into the complex realm can be proven anew, usually in a more cumbersome manner, without appeal to the complex numbers. Does the analogy go this far? According to Hilbert, To be sure [freilich] one can presumably prove a finitistic statement also without application of transfinite means of proof. . . but this claim is of the sort of claim that in general every mathematical assertion must allow itself either to be verified correct or refuted.

He illustrated this with the example of his theorem on invariants: after his first transfinite proof, he had given his second finitistic proof. Incidentally, in citing this example, Hilbert went on to recall how Gordan had distrusted the first proof and labelled it “theological”; it would appear that the old bitterness was resurfacing. If Hilbert was not yet ready to tackle the problem of the solvability of every mathematical problem or the conservation of transfinite methods over finitary ones, he was ready to commit himself as to the consistency of the transfinite. Indeed, and as I have already said, he sketched the outline of his approach. It is to this that we now turn. In presenting Theorem 1, I took the liberty of simplifying the system Hilbert proved consistent merely to give an example of a consistency proof. Because the present system offers an “exact” description of finitistic arithmetic, I shall present it in its original form. By way of preliminary explanation, I note that because of the equivalences, ϕ ∨ ψ. ↔ .¬ϕ → ψ

ϕ ∧ ψ. ↔ .¬(ϕ → ¬ψ),

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Hilbert’s logic included only the two connectives, ¬, →. Moreover, Hilbert used variables for formulæ as well as for numbers. Variable formulæ were used in this paper merely as place-holders, allowing the axioms to be stated as single sentences rather than as schemata; such variables do not appear in the theorems of the system and can be dispensed with. In line with Hilbert’s general ambivalence concerning the meaning of universal assertions, one might ask whether the same was to hold for the number variables in the finitistic system: was the axiom, a = a, with variable a, to be taken as an abbreviation for the schema t = t, for closed terms t, or was it implicitly to be taken as the universal assertion ∀a(a = a)? Given that consistency is itself a universal assertion (“for any proof p, the end formula of p is not a contradiction”) and that it was to be proven finitistically, we should probably assume even the finitist system to be intended to prove general assertions with free variables. Such an assumption does, however, require care in interpreting Hilbert’s remark on establishing provable formulæ by considering finite universes. We will return to this question later. The non-logical primitives of Hilbert’s initial system are the constant 0 (with no overbar), the successor function (·)+1, and the predecessor function δ. Initially, the only rule of inference is modus ponens: from ϕ and ϕ → ψ infer ψ . There are ten axioms broken into four groups: axioms of consequence (1-4), axioms of negation (5-6; axiom 6 is his formulation of the Law of the Excluded Middle), axioms of equality (7-8), and axioms of number (9-10). These axioms follow: A1. A → (B → A) A2. (A → (A → B)) → (A → B) A3. (A → (B → C)) → (B → (A → C)) A4. (B → C) → ((A → B) → (A → C)) A5. A → (¬A → B) A6. (A → B) → ((¬A → B) → B) A7. a = a A8. a = b → (A(a) → A(b)) A9. ¬(a + 1 = 0) A10. δ(a + 1) = a. Hilbert overlooked the axiom, ¬a = 0 → δ(a) + 1 = a, as well as some determination of a value for δ(0), e.g. δ(0) = 0. Even with such additional axioms, one has at best a fragment of finitist arithmetic. To obtain his full metamathematical arithmetic, Hilbert said one should add “recursion and intuitive induction.” With no quantifiers yet allowed, induction would have to be added as a rule of inference: from A(0) and A(a) → A(a + 1) infer A(b). The question of which recursions were to be allowed is premature: Hilbert was familiar with Dedekind’s recursive definitions of addition and multiplication, as well as Dedekind’s general theorem allowing definition of functions by recursion, but in 1922 the class of functions generated by recursions had not yet been subjected to any general study. By coincidence, Thoralf Skolem was already developing a quantifierfree number theory based on recursion, but his paper would first be published only in 1923—the year in which Hilbert’s Leipzig lecture would be published. The next

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few years would see the beginning of a general study—by the Hilbert school—of recursions, and in 1925 Hilbert would report that they had examples of recursively generated non-primitive recursive functions. In short, at the time of his Leipzig lecture Hilbert’s conception of recursion, and hence of finitism, was only partially formed, and I can be no more specific in clarifying the situation than to say that finitist arithmetic is obtained from the initial fragment already described by adding “recursion and intuitive induction”. [If the historical question of what Hilbert meant by finitism is replaced by the philosophical question of what he could or should have meant, an answer is at hand: Allow only primitive recursions. This is what Ackermann did in 1924, and there is today a consensus of opinion that finitistic arithmetic is adequately captured by a quantifier-free theory of primitive recursive functions. Through encoding, such a theory is taken to embody finitism itself.] Before discussing the addition of quantifiers through the adjunction of a “transfinite axiom”, I would like to give Hilbert’s proof-sketch of the consistency of the theory, say T0 , given by axioms 1-10 above and the rule modus ponens. To do this, I must first carefully define what a formal derivation is. This is easy: A formal derivation of a formula ϕ is a sequence of formulæ ϕ0 , ϕ1 , . . . , ϕn = ϕ (say, containing no variable formulae) such that each ϕi is an instance of an axiom, e.g. ϕ → (ψ → ϕ) or t = t, or follows from two earlier formulæ ϕj and ϕk = ϕj → ϕi of the list (i.e. j, k < i) by modus ponens. With some effort it can be shown that axioms 1-6 are complete with respect to purely propositional reasoning, whence, in particular, the consistency of T0 reduces to the unprovability of ¬ 0 = 0. Theorem 2. The system T0 is consistent. Proof sketch: Suppose D = ϕ0 , ϕ1 , , . . . , ϕn is a formal derivation of ¬0 = 0. As a first step, we can modify D by throwing away any ϕi (other than ϕn ) which is not used as a premise of an application of modus ponens. Moreover, by repetition of formulae, we can assume each ϕi occurs only once as such a premise. Call the resulting derivation D . Second, we can omit number variables from D by substituting, say, 0 for each occurrence of a variable in D . Call the result D . Third, we can simplify the terms to the point that each formula is a propositional combination of equations of numerals, 0, 0 + 1, 0 + 1 + 1, . . . . Fourth, every formula can be brought into a logical normal form (of some sort— Hilbert doesn’t say which kind). Each formula of the derivation is now subject to a control; i.e. one can check each formula for “correctness” or “falsity”. But it can be shown that each formula of this final derivation is correct, whence the end formula ¬0 = 0 is correct—a contradiction. Thus, there was no derivation D of ¬0 = 0.  This proof is as Hilbert gave it, and the reader should not fault me if he finds it less than clear. I myself do not find it completely clear. The first step has the character of something one finds one has to do to carry out the real work of the proof-transformations, and we can probably ignore it. The fourth step is an enigma to me—I simply don’t know what “normal form” Hilbert has in mind. The

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second and third steps, however, seem crucial. The second step guarantees that all the axioms of the derivation hold in some finite universe, thereby showing in principle the finitary nature of the derivation (in the sense of our earlier quote from Hilbert). More to the point, it turns every term into a computable term capable of reducing to a numeral. [This last is only true if one adds the axioms δ(0) = 0 and ¬a = 0 → δ(a) + 1 = a.] Hilbert’s subsequent partial extension of Theorem 2 to the “transfinite case” suggests the following interpretation of step 3: Calculate the value of each term t occurring in the given derivation D and replace t in each occurrence by the numeral for this value. Every formula is now a propositional combination of equations of numerals. The “correctness” or falsity of each equation is simply a matter of comparison, and the calculation of truth values of propositional combinations is a simple matter. One can now show by induction on the length of D that the formulæ occurring in the latest transformation are all true. [If this is indeed what Hilbert had in mind, then the reference to putting formulæ into a “logical normal form” must refer to the simple calculation of the truth value of a propositional combination of sentences once the individual truth values are known.] [Digression: Another plausible interpretation of step 3, which may shed a little light on Hilbert’s idea of transforming derivations, is this: transform the derivation D by repeatedly replacing terms δ(t+1) by t. With a little luck, the result is (or is easily made into) a derivation itself. For example, consider the following derivation of 0 = δ(0 + 1): (1) (2) (3) (4) (5) The

δ(0 + 1) = 0 by A10 δ(0 + 1) = 0 → (δ(0 + 1) = δ(0 + 1) → 0 = δ(0 + 1)) by A8 δ(0 + 1) = δ(0 + 1) → 0 = δ(0 + 1) by (1), (2) δ(0 + 1) = δ(0 + 1) by A7 0 = δ(0 + 1) by (3), (4). replacement of δ(0 + 1) by 0 results in the correct derivation:

(1) (2) (3) (4) (5)

0=0 0 = 0 → (0 = 0 → 0 = 0) 0=0→0=0 0=0 0=0

by by by by by

A7 A8 (or A1) (1), (2) A7 (3), (4).

The point to notice is that the instance of A10 has been replaced by an instance of A7. Similarly replacing δ(t) + 1 by t for t = 0 will transform an instance of the auxiliary axiom, ¬a = 0 → δ(a) + 1 = a, into the formula ¬a = 0 → a = a, which, although not an axiom, is readily given a quick logical derivation: (1) a = a (2) a = a → (¬a = 0 → a = a) (3) ¬a = 0 → a = a

by A7 by A1 or A5 by (1), (2).

Replacing δ0 by 0 changes the axiom δ0 = 0 into 0 = 0. It is tempting to jump to the conclusion that such replacements will eliminate the arithmetic axioms other than ¬(t + 1 = 0) from the derivation and yield a (nearly) purely logical derivation.

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I haven’t checked this, but I do note that it is not as trivial a matter as the above examples indicate.] Accepting the pre-digressive consistency proof for T0 as the one Hilbert had in mind, let us see how Hilbert intended to extend the proof to obtain the consistency of transfinite arithmetic, i.e. arithmetic with quantifiers. To begin with, Hilbert decided to go a step beyond quantifiers by dealing with choice functions: Given a formula ϕ(a) with free variable a, Hilbert added a new term τa (ϕ) attempting to choose a counter example to ϕ. Together, τ and ϕ satisfied the transfinite axiom: ϕ(τa (ϕ)) → ϕ(a). Formally, in terms of variable formulae, the axiom was A11. A(τ (A)) → A(a). [Note that since A is a variable for formulae, it has no numerical variable a occurring in it, whence τ (A) has no subscript indicating which variable is being bound. Although there are no quantifiers the new abstraction operators τa bind variables and the familiar problems with substitution arise; however, Hilbert was not yet aware of this.] Using τ, quantifiers can be introduced as abbreviations: ∀aϕ(a) : ∃aϕ(a) :

ϕ(τa (ϕ)) ϕ(τa (¬ϕ)).

With these, the usual laws for quantifiers can be derived: ∀aϕ(a) → ϕ(a), ϕ(a) → ∃aϕ(a) ¬∀aϕ(a) ↔ ∃a¬ϕ(a), ¬∃aϕ(a) ↔ ∀a¬ϕ(a). Hilbert did not claim that he had already extended his consistency proof to include the transfinite axiom. There were, as he noted, difficulties with the nestings of τ ’s. What he did do was to indicate how the proof of Theorem 2 could be extended to cover one special instance of the new axiom. Let f be a function variable and define τ (f ) = τa (f (a) = 0). The corresponding instance of axiom A11 reads A12. f (τ (f )) = 0 → f (a) = 0. As Hilbert quickly pointed out, the function τ (f ) would not be allowed by Brouwer and Weyl (Brouwer’s name now came first). Following the sketch of his consistency proof for A12, Hilbert gave a couple of examples of what this consistency proof would yield that Brouwer and Weyl forbade. First of all, we could talk about decimal expansions of numbers for which we cannot compute the expansion, contrary to the conclusion drawn by Brouwer in his 1920 paper cited a few pages back. Indeed, using τ one could define the function  √ 0, n n is rational, √ F (n) = 1, n n is irrational

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and then consider the number with dyadic expansion .F (2)F (3) . . . , i.e. the number ∞  F (n + 2) r= . 2n+1 n=0 Moreover, one could do this despite the fact that one could not (in 1922) even calculate the first bit of r—i.e. one did not even know F (2). Second, Hilbert showed in some detail—this time in response to Weyl—how using the τ ’s made it possible to prove the existence of the least upper bound of a bounded set of real numbers. The question, then, is the proof of the consistency of A12 with A1 − A10 (and the recursions). To simplify matters, we allow only one fixed function F (obtained by recursions) in A12. Thus, A12 becomes A12 .

F (τ (F )) = 0 → F (a) = 0.

Moreover, F is assumed unary. Let the resulting system with axioms A1−A10, A12 , and whatever recursion equations are necessary to compute F be called T1 . Theorem 3. The system T1 is consistent. Proof sketch: Using the first few steps of the proof of Theorem 2, we can transform any derivation D into a derivation D which consists solely of propositional combinations of equations involving numerals and the symbol τ (F ). [Not true, but let’s overlook this.] We now attempt a control of the proof by assigning τ (F ) the value 0. That is, we replace all occurrences of τ (F ) in D by the numeral 0 to obtain a new “derivation” D . As in the remarks following the proof of Theorem 2, we would like to show that all sentences in D are correct. Those that come from axioms A1 − A10 or the recursion equations defining F are correct, and modus ponens preserves correctness. Thus, any possible false sentences must have been introduced by A12 : F (0) = 0 → F (z) = 0, for some numerals z. If all of these are correct, then every sentence in D is correct and the original D could not have derived ¬0 = 0, as this formula is false and is left unchanged by the proof-transformations. If, on the other hand, an instance F (0) = 0 → F (z) = 0 is incorrect, we simply go back to D and replace all occurrences of τ (F ) by the numeral z. The instances of A12 now become instances like F (z) = 0 → F (x) = 0, for various numerals x. But these implications are correct formulæ because F (z) = 0 is false! Again, all sentences in D are correct, whence D could not have been a derivation of ¬ 0 = 0.  Hilbert ended his lecture with the remark that one had but to carry through the details of his proof sketch in order to complete the laying of foundations for analysis and therewith begin the corresponding work on set theory. That there would be difficulties is something Hilbert was aware of; the extent of these difficulties would not become clear until 1930. Indeed, only after that would it become clear that Theorem 2 itself was not without hidden difficulties. So we finally come to the end of Hilbert’s Leipzig lecture of 1922. Three things of a more personal nature happened in 1922. I have already cited Hilbert’s offer

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of a position in G¨ ottingen to Weyl and Weyl’s refusal. Bernays was appointed extraordinarius without tenure, but was able to keep his job until the Nazis came to power. And, as Reid reports in her biography, Hilbert’s new physics assistant was unimpressed: Hilbert was showing signs of aging. Hilbert was only 60 and was acting like someone much older. At first it was just assumed Hilbert was suffering from an early, but natural, decline. By the fall of 1925, however, it was known that he was suffering from pernicious anaemia, in those days a generally fatal illness. This illness would play a major rˆ ole in Hilbert’s battle with Brouwer. 9. More on Brouwer Before getting too involved in personal matters, however, let us return briefly to 1923 and Brouwer. In 1921 Bernays had spoken to the DMV in Jena, and in 1922 Hilbert lectured to the DMV in Leipzig. The 1923 meeting of the DMV was in Marburg, and on 21 September Brouwer spoke on “Die Rolle des Satzes vom ausgeschlossenen Dritten in der Mathematik” [On the rˆ ole of the principle of the excluded third in mathematics] and, in addition to a brief summary in the Jahresbericht, a version of the lecture was published in another German journal ¨ under the title “Uber die Bedeutung des Satzes vom ausgeschlossenen Dritten in der Mathematik, insbesondere in der Funktionentheorie” [On the significance of the principle of the excluded third in mathematics, in particular in the theory of functions]. As we have already seen, Hilbert’s papers on the foundations of mathematics were long on commentary and short on mathematical detail. Brouwer’s papers on intuitionism during this period had, on the other hand, more mathematics than philosophy. As the mathematics does not concern us here, I need not say much about Brouwer’s papers other than that they exist. The Marburg lecture is an exception, for Brouwer prefaces it with comments on the validity (in finite domains) and the invalidity (in infinite domains) of the Law of the Excluded Middle. Moreover, he finishes this preface with a few comments against formalism, ending with a pungent remark on the pointlessness of giving a consistency proof: An incorrect theory unrestrained by any refuting contradiction is thereby no less incorrect, just as a criminal policy unrestrained by a reprimanding court is thereby no less criminal.

Such a remark is certainly enjoyable, especially in the midst of Brouwer’s generally dry prose of the period. But it is still hardly an indication of a battle with Hilbert; it is more of a playful continuation of the political metaphor earlier used by Weyl and Hilbert. A real fight would soon be breaking out—the high points being the years 1925, 1927, and 1928. Through these next few years Brouwer continued to publish papers on topology and on his intuitionistic mathematics, the most important of the latter being a three-part reworking and extension of his 1918/1919 paper, published from 1925 to 1927 in Mathematische Annalen, and another paper published in the same jour¨ nal: “Uber Definitionsbereiche von Funktionen” [On the domains of definition of functions]. In this latter paper, published in 1927, Brouwer devotes a footnote to his two main objections to Hilbert’s formalism. The first of these would take us too far afield to be discussed intelligibly. The second reveals the great difference in Hilbert’s and Brouwer’s perspective. To Hilbert the consistency of a theory implied the existence of the objects of the theory; to Brouwer the objects were constructed

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and a consistent false theory proved no truths. To go from consistency to truth required, Brouwer said, the “Principle of the Reciprocity of Complements”, i.e. ¬¬ϕ → ϕ, which was equivalent to the Law of the Excluded Middle. Assuming Hilbert proved the consistency of the Law of the Excluded Middle, he could not conclude its truth without assuming its truth; i.e. Hilbert’s programme, once carried out, would rest on a circularity. The point here is philosophically subtle, and this is not the correct place to go into it. I will simply say that it seems to me Hilbert was misled by Weyl’s remarks on the meaninglessness of quantified formulæ into believing them to be meaningless to Brouwer. Thus, to Hilbert “transfinite formulae” did not have to be true, merely consistent, while to Brouwer the issue was truth. One of the many ironies of the dispute between Hilbert and Brouwer was that anyone who agreed with Hilbert did not need this programme, and anyone who agreed with Brouwer would not be convinced by its positive outcome should it occur, but yet Hilbert pushed on. In citing Brouwer’s 1927 paper I have once again deserted the chronology. Let me return to 1923—or, even better: 1924. On 8 January, Bernays lectured in Berlin on “Neuere Untersuchungen u ¨ber Hilberts axiomatische Methode” [New investigations on Hilbert’s axiomatic methods]. Perhaps he spoke on Ackermann’s dissertation, which Ackermann himself spoke on in G¨ ottingen on 12 and 26 February and in which Ackermann thought he had completed Hilbert’s above-sketched consistency proof. Ackermann, however, found an error and eventually claimed the consistency of only a fragment of “transfinite” arithmetic. On 22 July 1924, Brouwer spoke at G¨ ottingen on intuitionism. Is this a sign that he and Hilbert were still on friendly terms? In her biography of Hilbert, Reid describes the end of a talk Brouwer gave in G¨ ottingen. She says, “After a lively discussion Hilbert finally stood up. ‘With your methods,’ he said to Brouwer, ‘most of the results of modern mathematics would have to be abandoned, and to me the important thing is not to get fewer results but to get more results.’ He sat down to enthusiastic applause.” Was this the 1924 talk, an earlier one, or a talk Brouwer gave in 1926? Is there hostility in Hilbert’s stereotyped remark—a remark that applied to Kronecker and Weyl, but not quite as fairly to Brouwer? In the next year, the mood would indeed be hostile—albeit over matters more political. 10. Outbreak of Hostilities The year 1925 was a major one in the dispute, for now there was a real dispute and not just two competing mathematical philosophies. Hilbert gave a major lecture which, unfortunately, confuses rather than clarifies his views. And Weyl published an article reviewing the debate on the nature of the continuum (i.e., real number line) from the time of the ancient Greeks to Hilbert’s Leipzig lecture. I have been promising a fight for so long now that I feel compelled to begin with the real battle between Hilbert and Brouwer. This battle was political and nationalistic: The reader will recall that Europe had lain locked in war from 1914 to 1918 and that Germany had been declared the villain. Perhaps because this was the first mechanised war to be held in Europe and the damage so great, French mathematicians forgot that, under Napoleon, the French had been the villains only a century earlier. With some vindictiveness, in organising the first post-war

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International Congress of Mathematicians in Strasbourg in 1920, they ruled that Germans could not attend. The bar to German attendance at the ICM was still in effect in Toronto in 1924. It so happened that 1926 would mark the century since Riemann’s birth, and the editors of Mathematische Annalen wanted to commemorate the occasion with a special volume. Hilbert, although proud of German achievements in mathematics, was something of an internationalist. He felt that international co¨ operation was good for mathematics and wanted French contributions to the volume. Brouwer felt otherwise and sought to bar the French, particularly Paul Painlev´e, who had issued a number of anti-German statements during the war and who was becoming involved in the project. Ludwig Bieberbach, a Berlin mathematician and by then a close friend of Brouwer’s, managed a compromise—Einstein, also an editor, would contact friendly French mathematicians who could be asked to contribute to the volume. In suggesting this compromise, however, Bieberbach demanded unanimous agreement by the editors (including Brouwer and himself) and not just the agreement of the main editors (Hilbert, Blumenthal, and Constantine Carath´eodory). Hilbert suspected Brouwer was behind this demand. Ultimately, the volume was published with French participation. Such, in brief, was the nature of the first serious clash between Hilbert and Brouwer. It wouldn’t appear to deserve more than a footnote were it not for the fact that the French vs. German issue would again come between Hilbert and Brouwer. The one question it does raise is why Brouwer, a Dutchman, should go to the trouble of opposing Hilbert and Blumenthal on such a matter. To me the most compelling answer is this: The two leading groups of mathematicians in those days were the French and the Germans. Brouwer moved primarily in the German mathematical circle—he had been a member of the DMV since 1909 and had many German friends, particularly in G¨ ottingen, which, as already noted, was his second scientific home. If Brouwer himself was not being discriminated against, his friends and closest colleagues were. Moreover, his priority dispute with Lebesgue was still rankling him, and he probably hadn’t forgotten that the French mathematicians had sided with Lebesgue in this matter. 11. The Formula Game Weyl’s 1925 paper is an impartial account attempting to do justice to both sides. He repeats his earlier account of Brouwer, but acknowledges that his description is not true so much to Brouwer’s views as to his own reworking thereof. Although one can detect the beginning of Weyl’s transition from supporting Brouwer to supporting Hilbert, his treatment of Hilbert’s programme is none too flattering. Weyl compares Hilbert’s formalised mathematics to the game of chess: once the axioms and rules of inference are set up, one merely follows the rules. The consistency proof is comparable to a proof that there can never be 10 queens of a given colour on the chessboard during a game. Weyl even went so far as to call Hilbert’s “actual mathematics” a “Formelspiel”, a “formula-game”. The “reproach”, as Hilbert would eventually call it, that Hilbert was trying to turn mathematics into a meaningless game had long been a criticism of formalism in mathematics and, except perhaps in direct reference to Hilbert’s programme, did not originate with Weyl’s remarks. Hilbert himself would, two years hence, blame Brouwer for the phrase “formula-game”, although it seems to have been used by

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everyone except Brouwer. (At least, I haven’t been able to find such a remark in Brouwer’s published papers.) That Hilbert did indeed deny any meaning to “actual mathematics”, whether by design or accident, dismayed some. Weyl himself said, Without doubt: For mathematics to remain a serious cultural concern, some sense must attach itself to Hilbert’s formula-game; and I see only one possibility of ascribing an independent intellectual meaning to it, including its transfinite component.

To Weyl, Brouwer had demanded every mathematical assertion to be contentual and, in so doing, had revealed how little content most of mathematics had. Hilbert, on the other hand, he saw as denying any content at all to mathematics. Weyl saw room for compromise: Mathematics would become a theoretical science like physics, where not every assertion had intuitive content. Was Hilbert really turning mathematics into a formula-game? Was this his intent? Was Weyl’s representation of Hilbert’s views as liberally re-interpreted as his representation of the views of Brouwer? I would answer these questions yes, no, and yes, respectively. It is nowadays a commonplace among mathematicians that mathematics is a meaningless game of formal symbols, and Hilbert’s authority is invoked in support of this view. Hilbert’s contemporaries were more subtle thinkers. G.H. Hardy, perhaps the quintessential purist in mathematics, is quoted by Reid: is it really credible that this is a fair account of Hilbert’s views, the view of the man who has probably added to the structure of significant mathematics a richer and more beautiful aggregate of theorems than any other mathematician of his time? I can believe that Hilbert’s philosophy is as inadequate as you please, but not that an ambitious mathematical theory which he has elaborated is trivial or ridiculous. It is impossible to suppose that Hilbert denies the significance and reality of mathematical concepts, and we have the best of reasons for refusing to believe it: “The axioms and demonstrable theorems,” he says himself, “which arise in our formalistic game are the images of the ideas which form the subjectmatter of ordinary mathematics.”

[“Formalistic game” is a mistranslation of “Wechselspiel”. The passage from Hilbert’s Leipzig lecture quoted by Hardy directly follows Hilbert’s remark on how mathematics grows alternately by deriving theorems and adding new axioms. It was this “interplay” Hilbert described as “Wechselspiel”.] Hardy may not have chosen the most convincing quotation; I am inclined to point to the earlier cited remark that “one can presumably prove a finitist statement also without application of transfinite means of proof” and the contentual nature of finitist truths. This and the comparison with the use of ideal elements in algebra and geometry suggests that implicitly, if not yet explicitly, Hilbert’s programme was already a rendering of mathematics as the theoretical science suggested by Weyl. By the end of 1927 this would be clear. 12. On the Infinite In Hilbert’s M¨ unster lecture of 1925, published in two slightly different versions in Mathematische Annalen and Jahresbericht in the next two years, one can find remarks that seem to support the claim that Hilbert was trying to create a theoretical science out of mathematics. One can also find evidence supporting virtually any philosophy of mathematics other than the extreme sort of formalism (mathematics is a meaningless activity) generally attributed to Hilbert and the platonism implicit

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in his equation of consistency and existence—a platonism which would explain his blindness to the realisation that consistency alone might be unconvincing to others and which might also explain the horrible mess he made of explaining the “finitist point of view” in this lecture. Another explanation of the latter could be his illness: Hilbert was ill at the time of this talk. The M¨ unster lecture divides into two parts. The second part outlined an incorrect proof of Cantor’s continuum hypothesis (any uncountable set of real numbers can be put into one-one correspondence with the set of all real numbers) and is not particularly relevant to our discussion. I could propose it as an explanation of why, when Hilbert got down to details, he did not say anything about the actual workings of his consistency programme, or I could cite it as additional evidence of the effect his illness was having on his powers of concentration. Let us leave both interpretations in the realm of superficial asides and consider the more pertinent first part. Instead of building his exposition on his earlier two lectures, Hilbert explained anew the whole situation—finite vs. infinite, the ideal nature of existential assertions, etc. At the level of high generality, it is a good discussion and merits Jean van Heijenoort’s description as a “clear and forceful presentation of Hilbert’s ideas at the time on the foundations of mathematics”, and one can debatably agree with van Heijenoort’s further remark that among Hilbert’s papers on foundations it “stands out as the most comprehensive presentation of Hilbert’s ideas.” When it gets down to details, however, only the adjective “forceful” still applies. It is detail that I wish to discuss—specifically the detail that is new. Unfortunately, it is on just this point that Hilbert is not very clear, and I have had to consult both German and English versions of the paper and both past and future papers (“future” relative to 1925) in order to come to a clear understanding—one so clear and fitting his remarks so well that I declare it the correct interpretation: No one, though he speak with the tongues of angels, will convince me otherwise. This detail is a trichotomy replacing the old dichotomy between finitary propositions and transfinite, or ideal, formulæ. There are now real propositions, finitary general propositions, and ideal propositions. The real propositions—the term “real” being added first in 1927—are, Hilbert says, “of no essential interest” in themselves. They are simple propositional combinations of equations involving primitive recursive functions and fixed numerals: 2 + 3 = 3 + 2,

1 + 1 = 2,

etc.

They are directly contentual assertions verifiable by direct computation. Their importance lies primarily in affording a control on the results of formal mathematical proofs: The science of mathematics is by no means exhausted by numerical equations and it cannot be reduced to these alone. One can claim, however, that it is an apparatus that must always yield correct numerical equations when applied to integers.

I think we can read in these lines just the sort of theoretical view Weyl demanded: Mathematics is an abstract theoretical science subject to numerical control just as physics is an abstract theoretical science subject to experimental control. Real propositions do not exhaust the class of finitistic propositions. There are also what I shall call here the finitary general propositions—assertions of the form “for every numeral n, n + 1 = 1 + n”, which Hilbert would have written in Leipzig

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as the free-variable formula x + 1 = 1 + x. The general assertion “is from the finitist point of view incapable of being negated ” because the negation (“for some numeral n . . .”) “cannot be interpreted as a combination, formed by means of ‘and’, of infinitely many numerical equations.” [It may seem odd that infinite conjunctions are finitistic assertions. However, (i) one cannot seriously consider any science which does not propose universal laws; (ii) there are finitary schematic methods of proof; (iii) consistency, which must be proven finitistically, is such a universal assertion; and (iv) it was his existential theorems that Hilbert had been criticised for.] Finally, there are the ideal propositions that are not really propositions at all, but formal symbols manipulated according to pre-determined rules—those of what Hilbert did not like being called the “formula-game”. Though it is unnecessary for our general discussion, I would consider myself remiss in my responsibility as a chronicler if I didn’t mention the delightful irony and famous quotes to be found in Hilbert’s M¨ unster lecture. The former is given by one of his polemics: the literature of mathematics is replete with absurdities and inanities. . . for example, some stress the stipulation, as a kind of restrictive condition, that, if mathematics is to be rigorous, only a finite number of inferences is admissible in a proof—as if anyone had ever succeeded in carrying out an infinite number of them!

The irony is not so much that he is here criticising himself, but that he himself would in 5 years’ time be considering proofs with infinitely many inferences. His quotable remarks include “mathematical analysis is but a single symphony of the infinite”; “no one shall be able to drive us from the paradise that Cantor created for us”; “no one, though he speak with the tongues of angels, will keep people from negating arbitrary assertions, forming partial judgments, or using the principle of the excluded middle”; and, concerning what to do when confronted with a foundational problem, “let us remember that we are mathematicians” and simply solve the problem. Moreover, he reached all the way back to 1930 to recall “that within us we always hear the call: here is the problem, search for the solution; you can find it by pure thought, for in mathematics there is no ignorabimus.” If the year 1925 saw Hilbert and Brouwer fighting, Hilbert giving a confused lecture attacking even himself, and Weyl inadvertently sowing the seeds of future discord with his caricature of Hilbert’s programme, the year 1926, on the other hand, brought good news. Hilbert discovered that a treatment for pernicious anaemia of some promise was being experimented with in America and, after some maneuvering, he was receiving the treatment and his health was improving. And Hilbert and Brouwer had a reconciliation, albeit a short one. 13. A Fragile Truce The occasion of the reconciliation was a summer meeting of topologists in G¨ ottingen. By 1926, Brouwer was no longer actively working in topology, but he did keep in touch with the subject throughout the 1920s. Not only did he have young assistants in topology in Amsterdam, but his home was a Mecca for young topologists— and occasionally other mathematicians—from Germany and Russia. Moreover, despite his differences with Hilbert, Brouwer was still friends with many of the

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other G¨ ottingen mathematicians. Thus, the mathematicians at G¨ ottingen wanted Brouwer to come and they wanted a reconciliation. The method of reconciling Hilbert and Brouwer was simplicity itself. Emmy Noether, whom Weyl once described as “warm like a loaf of bread,” supplied the necessary Gem¨ utlichkeit: dinner was at her home. Sitting at the table with Hilbert and Brouwer were also Richard Courant, Edmund Landau, and a host of younger mathematicians, among them P.S. Alexandrov and Heinz Hopf, later joint authors of a famous topology book. The task of getting the conversation going fell to Alexandrov. According to him, the best way of getting two warring factions together is to find a common enemy. Thus, Alexandrov brought up the subject of a “famous Luckenwalder function theorist,” P. Koebe. His success exceeded all expectations: Hilbert and Brouwer were soon “falling all over themselves in a spirited exchange of opinion, during which they agreed ever more in their views of that function theorist.” The two became progressively friendlier and finished toasting each other. These good feelings lasted the entire period of Brouwer’s stay in G¨ ottingen. Unfortunately, Brouwer eventually had to return to Amsterdam, and the two were publicly attacking each other the following year. Hilbert’s attack may have been occasioned by the heavy interest in Brouwer’s lectures, in March of that year, in Berlin. Success in lectures on intuitionistic mathematics might be a bit hard to believe nowadays. Most mathematicians have little understanding of, hence little patience with, matters philosophical; and if they comment on such matters at all, it is usually in derision, as in the following opening remark in a paper of Oskar Perron of 1926: A set of numbers bounded from below (above) has a greatest lower (least upper) bound. Despite Brouwer and Weyl this fact, fundamental and indispensible for analysis, is known to have been clearly enunciated by Bolzano in 1817; it can be found just as clearly already in a posthumous manuscript of Gauss, which was written around 1800.

Berlin, however, was an exception. Under the leadership of Klein, and continuing under Hilbert, G¨ ottingen—not Berlin—had become the capital of German mathematics. The rivalry between the two universities was intense, and much of the interest in Brouwer’s lectures may well have been in Brouwer himself, the thorn in Hilbert’s side, and not in intuitionistic mathematics itself. Brouwer spoke to an overflowing hall, and his lectures were mentioned in the newspapers. Perhaps Hilbert, who had been roused to action at Weyl’s 1920 conversion, now saw another clear and present danger. In any event, in July he was in Hamburg again firing off a fresh salvo of polemics: it is part of the task of science to liberate us from arbitrariness, sentiment, and habit and to protect us from the subjectivism that already made itself felt in Kronecker’s views and, it seems to me, finds its culmination in intuitionism.

Probably the most famous line of the paper is: Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or the boxer the use of his fists.

Apparently Hilbert was aware of his image as a polemicist, for he prefaces his criticism of Brouwer with the disavowal, “Not because of any inclination for polemics, but in order to express my views clearly and to prevent misleading conceptions of my own theory, I must look more closely into certain of Brouwer’s assertions.” He

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then proceeds to compare Brouwer to Kronecker in declaring existence statements to be meaningless (fair enough, if a bit insulting) and then attributes to Brouwer Weyl’s paper economy and formula-game metaphors. Hilbert was, to put it bluntly, angry; this was the lecture about which Weyl, who was in the audience, would say that there were “anger and determination in Hilbert’s voice.” The anger, if not the determination, was irrational, for progress on the consistency proof was so great that everyone in Hilbert’s camp was certain success was forthcoming, and, moreover, Hilbert had a trump card to play—an application of consistency proofs that even Brouwer would have to admit the significance of: a method of obtaining finitistic proofs of finitary general propositions from formal proofs of their ideal representations. 14. “Hilbert’s Programme” Is Born I have offered an awkward statement of this last in order to remain as faithful as possible to Hilbert’s views. In a lecture in December 1930—again in Hamburg— Hilbert would point out the difference between a finitary general proposition, say, 1 + x = x + 1, and the similar ideal proposition, ∀x(1 + x = x + 1). The former is the assertion that 1 + n = n + 1, for all numerals n; the latter also asserts the equation for meaningless infinitary constructs involving the τ -function—or, rather, a replacement to be cited shortly. If we agree to ignore such subtleties, we can say that Hilbert said the following: Let S be a formal system of finitary arithmetic and let T be some system of transfinite mathematics. Suppose S proves the consistency of T . Then: for any universal assertion ϕ, if T ϕ, then S ϕ. I quickly note that it is the finitary general propositions that Hilbert is talking about here. For the more restricted class of real propositions, i.e. simple closed instances of quantifier-free formulae, Hilbert had already shown, if not mentioned (after all, such is of no essential interest), that the corresponding conservation result follows from the controllability of the results of mathematical derivations—which controllability follows from the method of his consistency proof. What Hilbert was doing now was settling the epistemological question he had brought up but dared not answer in his Leipzig lecture—namely, the question of the finitistic derivability of any finitistic statement obtained via transfinite means of proof. It was just this breakthrough that transformed Hilbert’s programme into Hilbert’s Programme and gave rise to the myth that, all along, Hilbert had known what he was doing. We shall discuss this new version of the programme when we reach the year 1930, when this new version of the programme finds its clearest (?) statement. Hilbert proved his method by means of an example, more or less in the way in which one proves theorems in Euclidean geometry by drawing specific triangles. The example he chose was Fermat’s theorem, FT :

∀xyzw(x > 1 ∧ y > 1 ∧ z > 1 ∧ w > 2 → xw + y w = z w ).

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We reason finitistically: Suppose we are given numerals k, m, n, p such that (*)

k > 1 ∧ m > 1 ∧ n > 1 ∧ p > 2 ∧ k p + mp = n p .

This assertion can be verified by a simple computation which translates directly into a proof in the transfinite system. But, by the provability of F T within the transfinite system, we see that, (**)

k > 1 ∧ m > 1 ∧ n > 1 ∧ p > 2 → kp + mp = np ,

is provable within the system. This means that kp + mp = np and kp + mp = np are both provable transfinitely, contrary to the consistency of the transfinite system. Hence (*) is false and we have proven that for any numerals k, m, n, p, (**) holds; i.e. we have proven F T . [Actually, we have only proven ¬¬(∗∗), but the validity of ¬¬(∗∗) → (∗∗) is intuitionistically acceptable because of the simple nature of (**). In other words, Hilbert was right that Brouwer would agree to the validity of Hilbert’s method—once the finitistic consistency proof was given.] As we shall see, Brouwer never had a chance to accept Hilbert’s method. Weyl, on the other hand, was in the audience and made some remarks immediately after the lecture. Despite his occasional stylistic excesses, Weyl was perhaps the only one of the major participants who could see clearly and dispassionately all the issues involved. He began with the words, “Permit me first to say a few words in defence of intuitionism.” He emphasised that the predominant view of mathematics had been that it was “a system of contentual, meaningful, and evident truths.” Brouwer, he said, was the first to realise that this was no longer the case, that mathematics had transcended content, meaning, and evidence. Brouwer’s solution was the obvious one and “it does not seem strange to me that Brouwer’s ideas have had a following.” Weyl did not say that Hilbert had been a bit unfair to Brouwer in his talk when he said, “To make it a universal requirement that each individual formula then be interpretable by itself is by no means reasonable; on the contrary, a theory by its very nature is such that we do not need to fall back upon intuition or meaning in the midst of some argument.” Rather more tactfully, Weyl pointed out that Hilbert was proposing a radical re-interpretation of the meaning of mathematics, in effect— though Weyl did not say this explicitly—turning the subject into the theoretical science Weyl had suggested in his 1925 paper. Following this, Weyl noted that “as I am very glad to confirm, there is nothing that separates me from Hilbert in the epistemological appraisal of the new situation thus created,” which I think means he was now supporting Hilbert—maybe. [Lecturing before an American audience in 1930, Weyl would say: My opinion may be summed up as follows: if mathematics is taken by itself, one should restrict oneself with Brouwer to the intuitively cognizable truths. . . nothing compels us to go farther. But in the natural sciences we are in contact with a sphere which is impervious to intuitive evidence; here cognition necessarily becomes symbolical construction. Hence we need no longer demand that when mathematics is taken into the process of theoretical construction in physics it should be possible to set apart the mathematical element as a special domain in which all judgments are intuitively certain; from this higher viewpoint which makes the whole of science appear as one unit, I consider Hilbert to be right.

Query: From this higher viewpoint, is it necessary to have an intuitively certain consistency proof?]

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Taken by itself, transfinite mathematics could be rescued only by the consistency proof. How was the search for this Holy Grail of proof theory progressing? In 1922, Hilbert had sketched his proof of consistency for an exceedingly limited case of his transfinite axiom for the counterexample-seeking τ -function. The τ -function was soon replaced by a choice function  satisfying A(a) → A(a (A(a))). Following this cosmetic change, Ackermann published a consistency proof in 1924. However, he discovered an error and tacked a footnote onto his paper restricting the construction of -terms. Soon he was working on an improvement and wrote to Bernays about his new proof. Johann von Neumann criticised the proof and worked out his own consistency proof—with restricted induction—and published it in 1927. However, in giving a sort of progress report on the consistency proof towards the end of his Second Hamburg Lecture, Hilbert briefly discussed Ackermann’s second proof, and the published version of this lecture, which came out in 1928, was accompanied by a note by Bernays explaining the proof in more detail. The Hilbert school firmly believed that the proof was almost complete. There was only the small matter of proving the finiteness of the number of substitutions of numerals for -terms before a given derivation would be transformed into a derivation-like collection of correct real propositions. Indeed, the following year Hilbert would announce that Ackermann and von Neumann had already proven the consistency of the arithmetic of the integers and that Ackermann had just to prove the corresponding finiteness result for analysis to complete his consistency proof for this latter theory.

15. Brouwer Takes Up Arms If Hilbert broke the truce in 1927, Brouwer was not sitting idly himself. On 17 December in Amsterdam and again a couple of months later in Berlin, Brouwer ¨ber den made his own attack in a lecture entitled “Intuitionistische Betrachtungen u Formalismus” [Intuitionistic reflections on formalism]. The contents of this lecture were published early in 1928, both in the proceedings of the Dutch Royal Academy and, under the sponsorship of his friend Bieberbach, in the Sitzungsberichte der Preussischen Akademie der Wissenschaften [Reports of the meetings of the Prussian Academy of Sciences]. The lecture begins with a list of those papers of Brouwer and Hilbert that Brouwer wished to discuss. This list did not include Hilbert’s Second Hamburg Lecture, which had not yet been published. The first section of this new paper has a list of 4 insights which would, upon their ultimate acceptance, render the choice between formalism and intuitionism a matter of taste. Briefly, these were: First Insight. The formalist distinction between transfinite mathematics and finitary mathematics is indispensable, as is the recognition of the need of the intuitionistic mathematics of the set of natural numbers for this finitary mathematics. Second Insight. One must not use the Law of the Excluded Middle thoughtlessly, but ought rather to investigate where it can be used. For intuitive (contentual) mathematics it is valid only in finite systems. Third Insight. The Principle of the Solvability of Every Mathematical Problem is to be identified with the Law of the Excluded Middle.

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Fourth Insight. The justification of the formal system of transfinite mathematics via a finitistic consistency proof contains a circulus vitiosus. According to Brouwer, Hilbert had already accepted the First and Third Insights and it was just a matter of time before he would accept the other two. Actually, Hilbert had accepted the First, but only half of the Third Insight—this latter because he did not yet accept Brouwer’s equation of the Principles of the Solvability of a Problem in principle and in practice. In any event, there was no hope of Hilbert’s accepting the Second Insight until he had accepted the Fourth and here, unbeknownst to Brouwer, Hilbert had shown that in a special case the justification was not circular. It is in his discussion of these insights that Brouwer, for the first time since Hilbert began attacking him in Hamburg in 1921, responded in kind in print. For example, with respect to the First Insight, he notes that it was hinted at by Poincar´e, appeared first in print by Brouwer in his dissertation in 1907, and had been discussed by Brouwer with Hilbert in the fall of 1909. (This last was noted in a footnote; it is the only reference in print by either party to their conversations in the dunes of Scheveningen when their acquaintance/friendship began.) Brouwer added that Hilbert subsequently published this distinction under a new nomenclature. In case the meaning of this should not be clear, he followed the individual discussions with a few closing remarks. These begin with the observation that formalism had got nothing but good from intuitionism and could expect more. There follows: Accordingly, the formalist school should afford some recognition to intuitionism instead of polemicising against it in a jeering tone and in the process not once keeping proper mention of authorship. Moreover, the formalist school should reflect on [the fact] that up till now nothing of mathematics proper has been secured in the frame of formalism. . . If thus the formalist school, according to its remarks. . . , has noticed modesty in intuitionism, then it should find cause therein not to take second place with respect to this virtue.

These are strong words, but apt. 16. Hilbert Finishes Off Brouwer Hilbert’s and Brouwer’s mutual attacks were published in 1928. The real fight that year, however, was far less public. It began with the preparations for the coming International Congress of Mathematicians in Bologna. As already noted, German mathematicians had been barred from attendance in 1920 in Strasbourg and in 1924 in Toronto. The Germans were allowed to attend in 1928, but not quite on a level of equality with the others. Moreover, the programme of the meeting included an outing to “liberated” areas, a direct slap in the German face as it were. Bieberbach, who had partially sided with Brouwer in the affair of the Riemann volume, announced that it would be a shame if a large number of Germans attended such a meeting, and Brouwer called for an all-out boycott. Hilbert, who felt international contact more important than nationalistic feeling, used the full force of his prestige to break the boycott and personally led the German delegation to Bologna, his health much improved by the American treatment. His lecture was met at beginning and end with thunderous applause. The content of this lecture will be discussed later. First, I shall dispose of Brouwer—something Hilbert probably would have liked to have done as he discussed the Brouwer problem with colleagues that August in Bologna. He soon came up

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with a solution: In October, he wrote to Brouwer telling the latter that he was fired from the editorship of Mathematische Annalen. The actual dismissal was a complicated affair that went on for several months while Hilbert’s forces tried to muster the proper authority to effect Hilbert’s pogrom. The Mathematische Annalen was, as I said earlier, one of—if not the—leading mathematical journals of the day. To be an editor thereof, as Brouwer had been since 1915, was a mark of great distinction. To be fired therefrom was a great insult, particularly if the editor being fired was the most conscientious and hard working of the lot, as Brouwer apparently was. Of Hilbert’s forces, the only one who appears to have had any qualms about firing Brouwer was Constantine Carath´eodory, who did his duty to Hilbert and then resigned as editor. The others—particularly Blumenthal and Courant—did not share Carath´eodory’s moral reservation; indeed, Blumenthal executed his task with unbecoming zeal. Once he had set the wheels in motion, however, Hilbert remained aloof from the whole process. But for Hilbert’s aloofness, it would be tempting to compare the situation with the British Royal Society’s investigation into the charge that Leibniz had plagiarised Newton: Leibniz, having faith in the integrity of the society expected an impartial investigation, not realising that Newton himself was ghost-writing the report. Brouwer was now trying to prepare his defence for the ultimate tribunal of all the editors, unaware that there was to be no tribunal—the delay in his receiving official notification not being caused by any question of whether or not to fire him, but of how: the decision had been made by Hilbert and few questioned it. The details of the struggle are fascinating but hardly relevant to our purpose, and I refer the reader to van Dalen’s report3 on the affair. The effect on Brouwer was devastating. Brouwer had not merely been publicly humiliated, but he had been betrayed by his friends—not all of them. Most of his friends in G¨ ottingen had had nothing to do with the affair, Carath´eodory had tried to spare Brouwer’s feelings, and Bieberbach stood steadfastly by Brouwer during the struggle. Nonetheless, Brouwer was left a broken man. Throughout the next decade he hardly published anything, and when he did resume publication in the 1940s, he never reached the heights he had achieved in the two decades from 1908 to 1928; his creative life, as Walter van Stigt put it in his moving account4 of Brouwer’s philosophical activity of the period, was over. Alas, matters are never as black and white as we like to paint them. Bieberbach—the true and faithful—later became a Nazi, and he and Brouwer disagreed on the latter’s acceptance of Jewish editors for Compositio Mathematica, a journal founded by Brouwer in 1930. The photographs I’ve seen of Blumenthal, twice the villain of our story, show such a pathetic character that it is difficult to condemn him. A Jew, he was ultimately deprived of his position by the Nazis and, after a temporary escape to the Netherlands, died in a German concentration camp in 1944. That leaves us with Hilbert. It is doubtful that he ever realised the effect his firing of Brouwer had. He was, it seems, temporarily unbalanced. His illness had taken a drastic downward turn—apparently through some bad medicine taken sometime 3 Dirk van Dalen, “The war of the frogs and the mice, or the crisis of the Mathematische Annalen”, Mathematical Intelligencer (1990), volume 12, number 4, pp. 17-31. 4 Walter P. van Stigt, “L.E.J. Brouwer, the signific interlude”, in: A.S. Troelstra and D. van Dalen, eds., The L.E.J. Brouwer Centenary Symposium, North-Holland, Amsterdam, 1982.

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after the Bologna meeting—and he, and everyone around him, thought he might die. Should he die, he feared Brouwer would take over Mathematische Annalen. Was he afraid that, with such a tool in his hands, Brouwer would alter the course of mathematics? At this point it might be worthwhile to consider how successful Brouwer had been at conversion. In 1919, in a reprint of one of his 1908 publications, Brouwer added a note that “the opinions which it defends have as yet not found many supporters.” His major convert was Weyl in 1920, but Weyl was only a philosophical convert— not for long either—and did not do intuitionistic mathematics. To a somewhat lesser extent, one could add Bieberbach to the list of converts. Brouwer did have a few students—he cited one in his Marburg lecture of 1923—but the only one whose name is remembered today was Arend Heyting. [A small aside: In 1927, under the suggestion of Gerrit Mannoury, a self-taught Dutch mathematician, the Dutch Mathematical Association put up a prize question on formalising Brouwer’s mathematics insofar as such was possible. Heyting submitted a paper, and early in 1928 he was declared the winner. The paper was to have appeared in Mathematische Annalen, but appeared instead in the Sitzungsberichte of the Prussian Academy, of which Bieberbach was a member.] This was approximately the limit of his success. Hilbert’s fear of Brouwer was not only objectively irrational in the sense that he manifestly overestimated Brouwer’s influence—Perron’s derisive remark cited earlier is a typical mathematician’s reaction—but it was also irrational in that it was not Brouwer, but Kronecker, with whom he had been fighting all along. Bernays would eventually say as much, but we do not need such authority to state this as it is clear already from Hilbert’s often recalling his first solution of the invariant problem; his constant comparison of Brouwer with Kronecker; and his attribution to Brouwer of attitudes foreign to the latter’s character, such as the reproach that Hilbert’s formalism reduced mathematics to a meaningless game. [In his youth he might have said such a thing, but in the 1920s his opinion of Hilbert remained high. Despite the polemics of his last paper cited here, when Hilbert wrote to fire him, Brouwer wrote to Hilbert’s wife asking her to intercede, saying that Hilbert was too good a man to go forward with such action. I don’t believe this was mere rhetoric—Brouwer was no dissembler.] Upon reflection it appears that Brouwer was not the only tragic casualty of the affair. Hilbert himself was destroyed spiritually. Like Alexander the Great, who forgot his contempt for the Persian despots and died a more despotic ruler than any of them, Hilbert, in doing battle with the restrictive policies of Kronecker, was more effective than Kronecker in his own restrictive policy—this time not restricting mathematical freedom, but restricting philosophical freedom. Following such dramatic events, the rest of the story will appear something of a tame anticlimax. Hilbert stopped polemicising against Brouwer. Whether or not he realised that Brouwer was safely out of the way, his programme was so far advanced that it continued to roll along and, like a snowball, gain in volume. The extension of his list of proof theoretic desiderata had begun already in Bologna. 17. The Programme Expands Hilbert’s Bologna address consisted of a core of four problems embedded in discussion. For the first problem, Hilbert incorrectly noted that Ackermann and von Neumann had proven the consistency of the arithmetic of the integers. Hilbert

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now wanted to extend the proof to allow the choice function x to apply not merely to arithmetic formulae but to function variables. He remarked that Ackermann had almost proved this, there remaining only an arithmetically elementary finiteness theorem to be established. The second problem was to give a consistency proof for a stronger theory in which more advanced parts of analysis and some set theory could be carried out. The third and fourth problems were variants of the completeness of arithmetic, conjectured on the basis of Dedekind’s categoricity result. Problem III is stated in terms of consistency: III. If one can prove the consistency of ϕ with the axioms of number theory, then one cannot prove such consistency for ¬ϕ. The point to such an odd statement is simply this: Consistency implies existence. If ϕ and ¬ϕ were both consistent, there would be two non-isomorphic systems of arithmetic, contrary to Dedekind’s result. Problem IV is a more familiar statement of completeness: IV. If ϕ is not provable from the axioms of arithmetic, then adding ϕ as an axiom yields a contradiction [i.e. ¬ϕ is derivable]. Kurt G¨ odel’s refutation was two years away. Actually, G¨ odel has another connection with this paper. Immediately following the statement of Problem IV, Hilbert also raised the question of the completeness of predicate logic. G¨ odel would read a statement of this problem in Hilbert’s joint book with Ackermann (a revision of Hilbert’s lecture notes from a decade earlier), published in 1928, and prove the completeness theorem in his dissertation in 1929, publishing the result in 1930. The completeness theorem finally gave rigorous expression to Hilbert’s contention that consistency implied existence. This contention was such a commonplace that in the (unpublished) introduction to his dissertation G¨ odel felt it necessary to defend his having bothered to prove the result at all. He also explained that, unlike the consistency problem, the question of completeness could be “meaningfully” posed within the transfinite system. Thus, he saw no reason to restrict his methods. This would not be the case a year later with his proof of incompleteness. 18. G¨ odel’s Theorem The impromptu announcement of the First Incompleteness Theorem was the big non-event of 1930. This took place during the first of no fewer than 3 conferences held in September in the east Prussian town of K¨ onigsberg. This first conference was a meeting organised by philosophers to take advantage of the second—a meeting of the Society of German Scientists and Physicians. The third was a meeting of the DMV and doesn’t enter our story. The first conference was a three-day affair devoted to the foundations of mathematics. The three major competing philosophies of mathematics were presented by three major proponents thereof: Rudolf Carnap spoke on logicism, Heyting on intuitionism, and von Neumann on formalism ´ a la Hilbert. Von Neumann described not so much Hilbert’s actual programme as a variant of the mythic Programme of showing the use of the transfinite to be an unnecessary but generally efficient detour: The problems which Hilbert’s proof theory has to solve are the following:

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1. To enumerate all symbols which find application in mathematics and logic. . . 2. To characterise unambiguously all combinations of these symbols which stand for expressions classified as “meaningful” in classical mathematics. These are called “formulae”. . . 3. To give a constructive procedure that allows the successive generation of all formulæ which correspond to “provable” assertions of classical mathematics. This procedure is consequently called “proving”. 4. To show (in a finite-combinatorial way) that those formulæ corresponding to finitistically controllable (arithmetically checkable) assertions of classical mathematics can be proven (i.e. constructed) as in 3 if and only if the actual “check” just mentioned yields the correctness of the corresponding mathematical assertions. Were 1-4 secured, the absolute reliability of classical mathematics for the following purpose would be established: as a shorter method for calculation of arithmetical expressions for which the elementary working out would be too involved.

Von Neumann added that problems 1-3 had been solved by the work of Russell and his school, and the real problem was thus 4. For this he said that a consistency proof sufficed and he even sketched a proof of the correctness of any controllable assertion represented by a formula derived in a consistent formalism. It is not clear from these remarks, or indeed from the rest of his paper, exactly what constituted a “controllable assertion”. There is an implicit definition of controllability in his introductory remarks: An assertion is controllable if when we’ve made a mistake in its proof, we can detect this mistake through a finite procedure other than rereading the proof. The naturally controllable assertions would be the finitary general propositions as, when we’ve proven such an assertion, say, “for all numerals x, E holds for x,” we can make simple calculations to see if E holds for 0, if E holds for 1, etc.; and, if the assertion is false, we will see this when we check if E holds for the wrong n. However, when proving that problem 4 reduces to proving consistency, it is real propositions, i.e. numerical formulae, he considers. Moreover, the formulation of problem 4 and its ensuing comment suggests a restriction to decidable assertions—the obvious ones being the real propositions. Von Neumann was describing a general conservation programme of clear, if not definite, meaning, not just the intuitionistically unconvincing consistency programme. For this reason, during an organised discussion two days later, Brouwer’s disciple Heyting announced his pleasure with the conference and his full acceptance of Hilbert’s Programme. It was then that G¨ odel spoke up on the issue of the exact extent of conservation to be expected to follow from a consistency proof: According to the formalist conception one adjoins to the meaningful statements of mathematics transfinite (pseudo-)statements which in themselves have no meaning but only serve to make the system a well-rounded one just as in geometry one achieves a well-rounded system by the introduction of points at infinity. This conception presupposes that when one adds to the system S of meaningful statements the system T of transfinite statements and axioms and then proves a statement from S via a detour over statements from T , then this statement is also correct in its content so that through the addition of the transfinite axioms no conceptually false statements become provable. One commonly replaces this requirement with that of consistency. I would like to indicate that these two requirements cannot by any means be regarded as equivalent. For, if a meaningful sentence p

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is provable in a consistent formal system A (say that of classical mathematics), then all that follows from the consistency of A is that not-p is not provable within the system A. Nevertheless it remains conceivable that one could recognise not-p through some conceptual (intuitionistic) considerations which cannot be formally represented in A. In this case, despite the consistency of A, a sentence would be provable in A, the falsehood of which one could recognise through finite considerations. However, as soon as one construes the concept “meaningful statement” sufficiently narrowly (for example restricted to finite numerical equations) such a thing cannot occur. In contrast it would be, e.g., entirely possible that one could prove with the transfinite methods of classical mathematics a sentence of the form ∃xF (x) where F is a finite property of natural numbers (e.g. the negation of the Goldbach conjecture has this form) and on the other hand recognise through conceptual considerations that all numbers have the property not-F , and what I want to indicate is that this remains possible even if one had verified the consistency of the formal system of classical mathematics. For one cannot claim with certainty of any formal system that all conceptual considerations are representable in it.

This much is clear from G¨odel’s remarks: He knew of Hilbert’s Programme prior to von Neumann’s talk, as his terminology differs, and his knowledge is not based on a careful reading of Hilbert’s papers. The phrase “one commonly replaces. . . ” suggests that, like the description of Hilbert’s proof theory as turning mathematics into a formula-game, the new conservation Programme was being discussed, and the myth that it preceded the consistency programme consciously was being promulgated. For our purposes, it doesn’t really matter whether Hilbert wanted to prove consistency in order to establish some conservation result or whether Hilbert merely wanted to prove consistency because of some platonistic belief that consistency implied existence and merely realised by accident that the conservation result he had clearly believed in since the advent of his programme followed from the successful completion of the consistency programme. It doesn’t matter because G¨odel was only a few breaths away from destroying the conservation programme for finitary general propositions, and therewith the consistency programme. Mistaking the last sentence of G¨odel’s remark already cited for Brouwer’s doubts about the formal representability of our mathematical thoughts, von Neumann noted that “It is not settled that all modes of inference that are intuitionistically permitted can be represented formally.” To avoid misunderstanding, G¨ odel now stated explicitly his incompleteness result: One can (under the assumption of the consistency of classical mathematics) even give examples of statements (and even of the sort of Goldbach’s or Fermat’s), which are conceptually correct but unprovable in the formal system of classical mathematics. Therefore, if one adjoins the negation of such a statement to the axioms of classical mathematics, then one obtains a consistent system in which a conceptually false sentence is provable.

G¨ odel’s announcement of his First Incompleteness Theorem, being given as a critique—at first hypothetical—of Hilbert’s Programme and the inadequacy of consistency for Hilbert’s newly perceived purpose, did not make a big impression on everybody: Kurt Reidemeister finished off the discussion with a quick recapitulation— sans G¨ odel’s remarks—of the discussion. Von Neumann understood what G¨ odel said, and the two discussed G¨ odel’s work that day.

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What G¨ odel had done, though it was not immediately clear because of his doubts about what constituted finitary mathematics and the general unfamiliarity of the day with the formal systems involved, was this: he had given a finitistic construction of a sentence asserting its own unprovability. On assumption of consistency, it was finitistically verifiable that the sentence was unprovable. Transfinitely, it was clear that the sentence was true. Applying the construction to the finitary system S would produce a finitary general sentence asserting its unprovability in S. This unprovability is recognised in the transfinite system T , whence T proves the sentence. However, if S could prove the consistency of T , then by Hilbert’s 1927 argument, S could also prove G¨ odel’s sentence. Then T would prove both the provability and unprovability of G¨ odel’s sentence, and T would be inconsistent! It follows that Hilbert’s conservation Programme and consistency programme must both fail. All that remained were the fine-tuning of the result, the working out of further consequences of G¨ odel’s technique, and getting the message across. Evidently, Hilbert did not get the message until some time early in 1931—and then he more-or-less rejected it. In K¨onigsberg it seems no one told him about G¨ odel’s result, and apparently on the day after G¨ odel’s unexciting announcement, Hilbert gave his lecture—or lectures: This was the occasion of Hilbert’s retirement and K¨ onigsberg was the site of Hilbert’s youth and first professional position. Following his lecture to the German scientists and physicians, Hilbert was whisked off to the local radio station to deliver a shortened form of his speech to the general populace. His subject, natural science and logic, was a broad one and he spoke in generalities, finishing with the observation that the reason the French positivist philosopher Auguste Comte had been unable to find any unsolvable problems is that there aren’t any, emphasising this point with the motto, cited earlier, We must know; We will know.

On 24 December, Bernays wrote to G¨odel to ask for the proofs of the paper G¨ odel was writing on incompleteness. Courant and Issai Schur had told him that G¨ odel had obtained “significant and surprising” results. Hilbert, unaware of what G¨ odel had done, gave his final published talk on the foundations of elementary number theory that month in Hamburg. After describing his formal system of number theory, Hilbert noted incorrectly that Ackermann and von Neumann had proven the consistency thereof and added two open problems, versions of completeness from his Bologna talk (problems III and IV cited above). He could, he said, prove this in a special case by adding a new finitistic rule of inference: if, for every numeral n, the numerical formula ϕn can be checked to be a correct one, then conclude ∀xϕx. Hilbert noted quickly that ∀xϕx says more than that ϕn is correct for all numerals n, because it asserts the “truth” of ϕt for all terms t. Nonetheless, he accepted the rule and proceeded to show how the consistency proof extended to include it and how completeness for assertions ∀xϕx followed. By the early part of 1931 Hilbert had learned of G¨ odel’s Theorem. He was angry at first, but was soon trying to find a way around it. His solution was to extend the rule cited above. The simplest form of the ω-rule reads as follows: from ϕ0, ϕ1, . . . infer ∀xϕx. Under a variety of restrictions, the ω-rule allows the derivation of all true arithmetic sentences. None of them would be considered finitistic by anyone. Nonetheless, in

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his last full publication on foundations, Hilbert proved the Law of the Excluded Middle by deriving all true arithmetic sentences via the ω-rule. G¨ odel, who voiced caution in interpreting his incompleteness results and did not want to commit himself on the issue of whether or not he had destroyed Hilbert’s programme, now complained to his friend Olga Taussky-Todd, “How can he write such a paper after what I have done?” Hilbert would have one last thing to say. In 1934 the first volume of Grundlagen der Mathematik was published. Ostensibly a joint work with Bernays, the latter wrote the text and Hilbert wrote a short preface, in which he said, . . . the occasionally held opinion, that from the results of G¨ odel follows the nonexecutability of my Proof Theory, is shown to be erroneous. This result shows indeed only that for more advanced consistency proofs one must use the finite standpoint in a deeper way than is necessary for the consideration of elementary formalisms.

If we accept Hilbert’s simple dichotomy of actual mathematics and finitary metamathematics, with the latter now “deepened” (i.e., strengthened), then Hilbert’s remarks lead either to nonsense (since the same refutation of the original programme obtains) or to the conclusion that the formal codification of actual mathematics inadequately represents finitary mathematics, for G¨ odel had already proven his Second Incompleteness Theorem, according to which no sufficiently strong consistent formal theory could prove its own consistency. In particular, the formal system of actual, transfinite mathematics cannot prove its own consistency. It therefore cannot contain the deepened finitary mathematics. A second reading of Hilbert’s remarks, a more sensible and more commonly accepted one, is this: there is a never-ending hierarchy—arithmetic, analysis, set theory, . . . —of formal systems of actual mathematics, and a corresponding hierarchy of deepenings of metamathematics in which to prove the desired consistencies. As with the early Hilbert, the practitioners of this modified Hilbertian programme have given no thought to what the newly constructive consistency proofs are supposed to accomplish.5 If this new programme is philosophically suspect, resembling more a continuation through force of habit than a reasoned course of action, it has nevertheless been mathematically a successful development. On this point, it may be well to remember that our rather long discussion of Hilbert’s programmes has not been made for the sake of these programmes themselves, but to give the background to Gentzen’s work on consistency proofs. 19. Concluding Remarks I wish to finish this appendix with a few comments on the significance of G¨odel’s Incompleteness Theorems and the philosophical demise of Hilbert’s programme. In this I offer merely a few quick remarks and not a carefully reasoned philosophical discussion. The chief issue of the philosophy of mathematics is generally taken to be the question of the truth of mathematical laws. In what sense are they true? The traditional platonic answer that they are truths about some non-physical reality, although never refuted philosophically, is considered hopelessly na¨ıve, especially in these materialistic times. Logicism attempted an unconvincing reduction of 5 The three lectures by Gentzen reproduced in Appendix C attempt to supply some rationale to the extended programme.

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mathematics to logic but never had any strong mathematical adherents and died a natural death. Intuitionism and formalism offered the most convincing explanations of this truth during the twentieth century. Barring a platonic belief in the actual existence of mathematical entities, intuitionism currently offers the only acceptable philosophy of mathematics under which mathematical truths are genuinely true. The honest intuitionist must find much of mathematics to be meaningless. He has the option of rejecting the meaningless material as simply mistaken (in much the way the modern physicist ignores questions about the æther, or modern scholars no longer attempt to read the book of nature for the messages God has written in it—as, e.g., when Physiologus tells us that the fact that lion cubs are born dead and only come to life three days after being born is one of the ways God chose to remind us of the death and resurrection of Jesus Christ) or of attempting to explain how classical, non-constructive mathematics has been so successful. Intuitionism is often rejected as a philosophy of mathematics for not adequately explaining all of mathematics, i.e. for not choosing this second option. Such a criticism of intuitionism as a philosophy of mathematics (as opposed to intuitionism as the philosophy of mathematics) is unfair, for, as Weyl pointed out, nothing compels the intuitionist to go beyond constructive mathematics. Hilbert’s Programme may be viewed as an attempt to offer an intuitionistic justification of classical mathematics. G¨ odel proved that Hilbert’s approach, that of proving the outright consistency of classical mathematics, was too na¨ıve. Since G¨ odel proved his Incompleteness Theorems, there have been successful partial attempts: G¨ odel and Gentzen independently proved finitistically that Hilbert’s transfinite arithmetic is conservative over intuitionistic arithmetic with respect to finitary general propositions, and this result has been much improved. Thus, for some fragments of mathematics there is nothing to fear. Currently, there is much research that can be viewed as seeing how much classical mathematics can be rescued through such a modification of Hilbert’s Programme. The success of a more realistic version of Hilbert’s Programme cannot, however, go all the way. Modern set theory far transcends the mathematics that is conservative over constructive mathematics with respect to, say, finitary general propositions. Thus there arises again the problem of choosing between demonstrably safe fragments—rejecting the higher infinite—and accounting for the latter. Weyl’s suggestion of mathematics as a theoretical science, with its meaningful finitary core and meaningless transfinite theoretical component, offers the only coherent philosophy of modern, classical, infinitary mathematics that I know of. On this account, as with Hilbert’s programme, the transfinite mathematics is significant only as a means of deriving, say, finitary general propositions. Hilbert believed that every true such proposition had a finitary proof. Hence, the justification of transfinite mathematics was its mere consistency. By G¨ odel’s First Incompleteness Theorem, no attempt to formalise mathematics can prove all finitary general propositions. Thus, when one views mathematics as a theoretical science, the justification of transfinite mathematics must become, like the justification of scientific theories in general, not the fact that it yields nothing new, but the fact that it does yield something new—and the ease with which it does so. Indeed, in 1946 G¨ odel explicitly called for an effort to use progressively more powerful transfinite theories to derive new arithmetical theorems.

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In short (if we ignore the unjustly maligned platonism) we are presented with two reasonable but antagonistic philosophies of mathematics, and G¨ odel’s theorems tell us this mutual antagonism is necessary. In broader terms, we see here the familiar opposition of caution and daring, of the search for TRUTH and the search for truths. G¨ odel’s theorems rule out the na¨ıve hope that the two approaches will, upon closer examination, happen to co¨ıncide. The fight between Brouwer and Hilbert is just an episode in the long-running debate between the cautious mathematical conservative and the daring mathematical liberal. The debate continues today, but with less self-knowledge and only polemics coming from the conservatives, while the liberals have developed catastrophe theory, chaos, and large cardinals into coherent mathematical theories. To try to hide my bias, I quickly add that the Hilbert-Brouwer dispute shows that it is not always the overcautious that are the villains.

https://doi.org/10.1090/hmath/033/11

APPENDIX C

Three Lectures Gerhard Gentzen 1. The Concept of Infinity in Mathematics M¨ unster, 27 June 1936

The big fight over the foundations of mathematics which flared up in recent decades is above all a dispute over the nature of infinity in mathematics. In the following I will attempt to present in the most comprehensible manner the problems involved. First, I stratify mathematics into three levels according to the degree in which the notion “infinite” is applied in the various branches. The first and lowest level is represented by elementary number theory, that portion of number theory not using analytic tools. The infinite comes in here in its simplest form. We encounter it in the infinite sequence of the objects of the theory, in this case the natural numbers. A series of other branches of mathematics are logically equivalent to elementary number theory, namely all those theories whose objects can be placed in one-to-one correspondence with the natural numbers, i.e. which are countable. This includes almost all of algebra—one can for example verify the rational numbers, algebraic numbers, and polynomials to be countable. Further, combinatorial topology, that is, that part of topology dealing only with objects the properties of which can be described through finitely many particulars, affords another example. The well-known Four Colour Problem belongs here. All these theories are, viewed logically, completely equivalent; it suffices to treat only elementary number theory. The theorems and proofs of the other theories may be interpreted by number theoretic theorems and proofs via a correspondence between their objects and the natural numbers. Thus there corresponds, for example, to the Four Colour Problem an equivalent number theoretic problem, but of course this latter naturally interests us only through its intuitive topological formulation. The second level of branches of mathematics is represented by analysis. With regard to the application of the notion of infinity, there is something genuinely new here: the objects of the theory themselves can be infinite sets. This is because the real numbers, the objects of analysis, are defined as infinite sets, usually as infinite sequences of rational numbers. It matters not should one in particular choose to define real numbers by nested intervals or Dedekind cuts or what have you. To the second level also belongs all of complex analysis; here there is nothing essentially new. The third level of use of the concept of infinity finally brings us to general set theory. Here the objects are not only natural numbers and other finitely describable 343

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things as in the first level, nor are they infinite sets of such as in the second level, but rather they are infinite sets of infinite sets and again sets of such, etc., in the most extreme, conceivable generality allowable. All branches of mathematics can be placed among these levels. Geometry, for example, no longer presents any special problems with regard to the notion of infinity. What one may perceive as such either belongs to physics or appears in equivalent form in analysis; one can always embed the various geometries as models in analysis, to which they are logically equivalent. On the nature of the concept of infinity in mathematics there are two fundamentally different views, which I wish to describe in the sequel. I call them the “in-itself” conception and the “constructive” conception.1 The first is the viewpoint of classical mathematics as we all learned it in university. The constructive viewpoint may be represented by individual mathematicians—incidentally, not in equal proportion—I mention the names Kronecker, Poincar´e, Brouwer, and Weyl. These names alone tell us that we have to deal with a point of view that has to be taken seriously. I will attempt to clarify the essence of the constructive conception in its opposition to the in-itself conception. This is, in the space of a lecture, only partially possible, particularly when we consider that, through our long familiarity, the in-itself conception is in our blood and it won’t be easy to adapt oneself to an entirely different way of thinking. I begin with the antinomies 2 of set theory. Here is a case where the in-itself conception leads to nonsense, a situation that doesn’t arise with the constructive conception. If one takes the previously mentioned entirely general notion of set as a basis, then one can for example conceive of the “set of all sets”; this is a correctly defined set. This gives rise, as is easily seen, to contradictions: the set of all sets must contain itself as an element; it is in a certain sense easily made precise, larger than itself, which simply cannot be. If one examines the situation more closely, it is not hard to see where the nonsense comes from: the “set of all sets” cannot really be counted among the sets; it is as it were a posthumous formation, generated out of an already given totality of sets as an entirely new set. Therewith we have the constructive view of the situation: sets are in general only allowed to be formed constructively, in order, building atop each other continually. Counter to that, the in-itself conception claims that the totality of sets are accounted for from the start through the abstract set concept and therewith are already in “actual” existence, 1 Gentzen’s term for the former is an-sich Auffassung. Word for word, “an-sich” translates as “on itself”. The Guide for Translating Husserl offers “in itself” and “self-existent” as alternatives. I like the latter, as it is closer to the distinction being made, namely that between what are usually called actual and potential infinities. I was in fact tempted to use “actual-conception”. Manfred Szabo uses “actualist conception”. For the latter of the two concepts, Gentzen uses konstruktiv — constructive. This suggests simply using “classical conception” and “constructive conception”. However, “classical mathematics” is classical only in its use of Aristotelian logic; the genuinely classical notion of infinity is mostly of infinity as a potential infinity. This is certainly true of Aristotle and Gauss, if not of Democritus or Euler. Another possibility, in accordance with the phrase “na¨ıve set theory”, would be “na¨ıve”. As for Auffassung, I vacillate between “conception” and “point of view”, depending on the context. In all cases each choice is still a bit awkward, and I am sorely tempted to refer to the two sides simply as the “classical camp” and the “constructive camp”.—Transl. 2 This is an old Kantian term for a contradictory pair of statements which have equal justification and hence between which we cannot decide. At the beginning of the 20th century with the generation of so many set theoretic paradoxes, the word “antinomy” was resurrected to describe such.—Transl.

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entirely independently of how one might single out individual sets, for example through special constructions. This conception leads to antinomies. If one attempted to express the essence of the constructive viewpoint in a most general principle, one would formulate it approximately as follows: “Something infinite may never be considered as completed, rather only as something becoming, that is continually being constructed.” One can also say, “There is no actual, only a potential infinity.” I remember the famous words of Gauss, that “the use of an infinite quantity in mathematics [is] never permitted.” If one now assumes this principle of the constructive conception of infinity, one will find differences from the in-itself conception not only in set theory but also in all branches of mathematics, even already in the domain of elementary number theory. I’d like to go into these differences. In elementary number theory we encounter infinity first in its simplest form, namely in the image of the infinite sequence of natural numbers. From the initself point of view, one may consider this as a completed infinite totality; on the contrary constructive point of view one can only say: one can always go further, ever constructing new numbers; one may not, however, speak of a finished totality. For example, an expression “all natural numbers have the property g” has in the two cases quite different meanings. On the in-itself conception it means: each arbitrarily chosen number from the finished totality of numbers has the property g. On the constructive conception, one may only say: however far one may go in the formation of numbers, the property always holds. In practice this difference in conceptions comes to nothing here, because an assertion about all natural numbers would normally be proven by complete inducton, and this rule is obviously in agreement with the constructive point of view; it is inherent in the concept of the progression of the number sequence. The situation is different for existential assertions. The expression “There is a natural number with property g” says on the in-itself conception: “Somewhere in the completed totality of natural numbers such a number occurs.” On the constructive conception, however, such a claim is meaningless. This does not mean that under this conception one must in general reject existential assertions. If, in fact, a certain number n which has the property can be given, then one may, on this conception, speak of the existence of such a number, for the existential assertions do not refer anymore to the infinite totality of the numbers; it would suffice to speak only of the numbers from 1 to n. The existence proofs that occur in practice are mostly of this kind, by which an example can actually be produced. There are, however, proofs for which this is not the case, namely indirect existence proofs: one assumes that for all numbers the property g fails to hold. Should this assumption lead to a contradiction, one would conclude: there must be a number for which property g is valid. It could yet be the case that a procedure for actually discovering such a number is not to be had. If one assumes the constructive standpoint, one must reject the proof. Another method of proof that likewise is to be thrown out on this standpoint and which is most often emphasised in this connection is the application of the Law of the Excluded Middle for assertions about infinitely many objects. For example, on the constructive conception one may never say, “A property g holds for all natural numbers, or it does not hold for all natural numbers.” This rejection probably appears at first to be especially paradoxical, but it is only a necessary consequence of the principle of the conception of infinity

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as being only potential, because the law mentioned assumes one is presented with a finished number sequence. This is not to be interpreted as if the law would be considered false by constructivists; rather it is more correct to describe it as meaningless from this standpoint. It lacks, in this view, any meaning to speak generally of the totality of numbers as completed, since “in reality” the number sequence is never completed; rather one has only an arbitrarily continuable process of progression. These unreliable modes of reasoning on the constructive conception practically never arise in elementary number theory. It is different in analysis and set theory. Here the differences in the two conceptions are overall the same as I have described for the natural numbers; I won’t go further into it. However, here the practical significance of the differences is genuinely greater, to such an extent that from the constructive standpoint large parts of analysis and almost the entire theory of sets cannot be accepted. Here it should be mentioned that the demarcation between what the constructive point of view allows and doesn’t allow is in certain borderline cases not easily explicitly established and that on this the opinions of the various mathematicians representing this conception are not uniform. But these differences are not so essential for the general, overall picture that I have to go into their particulars. Words like “intuitionistic” (Brouwer) and “finitist” (Hilbert) indicate such somewhat different constructive standpoints. The cardinal question is: which of the two conceptions is correct? Both have their representatives. On the one side stand the intuitionists under Brouwer’s leadership maintaining the radical thesis that all mathematics not in accordance with the constructive standpoint is to be thrown out. On the other side, however, the majority of mathematicians understandably will not accept such a sacrifice. They admit that the antinomies are founded on unreliable concepts but believe these can be separated from the reliable ones; in particular, all of analysis and even more so all of number theory are fully unobjectionable. Unfortunately, the borderline of the impermissible inferences may be drawn in quite different ways without a definite position emerging as being to some degree inevitable, and I must say that to me the clearest and most fundamental separation appears to be that given by the principle of the constructive conception of infinity. Nevertheless, one does not wish to condemn the large non-constructive part of analysis, which has, among other things, proven its value through its many applications in physics. Hilbert tried to clear a path to the solution of these difficulties with his proof theory. This should make it possible to clarify the mutual relation between the two conceptions of the infinite through a purely mathematical investigation. How is that conceivable? The first and most important task is to prove as far as it goes the consistency of mathematics. That is the strongest argument of the constructivists: the in-itself conception leads in set theory to contradictions; who knows if one day in, say, analysis they cannot also appear? This objection would be removed by a consistency proof for analysis. It is in fact entirely conceivable that one can prove the consistency of a mathematical theory with precise mathematical means. To see this, remember that the expression of consistency may be formulated as a mathematical assertion; it says: there is in a theory no proof of a contradiction. The “proofs” in a theory may be made the objects of mathematical investigation— proof theory—exactly as the natural numbers are investigated in number theory.

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To this end, one formalises the proofs; i.e. one replaces the verbal expressions in the proofs by definite symbols and symbol combinations—the inferences correspond to certain formal transformations of symbol combinations—so that in the end one has certain figures consisting of symbols as images of the proofs. In order to formally delimit the concept of a “proof in a theory”, it is necessary, naturally, that one delimit beforehand the modes of inference that occur in the proof. It is in fact the case that in mathematical practice there is only a small number of (repeatedly occurring) modes of inference. If one is to carry out a consistency proof, then naturally one must again use some modes of inference in this proof. The correctness of these inferences must be presupposed from the beginning; otherwise the entire proof would be circular. An “absolute” consistency proof cannot be given. The type that these presupposed inferences must be is determined implicitly by the earlier-made considerations; the inferences must correspond to the constructive standpoint. Their reliability is assumed and not challenged. With the help of constructive inferences the consistency of the in-itself conception is to be proven. Such a proof has been obtained by me for elementary number theory, thus for the first of the three levels of the infinity concept.3 It must yet be extended to analysis and finally to set theory so far as this is consistent. Here it is to be expected that the proof theoretic investigation will at the same time give information on how far one can go without encountering antinomies as well as on related questions. How would the two conceptions of the infinite stand with respect to one another were the consistency proof completed? One encounters different opinions. One possibility would be that consistency is seen as still not sufficiently secured, in that one can yet raise doubt against the constructive inferences used in the proof. The danger of this objection I don’t consider particularly great. Something is always attained when the certainty of mathematical inferences is reduced to the least possibly objectionable inferences; more is then simply not possible. I definitely believe that this foundation will yield a truly greater certainty than the in-itself conception. More important is another objection raised by the intuitionists: even if consistency were proven, the assertions of the in-itself mathematics nonetheless remain meaningless and would thus still have to be rejected. So, for example, an indirect proof of an existential assertion would be meaningless: a genuine meaning is had by an existential claim only when it can really be given an instance. What is one to say to this? One must admit that an indirectly proven existential assertion has a different, weaker meaning than a constructively proven one, but a certain “meaning” remains. Further, even if one bestows on the nonconstructively proven expressions no immediate meaning, there still remains the possibility that one can prove simple and surely constructively meaningful, directly verifiable numerical equations via a detour over such [i.e., the nonconstructively proven] expressions; these must, on the basis of the consistency proof, be correct, and it can happen that a direct constructive proof for these expressions is much more laborious or even not at all to be obtained. Therewith the in-itself modes of reasoning would at least be secured a practical value, which even the constructivist must recognise. This whole question of “meaning” seems to me at the moment not to be ripe enough for a final decision. 3 “Die Widerspruchsfreiheit der reinen Zahlentheorie”, Mathematische Annalen 112 (1935), pp. 493-565.

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Particularly, from proof theoretic investigations it may be expected that genuine contributions to the solution of this problem will be given. A certain residue will eventually always remain a matter of opinion. The objection against the meaning of the in-itself expressions may not in all cases be easily overcome; there is already something to that. I believe that, for example, in the general theory of sets a careful proof theoretic investigation will finally confirm the opinion that all powers exceeding the countable are, in a quite definite sense, only empty appearances and one will have to have the good sense to do without these concepts. Following these general remarks, I wish now to say a little about certain difficulties that arise in the consistency proof; I will have to speak on G¨ odel’s Theorem and the significance of transfinite ordinals for the consistency proof. G¨ odel has proven the theorem: “The consistency of a mathematical theory, which contains elementary number theory, cannot—assuming that the theory really is consistent—be verified with the means of proof of the theory itself.” One can first believe that the possibility of a consistency proof will be illusory, because such a thing is supposed to use only more restricted means than is contained in the theory whose consistency is to be proven. But it remains perfectly conceivable that the consistency of, for example, elementary number theory can be proven with means which on the one hand are constructive, thus are not contained in the initself component of elementary number theory, and on the other hand nevertheless go beyond the framework of elementary number theory. The means used in my proof are the rule of “transfinite induction” applied to certain “transfinite ordinal numbers”. I will briefly indicate what is meant by this and what the concept has to do with the consistency proof. The concept of “transfinite ordinal number” originates with G. Cantor and really belongs to set theory. We use, however, only a very narrow part of these ordinals— what are referred to in set theory as a “segment of the second number class”4 —a part whose construction can be carried out strictly constructively and which shares nothing of the dubiousness of the in-itself conception which permeates set theory and must be avoided in a consistency proof. These transfinite ordinal numbers are formed as follows: first comes the series of natural numbers 1, 2, 3, etc. Now a new number ω is introduced, and it is determined that it should be placed after all the natural numbers. After ω follow ω + 1, ω + 2, ω + 3, etc. After all the numbers of the form ω + n follows ω2, then ω2 + 1, ω2 + 2, etc. After all the numbers of the form ωm + n follows the number ω 2 , then again ω 2 +1, ω 2 +2,. . . ω 2 +ω, ω 2 +ω +1,. . . ω 2 +ω2,. . . ω 2 +ω3,. . . ω 2 2,. . . , ω 2 3,. . . ω 2 4, etc., finally ω 3 ; and so one can continue on through the formation of ω 4 , ω 5 ,. . . , and finally to ω ω , and even further if one wishes. The whole procedure— which I have only hinted at here—may at first seem somewhat weird. It is however grounded in only two operations whose repeated application alone yields all these numbers: 1) to an already given number one can form a successor (addition of 1), 2) to an infinite sequence of numbers one can form a new number which may be placed after the entire sequence (limit formation). One could be concerned that this procedure is not constructive, as it certainly appears as if already with the formation of ω the in-itself conception of the finished sequence of natural numbers is involved. This, however, is not the case; one can view the concept of infinity here 4 The natural numbers make up the first number class, and the infinite countable ordinals the second.

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as potential throughout by saying: the number ω stands to each natural number n in the order relation n < ω however far we might go in constructing the number sequence. And it is exactly the same in the infinite sequences occurring in the formation of further ordinal numbers viewed in the constructive sense. Now, as to the concept of “transfinite induction”: this is nothing other than the extension of the method of inference of complete5 induction on the natural numbers to the transfinite ordinals. As we know, complete induction may be stated as follows: if an expression is valid for the number 1, and if one has proven that from its validity for all numbers preceding n its validity for n follows, then it is valid for all natural numbers. If one substitutes the notion “transfinite ordinal number” for “natural number” here, one obtains the statement of the rule of transfinite induction. The correctness of this rule can easily be made clear for the start of the transfinite number series as follows: suppose an expression is valid for the number 1, and one has further proved that if it is valid for all numbers preceding a given ordinal number, then it is valid also for this number. Then we conclude: the expression is valid for the number 1, thus also for the number 2, whence also for 3, etc., thus for all natural numbers. It follows that it is valid for ω because it holds for all its predecessors. Thus it is valid in the same reasoning for ω + 1 , and again for ω + 2 , etc., finally for ω2 ; and correspondingly one shows its validity for ω3, ω4, etc., finally also for ω 2 . So one can proceed step-by-step up the series of transfinite ordinal numbers, all the while convincing oneself of the correctness of the rule of transfinite induction. Mind you, the situation becomes outwardly somewhat complicated as the numbers grow larger, but the crux is always the same. Now I wish to explain how these concepts of transfinite ordinal number and the rule of transfinite induction come into the consistency proof. The connection is entirely natural and simple. In the consistency proof for elementary number theory one has to consider all conceivable number theoretic proofs and to verify that every single proof in a certain formally explained sense yields a “correct” result, in particular no contradiction. This “correctness” of a proof rests on the correctness of certain other, simpler, proofs which are contained as special cases or parts in the original. This circumstance allows us to linearly order the proofs in such a way that those proofs on whose correctness the correctness of another proof rests all precede the latter in this sequence. This ordering of proofs can be produced in such a way that one can assign a certain transfinite ordinal to each proof; the proofs preceding a given proof are those proofs whose ordinal numbers precede that of the given proof in the series of ordinal numbers. One might at first imagine that for such an ordering the natural numbers would already suffice. However, one needs in truth transfinite ordinals for the following reason: it can happen that the correctness of a proof rests on the correctness of infinitely many simpler proofs. An example: in the proof an expression is proven for all natural numbers by complete induction. Then the correctness of this proof clearly rests on the correctness of each of the infinitely many individual proofs arising through specialising the proof to a fixed natural number. In such cases, a natural number as the ordinal number of the proof would not suffice, because such a number has only finitely many predecessors. Thus, one uses transfinite ordinal numbers to represent the natural ordering of proofs by their complexity. 5 In

American texts this is referred to as the strong form of induction.

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Furthermore it is now clear exactly why one needs the rule of transfinite induction as the decisive rule for the consistency proof: one proves via this rule the correctness of each individual proof. Proof number 1 is trivially correct; and if the correctness of all proofs preceding a given proof in the ordering is secured, then so too is this proof correct, because the ordering was so chosen that its correctness depended on the correctness of certain preceding proofs. From this one can now conclude the correctness of all proofs by means of transfinite induction, and therewith one has in particular proven consistency. This transfinite induction is now exactly the rule in the consistency proof that, according to G¨ odel’s Theorem, of necessity can no longer be proven with the means available in elementary number theory. The proof of its correctness follows rather through a special consideration (reflection) of the sort that I carried out above up to the number ω 2 . Of course, one already uses for the natural numbers a whole chunk more of transfinite numbers, namely: as I have defined ω ω above, so one ω ωω obtains through a corresponding continuation of the procedure ω ω , then ω ω , etc.; after all of these numbers follows the number ε0 , the “first ε-number”. This number places the limit of the domain of transfinite ordinals which are required for the consistency proof for elementary number theory as it is usually formulated. I assume—but it is for the time being only a conjecture—that one can in the same manner also prove the consistency of analysis—and of set theory insofar as it is possible—where one must proceed a considerable distance further in the sequence of numbers of the second number class. Altogether, the following picture presents itself: in step with the stratification of the infinity-concept into three levels that I mentioned in the beginning of this lecture—elementary number theory, analysis, and set theory—goes the sequence of transfinite ordinals; just as the number ε0 belongs to elementary number theory as an upper bound, so there is a definite number of the second number class belonging to analysis as an upper bound and further for a formally delimited set theory—so far as such a thing is at all reasonably possible. One should not overestimate the absolute significance of such number bounds: already for elementary number theory it is the case that in solving certain of its problems one has to bring in new modes of inference through which the domain of elementary number theory would be further extended; this means that for the consistency proof one can require even higher ordinal numbers. There is no absolute bound here. G¨ odel has shown that every formally defined system of this sort is incomplete in the sense that certain problems belonging to it can be solved only through the addition of further resources. This doesn’t make any difference to the consistency proof; one needs only simultaneously to further extend this by the addition of corresponding new resources.

2. The Concept of Infinity and the Consistency of Mathematics Descartes Congress, Paris, 1937

The concept of infinity in mathematics has for a long time been the subject of a good many disputes. In recent decades discussion on the matter has again revived, and one can probably say has entered into a decisive stage. The various positions which one can assume with respect to the mathematical concept of infinity, or those recently represented by various authors, may be put into an approximate order as

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follows, according to the degree to which they accept the notion of infinity in its different complexities. First comes a purely finite mathematics, from which infinity is completely excluded. At the next level should be mentioned the original “finitism” of Hilbert. Then follows the “intuitionism” of Brouwer and Weyl. Next comes the “ramified type theory” of Principia Mathematica with the “axiom of reducibility”. Then the current favourite of “logicists”, simple type theory. At the next higher step there is for example “axiomatic set theory”, under which designation falls a number of versions of various scope. Finally, the conclusion is reached with classical Cantorian set theory, in which the unrestrained application of the notion of infinity led to antinomies. The sequence of the standpoints just enumerated may furthermore be divided into 3 groups. First again comes mathematics of the finite. I like to collect the succeeding finitism and intuitionism and their miscellaneous variants together under the concept of the “constructive conception” of the infinite. The common feature of these perspectives is that they consider the infinite as a potentiality, as something being built up constructively, as opposed to the conception of the infinite as something there from the start, given “in itself”, in the sense of an actual infinity. This last conception I prefer to designate the “in-itself ” conception of the infinite; it characterises the third group, in which the remaining above named systems, i.e. type theory and set theory, are categorised. One can draw a certain parallel between the two conceptions and the philosophical concepts of “idealism” and “realism”. Distinguishing between the constructive conception and the in-itself conception is on the whole feasible with reasonable certainty, although it can be a little difficult or dubious in borderline cases. Concerning the position of the different standpoints toward one another, the view of the radical constructivists is the most remarkable; it completely rejects the in-itself conception, in fact with two arguments: first, there is the legitimate fear that one day, exactly as in the unrestricted theory of sets, contradictions can arise in other areas, e.g. in classical analysis, for the final cause of the appearance of the antinomies is to be sought in the, in principle, dubious in-itself conception itself. Second, the propositions of the in-itself mathematics, even were contradictions excluded, would be meaningless claims. I prefer not to go into this last point, which is very important and can perhaps be discussed in detail sometime in the future; instead, in accordance with my topic, I shall occupy myself in the following only with the first. As is known, Hilbert had set himself the goal of conclusively settling this question by demanding the consistency of all questionable parts of mathematics be proven with the aid of a “proof theory” or “metamathematics” by elementary mathematical methods. In such a consistency proof, naturally, certain mathematical inferences and concept formations must already occur, and the reliability of these must be presupposed. The elementary finite mathematics does not suffice, because the proof theory must deal with infinitely many conceivable “proofs”. One thus must choose as a basis a domain in which the concept of infinity already appears, however only in the most harmless ways, and for this the domain of the constructive conception of the infinite presents itself, which domain is generally considered sufficiently secure. The task presents itself thus: the consistency of these parts of mathematics operating with the in-itself conception of the infinite must be proven via modes of inference in which the infinite is applied only in its constructive sense. Obviously, one will attempt to get by with a minimum application of the infinity concept; one

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can in this regard again distinguish within the domain of the constructive conception another series of increasing complications, as already hinted at by the earlier division. In particular it must here be mentioned that Brouwerian intuitionism, in my opinion, makes in a few points such extensive use of complicated applications of the infinity concept that if at all possible the consistency proof will have to be carried out with more restrictive means. If, however, this is impossible, then one will have to accept such far-reaching methods. “Finitism”, as Hilbert originally conceived it, has been shown to be too narrowly restrictive and needs a certain extension. The decisive factor remains after all that we are left with the greatest possible distance between the questionable proof methods whose consistency is to be established and the proof methods utilised in the proof itself. The certainty of the latter must be assumed and permit after all no further foundation by mathematical methods. Connected with these questions is G¨ odel’s famous theorem, by which the consistency of a mathematical theory cannot be proven with the tools of the theory itself, hence definitely not with more restrictive means. This theorem only apparently constitutes a refutation of Hilbert’s Programme, because it remains entirely conceivable that there are proof methods which from the constructive standpoint are reliable and nevertheless transcend in a certain way the domain of a formally delimited theory of the third group. Such proof methods I believe may be perceived in certain modes of inference which are closely connected with the so-called “transfinite induction” and which I have already applied in restricted form in my consistency proof for elementary number theory, from the generalisation of which I further hope for success with the currently outstanding consistency proofs for analysis and possibly for parts of set theory. I conclude with a few words on the connection between the consistency proof and transfinite induction: in my proof the number theoretic “proofs” whose consistency is to be proven are ordered in a series in such a way that at all times the consistency of any “proof” in the series follows from that of the preceding “proofs”. This series is easily mapped to the series of transfinite ordinal numbers less than ε0 . From this the consistency of all “proofs” follows via a transfinite induction up to ε0 . This suggests that a corresponding procedure will also be applicable to a more comprehensive theory, say analysis. Because every formally delimited theory consists of countably many proofs, should one succeed in ordering these proofs in any series after their mutual dependencies, this series must be mappable to a segment of the second number class, and one would again have a definite number of the second number class up to which to apply transfinite induction. It only matters that the mapping be carried out in a constructive manner, which in the case of number theory succeeded without difficulty. With that, transfinite induction remains as the only critical inference left. Naturally, it won’t do to include this among the unexamined assumptions, for transfinite induction is for the moment a highly questionable inference, which in classical set theory is proven with an essential reliance on the in-itself conception of the infinite. For this reason in my consistency proof transfinite induction up to ε0 was in no way assumed, but rather was grounded on a special proof of a constructive nature. The procedure is somewhat complicated, and I cannot go into it in more detail here, although admittedly it represents the essential point of the whole proof in view of the questions raised above. The procedure supplies, in accordance with its constructive character, the grounding of transfinite induction up to a definite ordinal number, in this case ε0 . If one wishes

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to include higher ordinal numbers, then one must extend the grounding procedure. I harbour the confident hope that in this or a similar way sooner or later Hilbert’s splendid Programme will find its completion in spite of all doubts. 3. The Current Situation in Research in the Foundations of Mathematics Bad Kreuznach, 21 September 1937

1. The Various Standpoints on the Question of the Antinomies and the Concept of Infinity 2. The Exact Mathematical Foundational Research: Axiomatics, Metalogic, Metamathematics, Theorems of G¨odel and Skolem 3. The Continuum 4. Possibility of the Union of the Various Standpoints 1. The Various Standpoints on the Question of the Antinomies and the Concept of Infinity. The antinomies of set theory were discovered around 40 years ago, and to this day their final clarification has not been achieved. They have greatly stimulated mathematical foundational research. The resulting clear revelation of the instability of the foundations of mathematics directly moved some of the most distinguished mathematicians—I mention only Brouwer, Hilbert, and Weyl—to have a good look at these questions, which otherwise normally lie far from the thoughts of most mathematicians, to whom they are frequently somewhat disagreeable because of their relation to philosophy with its uncertainty and multiplicity of opinion, in contrast to mathematical thinking. Several attempts have been made to find a “solution” to the antinomies, i.e. to show clearly where “the fallacy” is hiding. These attempts have led to no satisfying result, and one should expect no such solution in the future. The situation is rather thus that it cannot be a question of a flaw in reasoning that can definitely be pointed to. We can only say definitely that the materialisation of the antinomies is connected with the concept of infinity, because in a purely finite mathematics, as far as anyone can judge, no contradictions can arise, provided the mathematics is correctly constructed. Certain analogues in the finite rest on blatant vaguenesses in concept formation. To find a way out of the uncomfortable situation created by the antinomies, various courses have been charted. The simplest procedure is this, to draw a boundary between the permissible and impermissible modes of inference, whereby the inferences leading to antinomies are classed among the impermissible. There has been a whole series of such attempts; sometimes the proposed delineations are presented as being in some sense natural, and sometimes too such justification is done without. Examples are axiomatic set theory and the system of Principia Mathematica. In practice, this procedure is quite workable, though fundamentally it is not very satisfying. First, the delineation is somewhat arbitrary, a natural consequence of the circumstance that one cannot even determine “the error” behind the antinomies exactly. Second, the question suggests whether some day contradictions could not also arise in the circumscribed domain of permissible modes of inference. Certainly one can cite considerations which make it plausible that one has finally banished the antinomies, but this security is not particularly great. The possibility does not seem to me entirely excluded that contradictions can also be hiding in classical

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analysis. That one has not yet discovered them doesn’t mean much if one considers that the mathematician always gets by in practice with a relatively modest part of the, in themselves logically possible, manifold complexities of concept formation. The most logically consistent of the delineations is that given from the “intuitionistic” standpoint, which has chiefly been formulated by Brouwer and Weyl. Its standpoint is probably most simply understood from the following basic thesis: the notion of the infinite in mathematics must not be conceived as if infinite sets are in existence in themselves from the start and are discovered by mathematicians—a conception that I refer to briefly as the “in-itself” conception—rather merely in the sense that an infinite totality can be built up stepwise in a constructive manner from finite sets, whereby the infinite is never completed and is only an expression of the possibility of an unbounded extension of the finite. There is without doubt something to this principle, and even before the time of the antinomies there were endeavours with similar objectives. As soon as one adopts it, the antinomies disappear, because they obviously must use the in-itself conception of infinite sets. On the other hand, from this principle of the constructive conception there follows immediately the intuitionistically enforced taboo against certain familiar modes of inference of modern mathematics.6 To take an example, we consider the indirect existence proof, in practice the most important of such cases. With the classical conception one can indirectly prove the existence of a number with, say, a property E by deriving a contradiction somehow from the assumption that no number possesses the property E. Such a proof is to be rejected with the constructive conception, for one assumes in such a proof the infinite totality of all natural numbers; this is—on the constructive standpoint—meaningless, in that these can never be given as a completed totality; rather the numbers may only be considered an incomplete, ever extendible sequence. Nevertheless, on this standpoint, one can also prove the existence of a natural number with property E so long as one can directly give such a number or show the way to its calculation. Then the concept of the totality of all numbers no longer enters into the proof. A clear, easily readable summary exposition of the intuitionistic standpoint by Heyting has recently appeared.7 One must acknowledge that intuitionism has in fact drawn the most thoroughgoing conclusions from the difficulties raised by the antinomies. But serious objections have been raised against a radical intuitionism which categorically rejects as meaningless everything in mathematics that does not comply with the constructive point of view. I will go more deeply into this in section 4. At this point only one objection must be mentioned: if one assumes this standpoint, the entire classical analysis is reduced to a field of rubble. Many, and even some fundamental theorems, become invalid; other results must be taken and proven by other means. Moreover, the formulations become most awkward and the proofs much longer. Existence proofs such as, for example, that of the “Fundamental Theorem of Algebra” must now be so transformed in such a way that the number whose existence is claimed

6 A detailed account of these matters can be found, as it relates to number theory, in the third part of my work cited in note 8. Cf. also section 3 of the present paper. 7 Arend Heyting, Mathematische Grundlagenforschung—Intuitionismus—Beweistheorie, Erg. Math. Grenzgeb. 3 (1935), number 4.

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must be given a procedure for its calculation, and any exceptions where this cannot be done must be excluded. Certainly one may not shrink from this sacrifice if it is really necessary. But is such a sacrifice necessary? Therewith I come to Hilbert’s conception. He put forward the programme of rescuing the whole of classical mathematics from its evolved dubious situation by verifying its consistency by exact mathematical methods. The carrying out of this programme is unfortunately for the most part incomplete. It has turned out that the difficulties of such a consistency proof are greater than one was at first inclined to assume. (Cf. §2, G¨ odel’s Theorem.) In 1936 my proof of the consistency of elementary number theory appeared8 ; earlier partial results are due to Ackermann, von Neumann and Herbrand; however the (for practice above all important) proof for analysis has yet to appear. To carry out a consistency proof, one needs of course certain mathematical means of proof, the unobjectionability of which cannot further be grounded by these methods and so must be assumed. An absolute, i.e. assumption-free consistency proof is evidently impossible. One may now ask: what sort of proof means can be regarded as a foundation in this sense? The answer is given by our earlier comments. One may use such proof means in which the concept of infinity is applied only in the constructive sense, all the time paying strict attention to avoid everything that rests on the in-itself conception and is thus of a dubious nature. This restriction is something like what Hilbert had called the “finitist standpoint”. It appears though that for the consistency proof one needs somewhat broader means than those Hilbert originally envisioned and understood under the concept of “finitary proof means”. But in any case these means remain in compliance with the constructive conception of infinity, and that is the essential thing that differentiates them fundamentally from the dubious means of proof. It seems to me that a key feature of Hilbert’s standpoint is the attempt to remove the mathematical foundational problem from philosophy and handle it so far as it is possible with the proprietary methods of mathematics. Clearly one cannot solve the problem entirely without extramathematical presuppositions. Hilbert’s plan reduced these to a minimum: the fundamental difference between the constructive and the in-itself conception of the infinite must be brought into view and clarified— how it is that a genuinely greater measure of certainty befits an inference that is in accord with the constructive conception—so that one can choose this as a secure basis to which to reduce the consistency of that portion of mathematics using the in-itself conception. In the following I will not go into all the philosophical disputes, the responses to which have no influence on mathematical practice and which make the problem situation appear unnecessarily complicated and difficult. 8 G.

Gentzen, “Die Widerspruchsfreiheit der reinen Zahlentheorie”, Math. Ann. 112 (1936), pp. 493-565. It should be noted that in the 4th section of that work, in contrast to the remaining parts, due to lack of space and time, the intelligibility of the coherence of the proof did not fare well. A new version of the proof with detailed exposition of the underlying ideas forms the second part of the present number.—NB (EMT): in the printing in Deutsche Mathematik 3 (1938), pp. 225-268, specifically on page 257, one reads, “A new version of the proof with detailed exposition of the underlying ideas will appear soon with the present report on foundational research as number 4 of Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften.” From this it seems that the printing in Deutsche Mathematik was the first printing.

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Let me comment briefly on the so-called “logicism”, which is usually named along with intuitionism and the Hilbert conception as the third fundamental standpoint on the foundations of mathematics. Its theses are grounded on definite philosophical intuitions, which, in accordance with what I just said, I do not want to enter into. As to the, for mathematics of paramount importance, antinomies and the infinity problem, this direction has till now taken an indecisive, wait-and-see position and supplied hardly anything to its decision, since its interest in the main is aimed at other questions, e.g. the foundation of the number concept. 2. The Exact Mathematical Foundational Research: Axiomatics, Metalogic, Metamathematics, Theorems of G¨ odel and Skolem. In the following I shall say a few things about new results and especially about older particularly important results of the exact mathematical foundational research, i.e. of that branch of mathematics which does mathematical investigations on the foundations of mathematics. The objects of this research are, for example, axiom systems for mathematical theories—such investigations are known since antiquity; in recent times, however, the logical inferences as well as the proof methods of mathematics have also become objects of such study. In the last decades a growing number of researchers from all countries have occupied themselves with these questions and obtained a body of results. In Germany this metalogical and metamathematical research was for a while probably only conducted regularly in M¨ unster ; abroad, mainly America and Poland should be named as the main centres for this branch of mathematics. One of the main tasks of metamathematics is the carrying out of the consistency proofs required by Hilbert’s programme. Further major problems are: the Entscheidungsproblem (decision problem),9 i.e. the problem for a given theory to find a procedure which permits one to decide whether any conceivable sentence on the area in question is true or false; further, the question of completeness, i.e. the question whether a given system of axioms and inference rules is complete for a given theory, thus whether for any conceivable sentence of the theory one can prove with the help of these inference rules either the correctness or incorrectness of the sentence. odel Closely related to these main problems are some important theorems which G¨ proved about 8 years ago10 and which caused a sensation and which, to some extent, have been misinterpreted. First there is the theorem on consistency proofs, which says that the consistency of a mathematical theory which contains elementary number theory and is in fact consistent cannot be proven by the means of the theory itself; in particular, not with a fraction of these means. This theorem is often interpreted as if by it Hilbert’s programme has been proven unexecutable. One assumes, on several points of view, that the permissible “finitistic”, respectively, “constructive”, modes of inference to be used in a consistency proof represent only a portion of those available in elementary number theory and are precisely formulable rules of inference. Were this the case, then by G¨ odel’s Theorem the consistency of number theory would not be provable by such means. I am of the opinion, however, that there are modes of 9 As seen in the titles of Church’s and Turing’s papers cited in footnote 12 below, at the time it was fashionable to leave this untranslated.—Transl. 10 K. G¨ ¨ odel, “Uber formal unentscheidbare S¨ atze der Principia Mathematica und verwandter Systeme, I”, Monatsheft. Math. Physik 38 (1931), pp. 173-198.

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inference which are in complete agreement with the constructive conception of infinity and, on the other hand, do not belong in the framework of formalised number theory, indeed which conjecturally can be extended in general beyond the framework of any formally delineated theory. I have provided the relevant inferences, so far as they are required for the consistency proof of elementary number theory in my treatment thereof.11 They are connected with the “transfinite induction” of set theory, which does not mean that they share the sticky dubiousness of the latter; they can rather be proven constructively by methods entirely independent of set theory. Naturally, G¨ odel’s theorem remains highly significant regardless of these facts. A particular value is its service in the discovery of consistency proofs, for it tells one with which means one cannot attain the goal. Another of G¨ odel’s Theorems concerns the Entscheidungsproblem, in particular for the so-called “predicate logic”. It asserts that certain statements of the system cannot be decided with certain very far-reaching mathematical means. This theorem has recently been sharpened by Church in such a way as to show that, taking as a basis a very general concept of “procedure”, there is no general decision procedure for predicate logic, that the Entscheidungsproblem in this case is therefore not at all generally solvable.12 The situation now is that if the Entscheidungsproblem for predicate logic were solved, then, for example, the validity or invalidity of Fermat’s Last Theorem and similar number theoretic problems could in principle simply be calculated, and one would likely say that it doesn’t at first seem possible that such a decision procedure could be found. Nonetheless, it is naturally valuable to see this conjecture confirmed by a definite proof. Admittedly, Church’s proof rests on the assumption that an established concept of “computation procedure” is the most general possible. Should one succeed in discovering yet another sort of computation procedure, then it would be conceivable that one could attain thereby a more general decision procedure. One can say, however, that the concept provided by Church is so general that one cannot properly display any procedure that doesn’t fall under his definition. Further support for his formulation is the circumstance that attempts to isolate the notion of “computation procedure” have always resulted in this or an equivalent concept. A third result of G¨ odel’s concerns the completeness problem. It asserts that every formally delimited consistent mathematical theory is incomplete in the sense that one can give number theoretical sentences which are correct but which are not provable by means of the theory. This is without a doubt very interesting, but in any case it is not a disquieting result. One can also express it so: for number theory no once-and-for-all sufficient system of modes of inference can be given, but rather, sentences can always be found, the proofs of which again require new modes of inference. That this is the way things are one might not expect at first, but in any case it is not implausible. The theorem naturally reveals a certain weakness in the axiomatic method. 11 See

note 3. I had to be brief in the treatment and believe that a more thoroughgoing presentation clarifying this point as the crucial point of the entire proof would be of use: I hope on occasion to publish such, if possible, including the same for the consistency proof for analysis. 12 A. Church, “An unsolvable problem in elementary number theory”, Amer. J. Math 58 (1936), pp. 345-363; “A note on the Entscheidungsproblem”, J. Symbolic Logic 1 (1936), pp. 101-102. Cf. also: A.M. Turing, “On computable numbers, with an application to the Entscheidungsproblem”, Proc. Lond. Math. Soc. ser. 2, vol. 42 (1937), pp. 230-265.

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As consistency proofs generally secure only a limited system of proof means, one must extend them whenever one extends the proof means. It is remarkable that the sum of mathematics till now has applied only a small number of easily enumerable, constantly recurring, modes of inference and that, thus, the necessity of extension which exists theoretically is not relevant in practice. Now, in fact, the unprovable number theoretic sentences of G¨ odel were deliberately constructed for this purpose and in practice are without significance—mind you, with an essential exception: the expression of the consistency of a theory, which of course does not belong to the collection of provable theorems of the theory. For a proof of this the application of novel modes of inference is necessary, which in this case must moreover be of a constructive nature; on this I have earlier reported. I wish now to mention some results concerning set theory. To rescue set theory from the disaster of the antinomies, one has to set up certain restrictive conditions by which the inconsistencies will be excluded. To this end, various axiom systems for set theory were developed; most famous is the system of Zermelo and Frænkel. For a fragment of this system, the so-called “general set theory”,13 Ackermann will shortly have carried through a consistency proof, or, rather, the consistency of this system will be reduced to that of elementary number theory.14 “General set theory” arises if one removes from the full axiom system the “axiom of infinity” which asserts the existence of infinitely many objects of the theory.15 Ackermann’s proof rests on the fact that one can produce for this part of set theory a model consisting of natural numbers, a fact that is already long known. If one adds the axiom of infinity, one can no longer expect the same possibility, for therewith the existence of uncountable powers enters straightaway into set theory. In this connection, however, one should mention a theorem which on first view appears very peculiar and has some interesting consequences. This was first formulated by Skolem about 15 years ago and is termed the “Theorem on the Relativity of the Concept of Set”.16 Incidentally, in contrast to the earlier discussed metamathematical results, this theorem no longer belongs in the domain of the constructive conception, but rather is to be counted in the domain of the in-itself conception just because it concerns uncountable powers. (This circumstance, that it refers directly to the in-itself conception, in no way lessens its significance.) Skolem’s Theorem says: if for an axiom system of a certain sort any model of however large a cardinality exists, then there already exists a countable model which satisfies the axiom system. To the sort in question belong all prior customary axiom systems, or they may be transformed so that they are of this sort, and it is not obvious how one can formulate an axiom system that would not fall under Skolem’s scope. If one applies this theorem to any axiom system of set theory, one concludes that if the system is satisfiable at all, as one would naturally assume, then it can already be satisfied by a countable model. One could probably say that this result is not 13 This is not the general set theory referred to in the first lecture, where the phrase indicated ordinary set theory.—Transl. 14 W. Ackermann, “Die Widerspruchsfreiheit der allgemeinen Mengenlehre”, Math. Ann. 14 (1937), pp. 301-315. 15 Badly stated. The axiom of infinity asserts the existence of one infinite object, i.e. one object consisting of infinitely many objects as elements.—Transl. 16 Thoralf Skolem, “Einige Bemerkungen zur axiomatischen Begr¨ undung der Mengenlehre”, ¨ Verh. V. Skand. Math. Kongr. (1922), pp. 217-232; Th. Skolem, “Uber einige Grundlagenfragen der Mathematik”, Skr. Norske Vid.-Akad., Oslo, I., mat. - nat. Kl. (1929), no. 4.

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especially agreeable for axiomatic set theory. It asserts that all uncountable powers one speaks about in set theory are, in a certain sense, only appearances, insofar as one can substitute certain countable sets for them without altering the validity of any theorems. One’s first impression is that an inconsistency must arise from this. One proves in axiomatic set theory, for example, that the set of all real numbers is not countable. To be precise, one proves the theorem: there is no one-to-one correspondence between the natural numbers and the real numbers. Consider what this means in Skolem’s countable model of set theory. This model contains objects which represent the natural numbers of the axiom system, other objects which represent the real numbers, and still others representing the correspondences which are possible on the basis of the axiom system; and each sort consists of at most countably many objects. Nevertheless the theorem mentioned remains valid in this model, for among the correspondences available in the model, there are none mapping the countably many representatives of the “natural numbers” one-to-one onto the countably many representatives of the “real numbers”. If we assume a one-to-one correspondence, it will not be represented among the correspondences available in the model. Perhaps these not easily understood facts become somewhat clearer if one gives them the following expression, whereby I restrict myself to the continuum of real numbers as the prototype of an “uncountable set”: one puts oneself in the standpoint of the in-itself conception, that the continuum in and of itself is given beforehand, say as the set of all arbitrary infinite decimal fractions. Then following Cantor one can prove the uncountability of this system. Now, however, the following may be said: every axiom system for analysis that one may propose is in a certain sense inadequate for the complete capture of the continuum so conceived, for the theorem of Skolem tells us that, taking a definite axiom system, this continuum can be replaced by a countable model which similarly satisfies all the properties of the continuum laid down in the axiom system. By this conception, Skolem’s result would so to speak be demonstrated not to be a defect of the continuum or of higher cardinalities, but rather a defect of human thought with respect to characterising these powers. How abstract set theory will allow itself to be saved from the Scylla of the antinomies and Charybdis of the Skolem Relativity Theorem—if it will allow itself to be saved at all—is for the future to decide. One should emphasise that other parts of set theory (point sets, the second number class) are only in small measure affected by these difficulties and, in any event, always retain a certain significance. If one compares Skolem’s Theorem with G¨ odel’s Incompleteness Theorem, one can say that both illuminate certain imperfections of formally delimited axiom systems (including too their authorised proof methods). The perhaps at first view surprising circumstance of G¨ odel’s Theorem, that even when choosing the most complicated axiom systems of analysis, etc., unprovable number theoretic propositions constantly remain, receives a certain clarification from Skolem’s theorem: from this even the most complicated axiom system has a countable model and is thus referable to the natural numbers; its propositions may all be interpreted by number theoretic propositions: all these axioms are thus basically only number theory.

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Another result of Skolem17 illuminates this imperfection of the axiomatic procedure with respect to number theory; it states: given any axiom system for the natural numbers, of the above wholly general kind, then these numbers may be replaced by a model not isomorphic to the number series, which likewise satisfies the axiom system. 3. The Continuum. In this and the following paragraph I wish to go more deeply into the contrasts between the in-itself conception and the constructive conception as they relate to the most important domain for practical mathematics, namely analysis. To be precise, in this paragraph the difference between the two conceptions in the formation of the concept of real numbers and real functions will be explained, while in §4 a path to the unification of the different standpoints will be indicated. The treatment of irrational numbers commonly goes as follows: one divides the interval from 0 to 1 into two parts, each part again into two parts and so on, so one obtains successively finer subintervals. A sequence of such intervals where each is a subset of its predecessors contracts, the further one goes, more and more to a single point. Now in the framework of the in-itself conception of the leap into a completed infinity, here follows the declaration of an infinitely long such sequence as a “real number ”. Some peculiar consequences emerge from this conception which one could cite, quite apart from the general dubiousness of the in-itself conception, as arguments against it. On the one hand, one can prove in the usual manner that the real numbers form an uncountable set; on the other hand, all the theorems, definitions, and proofs which may ever be listed, respectively carried through, are countable, as they may always be represented via finitely many symbols. Thereby one has the consequence that there are real numbers which cannot be individually defined, valid theorems which no one can express and thus no one can prove. Add to this Skolem’s Relativity Theorem, and it further follows that the entire analysis up till now remains correct in all its points if one applies it to a certain countable model. The question arises: if the “uncountable continuum” as such remains completely inaccessible to our thoughts, is there any sense at all to speak of it as something real? In §4 I will show how one can—in a restricted sense—obtain an affirmative answer to this question. Next I attempt to explain what the constructive standpoint has to offer as a substitute for the in-itself conception of the irrational number. The sequence of nested intervals can be started as before. But the concept of a completed infinite sequence of intervals must be rejected as meaningless. The infinite is to be viewed only as a possibility, as an expression for the unboundedness of the finite. One may thus say it is possible to carry the subdivision on and on. One achieves no irrational number, but rather only in each stage of the subdivision a number of eventually very closely lying rational numbers. In this sense Kronecker said that “there are no irrational numbers.”18 The constructivist needn’t 17 Th.

¨ Skolem, “Uber die Unm¨ oglichkeit einer vollst¨ andigen Charakterisierung der Zahlenreihe mittels ein endlichen Axiomensystems”, Norsk mat. forenings skr., ser. II, nos. 1-12 (1933), ¨ pp. 73-83; Th. Skolem, Uber die Nicht-charaktersierbarkeit der Zahlenreihe mittels endlich oder abz¨ ahlbar unendlich vieler Aussagen mit ausschließlich Zahlenvariable”, Fund. Math. 23 (1934), pp. 150-161. 18 Compare H. Poincar´ e, Wissenschaft und Hypothese, German edition with remarks by F. Lindemann, p. 246 of the first and second editions.

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be so hidebound; there is the possibility to go further. Namely, one can consider an irrational number to be given if a law is at hand permitting one to compute a sequence of intervals of the given sort arbitrarily far. (It is advisable, to avoid certain formal difficulties, to take double dyadic intervals as one’s basis, i.e. sequences  (I0 , I0 ), (I1 , I1 ), . . . , for which In+1 ∪ In+1 = In .) √ Such a law is easily produced, for example for 2, the general n-th root of m, and also for transcendentals like π and e, indeed generally for almost all numbers which one needs as individually defined numbers: namely, whenever one can calculate the number concerned to any desired degree of accuracy. To remain true to the constructive standpoint, one must proceed cautiously with these so-called “numbers”. One must not be led astray and view such a number as a complete, infinitely long dyadic fraction; what is given is not really the entire number in this sense, but rather only the law that makes possible its stepwise determination. This itself is something finite, and it is merely proper to substitute it in certain connections for the infinitely long number which one can still say doesn’t really exist. In his work Das Kontinuum,19 Hermann Weyl attempted to construct an analysis on the basis of this sort of number concept. (Mind you, he did not carry this fundamental constructive philosophy through to the utmost with respect to the natural numbers, but did this later.20 ) A difficulty that arises here is the delineation of the permissible means of computation and therewith the definition of numbers. Weyl initially assumed a definite delineation; usually on the intuitionists’ part one would do without such, a standpoint that is thoroughly proper in that a generally valid delineation has fundamental difficulties analogous to those mentioned above for axiom- and inference-systems, namely: they are always capable and in need of extension. This is in no way a defect; in the, for the time being, most important cases in practice, it is immediately clear what is meant by the concept of “calculability”. One precise concept of calculability is represented by Church’s computability concept already mentioned in §2; independently of this, an equivalent concept was set up by Turing and applied to the computability of real numbers.21 These precise definitions are accomplished by providing an encompassing collective notion of “computable procedure” by means of which an impossibility proof as mentioned in §2 can be carried out; however, it doesn’t allow one to decide whether everything falling under its scope is a “computable procedure” or not. Brouwer introduced an extension of the constructive concept of real numbers with his “free choice sequences”. One can do this consistently as soon as one introduces the concept of a function of real numbers. The in-itself mathematics clarifies this familiarly as a correspondence through which each arbitrary real number is associated with a second real number as its function value. The concept of a completed infinity enters three times in this: namely, in both real numbers and in the universal abstract “assignment”. The constructivist thus cannot begin with this. He can proceed in such a manner that a function is defined as a law that assigns to each law defining a real number a second law that also defines a real number. It is 19 Kritische 20 Cf.

Untersuchungen u ¨ber die Grundlagen der Analysis, Leipzig, 1918—EMT. ¨ H. Weyl, “Uber die neue Grundlagenkrise der Mathematik”, Math. Z. 10 (1921),

pp. 39-79. 21 Cf. footnote 12.

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easy to see that the following less restrictive version which comes close to the function concept of the in-itself conception is entirely consistent with the fundamental constructive thesis. One works now with the concept of a real number determined by a law which satisfies the following: if, however, one begins a sequence of intervals of the above described kind, then at a certain finite initial segment of this sequence the function law will associate a first interval of the “function value”, continuing the sequence to a certain further position, a second such interval is associated, etc. The associated intervals must again form a nested chain. Briefly put: one must at any time be able to calculate by a function law a desired finite number of initial positions of the function value from a sufficiently large number of initial positions of the argument value. That this function concept is still narrower than the in-itself concept one can recognise already from the easily seen fact that such a “function” is always a continuous function. Brouwer proved moreover the uniform continuity of these functions using a considerably far-reaching application of “transfinite induction”.22 The argument values for this function concept are those that Brouwer called “free choice sequences,” that is, sequences of intervals for which one can choose the next interval freely—provided only that it satisfies the given restrictions for sequences of nested intervals. One must use this number concept with caution; it has no independent meaning, rather a meaning only in the appropriate context, for the completed infinite sequence is, as before, a meaningless concept. The free choice sequence may be used only in those contexts where one speaks only of a finite initial segment of it, or at most of the possibility of an arbitrary further extension. This is the case with the given definition of function. With this Brouwerian function concept one can constructively define the most commonly used functions of analysis without difficulty, for these are generally of the sort for which one can calculate progressively more exact values of the function by progressively narrowing the range of the argument values. Noticeable differences between intuitionistic analysis and the classical in-itself analysis reveal themselves in the further development of the theory, particularly with existence theorems, as already remarked in the first paragraph. This is because the constructivist must provide instruction for the calculation-law for the number for which existence is claimed and which the in-itself existence proofs usually do not give. Intuitionistic analysis is thus much more complicated than the classical. One has already observed something of this from the above given definitions of the basic concepts. The constructivist uses for example different concepts of real numbers for different purposes, while the in-itself conception gets by with a single simple concept. Admittedly because of the fundamental dubiousness of the in-itself conception a constructive consistency proof for classical analysis is urgently desired. I assume that this can be accomplished through further development of the means which made the consistency proof for number theory possible. One might perhaps think that because of the uncountability of the continuum, a fundamentally new 22 L.E.J. Brouwer, “Beweis, daß jede volle Funktion gleichm¨ assig stetig ist”, Proc. Akad. Wet. Amsterdam 27 (1929), pp. 189-193 and 644-646.

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difficulty will appear. But one can reply to this that by the Skolem Relativity Theorem every definite formally delimited system of analysis—and only for such a circumscribed system does one need to prove consistency—is already satisfied in a countable model, so that also for the question of consistency the uncountable is judged as only apparent. 4. Possibility of the Union of the Various Standpoints. I’d like now to express the opinion that upon the successful outcome of a consistency proof for analysis, the representatives of the various directions—that is the constructivists, that is to say the intuitionists on the one side and the Hilbert-followers as well as representatives of a pure in-itself conception on the other—can unite on the retention of classical analysis in its traditional form. Momentarily, to be sure, it is the case that the radical constructivists do not agree with this conclusion, and here lies the single, fundamental difference of opinion between Brouwer and Hilbert. Namely, the intuitionists declare all propositions depending on the mathematics of the in-itself conception of the infinite to be meaningless, their modes of inference to be an empty game of symbols without any significance. In the preceding paragraphs I have reported on various circumstances which lend this view a certain support. Against it, on the other hand, stands the immense richness of the successful applications of classical analysis in physics—to name only one perspective of greatest importance. In the following I wish to attempt to lay clear how one, even if he accepts the fundamental constructive thesis, can nonetheless succeed to an approval and continuation of the in-itself analysis. Hilbert himself showed the way to do this: namely, the method of ideal elements.23 That is: one regards those expressions involving the infinite in the sense of the in-itself conception as “ideal expressions”, as expressions which basically don’t mean what they say but which serve to round out the theory and can be of the greatest utility in making proofs easier and giving more convenient formulations of their results. In such a way for example one introduces the imaginary points into projective geometry and gains therewith the advantage of simplifying many theorems which would otherwise be loaded down with exceptions. Mind you one pays the price that the meaning of a proposition is in some cases not the familiar one. One says for example: “Two lines always have a point in common.” If the lines happen to be parallel, then in reality they have no common point. The procedure is completely harmless, because one has laid down exactly what one will understand in such exceptional cases by the concept “point”, which now has a broader meaning. Or let us take another example that seems to show with respect to physics even clearer analogies to the relation between constructive mathematics and the in-itself mathematics: I think of the occasional attempts to build a “natural geometry”, i.e. a geometry that fits physical experience better than the usual Euclidean geometry.24 In the former, for example, the proposition “through two distinct points runs exactly one line” is valid only if the points do not lie too close to each other. If in fact they lie very close together, then one can obviously draw several neighbouring lines through both points. The draughtsman must consider in such circumstances that this doesn’t happen in pure geometry because in it the points are idealised. It 23 Cf.

¨ D. Hilbert, “Uber das Unendliche”, Math. Ann. 96 (1926), pp. 161-180. J. Hjelmslev, “Die nat¨ urliche Geometrie”, Hamb. Math. Einzelschriften 1 (1923); also: Abhand. Math. Sem. Hamburg 2 (1923), pp. 1-36. 24 Cf.

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replaces the extended points of experience by the ideal, extensionless, in reality not occurring “point” of mathematical theory. And it does well thereby, as its success shows: it yields a mathematical theory of much simpler and better rounded form than does the natural geometry, which always has to deal with inconvenient exceptional cases. The relationship between the in-itself mathematics and the constructivistic mathematics corresponds completely: the in-itself mathematics idealises for example the concept of “existence”, in that it says: a number exists if its existence can be proven with the help of a proof in which the logical inferences in like manner as is valid for finite totalities is applied to completed infinite totalities; entirely as if this were a really existing creation. Thereby this existence concept acquires both the advantage and disadvantage of an ideal element: the advantage, namely, is that a considerable simplification and rounding off of the theory is achieved, because the intuitionistic existence proofs are, as mentioned, mostly very complicated and encumbered with disagreeable exceptions; the disadvantage in comparison is that this ideal existence concept is no longer applicable in the same measure to physical reality as is the constructive existence concept. Consider for example the equation a · x = b in real numbers. On the in-itself conception it means simply: the equation has a root so long as a is not 0. As opposed to this the intuitionist says: the equation has a root if I’ve established that a is different from 0. It can happen that from the form in which a is given one cannot determine that a equals 0 nor that a differs from 0. In this case, the question of the existence of a root remains open. One could probably agree that this conception better corresponds to the situation of the physicist, who has approximately determined the coefficient a from an experiment, but not exactly enough to determine a difference between a and 0 with certainty. One could now object: what use are pretty rounded off systems of theories and particularly simple propositions to us if they are not applicable in their literal sense to physical reality? Should one not prefer then a procedure that is more laborious and brings about more complicated results but has the advantage that these results possess immediate significance for reality? The answer is given by the success of the first procedure; think only of the example of geometry. The great achievements of mathematics for the furtherance of physical knowledge rest directly on this method, to idealise physical reality and thereby simplify its study. Admittedly, in application of general results in the real world, one must be aware of the qualified difference occasioned by the idealisation and carry out the corresponding translation. Applied mathematics has its field of activity here. For comparison, I cite the words of Heyting and Weyl. The intuitionist Heyting at one point says:25 “From the formalist standpoint the mastery of nature can be emphasised as the goal of physics. If his goal is achieved by means of formal methods”—i.e. the in-itself mathematics—“no argument against it is sound.” Weyl formulated his position in the fight between Brouwer and Hilbert:26 If mathematics is taken by itself, one should restrict oneself with Brouwer to the intuitively cognisable truths and consider the infinite only as an open field of possibilities; nothing compels us to go further. But in the natural sciences we are 25 In

the book cited in footnote 7, p. 65. Weyl, “Die Stufen des Unendlichen”, Jena (1931), p. 17.

26 Hermann

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in contact with a sphere which is impervious to intuitive evidence; here cognition necessarily becomes symbolical construction. Hence we need no longer demand that when mathematics is taken into the process of theoretical construction in physics it should be possible to set apart the mathematical element as a special domain in which all judgments are intuitively certain; from this higher viewpoint which makes the whole of science appear as one unit, I consider Hilbert to be right. I have the impression that certain fundamental intuitionistic concepts, e.g. the concept of existence or that of the real numbers, taken rigorously, are already “ideal elements”. But let that remain undecided; it is difficult to discuss and not so important. In any case it would not mean that the application of such concepts required a consistency proof ; one applies them only in such a way that one always remains aware of their exact constructive sense (§3). Just so is the situation in projective geometry with the “imaginary points”; otherwise, however, it is different with the ideal concepts of the in-itself mathematics, which—seen from the constructive standpoint—have absolutely no circumscribing “meaningful” content, but which nonetheless are applied as if they possessed a literal meaning. If on the one hand the in-itself mathematics receives a justification for the constructivist, so on the other hand a bigger place should be made in mathematics for the constructive standpoint than has previously been given to it. In foundational research it is already usual to lead proofs as much as possible down constructive paths, not only for their greater security—one hasn’t always relied on them—but rather because of the greater practical content of the results, for it is clear that a constructive existence proof means more than an indirect in-itself proof. Above all in elementary number theory, generally in all theorems having to do with finitely describable objects, it is natural to assume the constructive standpoint.27 One had always done that automatically: the wholly na¨ıve conclusion that one makes with no special thoughts about proof methods is by nature at first constructive; i.e. it shuns the “infinite”. Moreover in these areas the application of transfinite in-itself modes of inference are of practically no use. Alternatively, in the realm of the continuum, in analysis and geometry: here the in-itself conception celebrates its triumphs; here the constructive conception is inferior to it in practice. In conclusion let me say the constructive (“intuitionistic”, “finite”) mathematics represents an important part of the whole of mathematics through its great evidence and the special meaning of its results. As to its radical rejection of the parts of analysis underlain by the in-itself conception, however, there are no compelling reasons; on the contrary, this latter conception has its own major significance, above all in view of its physical applications. Whether one finally considers the continuum as a mere fiction, an ideal creation, or one understands it in the sense of the in-itself conception, that it possesses a reality independent of our means of construction, is a purely theoretical question, the decision of which is a matter of taste; for mathematical practice it has hardly any significance.

27 Compare also the foreword to the second edition of the first part of van der Wærden’s Moderne Algebra.

https://doi.org/10.1090/hmath/033/12

APPENDIX D

From Hilbert’s Programme to Gentzen’s Programme Jan von Plato 1. Mathematical Proof 1.1. The idea of proof. One of the earliest geometrical theorems is that of Pythagoras. In modern terms, Pythagoras had proved that the sides a, b, c of a right-angled triangle obey the equation a2 + b2 = c2 . Archæological and historical discoveries have shown that this relation was known much earlier, in Babylonian mathematics. There we find, written on clay tablets in cuneiform script, computational problems, the solution of which required a knowledge of the relation. For example, a straight piece of wood of known length stands against a perpendicular wall. The distance of the base of the wood to the wall is also known. At what height does the other end of the wood touch the wall? This particular example is from ca. 1800 BC. No trace of an idea of proof has been found in the sources of Babylonian mathematics. Moreover, general solutions to problems are always shown through particular examples. So it seems that geometry was not an invention of the ancient Greeks, contrary to what they themselves claimed, but the idea of mathematical proof was. Perhaps the earliest tale to this effect is about the pre-Socratic philosopher Thales, who is credited with having proved that the diameter of a circle divides it into two equal parts. This property is so obvious that one is naturally led to ask what can be meant by proving it. The art of proofs in geometry culminated in the great synthetic presentation of Euclid, the Elements. Geometry is organised as an axiomatic science, with a number of given basic concepts assumed understood without further explanation and all the other concepts defined in terms of these. Next, there is a number of geometric axioms the truth of which is to be immediately evident. The truth of the theorems of geometry, in turn, is to be reduced to the truth of the axioms through mathematical proof. To this organisation can be added construction problems: the basic constructions are those of a line segment connecting two given points, of a line continuing indefinitely a given line segment, of a circle with given radius and origin, and of a parallel to a given line through a given point. Arbitrary constructions are to be reduced to combinations of the basic ones. The Greek geometers did not produce any general theory about proofs in mathematics. They said many interesting things about proofs, such as that the search for proofs proceeds through an analytic method and that proofs are instead presented in a synthetic manner. Still, the principles of proof had to be learned intuitively, and this remained the situation in geometry until close to the year 1900. 367

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A general science of logic was proposed in Aristotle’s “theory of demonstrative arguments.” The pattern of logical arguments is that “given” assumptions lead to a determinate conclusion. Thus, these arguments are hypothetical, and their general pattern fits well the proofs of ancient geometry. In each theorem, it is spelt out what the assumptions are in the form of a number of given objects with given properties. Then a number of “sought” objects have to be found with determinate properties. The specific form of Aristotle’s logical theory, in contrast to its general pattern of hypothetical arguments, seems to have little in common with the proofs found in Greek mathematical texts. Aristotle singled out a great number of distinct forms of argument, his famous “syllogisms”, that were supposed to exhaust the possible logical steps of proof. These have forms such as: if each A is B and each B is C, then each A is C; if some A is B and each B is C, then some A is C; and so on. No sensible general theory of mathematical proofs was developed before the latter part of the 19th century. The German mathematician Gottlob Frege gave, in a little book of 1879 called Begriffsschrift (“concept writing”), the logic of the connectives “and, or, if. . . then . . . ”, and “not”, and of the quantifiers “for all” and “there exists”. Moreover, he made explicit the principles of proof of his logical language: if an implication “if A, then B” has been proved and if its condition A has also been proved, B can be concluded. Secondly, if a property C has been proved for an “arbitrary” individual x—let us denote this by C(x)—then the universal statement “for all x, C(x)” can be concluded. Frege’s main objective was to use his logic in the study of arithmetic. The subtitle of his book reads in fact “a formula language for pure thought, built upon the model of arithmetic.” Frege had very few readers. One reason was his forbidding notation: the formulas of his symbolic language were written as two-dimensional figures consisting of horizontal and vertical line segments and some letter symbols. In completely different quarters, namely in the school of the Italian geometer Giuseppe Peano around 1890, a symbolic notation was being developed for the formalisation of mathematics, in particular geometry and arithmetic. However, Peano did not give an explicit set of principles of proof; only the statements of mathematics got a formal representation, not what one does with them. This sounds quite amazing today. In Peano’s proofs, formulas appear in numbered lines one after the other. What they follow from is written as an implication: on each line, there appear first axiom instances and numbers of previously proved formulas, then an implication sign, and last the formula that is proved. A close look at Peano’s proofs shows that they consist of repetitions of the rule: if axioms and previously proved formulas imply that A implies B and the same for A, then B can be concluded. Young Bertrand Russell combined Frege’s logic with Peano’s notation in a work that culminated in the massive Principia Mathematica, written with A. N. Whitehead and published in three volumes in 1910–1913. Russell’s foundational aim was the same as Frege’s: to show that mathematical truths are ultimately logical truths. This doctrine came to be known as logicism. Frege’s foundational system of mathematics contained, in addition to the logic of the connectives and quantifiers, also principles about sets of mathematical objects. In particular, given a possible property P (x) of mathematical objects, all the objects that have the property can be gathered together into a set. This set is

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denoted by {x|P (x)}, “the set of all x such that P (x)”. In a famous letter to Frege dated 1902, Russell showed that elementary reasoning about sets of objects leads to contradictions. The famous paradox of Russell will be explained in Section 3. The turn of the century marks the beginning of an intense period of foundational studies in mathematics. Frege’s logic was the crucial tool by which intuitive mathematical reasoning could be made precise. Russell’s paradox did not result from this logic, which is faultless, but from Frege’s principles for forming sets. 1.2. The language of logic. Frege had proudly announced that in his new logical language, “everything necessary for a correct inference is expressed in full, but what is not necessary is not generally indicated; nothing is left to guesswork.” Variation in expressions, such as in “if A, then B” and “A is a sufficient condition for B”, etc., introduces grammatical distinctions that have no logical bearing. A symbolic notation divests the language of logic of such aspects of synonymity, as well as of ambiguity. The basic logical languages are those of propositional logic and its extension, predicate logic. The latter is expressive enough to function as the language of arithmetic, elementary algebra, geometry, and many other fields of mathematics. In propositional logic, we abstract from the standard informal language of mathematics such components as “A and B, A or B, not A, if A then B, A if and only if B.” Symbols are introduced for these “propositional connnectives”, and we write A&B for the conjunction “A and B”, A ∨ B for the disjunction “A or B”, ∼ A for the negation “not A”, A ⊃ B for the implication “if A, then B”, and A ⊃⊂ B for the equivalence “A if and only if B.” The symbols are like those used by Gentzen; others have used different symbols such as A ∧ B for A&B, A → B for A ⊃ B, and A ↔ B or A ≡ B for A ⊃⊂ B. In addition, parentheses need to be used for the unique readability of formulas. For example, A&B ∨ C could be either of (A&B) ∨ C or A&(B ∨ C). The rule for the use of parentheses is as follows: formulas that contain connectives are called compound; others are called simple. If a compound formula is an immediate part of a larger formula, it is put in parentheses. As an example, consider the formula (1)

(∼ A ⊃ (B ∨ C)) ⊃ ((∼ A ⊃ B) ∨ (∼ A ⊃ C)).

A little reflection shows that parentheses are used for coding in a linear sequence of symbols a structure that would be best seen as a two-dimensional one. Linguists call it a “syntax tree” of an expression. Such a tree unfolds the way in which a sentence has been constructed. Looking at (1), its “outermost” form is an implication. The left component is the implication ∼ A ⊃ (B ∨ C) and the right the disjunction (∼ A ⊃ B) ∨ (∼ A ⊃ C), and so on. Old Frege needed no parentheses, because he wrote his formulas in two dimensions to start with. To cut down the number of parentheses, it is agreed that & and ∨ “bind” more strongly than ⊃ and ⊃⊂, so we write A&B ⊃ C ∨ D instead of (A&B) ⊃ (C ∨ D), etc. This practice is analogous to that in algebra, where one writes, say, a · b + c instead of (a · b) + c. Moreover, we can leave parentheses out of conjunctions and disjunctions of more than two formulas. We have not yet said what the simple formulas are, those without connectives. In propositional logic, this is left unspecified: simple formulas are just assumed to be formal representations of “complete declarative sentences that state a possible state of affairs.” In predicate logic, simple formulas are analyzed into “terms”

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and “predicates”. Terms are expressions for individual objects, and predicates are expressions for possible properties of these objects. Say, for example, the objects can be the natural numbers 0, 1, 2, . . . with possible properties such as evenness, being a prime number, one number being a factor of another one, etc. In the language we thus have expressions for the objects 0, 1, 2, . . . , called constants. Initially one has to keep in mind the distinction between an object and an expression for an object, but in the long run it is not convenient to repeat “expression for. . . ” all over when talking about the language of arithmetic. We talk about objects and properties and so on, and the context determines if it is direct talk or talk about a language. The terms of a logical language include, besides the constants, also variables that are used for expressing generality and existence. Let P (7) express “7 is prime”. Then the existential formula ∃xP (x) says “there is an x such that x is prime.” Likewise, the universal formula ∀xP (x) says “for all x, x is prime.” (An obvious falsity, but even falsities need an expression.) The two quantifiers ∀, ∃ are applied in situations in which a “domain” D (or set) of individual objects has been specified. So variables take values in this set D. To be a prime number is a “one-place” predicate, divisibility a “two-place” predicate, that is, a relation between two numbers, and so on. As an example, consider the arithmetic formula ∀x∀y∀z∀v(v > 2 ⊃ ∼ xv + y v = z v ). This formula expresses Fermat’s theorem. Now consider ∼ ∃x∃y∃z∃v(v > 2 & xv + y v = z v ) and

∀v(v > 2 ⊃ ∼ ∃x∃y∃z xv + y v = z v ). A little reflection shows that these should express the same theorem, albeit in different forms. In predicate logic, we can prove that the formulas ∀x∀y∀z∀v(A(v) ⊃ ∼ B(x, y, z, v))

and ∼ ∃x∃y∃z∃v(A(v)&B(x, y, z, v)) and ∀v(A(v) ⊃ ∼ ∃x∃y∃zB(x, y, z, v)) are equivalent on the basis of their general logical form. Here, in place of the specific simple formulas in Fermat’s theorem, any one- and four-place predicates can occur. As another example of formalisation of mathematics with the language of predicate logic, we consider elementary plane geometry. The domain consists of two kinds of objects, points a, b, c, . . . and lines l, m, n, . . . . We have a number of basic relations: equality of two points a = b, equality of two lines l = m, and incidence of a point on a line a ∈ l. These relations suffice for projective geometry. In affine geometry, we add the relation of parallelism of two lines l  m. It is natural to enrich the language by adding geometric constructions, that is, objects constructed out of other given objects. We have the connecting line of two points a and b, denoted by ln(a, b); the intersection point of two lines l and m, denoted by pt(l, m); and in affine geometry the parallel to line l through point a, denoted by par(l, a). We have now defined the formal languages of elementary plane projective and affine geometries. An example of a formula in these languages is a ∈ l & b ∈ l & ∼ a = b ⊃ l = ln(a, b)

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in which we state formally that if a and b are distinct points incident with a line l, then l is equal to the connecting line of a and b. Another example, from affine geometry, is ∼ a ∈ l ⊃ ∼ (b ∈ l & b ∈ par(l, a)). In words, this is the Euclidean parallel postulate: “Given a point outside a line, no point is incident with that line and its parallel through the given point.” The geometrical examples were given with arbitrary objects a, b, and l. We could have used, equivalently, universally quantified variables as in ∀x∀y∀z(∼ x ∈ y ⊃ ∼ (z

∈

y&z

∈

par(y, x))).

Here the types of the variables have to be read from the places they take in the incidence relation. Predicate logic, or first-order logic as it is often called, is rich enough to work as the formal language of elementary geometry. Such geometry, in fact, was one of the first fields to which the nascent science of mathematical logic was applied towards the end of the 19th century. To cite an example, Peano’s student Mario Pieri considered in 1894–95 an axiomatisation of the basics of geometry. He realised that formalisation requires an explicit rule or axiom for the “substitution of equals by equals”, as in a ∈ l&a = b ⊃ b ∈ l. Is there a formal language for arithmetic within predicate logic? Again, as with geometry, the answer is yes, though this time with a qualification: the principle of arithmetic induction cannot be expressed in one formula, but only as a schematic axiom that has an infinity of instances, one for each property considered in a proof by induction. 1.3. Towards a crisis in foundations. The “crisis in foundations” in mathematics was brought forth by the paradoxes found around 1900. The most famous of these was Russell’s paradox of the “class of all classes not containing themselves.” Frege had put down the principle, mentioned already, by which all objects sharing a common property can be brought together into a set. Moreover, sets are considered to be objects in their own right. Now, “to be a set” is a property, and the collection of objects with this property is a set, the “set of all sets”, to be denoted V. Being a set, it has the defining property of V, in symbols, V ∈ V. To belong to itself is therefore a property that a set can have, and so is the opposite property, “not to belong to itself”. Let the set of all sets that have this property be denoted by R. Symbolically, we have R = {x| ∼ x ∈ x} with x a variable ranging over sets. If R ∈ R, then R has the property of not belonging to itself, i.e., ∼ R ∈ R. If ∼ R ∈ R, similarly R ∈ R. It is not clear when the development leading to the crisis of 1900 began. Still, it is clear that the development had to do with the levels of abstraction that are permitted in mathematics. Traditional mathematics began with some rather concrete and familiar objects such as the points and lines of school geometry or the sequence of natural numbers. The latter, in particular, is exemplary of the “genetic” structure of traditional mathematics: on the basis of the natural numbers, one introduces in succession the integers, the rational numbers, then the real numbers, and finally the complex numbers. Next one considers functions in each of these classes: the arithmetic, real, and complex functions.

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The natural numbers, integers, and rational numbers are all generated through an inductive definition: 0 is a natural number, and if n is a natural number, so is n + 1, and so on. These are finite objects, but not so for the real numbers which cannot be defined inductively on a finitary basis. The founder of set theory, Georg Cantor, found in 1873 a proof that the totality of real numbers cannot be put in a one-one correspondence with the inductively defined totality of the natural numbers. He then used the existence of the “nondenumerable infinity” of the real numbers in a proof of the existence of transcendental numbers. By definition, a number is transcendental if it cannot be given as a root of a polynomial axn + bxn−1 + · · · + dx + e = 0 with integer coefficients. Otherwise it is an algebraic number. Assume that there are no transcendental numbers, i.e., real numbers that are not roots of any polynomials. All possible polynomials can be put in a one-one correspondence with the natural numbers and form therefore only a denumerably infinite totality. (Say, count the number of symbols needed for writing the polynomial in standard notation. Then for any given number of symbols, there is only a bounded number of distinct polynomials.) The real numbers, Cantor’s argument continues, cannot instead be numbered: there is an excess of transcendental real numbers over and above the denumerably infinite algebraic ones. The assumption that there are no transcendental real numbers turned out to be contradictory. We can represent the logical form of Cantor’s argument in predicate logic as follows: Let T r(x) express that x is transcendental. Then, assuming ∼ ∃xT r(x), Cantor deduced a contradiction, by which ∃xT r(x) followed. Such was one of the first genuine indirect existence proofs mathematics has known. A direct proof instead would proceed by exhibiting some real number and a proof that it is not algebraic. Even propositional logic is sufficiently expressive to isolate the indirect inference. If a negative assumption ∼ A turns out to be impossible, A can be inferred. That ∼ A is impossible can be written as ∼∼ A, and indirect inference is captured by a logical law known as the “law of double negation”, ∼∼ A ⊃ A. There is a subtle difference between A and its double negation. It can be hinted at by reading ∼∼ A as “eventually ∼ A will turn out impossible.” Say, we have a method for computing the decimal expansion 0.000 . . . for a number, as well as a proof of impossibility of the assumption that it has no decimal distinct from 0. By indirect inference, some decimal is distinct from 0, but computationally we can only say, with no upper bound on how long it may take, that “eventually a nonzero decimal will be found.” Cantor’s problematic proof used, besides indirect inference, also Cantor’s proof of the nondenumerable infinity of the continuum of real numbers. Cantor had created a theory of a “transfinite” ordering of infinite sets: two sets have “the same cardinality” if there is a one-one correspondence between them. If there is such a correspondence between a set S and a subset T of U, but not between S and U itself, then U has a strictly greater cardinality than S. In particular, Cantor showed that the set of all subsets of a given set always has a greater cardinality than the set itself. The set of subsets of the set of natural numbers can be brought into a one-one correspondence with the real numbers. The nondenumerability of the latter was thus a specific case of a more general result that Cantor had established by his set theory. The most burning problem of set theory of the time around 1900 was to determine whether the infinity of the real numbers is the next one after that of the denumerable

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infinity of the natural numbers. In other words, does there exist a subset of the reals the cardinality of which lies strictly between the mentioned two infinities? Cantor’s famous continuum hypothesis assumes that there is no such set, but nobody has been able to settle the question in the more than one hundred years since it was formulated. What is worse, there are many who maintain that the whole problem is ill-defined. We have now seen the two ends of the foundational debate. One is: the construction of mathematical structures through definitions that start with a concrete basis, usually the natural numbers, with objects brought into existence through a direct construction and their properties proved by concrete arguments. The other extreme is: a set of abstract postulates implicitly defines a structure and its objects of study, with existence proved by showing nonexistence impossible. 2. Hilbert’s Programme 2.1. Hilbert’s Foundations of Geometry . David Hilbert’s worries about the foundations of mathematics became public around 1900, especially through his famous list of open mathematical problems of 1900. The first problem was to establish Cantor’s continuum hypothesis, and the second the consistency of the theory of real numbers. In his books and papers around 1900, we can trace a rapid transition from the traditional “genetic” approach to a “postulational” one. These are his own terms from a short article published in 1900. In the 1890s, Hilbert worked on the foundations of geometry, giving several series of lectures. Previous important work included that of Moritz Pasch and the Italian school led by Peano. The fruit of Hilbert’s efforts was the book Grundlagen der Geometrie (Foundations of Geometry) of 1899, a work that has been seen and is still seen as a turning point in foundational studies. Hilbert and his followers liked to describe formalised mathematics as an “empty game with symbols.” In the case of geometry, no concrete interpretation of the basic objects and properties is needed. This does not mean that the axioms would not have arisen from some intuitive basis; Hilbert himself states as much in the preface of his book: the erecting of a system of axioms for geometry is “a task that begins with the logical analysis of our spatial intuition.” Rather, once the axioms have been determined, they can have interpretations other than the one from which they were abstracted. Hilbert ended up with a symbolic axiomatisation, a system of synthetic geometry that, if it has to be placed in a mathematical tradition, belongs more to the Euclidean one than the tradition of geometry after Descartes. Hilbert’s book is widely cited as the beginning of a new era in foundations of mathematics. A reading of the book and of the literature on foundations of geometry gives a somewhat different picture. First of all, the idea of formalising geometry to the extent that it can be studied purely symbolically was by no means original to Hilbert. Also, previous axiom systems had already covered many of the intuitive passages of geometrical reasoning. What is worse, it turns out that even Hilbert’s own reasoning resorts to intuition and that the conceptual basis was incompletely formulated until at least the seventh edition of his book in 1930. Still, critical voices on Hilbert’s geometry have been few, the most notable being Hans Freudenthal’s long review of 1957. How is this to be explained? If we study systematically the literature on logic and foundations of mathematics of the past hundred years, we soon notice that the foundations of geometry, especially of the synthetic kind as in

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Hilbert, has been an absolutely marginal field of study. There are perhaps a dozen, and certainly not more than twenty, papers with contributions of any importance. Let us now discuss a bit more in detail Hilbert’s geometry to substantiate the claims made. Hermann Weyl has written that Hilbert formalised the geometrical part but left the logical part of geometrical proofs on an intuitive level. Yet even the geometrical part has many “gaps” that inevitably turn up when one starts formalising Hilbert. In the past couple of decades, formalisation has often been done with the help of a computer. There exist several “proof editors” for this purpose, computerised systems for formalised mathematics. These systems are usually interactive in many ways: the system, first of all, checks the correctness of each step suggested to it. Next it can complete parts and gaps in proofs and constructions in the solution of problems. A person who takes up Hilbert’s book and starts formalising should soon notice that something is wrong with Hilbert’s proofs. This occurred to the present writer even without the help of a proof editor. Keeping track of what axioms are used at which points in the proofs of Hilbert’s theorems, it turned out that the first axiom is never mentioned in the proofs. The two first axioms state: I1 For two points A, B there exists always a line a such that both of the points A, B are incident with it. I2 For two points A, B there exists not more that one line with which both points A, B are incident.

Hilbert adds that one always intends expressions such as “two points” as “two distinct points.” Instead of mentioning axiom I1 in proofs, Hilbert just writes “AB” for a line whenever two points A and B are given. By axiom I1, there exists such a line, but the notation is never explained. A comparison with the first edition shows why: I1 Two points A, B distinct from each other determine always a line a; we shall set AB = a or BA = a. I2 Any two distinct points of a line determine this line a; that is, if AB = a and AC = a, and B = C, then also BC = a.

The second edition, of 1903, is like the first one but with the notation for the constructed line AB left out: I1 I2

Two points A, B distinct from each other determine always a line a. Any two distinct points of a line determine this line.

Between 1899 and 1903, the construction of a connecting line has been lost, even if it is indirectly present in proofs that keep the notation AB. The reader is not warned. More serious gaps in Hilbert are revealed by his very first theorems. Theorem 1 states that “two (distinct) lines in a plane have one or no points in common.” No proof is given, but the idea must be that if the lines a and b had more than one point in common, say A and B, then axiom I2 would give a = AB = b, which goes against the assumption that a and b are distinct. Interestingly, the argument is indirect and shows that there is at most one point in common. In Hilbert’s notes for his lectures in geometry from 1893, the following passage is found: It escapes our experience whether one can always find a point of intersection of two lines. We therefore leave the matter preliminarily undecided and state only: 2 lines of a plane have either one or no points in common.

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Hilbert’s first theorem serves to prove the uniqueness of the parallel line construction, which is one of the geometric constructions in the first edition. Parallelism of lines, however, is explicitly defined only in the seventh edition of 1930. Before that, parallelism entered through the parallel line construction, turned into an existential axiom in 1903, which made the construction useless as a basis for a notion of parallelism. Hilbert’s fifth theorem assumes without proof, that two points on different sides of a line are distinct. One can only wonder how Hilbert may have proved this. Let us call the sides “side one” and “side two”, and let point A be on side one and B on side two. If A and B were equal, then, since A is on side one, also B would be on side one, and by the same reasoning, both would be on side two. The unstated geometric assumption seems to be that no point can be on both sides of a line. Moreover, the substitutability of equal points in geometric statements about the two sides of lines is assumed (see von Plato 1997 for details of these criticisms). The details of Hilbert’s geometry are incoherent. The reason lies in part in the imprecision of the formalisation that requires implicit appeal to intuition, as with Hilbert’s theorem 2. The incoherence is also due to Hilbert’s changing ideas about mathematical existence. These new ideas led to changes that were not consistently applied throughout the book. As said, synthetic geometry never turned into an active field of research after Hilbert. The permanent effect of his Grundlagen der Geometrie lies rather in its identification of some of the central foundational problems that a mathematical theory faces. 2.2. Hilbert’s programme. Hilbert’s central foundational problems, after his book on geometry, were the following: 1. The formalisation of a mathematical theory. This includes a choice of its basic objects and relations, and a choice of the axioms. 2. A proof of consistency of the axioms. 3. The question of the mutual independence and completeness of the axioms. 4. The decision problem: is there a method for answering any question that could be raised within the theory? Another list of problems of Hilbert’s is much better known: his list of open mathematical problems presented at the International Mathematical Congress in Paris in 1900. In it he writes: If we succeed in proving that the properties given to our objects never can lead to a contradiction in a finite number of logical inferences, I will say that the mathematical existence of an object, say a number or a function fulfilling certain properties, has been demonstrated.

In another paper of the year 1900, on the concept of number, Hilbert compares the “genetic” and “axiomatic” approaches: the former starts with the number 1, then builds up the sequence of natural numbers, and so on. The axiomatic approach, instead, postulates the existence of sets of objects the axioms talk about and poses the requirements of consistency and completeness on axiom systems. Both papers indicate that by 1900, Hilbert had definitely chosen the abstract and axiomatic approach to mathematics. To justify an axiomatic system, questions of consistency and completeness had to be settled. The next natural question was

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how this should be done. That is, what principles are permitted in the “metamathematical” study of mathematical systems? Hilbert’s main idea was that these methods should be “absolutely reliable”, for that would be the only way of clearing the foundational problems once and for all. Absolute reliability was to be achieved by reducing mathematics to a formal game with symbols, such that the correctness of each “move” can be checked mechanically. These “moves” make up a formal proof, so one also needs to show that the composition of steps of formal proof never leads to contradictions. At this point, it is clear that the study of formalised systems of mathematics is itself a mathematical task: in the first place, a study of combinatorial structures generated by inductively defined processes of construction. The threat of circularity in metamathematics was avoided by requiring that its methods of study be “finitary.” Since Thoralf Skolem’s introduction of what is known as primitive recursive arithmetic in 1923, it is almost universally agreed that finitary methods are precisely those that can be coded into principles valid in primitive recursive arithmetic. Hilbert had devised a programme for solving the foundational problems. The first task was to study proofs in pure logic. Hilbert’s way of formalising propositional logic is an instructive example of his approach. Let us call formal proofs “derivations”. These are defined inductively by giving the base case and the inductive steps. The base case is any instance of a finite number of axioms. The inductive case is just the familiar rule of inference known since mediæval times as “modus ponens”: If A ⊃ B and A have been derived, B can be concluded. Behind Hilbert’s approach there is a development arising from Frege’s and Russell’s philosophy of logic and mathematics. Logic is the same as the study of “logical truths”. The simplest logical truths are given by the axioms of logic. More complicated logical truths are reduced to the simplest ones through proof. So far so good, one might think. However, it turned out that a working set of axioms for propositional logic is something like: Hilbert’s axioms for propositional logic 1. A ∨ A ⊃ A, 2. A ⊃ A ∨ A, 3. A ∨ B ⊃ B ∨ A, 4. (A ⊃ B) ⊃ (C ∨ A ⊃ C ∨ B). The rule of inference is, from A ⊃ B and A to deduce B. In the axioms and the rule, implications A ⊃ B are just abbreviations for ∼ A ∨ B, and negation and disjunction are the only primitive connectives of the language. What guarantees that the axioms, especially the last one, are “simple logical truths”? What is worse, should not a formula such as A ⊃ A be such a truth? In Hilbert’s logic, it has a rather complicated proof. The overall conclusion is that Hilbert’s axiomatisation captures propositional logic, but the price for having just one rule of inference is that the system is next to useless for the actual proving of theorems of propositional logic. Next to propositional logic, Hilbert also formulated predicate logic, with two additional axioms and rules of inference for the universal and existential quantifiers. Hilbert wrote with his assistant Wilhelm Ackermann a short book on logic in 1928, called Grundz¨ uge der theoretischen Logik (The Basics of Theoretical Logic) that

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presented the above axiomatisations and summarised the situation: propositional logic had been formalised, proved consistent, complete, and decidable. Hilbert and Ackermann believed the same to be true of predicate logic. Consistency was easy, and the completeness of predicate logic was given as one central open problem in Hilbert and Ackermann. Kurt G¨ odel in his first work, published in 1930, gave a positive answer to this problem. Six years later, Alonzo Church showed that there is no general decision method for predicate logic. Thus, the central foundational questions had all been answered as far as pure logic goes, even if the answer to decidability was not the one Hilbert and Ackermann expected.

2.3. The cost of formalisation: relativity and incompleteness. The study of pure logic was preparatory for the greater aims of Hilbert’s proof theory: finitary proofs of consistency and completeness for arithmetic and for analysis. With the first of these began the ordeals of Hilbert’s axiomatic proof theory, for G¨ odel showed in a paper of 1931 that even a formalisation of elementary arithmetic can never be complete. Further, he showed that the consistency of arithmetic cannot be settled by such finitary means as required by Hilbert. Hilbert’s school took some time to accept the second conclusion: at the time of G¨odel’s results, Paul Bernays was writing a comprehensive presentation of Hilbert’s proof theory, titled Grundlagen der Mathematik. The first of two projected volumes was all but finished when G¨ odel’s results hit the enterprise. When, after a three-year delay, the first volume came out, it contained a preface by Hilbert in which he expressed his continued belief in the possibility of a finitary consistency proof of arithmetic. Bernays, instead, had accepted G¨ odel’s incompleteness results and started modifying Hilbert’s original proof-theoretical programme. G¨ odel’s theorem showed that the central aim of Hilbert’s programme is an impossibility. G¨ odel himself did not first make this conclusion in any categorical way, at least not in public, but Johann von Neumann instead did after he had found that the consistency of arithmetic is among the unprovable G¨ odelian propositions. He conjectured that the consistency of those parts of mathematics that use infinitistic principles is unprovable in an absolute sense. However, a way out was found in 1932, namely the translation from a classical to a constructive or intuitionistic system of arithmetic. It showed that constructive methods surpass finitistic methods, and led Gerhard Gentzen to develop a proof theory of arithmetic by which he succeeded in proving its consistency. It is today generally accepted that the first aim of Hilbert’s proof theory has in principle been achieved: mathematics can be represented as a formal system, or at least each part of mathematics can be. There are the grand attempts at a unified formalisation that use set theory as the language. These suffer, however, from the following defect: well before G¨odel’s result, Skolem had noticed a puzzling effect of formalisation, now called the L¨owenheim-Skolem theorem. It says that if a truly formal system is consistent, it has an interpretation in an at most denumerably infinite domain of individuals. Thus, a formal theory of real numbers, for example, can be interpreted as a theory about some denumerably infinite totality, and the same for formalised set theory. This situation came to sometimes be called “Skolem’s paradox”. For Skolem, it was no paradox but a refutation of the notion of an absolutely nondenumerable infinity. Skolem presented his own conclusions in a talk in Helsingfors (Swedish for Helsinki) in 1922:

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The most important result above is the relativity of set-theoretic concepts. . . I believed that it would be clear that this set-theoretical axiomatics could be no satisfactory ultimate foundation of mathematics, so that mathematicians for the most part would not care especially about it. In recent times I have instead seen, to my astonishment, that very many mathematicians consider these axioms of set theory an ideal foundation of mathematics. It therefore seemed to me that the time had come to publish a criticism.

Skolem took a strict finitist position in foundational questions, exemplified by his development of the theory of primitive recursive arithmetic in 1923 and later. He also foresaw the effect of “relativism” on arithmetic. The notion of an “arithmetical property” cannot be captured once and for all, but remains open-ended. Therefore, as Skolem wrote in a paper of 1931 published right before G¨ odel’s incompleteness paper, “the possibility of a complete characterisation of the sequence of natural numbers is dubious.” Two years earlier, Skolem had written that “it would be an interesting task to show that each collection of first-order formulas on the natural numbers remains valid when certain changes in the meaning of a ‘number’ are made” (see Skolem’s Selected Works in Logic, p. 269). Here we have a clear pre-G¨ odelian view that arithmetic has no complete formalisation. Set theory is remarkable for the simplicity of its basis. Its basic relation states that an object a is the member of a set S, symbolically a ∈ S. An equality relation for objects that are not sets is also needed, but all the other relations can be defined in the language of logic. Simplicity of the language is paid for by the lack of “expressive power”, as in the case of functions. Functions in set theory are defined as sets of “ordered pairs”. If the argument of a function f is x, and the value y = f (x), the idea is that all we need to know about f is which ordered pairs (x, y) “belong” to the function conceived as a set. What we cannot express is the simple idea of the application of a given function to an argument, nor do we have a representation of the computation of the value of a function. A final drawback is that interesting functions should permit an infinity of different arguments, but then, the function-asset is an infinite set. In order to define even one such function, we need a language in which it can be defined in some finitary manner, but pure set theory contains no such language. Thus, parts of mathematics that deal with the computability of functions have not been formalised within set theory. A general framework for such areas of mathematics was developed by Alonzo Church in the 1930s, known as the simple theory of types (see Negri and von Plato 2001, appendix A, for a short presentation). In conclusion, Hilbert’s first problem about formalisation can be met if we allow different formal languages for different parts of mathematics. His second problem about consistency was solved for the case of arithmetic by Gentzen in 1935, with a full clarification of the situation by 1943. We shall discuss these matters below. The consistency of analysis has been one of the major fields of study in proof theory, but no final answer is in sight. The final natural problem would be to study the consistency of set theory. As regards the completeness problem, G¨ odel’s theorem brought a fundamental limitation to formalisations of mathematics that use arithmetic as a basis. With the fourth foundational problem, it has been felt that by the undecidability of predicate logic, the “general case” already with mathematical theories formalisable within the mathematically weak language of predicate logic should be that they are undecidable. There are exceptions to this, the best-known being perhaps the theory

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of real-closed fields. A vast literature of studies exists on the decision problem in specific cases. Thus, one can study classes of formulas in predicate logic and locate with exactness a certain point of quantificational complexity beyond which it cannot be decided by any algorithm whether a formula is a theorem or not. Similarly, theories formalisable within predicate logic have been shown undecidable, one of the simplest examples being group theory. Remarkably, if one adds that the group operation, write it a ◦ b, obeys the commutative law a ◦ b = b ◦ a (an Abelian group), the theory becomes decidable. Not so long ago, decidable theories were thought uninteresting: why prove things if it can be done by a machine? Today, there are researchers who find the exact opposite to be the case. First of all, decidability is an “in principle” property. In practice, the decision might require computations that are exponential in the size of the input. Thus, one is led to introduce concepts that try to capture notions of “feasible computation” that are weaker (or stronger, depending on the point of view) than decidability. The prime example is polynomial time computability in which the number of steps to be taken in a computation is given by a polynomial function of the data of the computation. Such a notion is weaker than decidability, because each problem decidable in polynomial time is decidable but not vice versa. On the other hand, it is a stronger notion in the sense that it has stronger properties than mere decidability. The main open question in the field is if propositional logic has a polynomial time decision algorithm. Hundreds of equivalent formulations for this “P=NP” problem have been found. 2.4. Constructive and classical reasoning. We have seen that some mathematicians preferred a “genetic” structure of mathematics, or perhaps we should say more precisely, an inductively defined structure, with direct existence proofs. Others chose an abstract axiomatic approach and accepted indirect existence proofs. Prominent among the former around 1900 were several French mathematicians, including Henri Poincar´e and Emile Borel. Mathematical objects were to be defined “in a finite number of words.” Borel especially had some very deep insights that anticipated later developments. He saw that the equality of two real numbers is not a decidable property. It follows that not all real numbers can be expressed as decimals. Another consequence is that the behaviour of real functions at points of discontinuity cannot be computed. Thus, computable real functions must be continuous (see von Plato 1994, pp. 37–41, for these ideas of Borel’s). Borel’s early intimations became a clear-cut philosophy in the work of the Dutch topologist L. Brouwer. In 1907, he located the logical source of indirect proofs in the “law of excluded middle”, the logical law A∨ ∼ A. It has the same effect on proofs as the law of double negation. To see this, take an instance of the law of excluded middle: ∃xP (x)∨ ∼ ∃xP (x). Assume the latter impossible, i.e., assume ∼∼ ∃xP (x). We have now denied the second of the two alternatives given by our instance of the law of excluded middle, so only the first alternative ∃xP (x) remains. At the same time, we have derived ∃xP (x) from ∼∼ ∃xP (x) so the law of double negation follows from that of excluded middle. One may ask why it took so long to locate the logical source of indirect arguments in mathematics. Brouwer himself believed that the law of excluded middle leads to problematic conclusions only in the world of mathematics. He thought the world of everyday experience to be finite: empirically, we can discern only a bounded number of alternatives. Given a possible property P , we can decide between P

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and ∼ P for each alternative. If instead the alternatives are infinite in number, such as the possible objects of arithmetic, the law of excluded middle becomes, in Brouwer’s words, “unreliable.” Next to the logical aspect, Brouwer also undertook a revision of existing mathematics along constructivist lines. He called this “intuitionism”, a terminology that in philosophy relates to Kant. Of the Kantian intuitions, such as that of space and time, or of causal relations, Brouwer accepted only the intuition of discrete moments of time that follow each other. This intuition of an ever-repeating “present” that divides time into a “past” and a “future” gives rise to the sequence of natural numbers. At each stage, the sequence remains incomplete, and it is not permissible to contemplate properties that may depend on a completed infinity. Thus, to assume for a property of natural numbers P the law ∀xP (x) ∨ ∃x ∼ P (x), a version of excluded third, is not legitimate: a decision on the two alternatives may require something impossible, namely the inspection of an infinity of cases. Brouwer went on to develop the theory of real numbers on the basis of his philosophy. The structure turned out rather complicated. It was also felt that too much in mathematics depended on indirect proofs to give up such proofs. Hilbert, in particular, feared that intuitionistic restrictions on proofs would “mutilate” existing mathematics. Brouwer’s intuitionism contained aspects not inherent to its basic constructivism that made it look unattractive. Thus, Brouwer referred to the “primordial intuition” of mathematics as a “mental act” that creates the sequence of natural numbers. All mathematical objects are based on these and are “creations of the mind.” Further, proper communication with others is an impossibility, for language is not able to capture what a languageless pure mind has constructed. Instead, language “corrupts” the purity of thought. The use of language, like any social activity, is just a means for trying to win control over others. Many mathematicians accepted fully Brouwer’s criticisms of classical existence proofs. The most prominent of these was Hermann Weyl, a student of Hilbert. His paper of 1921 on “the new foundational crisis in mathematics” was widely read. As far away as in Moscow, young Andrei Kolmogorov and Alexander Khintchine were inspired by Weyl’s text, so much so that by 1925 they took Brouwer to be the in-principle winner of the foundational debate. Hilbert was alarmed by these developments. It seems that he was not fully able to distinguish between the objective scientific dispute and the aspirations to power in the world of mathematics that Brouwer held. The culmination of their “Grundlagenstreit” was the famous episode in 1928 in which Hilbert threw Brouwer off the board of four editors of the leading mathematics journal, the Mathematische Annalen. Brouwer had requested that only he be in charge of papers submitted to the journal by Dutch mathematicians, and that was too much for Hilbert. The upshot of the developments of the 1920s was that intuitionism got a bad reputation in mathematics and has kept it ever since. It is true that Brouwer’s programme was not successful in all respects. It actually came to a halt around 1928 with the theory of real functions and a principle called “bar induction”, or “induction on well-founded trees”. As background, we recall a principle called “K¨ onig’s lemma”: Assume we have a finitely branching tree and know that it has an infinity of nodes. By K¨ onig’s lemma, the tree has an infinitely long branch. Why is this not constructively acceptable? Imagine starting to climb up along all possible

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branches of the tree. Some turn out to be dead ends, but others continue, as the tree is infinite. There is no way of identifying an infinite branch in this way. Brouwer had turned K¨ onig’s lemma “upside down” by considering the contrapositive: if all branches in a tree are finite, the tree itself is finite. Brouwer called this “the fan theorem”. Next one considers a tree that can be infinitely branching, and bar induction is a similar contrapositive of a classical principle about trees. Let us keep in mind at this point that Hilbert had limited the methods of study in his proof theory to the finitistic ones. Even though Brouwer did not continue his intuitionistic programme, he had gone beyond the Hilbertian methods. It is therefore rather ironical, and indicative of Hilbert’s misunderstandings, that he took Brouwer to be a scientific threat. Brouwer himself saw the relation in a more proper light, actually suggesting that he was the real father of metamathematics by a paper of his of 1912. When in the 1930s proof theory borrowed its methods from intuitionism, Brouwer’s position was fully corroborated. G¨ odel’s incompleteness results for arithmetic showed, only three years after the Hilbert-Brouwer clash, that finitary methods are not sufficient. A very specific scientific discovery showed the way out. G¨ odel and Gentzen found out independently that the consistency problem of arithmetic is independent of the question of classical versus constructive existence proofs. 2.5. From Peano arithmetic to Heyting arithmetic. Around 1925, the Dutch philosopher G. Mannoury proposed the problem of making formally precise the logic behind Brouwer’s intuitionism, one that does not contain the classical law of excluded middle, A∨ ∼ A. He also proposed the dual problem of a logic that does not contain the law of contradiction, ∼ (A& ∼ A). The first problem was solved by Arend Heyting, a student of Brouwer’s, and published in 1930. Heyting proposed an axiomatic system of logic that would capture the logical side of Brouwer’s intuitionism. He also presented an axiomatisation of elementary arithmetic based on this logic. Soon, Heyting’s logic came to be called intuitionistic logic, and the intuitionistic system of arithmetic Heyting arithmetic. Heyting’s axiomatisation was, if possible, even more ugly than Hilbert’s axiomatisation of “classical logic”, a terminology invented for accentuating the difference and now standard. With G¨ odel’s incompleteness theorem, the conclusion was soon made that a finitary proof of consistency of arithmetic is impossible. The same followed for any theory built on arithmetic, such as the theory of real numbers and their functions, i.e., mathematical analysis. As mentioned, Johann von Neumann conjectured the absolute unprovability of the consistency of mathematics. G¨ odel and Gentzen found independently of each other a path that led out of this seemingly hopeless situation. The first indication is in a little theorem by the Moscow logician and probability theorist Valeri Glivenko. A lively debate had begun in the late 1920s on the proper axiomatisation of intuitionistic logic, conducted on the pages of the Bulletin of the Royal Belgian Academy. Glivenko observed in this connection in 1929 that in propositional logic, whenever a negative formula is provable in classical logic, it is already provable in intuitionistic logic. On the other hand, a formula in classical logic is equivalent to its double negation. Thus, a formula A is provable in classical propositional logic if and only if ∼∼ A is provable in intuitionistic propositional logic.

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G¨ odel and Gentzen worked through an idea similar to Glivenko’s for the whole of predicate logic and for an axiomatisation of arithmetic within the language of predicate logic. They defined a “translation manual” for formulas, starting from the simple formulas without logical structure and showing how the translation works on compound formulas. In the table below, A and B stand for given formulas of arithmetic, and A∗ and B ∗ for their translations: The G¨ odel-Gentzen translation 1.

(a = b)∗

2.

(∼ A)∗

3.

(A&B)∗

4.

(A ⊃ B)∗

5.

(∀xA)∗

6.

(A ∨ B)∗

7.

(∃xA)∗

;

a = b,

; ∼ A∗ , A∗ &B ∗ ,

; ; ; ; ;

A∗ ⊃ B ∗ , ∀xA∗ , ∼ (∼ A∗ & ∼ B ∗ ),

∼ ∀x ∼ A∗ .

G¨ odel’s translation removes also implications A ⊃ B by translating them into ∼ (A∗ & ∼ B ∗ ). The crucial points of the translation are disjunction and existence. The former is translated into the intuitionistically weaker ∼ (∼ A∗ & ∼ B ∗ ), and the latter into ∼ ∀x ∼ A(x)∗ . Even though there is a difference in the translations G¨ odel and Gentzen defined, the translation theorem that they established was the same: a formula A is a theorem of Peano arithmetic if and only if A∗ is a theorem of Heyting arithmetic. From the point of view of the “classicist”, the abandonment of indirect proofs has no effect: formulas that are provable only indirectly can be substituted by classically equivalent intuitionistically provable ones. The formulas are not intuitionistically equivalent, so from the intuitionistic point of view one can make a distinction that is not available for the classicist—that between directly and indirectly provable existential statements. Thus, even if every theorem of Heyting arithmetic is also a theorem of Peano arithmetic but not vice versa, this added “deductive strength” of Peano arithmetic is only apparent and is offset by an expressive weakness: disjunction and existence have no independent meaning in classical arithmetic, contrary to intuitionistic arithmetic. As a special case of the G¨odel-Gentzen translation, and with atomic formulas double-negated, a translation of classical predicate logic into intuitionistic predicate logic is obtained. The overall conclusion, regarding the fearful “limitations” set by intuitionism, can be concisely put in the words of G¨ odel himself: “Nothing at all is lost by dropping the law of excluded middle, but only the interpretation of the theorems has to be changed” (from G¨odel’s “Yale lecture” of 1941). Gentzen drew the same conclusions, but the fate of his paper, titled in English translation On the relation between intuitionistic and classical arithmetic, was unfortunate. It was submitted in March 1933, a fact testified by galley proofs that have remained, but was withdrawn when Gentzen heard of the publication of G¨ odel’s paper. The latter came out in the Ergebnisse eines mathematischen Kolloquiums, a journal certainly much less read than the Mathematische Annalen, to which Gentzen had sent his thoroughly written paper. Besides the formal result, it drew conclusions about the methodological position of proof theory after G¨ odel’s

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incompleteness theorems. Its publication in the leading mathematics journal of the time would certainly have clarified many misunderstandings. Afterwards, it seems next to miraculous how well both G¨ odel and Gentzen were able to navigate in unknown seas: Heyting’s brand new intuitionistic logic was accepted in its essentials, irrespective of its clumsy axiomatic formulation. It is told that Heyting arrived at his axioms as follows: he took up Russell’s Principia Mathematica and judged which of its axioms and theorems would be intuitionistically acceptable, and then he put down a list of axioms sufficient to guarantee the provability of these. (I owe this story to Johan van Benthem.) The foundational motive of G¨ odel’s translation was the following. Intuitionistic reasoning should be “more reliable” than classical, and by the translation theorem, it follows especially that if a contradiction is provable in Peano arithmetic, it is already provable in Heyting arithmetic. Thus, the consistency of the latter is sufficient to guarantee the consistency of the former. Gentzen arrived independently at the same conclusion. The good news for Hilbert’s programme was that indirect inferences are justified: their addition to intuitionistic arithmetic will not lead to contradictions. On the other hand, by G¨ odel’s incompleteness theorem, intuitionistic arithmetic must go beyond finitistic reasoning. 3. Gentzen’s Programme Gentzen found the translation from Peano arithmetic to Heyting arithmetic in 1932. He had already written one paper that we shall discuss briefly later, but now present its topic. The new discovery showed that the consistency of arithmetic, even if unattainable finitistically, is still within the reach of constructive methods as formulated in intuitionistic mathematics. Thus, Gentzen launched on a programme, the aim of which was to establish the consistency of classical mathematics, in the first place arithmetic and analysis, by intuitionistically acceptable methods. A further possible extension was set theory, but Gentzen was not sure that it could be done. As a premilinary to his programme, Gentzen set himself the task of studying pure logic. 3.1. Natural deduction. Gentzen started at once with his programme, and within one year he had covered pure logic and its application to arithmetic with the principle of arithmetic induction excluded. The results were contained in his doctoral dissertation Untersuchungen u ¨ber das logische Schliessen (Investigations into logical inference), accepted in G¨ ottingen in June 1933 and published in two parts in 1934 and 1935. Gentzen must have been dissatisfied with the formulation of logic in the axiomatic way of Frege, Russell, and Hilbert. Heyting’s intuitionistic axiomatics, reproduced in Gentzen’s paper, is even more unmanageable and may have contributed to his dissatisfaction. Anyway, Gentzen took the language of predicate logic as given, but none of its axioms. Instead, through a study of how logical arguments are really made in mathematical proofs, he arrived at a system of logic he called “nat¨ urliches Schliessen” (natural deduction). It is natural in the simple sense of “being a formalism as close to actual reasoning as possible” (Gentzen 1934–35, Introduction). The structural analysis of proofs through natural deduction was successful for intuitionistic logic, but for classical logic, Gentzen had to invent a more general logical calculus, known as “sequent calculus.”

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Gentzen’s main observation about “actual reasoning” in mathematics was that it is hypothetical, based on the making of assumptions. In axiomatic logic, instead, derivations start with instances of axioms from which conclusions are made with one or two rules of inference. These conclusions are categorical; i.e., they depend on no assumptions. As an example of a formula to prove, consider A ∨ B ⊃ C&D. According to Gentzen, the natural way to proceed is to assume A ∨ B to be the case and then to consider what can be done under this assumption. Likewise, one considers what would be sufficient for the conclusion. For the given formula, the assumption A ∨ B can be analyzed into: (1) A is the case: it is sufficient to prove C from assumption A and also D from assumption A. (2) B is the case: the task is to prove C from assumption B and D from assumption B. If all parts of 1 and 2 succeed, we have proved A ∨ B ⊃ C&D with no assumptions left open. Gentzen suggested that to each form of proposition, A&B, A ∨ B, A ⊃ B, and so on, corresponds a principle of proof: namely, one that gives the sufficient conditions for concluding the proposition. The rules for the above propositions are, with a line separating the premises (one or two) and the conclusion of the rule: 1. The introduction rules of natural deduction

A B &I A&B

A ∨I 1 A∨B

B ∨I 2 A∨B

[A] .. .. B ⊃I A⊃B

There are signs next to the inference line to indicate what rule has been applied. Rule &I, called conjunction introduction, says that to prove A&B, it is sufficient to have proved A and proved B. Next, to prove A∨B it is sufficient to have proved one of A and B. To prove A ⊃ B it is sufficient to have proved B from the temporary assumption A. The square brackets indicate that the conclusion A ⊃ B does not depend on the temporary assumption A that has been “discharged”. The rules for the universal and existential quantifiers are : 2. The introduction rules for the quantifiers A(y) ∀I ∀xA(x)

A(t) ∃I ∃xA(x)

Rule ∀I has the standard variable restriction: the eigenvariable y of ∀I must not be free in any assumptions A(y) depends on. Thus, the meaning of a universal quantification ∀xA(x) in terms of proofs is that A(y) can be proved for an arbitrary y, i.e., one of which no assumptions have been made. For rule ∃I, the explanation is that the property A has been proved for some individual t. The object of logic, in Gentzen’s view, is to study the general structure of proofs. These proofs proceed from assumptions to conclusions, following the general pattern that we found already in Aristotle. It was a complete break with the logicist tradition of Frege, Peano, and Russell that Hilbert and his school had been pursuing and in which the notion of logical truth is basic. Proofs that follow a precise set of rules are called derivations, to distinguish them from most of the informal proofs found in mathematics. The combination of several steps of inference produces a tree-like figure. Given a derivation with a conclusion

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C, those top formulas in the derivation tree that have not been discharged are the open assumptions. If there are none, the conclusion is a theorem. In addition to the above introduction rules for each of the connectives, Gentzen gives elimination rules that are the reverse of sorts to the introduction rules, where the “major premises” are the formulas with the connective: 3. Gentzen’s elimination rules for natural deduction

A&B &E 1 A

A&B &E 2 B

[A] [B] .. .. .. .. A∨B C C ∨E C

A⊃B B

A ⊃E

The rule for disjunction elimination is entirely natural, and we used it already informally above, where we had an assumption of the form A ∨ B. When A ∨ B appears as the major premise of an elimination step, there are two cases in the proof, A and B; and if both lead to the same conclusion C, that conclusion is obtained irrespective of whether A or B was the case. This is what the discharge brackets indicate in the rule. The elimination rule for implication is the same as modus ponens. The elimination rules for the quantifiers are: 4. The elimination rules for the quantifiers

∀xA(x) ∀E A(t)

[A(y)] .. .. ∃xA(x) C ∃E C

The same variable restriction holds for rule ∃E as for rule ∀I; i.e., the eigenvariable y of ∃E must be arbitrary. Rule ∀E has a term t (a constant or variable) similarly to rule ∃I. We add to the above system of introduction and elimination rules what is sometimes called “the rule of assumption”, namely, that derivations can be started with any formulas A that act as open assumptions. The system of rules of introduction and elimination is called intuitionistic natural deduction. Gentzen’s main result about natural deduction is what is today called the “normal form theorem”. For reasons to be explained soon, he indicates the result by only one example in (1934–35, II.5.1 3). By the theorem, derivations in natural deduction can be transformed into a certain transparent form in which no step of introduction is followed by a corresponding elimination. As an example, assume there is an instance of &I followed by &E1 : .. .. .. .. A B &I A&B &E 1 A .. ..

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This can be converted into the simpler form .. .. A .. .. and similarly in the other cases. The first derivation has a part, the derivation of B, that has disappeared in the second derivation. Inspecting the rules of natural deduction, the formulas above the inference lines of the I-rules are all parts of the conclusion. Similarly, disregarding rules ∨E and ∃E that we set aside in order not to complicate matters, the conclusions of the E-rules are all parts of premises. It can be shown that derivations in normal form contain no extraneous parts, but all formulas are parts of the conclusion or of the open assumptions. This is Gentzen’s famous “subformula property” of derivations in normal form, the central tool in the analysis of proofs. So far we have not shown any rules for negation. One way to handle negation in natural deduction, suggested by Gentzen (1934–35, II.5.2) and followed by Dag Prawitz (1965), is to assume that there is a proposition that is always false, denoted ⊥. Now ∼ A can be taken to be just A ⊃ ⊥, and negation introduction is a special case of implication introduction, with B = ⊥. The rule of ex falso quodlibet is added: it is ⊥ C (from falsity, anything follows). If a contradiction such as A & ∼ A is provable, by conjunction elimination also A and ∼ A are provable. Applying implication elimination to these two, ⊥ is provable. Then it is provable by a derivation in normal form, and by the subformula property all formulas in the derivation are parts of the conclusion ⊥. But ⊥ has no parts and there cannot be any such derivation, so that the consistency of the system of rules of natural deduction can be concluded. Before turning to Gentzen’s sequent calculus, we note the independent development of systems of natural deduction by Stanislaw Ja´skowski (1934). This work contains no profound results on the structure of derivations, in contrast to Gentzen. In Ja´skowski’s systems, the formulas are arranged in a linear numbered succession, a device that has been followed in many pedagogical presentations of natural deduction. Gentzen himself had some doubts about the tree-like arrangement of formulas in his natural deduction derivations, thinking that such derivations “deviate from actual inference in which there necessarily is a linear sequence of propositions, caused by the linearity of thinking” (1934–35, II.2.2). Towards the end of the 1930s, Gentzen found Ja´skowski’s work and considered its linear arrangement an improvement on the tree form. 3.2. Sequent calculus. Gentzen developed his central results on the structural analysis of proofs for a calculus that differs somewhat from the calculus of natural deduction presented above. In his sequent calculus, all open assumptions are collected into a list, let it be Γ, and the derivability of a formula C from assumptions Γ is written as Γ → C. Thus, the relation of derivability between a list of assumptions and a formula has been presented in a local way in one line, whereas in natural deduction this relation has to be read from the root and the leaves of the derivation tree. Sequent calculus has rules that correspond to the introduction rules of natural deduction, and they transform the conclusion. Secondly, there are rules that correspond to the elimination rules, and these transform

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the open assumptions. Both transformations are always such that the formulas in the premises of a rule are subformulas of the conclusion. Derivations start from sequents of the form A → A that correspond to the assumption of a formula A, and each rule adds some complexity to the left or right of the arrow. The rules of sequent calculus can be motivated from those of natural deduction. The “arrow” notation can be taken as one that replaces the “vertical dots” in the natural deduction rules for ⊃ I and ∨E. The former becomes: if A → B, then → A ⊃ B. For the latter, we have that if A → C and B → C, then A ∨ B → C. Thus, sequent calculus formalises the relation of derivability of a formula from other formulas. The normal form of natural deduction, or the eliminability of “IntroductionElimination-pairs”, is expressed in sequent calculus through Gentzen’s famous “cut rule”. In a typical case we have a derivation of some result of interest C from assumptions Γ and A, and in a second stage we succeed in establishing a lemma that shows A superfluous. By the lemma we have Γ → A, and by the main result we have A, Γ → C. (Here the comma in A, Γ indicates concatenation of A with the list Γ.) The cut rule combines these two sequents: Γ → A A, Γ → C Cut Γ→C Formula A has disappeared from the list of assumptions, and there is no a priori bound on the complexity of possible cut formulas that might be needed in order to derive Γ → C. Gentzen’s main result, his famous “Hauptsatz” (main theorem) is a procedure by which applications of the rule of cut can be eliminated from derivations. The subformula property is then straightforward. By the use of sequent calculus, Gentzen found a beautiful way of expressing the principles of proof of classical logic. He generalised sequents to the form Γ → Δ, in which also the succedent Δ is a list of formulas. An account of the origins of sequent calculus, in the work of Paul Hertz in the 1920s and in Gentzen’s first paper (1932), can be found in Bernays (1965). With arbitrary finite lists of formulas Γ, Δ, . . . on both sides of the derivability symbol → , Gentzen’s logical rules for his classical sequent calculus LK are: 5. Gentzen’s rules for classical sequent calculus Γ → Δ, A Γ → Δ, B R& Γ → Δ, A&B

A, Γ → Δ L&1 A&B, Γ → Δ

B, Γ → Δ L&2 A&B, Γ → Δ

A, Γ → Δ B, Γ → Δ Γ → Δ, A Γ → Δ, B R∨1 R∨2 L∨ A ∨ B, Γ → Δ Γ → Δ, A ∨ B Γ → Δ, A ∨ B A, Γ → Δ, B Γ → Θ, A B, Δ → Λ R⊃ L⊃ Γ → Δ, A ⊃ B A ⊃ B, Γ, Δ → Θ, Λ A(t), Γ → Δ L∀ ∀xA(x), Γ → Δ

Γ → Δ, A(y) R∀ Γ → Δ, ∀xA(x)

A(y), Γ → Δ L∃ ∃xA(x), Γ → Δ

Γ → Δ, A(t) R∃ Γ → Δ, ∃xA(x)

As mentioned, derivations start with initial sequents of the form A → A. The rule of ex falso quodlibet can be given by letting derivations start also with sequents of the form ⊥ → Δ. Gentzen himself did not do this, but used rules for negation:

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6. Rules for negation in classical sequent calculus Γ → Δ, A L∼ ∼ A, Γ → Δ

A, Γ → Δ R∼ Γ → Δ, ∼ A

In Gentzen’s thesis, the reading of sequents was not clear yet. Gentzen suggested a reading of Γ → Δ as: the conjunction of formulas in Γ implies the disjunction of formulas in Δ. The reading in terms of derivability of a number of cases Δ under the open assumptions Γ is suggested in Gentzen’s second paper on the proof theory of arithmetic (1938). In his (1936) paper on the consistency of arithmetic, Gentzen uses what is today known as “natural deduction in sequent calculus style”, with sequents of the form A1 , . . . , Am → B. The calculus has sequents with a single succedent formula, but the rules are those of natural deduction: instead of the left rules of sequent calculus one has, with Γ and Δ lists of formulas, two rules of conjunction elimination concluding Γ → A and Γ → B from the premise Γ → A&B, and a rule of implication elimination concluding Γ, Δ → B from the two premises Γ → A ⊃ B and Δ → A. In this calculus, says Gentzen, the sequents are “a formal expression of the meaning of a proposition in a proof in its dependency on some assumptions” (1936, 5.21). The passage points to the possibility of a proof-theoretical reading of single-succedent sequents, used for “indicating fully the meaning of a proposition as it occurs in a proof” (ibid., 5.1). In Gentzen’s second paper on the consistency of arithmetic, in 1938, multisuccedent sequents are used (1938, 1.2): In the previous work I had introduced the concept of a sequent, with just one succedent formula, in its immediate connection to the natural representation of mathematical proofs (1936, SS5). It is possible to arrive at the new, symmetric concept of a sequent also from that same point of view, namely, by striving at a particularly natural representation of the division into cases (see SS4 of the previous work, and in particular 5.26). Namely, a ∨-elimination can now be represented simply as: from → A ∨ B one concludes → A, B, to be read as: ‘Both possibilities A and B obtain’.

Gentzen’s suggestion is that a sequent Γ → Δ gives a listing of the open cases Δ under the open assumptions Γ. Logical rules change and combine open assumptions and cases: for example, Gentzen’s left conjunction rules replace the open assumption A or B by the open assumption A&B, and his right disjunction rules change the open case A or B into the open case A ∨ B, and so on. There can also be an “empty case” representing impossibility, with nothing on the right of the sequent arrow. Some care is needed in the above reading, for the open cases are to be understood classically: it need not be decidable which formula of Δ is the case. Arnold Schmidt’s review of Gentzen’s thesis sheds some light on the interpretation of sequents. Schmidt writes (1935, p. 145) that a sequent A1 , . . . , Am → B1 , . . . , Bn means that “B1 or . . . Bn depends on the assumptions A1 and . . . Am .” The Bj are referred to as “claims”. The interpretation in terms of the truth of the implication A1 & . . . &Am ⊃ B1 ∨ · · · ∨ Bn is not explicitly given, but it is referred to as “the trivial interpretation” (ibid., p. 146). Schmidt was Gentzen’s contemporary in G¨ ottingen, and some of his reviews of Gentzen’s papers show that they had discussed the reviews before publication. The rules of natural deduction follow the standard pattern of introductions and eliminations. There has been little development in these since Gentzen: some

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generalisations of elimination rules have been proposed, as in Schroeder-Heister (1984), as well as different ways of handling the discharge of open assumptions (see Negri and von Plato 2001, ch. 8, for more details). The introduction and elimination rules give intuitionistic logic, but the rule of double negation elimination that leads to classical logic is of a different character as emphasised by Gentzen (1934–35, II.5.3). Prawitz (1965) was able to extend the normal form theorem from intuitionistic logic to that part of classical logic that does not use disjunction or existence. Normal derivations start with assumptions, followed by E-rules, then the classical rule of double negation or indirect proof, and last follow I-rules. On the side of sequent calculus, no comparable stability regarding the rules has been achieved. Gentzen’s choice of his particular set of rules is dictated by the requirements set by the proof of his cut elimination theorem. These rules have specific properties: in rule L⊃, the lists (contexts in today’s terminology) Γ, Θ, Δ, Λ in the two premises are independent and are added up in the conclusion. In the two other two-premise rules R& and L∨ instead, contexts are shared, or the same in both premises. Gentzen designed the proof of the Hauptsatz so that its proof for his intuitionistic sequent calculus, denoted LJ, is directly a special case of the proof for the classical calculus LK. The calculus LJ is obtained from LK by putting on the rules the requirement that each succedent of a sequent consist of not more than one formula. It follows that there must be two right disjunction rules and that the left implication rule must have independent contexts, for otherwise there would be no single succedent instances of these rules. Gentzen mentions as a second design principle that sequent calculus must display the duality of & and ∨ (1934–35, III.2.4). Since there must be two right disjunction rules, there must be dually two left conjunction rules. In fact, all the rules except L⊃ and R⊃ display the symmetry. Gentzen arranged the rules in his table (1934–35, III.1) in two columns that are dual mirror images of each other. The structural rule of cut is self-symmetric. The implication rules are an exception to symmetry and they are given last, separated from the other rules. (Unfortunately the layout of the rules is not reproduced in the English translation of Gentzen’s papers.) Gentzen was very much struck by the left-right symmetries of the classical sequent calculus LK. The emergence of the symmetry in the case of the classical negation rules L ∼ and R ∼ made him exclaim in (1938, 1.6): “The special position of negation, which makes for an annoying exception in the natural calculus, is lifted away as if by magic.” It is certain that Gentzen considered various forms of rules, as that would be the only way of arriving at the ones he had. For example, in his 1943 work on transfinite induction, written around 1940, Gentzen removed rule L⊃ by allowing instead also initial sequents of the form A ⊃ B, A → B, a trick he had used also in the revised form of his 1936 paper. Incidentally, this shows that the possiblity of invertible rules (see below for this notion) had not occurred to Gentzen. Next we consider the “structural” rules of Gentzen’s sequent calculi. These rules are those of weakening, contraction, exchange, and cut. Weakening is the addition of superfluous assumptions: if Γ → Δ is derivable, then A, Γ → Δ is. Secondly, by the rule of contraction, one concludes from A, A, Γ → Δ into A, Γ → Δ. Mirror image rules hold for the succedent parts. The rule of exchange permits one to change the order of formulas in the antecedent and succedent parts of a sequent.

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The most important structural rule is cut, which was informally motivated above as the combination of two results Γ → A and A, Δ → C. Gentzen’s Hauptsatz is proved by giving an algorithm for eliminating all applications of the cut rule in derivations. An inspection of the rules of sequent calculus shows that all formulas that appear in a cut-free derivation of a sequent Γ → Δ are subformulas of Γ or Δ. Gentzen was able to prove the normal form theorem only for intuitionistic natural deduction, which is why he developed the multisuccedent sequent calculus and proved cut elimination for both intuitionistic and classical logic (see 1934–35, Introduction). Gentzen’s thesis gives the rules of sequent calculus in two groups: the structural rules of weakening, contraction, exchange and cut as the first group, and the logical rules as the second. Two years later he calls the rules of weakening, contraction, exchange and change of bound variable “Struktur¨ anderungen” structural modifications (1936, 5.22). All of these latter except weakening “do not change the meaning of a sequent,... all these possibilities of modification are of a purely formal nature. It is only because of special features of the formalism that these rules must be expressly given” (ibid., 5.244). Gentzen’s view of the purely formal nature of structural rules should be based on a comparison between the situation in sequent calculus and in natural deduction. The latter has no explicit structural rules. He describes the left and right rules of sequent calculus as corresponding to the elimination and introduction rules of natural deduction, respectively, even if this correspondence is not quite perfect (1934–35, III.1.1; see also Schmidt 1935, p. 145). The reason for this discrepancy lies in the form of rules &E and ⊃E (see Negri and von Plato 2001, ch. 8). Gentzen’s thesis contained some of the first results on intuitionistic logic. If a formula of the form A ∨ B is derivable, already one of A or B is derivable. This follows from the cut elimination theorem, because the last rule has to be R∨1 or R∨2 . This “disjunction property” of intuitionistic logic was also mentioned in passing by G¨ odel (1932), without a proof or even indication of how he knew the result to hold true. A related result of Gentzen is a proof of the underivability in intuitionistic logic, by purely syntactic means of proof analysis, of the law of excluded middle. The Hauptsatz also has as a corollary a positive solution to the decision problem of intuitionistic propositional logic. The proof is not immediate but proceeds through a lemma showing that it is sufficient to have formulas repeated at most three times in the antecedents and succedents of sequents in derivations in the intuitionistic calculus LJ. With this restriction, there is only a bounded number of different possible cut-free derivations of a given sequent. As another application, Gentzen gives a proof of consistency of arithmetic with the induction principle left out. This result was known from earlier proofs that had been given by von Neumann and Jacques Herbrand among others. For Gentzen, a proof by his own methods of sequent calculus was a first step in the direction of a consistency proof of arithmetic. 3.3. The consistency of arithmetic. Gentzen’s greatest achievement was his proof of the consistency of arithmetic. This much is generally known, namely that he proved it by the use of a “transfinite induction” that is still constructively an acceptable principle of proof. His last discovery, published in 1943, adds a significant aspect to the proof, by which G¨ odel’s incompleteness theorem, and especially the unprovability of consistency within Peano arithmetic, arises in a completely natural

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way from a combinatorial analysis of formal arithmetical derivations. Gentzen’s induction principle can be expressed as an arithmetic formula, and Gentzen shows directly two things: 1. The formula is unprovable in Peano arithmetic. 2. Any induction principle weaker than the formula is provable in Peano arithmetic. The overall foundational conclusions include: from result 1, one sees that G¨ odel’s theorem is no “freak” phenomenon. The rather artificial construction of an unprovable formula in G¨ odel’s original paper had led to such thoughts, now refuted by a completely “normal” arithmetic formula that is unprovable. Consistency follows, for anything would be provable in an inconsistent system. Relating to result 2, the transfinite induction used by Gentzen is optimal and serves to measure the “proof-theoretic strength” of Peano arithmetic. We can say that Gentzen’s last paper settled completely the situation with G¨ odel’s second incompleteness theorem about the unprovability of consistency. It is sad how little this achievement is appreciated even today. There are entire books dedicated to the incompleteness theorems that do not so much as mention Gentzen. It would be out of place to try to explain in anything like comprehensive terms Gentzen’s result, but we can get an idea of it at least. If arithmetic were inconsistent, the empty sequent, or equivalently, the sequent → ⊥, would be derivable. Our task is to order all possible derivations according to a suitable criterion of simplicity. The criterion is to be such that given a derivation, there is a finite, bounded number of derivations that lie “below” it in the ordering. This would be easy if the ordering were one of length, say the number of steps of inference, and some lexicographic ordering if the number of steps is the same. There is already an infinity of one-step derivations, so that ordering will not do. Gentzen managed to define an ordering with the following property: assume there is a derivation of the empty sequent. On the face of it, it could have any complexity. But now Gentzen defined “reduction steps” by which any such hypothetical derivation can be transformed so that its position in the ordering of all possible derivations decreases. By applying the reductions, we arrive at a derivation of the empty sequent that does not use the induction principle, and now we see that there can be no such derivation. One common but ignorant remark about Gentzen is that he had proved the consistency of arithmetic, so essentially the consistency of the standard induction principle of arithmetic, by a much stronger transfinite induction principle. Gentzen, certainly, would never have done anything that stupid: his transfinite induction principle was restricted to apply to only “primitive recursive predicates.” In general, given a property P (n) to be proved inductively to hold for all natural numbers, we are not in the possession of a computational method that would verify or falsify P (n) in some bounded number of steps. With Gentzen’s transfinite induction, instead, each concrete instance of properties to which the induction is applied can be verified by a bounded computation. We shall now describe in some detail the altogether four different proofs of the consistency of arithmetic that Gentzen left behind. The first proof remained unpublished until an English translation in Gentzen’s Collected Papers in 1969. The second one is the original publication of 1936, the third the reworking published in 1938, and the fourth the one contained in Gentzen’s last paper of 1943.

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After G¨ odel’s announcement of the first incompleteness theorem in K¨onigsberg in September 1930, i.e., of the existence of true but unprovable statements of arithmetic, Johann von Neumann started to work on the topic and found within a couple of months the second incompleteness theorem, by which the consistency of arithmetic itself is a G¨ odelian unprovable sentence. He planned to publish this result soon but was stopped by G¨ odel’s simultaneous discovery. Von Neumann’s letter announcing the theorem reached G¨ odel a few days after the latter had sent his paper, the second incompleteness theorem included, for publication. Von Neumann conjectured that the unprovability of the consistency of arithmetic extends to mathematics in general and that consistency is in some sense absolutely unprovable. We have seen that this conjecture was too pessimistic, by the G¨odel-Gentzen translation of Peano arithmetic based on classical logic into Heyting arithmetic based on intuitionistic logic. The translation was seen as a first proof of the consistency of arithmetic. The reason is that indirect existence proofs were thought to be the part of arithmetic proofs that could lead to inconsistencies, whereas the rest was thought reliable. By the translation, the consistency of classical arithmetic is reduced to the consistency of intuitionistic arithmetic that does not use indirect existence proofs. The G¨ odel-Gentzen argument of 1932-33 left open the question of a direct, elementary proof of consistency. Gentzen’s programme had as its second aim, after the treatment of pure logic, the consistency of arithmetic, and he found a proof by the end of 1934. The fate of this first proof is quite peculiar. Bernays took a voyage by ship to the United States in September 1935 for a stay at the Institute for Advanced Study in Princeton and had Gentzen’s manuscript with him. G¨ odel was on board with the same destination, and we can be sure that they studied the proof, because a consistency proof for arithmetic was without doubt the next big step ahead after G¨ odel’s incompleteness theorems. Neither Bernays nor G¨odel were satisfied with Gentzen’s proof, as is shown by the letters from Gentzen to Bernays in the fall of 1935. At the time of the publication of Gentzen’s Collected Papers, Bernays returned to the first proof. It had been preserved in the form of galley proofs from which the translation into English was made. Some years later, the original German text was published. Bernays (1970) is an account of Gentzen’s first proof. There we read that the criticism was that the proof used implicitly the fan theorem that we met above. By Bernays’ admission in 1970, the criticism was unfounded, and it can indeed be shown that the fan theorem is not a principle that would make the consistency of Peano arithmetic provable, as remarked by Kreisel (1987, p. 173). Gentzen, even if the correspondence with Bernays shows that he had thought of the objections made, did not hesitate to change his original proof into one that uses the principle of transfinite induction. Parts of the paper published in 1936 got changed, but other parts remained unchanged, with an all but optimal presentation as a result. In particular, the logical calculus used in the crucial section was changed from that of the earlier ones. Gentzen’s style of writing proofs leaves a lot to be filled in or worked out in detail by the reader, on the basis of his verbal descriptions. Having done such work for the earlier parts of the paper, one finds that not all of it is needed in the revised proof. Gentzen thus returned to the proof a third time, in 1938, with the aim of making it as clear as possible.

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Gentzen’s original proof is the easiest one to follow of his four proofs. The basic idea can be described as follows: consider a possible derivation of a falsity from some assumptions Γ in arithmetic and show that in this situation at least one of the assumptions must be false. Thus, if none of the assumptions are false, neither is the thing derived from them. Gentzen gives a “reduction procedure” for any claimed derivability of a formula A from assumptions Γ, as follows. First, free variables can be instantiated by numbers in an arbitrary manner. Then, if the conclusion A is a conjunction B&C, one of B and C is chosen. If it is a negation ∼ B, the reduction continues with 0 = 1 as conclusion and B, Γ as assumptions. If A is ∀xB(x), any numerical instance B(n) is chosen. It is sufficient to have just conjunction, negation, and universal quantification because the other connectives and quantifiers can be defined in classical logic. Classical logic, on the other hand, is no problem by the G¨ odel-Gentzen translation. The above reduction steps act on the conclusion part of a claimed derivability of A from assumptions Γ. One can think of these steps as choices being made by an omniscient opponent who tries to show that your derivations in arithmetic lead to contradictions. The opponent’s task is to create a “worst possible” consequence of the assumptions Γ that you have made in concluding A. By the reduction steps, the consequence is simplified at each step until an equality is found. If it is true, no contradiction followed, and if it is false, it is your turn to show that the assumptions contained some falsity. At this point, Gentzen’s reduction procedure is not optimal, but it does its job. If one assumption formula is a conjunction B&C, you can replace it by one of the conjuncts. If it is a negation ∼ B, with Γ equal to ∼ B, Γ , consider the derivability of B from Γ ; and if it is a universal quantification, consider some instance. The process is not as simple as it may sound, because reduction steps on the consequence and the assumptions become mixed. This happens when a negation in the assumptions is met, because it brings a formula to the conclusion to which the opponent’s reduction steps apply. The order of these steps is such that the opponent always comes first, until there is an equation as a consequence. By the reduction procedure, in whatever way the opponent has produced a false equality as a consequence, you are able to produce a false equality as an assumption. The reduction procedure for sequents is applied in the consistency proof by showing that initial sequents are reducible and that derivability maintains reducibility. Gentzen’s first proof contains some redundancies and omits an important consideration, namely a proof that two derivations in the calculus that he uses can always be composed with each other. In von Plato (2004), a Gentzen style consistency proof for intuitionistic arithmetic is given with all the details and with a comparison to Gentzen’s original proof. The sketch for a proof in Bernays (1970) uses Gentzen’s original idea, but the logical calculus is the one of the published paper of 1936; thus, yet another proof. The logical calculus of Gentzen’s first proof is natural deduction with derivations written in “sequent calculus style”. The published proof of 1936 uses a variant of this calculus in which the number of logical rules is diminished. The general reduction procedure for sequents defined in the earlier parts of the paper is now not used at all. A special notation is developed in the 1936 proof for ordinal numbers, but this is dropped in the 1938 proof that uses the standard notation for ordinal numbers. It also uses a version of the multisuccedent sequent calculus of Gentzen’s thesis. Moreover, the reduction procedure of 1938 bears a close resemblance to the

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elimination of cuts in sequent calculus. As with the first proof, in the third proof simplifications are possible through an improved logical calculus. The reduction procedure of 1938 applies to derivations of the empty sequent → . First, all free variables are instantiated by 0. If the derivation is correct, it remains so after this instantiation. Next one considers the “endpiece” of the derivation: take the empty endsequent and climb up the derivation tree until you hit a sequent derived by a logical rule. The endpiece so obtained contains only structural rules (cut, weakening, and contraction) and the rule of induction. The last one is written in sequent calculus as: 7. The rule of induction in sequent calculus A(x), Γ → Δ, A(x + 1) Ind A(0), Γ → Δ, A(t) In the rule, x is the eigenvariable and has to be arbitrary. This is ensured by requiring that it does not appear in any of the other assumptions or cases Γ, Δ. The conclusion has the formula A(t) in which t is any term, i.e., any number or variable. In a typical case, it is a variable, say y, and the rule of universal generalisation can be applied to conclude ∀yA(y). Thus we have that if A(x + 1) follows from A(x), then ∀yA(y) follows from A(0). An instance of Ind with a number n instead of a variable in the conclusion is called a “numerical induction”. Such steps can be replaced by cuts. For this, assume the premise of Ind, namely A(x), Γ → Δ, A(x + 1) to have been derived. Since x is arbitrary, the instance A(0), Γ → Δ, A(1) is derivable. Also A(1), Γ → Δ, A(2) is derivable, and so on. Given a number n, we can derive A(0), Γ → Δ, A(n), i.e., the conclusion of Ind, by just the structural rules. Say n is 3, and we have A(1), Γ → Δ, A(2) A(2), Γ → Δ, A(3) Cut A(0), Γ → Δ, A(1) A(1), Γ, Γ → Δ, Δ, A(3) Cut A(0), Γ, Γ, Γ → Δ, Δ, Δ, A(3) The extra copies of formulas in Γ, Δ are removed by as many contractions. Note how the number of steps in the derivation grows with n in the conversion of a numerical induction into several cuts (not counting the contractions). Returning to Gentzen’s third proof, if there is an instance of Ind in the endpiece, it either is a numerical induction or becomes one after the instantiation of free variables. Next, cuts are used to remove the numerical induction, after which the endpiece contains just structural rules. A crucial lemma shows that there is at least one cut formula such that it is principal in a left and a right rule, respectively, of two sequents that begin the endpiece. Gentzen gives a procedure by which the cut with such a formula is replaced by cuts on shorter formulas and a cut in which the same formula is principal only in a left rule that delimits the endpiece. Disregarding the latter cut, which is a complication caused by Gentzen’s logical calculus, the limits of the endpiece are “pushed up” step by step by permuting cuts up to the premises of the last logical rules before the endpiece. If an initial sequent is met, the cut disappears. If a concusion of Ind is met, it is or becomes a numerical induction. In the end, there would have to be a derivation of the empty sequent without rule Ind, which is not the case. The crucial point of the proof is to show that the reduction procedure terminates. Numerical inductions up to any number can be met, and no upper bound can be given for the size of derivations that will have to be reduced,

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even though the original derivation was of some given size. This is the point at which finitistic reasoning in the consistency proof is superseded. Gentzen’s third proof has its awkward aspects, such as the purported proof of the lemma by which a cut formula principal in a left and a right rule at the top of the endpiece can be found. It consists of close to two huge pages of verbal arguments. Another point is the very complicated transformation of a cut as given by the lemma. A detailed presentation of the proof can be found in Takeuti’s book Proof Theory. We come now to Gentzen’s fourth and last proof of the consistency of arithmetic. Its description would require a knowledge of transfinite ordinal numbers, so we just make some comments indicative of its method. The approach is to extend the natural numbers by what are called constructive ordinals. The induction principle for arithmetic is also extended into a transfinite induction principle (call it TI-induction). Gentzen speaks of the “provability of transfinite induction.” By this is meant the following: if an induction principle is given as a formula, as in A(0)&∀x(A(x) ⊃ A(x + 1)) ⊃ ∀yA(y), it can be a provable formula. For example, if x ranges over the natural numbers, it is a provable formula in Gentzen’s system of ordinary arithmetic. If x ranges over the transfinite ordinals below a certain limit called 0 , TI-induction can be shown equivalent to ordinary induction, and the corresponding formula is provable. (This result seems to stem from Bernays sometime after Gentzen’s first proof of the consistency of arithmetic.) If the limit ordinal 0 is added, TI-induction can still be expressed in arithmetic, but Gentzen shows that it is unprovable. The ordinal 0 is called the proof-theoretic ordinal of Peano arithmetic. Much of the research on consistency problems after Gentzen has consisted in the determination of proof-theoretic ordinals of theories stronger than Peano arithmetic. Foremost among such results is the determination of the “ordinal of predicativity” around 1964 by Kurt Sch¨ utte and Solomon Feferman independently of each other. Gentzen’s last proof established directly an arithmetic formula that is unprovable in Peano arithmetic. Consistency follows at once, because anything would be provable in an inconsistent system. Furthermore, TI-induction up to 0 is a principle with a direct mathematical content, not a product of the coding of the provability predicate into arithmetic as in G¨ odel’s original proof. Much later, similar “mathematical” examples of incompleteness in arithmetic, in contrast to “logical” ones, were found in infinitary combinatorics. 3.4. The consistency of analysis. The next obvious step after the consistency of arithmetic was to prove the consistency of analysis, i.e., of the theory of real numbers. Gentzen did some work in this direction, but the difficulties were great, both practical and those intrinsic to the topic. Analysis was to be formulated as a system of second-order arithmetic, which means that quantification is extended over number-theoretic predicates or, equivalently, over sets of natural numbers. Second-order number theory is used in Gentzen (1943), in which it is shown that the principle of transfinite induction up to 0 is derivable in second-order number theory. The practical difficulties included a world at war and the interruption of Gentzen’s research through military service in 1939. These latter came, however, to an end by 1942 through a nervous breakdown.

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Traces of Gentzen’s work towards the consistency of analysis can be found in some of the published work and in a summary he had prepared in February 1945. This 11-page manuscript has page number references to “W.-A.”, which stands for “Widerspruchsfreiheit der Analysis”, the consistency of analysis. These references contain numbers running to well over three hundred. This could be the origin of the myth of the lost “black briefcase” with Gentzen’s presumed proof of the consistency of analysis. At one point, Gentzen himself estimates that a proof of consistency could take several decades. More than half a century has passed with no constructive proof of the consistency of full second-order arithmetic in sight. One has instead studied what are known as subsystems of second-order arithmetic. Current research in the proof theory of second-order arithmetic has arrived at the following: let X, Y, Z, . . . range over number-theoretic predicates. A formula such as X(x) states that x has the property expressed by X. We can now use first- and second-order logic to form compound formulas such as ∀X(X(x)∨ ∼ X(x)). The collection of natural numbers for which this formula holds is called a Π11 -set. Consider the consistency of such subystems of second-order arithmetic in which the collection of natural numbers into sets is restricted to formulas with a bound on the second-order quantifiers. It is known, through the work of Toshiyasu Arai and Michæl Rathjen, that the second-order quantifiers can have the structure ∀X∃Y , or what are known as Π12 formulas. A review of the present state of research is given in Rathjen (1995) and (1999). 4. Later Developments in Structural Proof Theory 4.1. Developments in sequent calculus. The first contributions to structural proof theory by others than Gentzen come from the late 1930s. In 1944, Oiva ottingen in 1938– Ketonen, a Finnish logician who had studied with Gentzen in G¨ 39, improved the rules of sequent calculus by replacing some of Gentzen’s original rules for classical propositional logic so that all rules became “invertible”, meaning that if a sequent of the form of the conclusion is derivable, the sequent, or sequents, of the form of the premise is also derivable. The changed rules are: 8. Invertible rules for classical sequent calculus A, B, Γ → Δ L& A&B, Γ → Δ

Γ → Δ, A, B R∨ Γ → Δ, A ∨ B

Γ → Δ, A B, Γ → Δ L⊃ A ⊃ B, Γ → Δ Each connective has just one left and right rule. Further, all two-premise rules have shared contexts, and Ketonen obtained a fully invertible classical propositional sequent calculus in which derivations of a given sequent are found by decomposing it in any order whatsoever: given a sequent Γ → Δ to be derived, choose from Γ or Δ any formula with a connective. The corresponding rule determines uniquely the premises. Repeating this “root-first proof search”, formulas are decomposed into parts until there is nothing to decompose. At this stage, it can be determined if the original sequent Γ → Δ is provable or not by checking if all top sequents are initial sequents. The process terminates in the case of classical propositional logic in a bounded number of steps as determined by the number of connectives in the given sequent.

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It was Ketonen’s good luck that Paul Bernays, living in neutral Zurich during the war years, wrote reviews of papers in logic in The Journal of Symbolic Logic, the only journal of the time devoted to logic. The long review of 1945 explains in good detail Ketonen’s invertible sequent calculus. This part was noticed and taken into use by Evert Beth and forms the basis of his “method of tableaux”, even if Ketonen’s key rˆ ole has not been noticed by researchers in tableau methods. Also Kleene’s influential book Introduction to Metamathematics of 1952 made extensive use of Ketonen’s calculus that Kleene knew only though Bernays’ review. Ketonen’s thesis was his first and only work in logic. Other parts of the work dealt with the problem of “proving unprovability”, i.e., of showing that a sequent is underivable in a given system. To this purpose, Ketonen used a result of Gentzen’s known as the “midsequent theorem”. He refined the result by showing that if a given sequent is derivable, there is a “weakest possible” midsequent that is derivable. Thus, if one is able to show that such a weakest possible midsequent is not derivable, no midsequent is derivable, and neither is the originally given sequent. Ketonen proved this result by showing that the order of quantifier rules with eigenvariables can be suitably permuted and the number of rules in a derivation reduced to a minimum. A third part of Ketonen’s work dealt with the application of sequent calculus to elementary geometry, in particular, to plane projective and affine geometry. The former had been studied by Thoralf Skolem (1920), but Ketonen is the only one who ever paid attention to the geometrical part of that otherwise very famous work (famous for the L¨ owenheim-Skolem theorem). Ketonen was able to show that the axiom of parallels is independent of the other axioms of affine geometry through the analysis of possible proofs and his general theorem about weakest midsequents. A limitation of the result is that the axiomatisation does not include the axiom of noncollinearity or the existence of at least three noncollinear points. Another minus, noted by Bernays, if it is one, is that the arguments and proofs are very difficult to follow. On the other hand, a proof of independence through proof analysis brings to light the workings of an axiomatic system. Earlier proofs had been based on the discovery of non-Euclidean geometries, i.e., geometries in which all axioms but the one of parallels hold. Ketonen’s proof of invertibility of the rules used the structural rule of cut. Later Kurt Sch¨ utte (1950) and Haskell B. Curry (1963) gave direct proofs of invertibility, the latter with the explicit result that the inversions are “height preserving”: if a given sequent is derivable in at most n steps, also the premises in any rule that can conclude that the sequent has a derivation in at most n steps. Gentzen had shown the disjunction property of intuitionistic propositional logic. The related existence property follows easily through proof analysis in intuitionistic sequent calculus and seems to have been known to Gentzen: if ∃xA(x) is derivable, there is some individual a such that the instance A(a) is derivable. Thus, intuitionistic logic corresponds on a formal level to the constructive notion of existence. Both properties were generalised by Ron Harrop (1960) to hold also under assumptions if these do not contain, overtly or covertly, any disjunctive or existential assumptions. As noted, Gentzen (1934–35) had proved the decidability of intuitionistic propositional logic. A direct terminating method of proof search for intuitionistic propositional logic, similar to Ketonen’s method for the classical calculus, was found as late as around 1990 independently by J¨ org Hudelmaier (1992) and Roy Dyckhoff

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(1992). These discoveries follow a line of research from Kleene (1952, pp. 480 ff. and 1952a), who found a way of avoiding also Gentzen’s structural rules of weakening and contraction in classical and intuitionistic sequent calculus. If the rule of cut were indispensable, one could try to derive a sequent Γ → Δ from two premises with an arbitrary cut formula. The rule of contraction has a similar effect: with it, a root-first proof search could go on forever by the multiplication of formulas in sequents. A minor modification concerns Gentzen’s exchange rules which permit the change of order of formulas in sequents. These rules disappear through the use of “lists without order”, or “multisets” as one says. Research in sequent calculi from Gentzen on has led to the remarkable sequent calculi known as G3-calculi which have no structural rules. The logical rules of the classical calculus G3c are, in their propositional part, the same as Ketonen’s. For intuitionistic logic, there are both single succedent and multisuccedent calculi G3i and G3im. The most remarkable property of these calculi is the height-preserving admissibility of contraction, meaning that if a sequent with a duplication of a formula is derivable, the sequent without the duplication is derivable and the derivation of the latter is not a bigger derivation than that of the former. This property and the exact form of the calculi G3c and G3im are due to Albert Dragalin (1988, Russian original 1979), and in the case of the intuitionistic single succedent calculus G3i, to Anne Troelstra in Troelstra and Schwichtenberg (1996). The G3-family of logical calculi offers the strongest known methods for the structural analysis of proofs. 4.2. Logic and computation. In the time of the beginning of intuitionistic logic, around 1930, Heyting and Kolmogorov suggested an explanation of the principles of intuitionistic logic in terms of the notion of proof. Also Gentzen makes in his doctoral thesis the suggestion that the introduction rules give the meanings of the various forms of propositions in terms of provability. These rules make more precise Heyting’s earlier discussion. In Kolmogorov (1932), it is suggested that intuitionistic logic is a “logic of problem solving”: the atomic formulas express the primitive problems that have no logical structure. A&B expresses a problem that is solved by solving A and B separately, A ∨ B is solved by solving at least one of A and B, and A ⊃ B is solved by reducing the solution of B to one of A. Falsity ⊥ is an impossible problem that has no solution. In Kolmogorov’s interpretation, the notion of a problem comes before the notion of a theorem: a theorem can be considered that special case of a problem in which the task is to find a proof. In 1969, William Howard made precise some of the ideas behind the interpretation in his paper “The formulae-as-types notion of construction”. The paper circulated as a manuscript and was finally published as Howard (1980). It established what came to be called the “Curry-Howard isomorphism” or “Curry-Howard correspondence”. Curry’s rˆole was that he suggested the idea for implication in (1958). The basic idea of Curry and Howard is that a formula corresponds to the set of its proofs. More precisely, to each formula A there is the set of proofs of A in the sense of formal derivation. The notation a : A stands for “a is a proof of A”. In terms of sets, the reading is “a is an element of the set A”. An introduction rule shows how to construct a proof from proofs of components: if a : A and b : B, then (a, b) : A&B. The operation of forming the pair (a, b) is the construction that gives a proof of the conjunction A&B. If a : A, then i(a) : A ∨ B; if b : B, then j(b) : A ∨ B. The two operations indicated by i and j carry the information from which component of the disjunction the proof is constructed, proof of A or proof of

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B. Implication is more difficult: assume an arbitrary proof x of A, so symbolically x : A. If you succeed in constructing from x a proof b(x) of B, then the proof of A ⊃ B is written as (λx)b. This is the lambda-abstract of the expression b(x) depending on the variable x, as invented by Alonzo Church (1932) (see also Church 1941 and Barendregt 1997). Set-theoretically, A&B is the Cartesian product of the sets A, B, A ∨ B their (disjoint) union, and A ⊃ B the set of functions from A to B. As to the elimination rules, they show how to pass from an arbitrary proof of a formula to its components: if x : A&B, then p(x) : A and q(x) : B are the projection constructions that do this. For disjunction, the rule is too complicated to be given here. For implication, using a suggestive symbol for a member of A ⊃ B, if f : A ⊃ B and x : A, then f (x) : B. In terms of sets, a proof f of A ⊃ B is a function f that transforms any proof x of A into some proof f (x) of B. Thus, rule modus ponens is the same as the application of a function. Implication introduction, then, is functional abstraction as invented by Church. The rules of natural deduction become the rules of typed lambda-calculus under the Curry-Howard correspondence. Truth of a formula A is established by a proof a : A. Thus, A is true corresponds to A being, when considered as a set, nonempty. Typed lambda-calculus shows the rules of intuitionistic natural deduction to be correct under the semantics given by the Curry-Howard correspondence: if the premises of a rule are assumed true, each of them has an element, and the rules show how to construct an element of the conclusion that thereby also must be true. In the natural representation of mathematical proofs, their characteristic form is that a claim B follows under some conditions A, and this is expressed concisely as the implication A ⊃ B obtained by the rule of implication introduction. If at some stage the conditions A are obtained, B follows by implication elimination. The Curry-Howard correspondence gives this latter step as a functional application: an argument a : A is fed into the function f : A ⊃ B and a value f (a) : B is obtained. Reasoning constructively, without use of the classical law of excluded middle, the function f : A ⊃ B is an algorithm or computable function. Gentzen’s idea of normalisation, which is basically the same thing as cut elimination, has the following specific meaning: given the function f : A ⊃ B and the argument a : A, normalisation consists in the computation of the value of f (a). The computation of the value of f (a) is the same as the conversion of the nonnormal derivation into normal form, which makes apparent the importance of strong normalisation and uniqueness. The former concept requires that a normal form is reached independently of the order of conversions, the latter that the result moreover must be unique. From the point of view of the Curry-Howard idea, formal proofs in intuitionistic natural deduction are computable functions. Constructivity, which used the philosophical principle behind intuitionistic logic and mathematics, now has the rˆ ole of guaranteeing that computations do not go on indefinitely, but terminate after some bounded number of steps. 4.3. Hilbert’s last problem. Hilbert presented his famous list of open mathematical problems at the International Congress in Paris in 1900. First in the list was Cantor’s continuum problem, the question of the cardinality of the set of real numbers. The second problem concerned the consistency of the arithmetic of real numbers, and so on, until a problem dealing with the calculus of variations, the

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23rd and last problem – or so it was until recently, when German historian of science R¨ udiger Thiele found from old archives in G¨ ottingen some notes in Hilbert’s hand, containing the following passage: As a 24th problem of my Paris talk I wanted to pose the problem: criteria for the simplicity of proofs, or, to show that certain proofs are simpler than any others. In general, to develop a theory of proof methods in mathematics.

Hilbert’s first thoughts for a theory of proofs are from the year 1904. At this stage, the representation of mathematics as a formula language and of mathematical proof as a purely symbolic manipulation of formulas without considering their meaning is clear. The central and almost only aim of the exercise of formalisation is to show the consistency and completeness of formalised arithmetic and analysis. When this aim is reached, the problems about the foundations of mathematics can be forgotten, left behind for good, thought Hilbert. Here Hilbert’s attitude seems quite different from what his statement of the 24th problem only four years earlier suggests. Now it would not matter at all in what way mathematics is formalised, if the two central aims of formalisation are reached. We know since the results of G¨odel that Hilbert’s original proof-theoretic programme failed: there is no complete formalisation even for the case of arithmetic. Secondly, the consistency of Peano arithmetic has no strictly finitary, hence no “absolutely reliable”, proof. However, it is surprising how soon the study of the foundations of mathematics recovered from this shock. Some, such as von Neumann, thought there was nothing to do, but others such as Gentzen, studying at the time with Hilbert’s closest associate Paul Bernays, soon regained hope. It took perhaps a year from G¨ odel’s paper of 1931 to realise that the consistency of arithmetic is not completely lost, even if it cannot be shown in any Hilbertian strictly finitary way, and that Hilbert’s finitistic approach can be extended into an intuitionistic or constructive one. To conclude this story, Gentzen had solved, naturally without knowing it, the general part of Hilbert’s 24th problem, as far as the general logical principles of proof and proofs in elementary arithmetic are concerned. After this success, Gentzen’s next natural aim was the proof theory of analysis, but this work was in its beginnings when he died in tragic circumstances in August 1945. Others have continued the programme, and therefore the attempts at solving Hilbert’s last problem also for analysis, even if the difficulties have been great. Added in 2007: After the above was written, I found an early handwritten manuscript version of Gentzen’s doctoral thesis. The original is in Zurich in the Paul Bernays collection. The most important novelty in it is a very detailed proof of normalization for intuitionistic natural deduction, some 14 pages if printed. The proof proceeds by first removing what are today called permutation convertibilities, after which detour convertibilities are removed. Thus, it is the same proof as today’s standard proof. The thesis manuscript begins, after a summary, with an exposition of natural deduction. It is followed by the normalization proof. Next there would be the translation of classical to intuitionistic arithmetic, only this part is missing because it was taken out and submitted for publication in March 1933. A list of corrections to this part is all that remains, corrections that are reproduced verbatim in the preserved text of 1933 that Gentzen in the end withdrew from publication.

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Gentzen’s original plan was then to show the equivalence of natural deduction and an axiomatic formulation of intuitionistic logic, an equivalence that is also found proved in the printed thesis. The groundwork was now done, for it is shown that natural deduction can work as a formalism through which the consistency of a standard axiomatic formulation of classical Peano arithmetic could be proved. The idea was to extend the normalization proof for natural deduction to intuitionistic arithmetic, the consistency of which implies the consistency of Peano arithmetic. A letter to Hellmuth Kneser shows that Gentzen still hoped, towards the end of 1932, to finish soon his original thesis plan. However, he must have realized around New Year that the subformula property of normal derivations would not carry over from pure logic to arithmetic in a natural deduction formulation. Thus, the grand plan for a consistency proof failed. The part with the translation of classical to intuitionistic arithmetic was then sent to Heyting as a separate work of its own standing by February 1933, as is shown by the correspondence between the two. The early thesis manuscript also contains the first beginnings for a sequent calculus. Its full development had to do for an Ersatz doctoral thesis after the grand consistency plan had failed. It seems safe to estimate that the whole of sequent calculus was developed in a few months in 1933, with the thesis finished by June. With the Hauptsatz for classical and intuitionistic sequent calculus secured, Gentzen had no use for the normalization proof for intuitionistic natural deduction, and Paul Bernays never realized what had been turned over to him. Curiously, the manuscript must have been inspected at least to some extent in connection with the publication of the English translation of Gentzen’s papers in 1969, because that edition reproduces photographically part of page two of the manuscript among the frontmatter of the volume. A detailed examination of Gentzen’s manuscript can be found in my essay “Gentzen’s logic”, to appear in volume V of the Handbook of the History and Philosophy of Logic. There also can be found a summary of contents of the stenographic Gentzen papers that Prof. Thiel is at present deciphering; I wish to thank him for having granted me twice the opportunity to study these materials for my chapter on Gentzen, even if their editing is still far from complete. Most of the Gentzen papers are taken up by two sets of notes: The first one is about the consistency of analysis, but there is another set of equal extent, namely notes on repeated but apparently failed attempts at proving the consistency of arithmetic through an intuitionistic sequent calculus. These latter are from between 1939 and 1943, thus well after Gentzen had worked through his “new proof of the consistency of arithmetic” based on classical sequent calculus. 5. References Barendregt, H. (1997), The impact of the lambda calculus in logic and in computer science, The Bulletin of Symbolic Logic, vol. 2, pp. 181–214. Beth, E. (1955), Semantic entailment and formal derivability, Mededelingen der Koninklijke Nederlandse Akademie van Wetenschappen, Afd. Letterkunde, vol. 18, pp. 309–342. Bernays, P. (1945), Review of Ketonen (1944), The Journal of Symbolic Logic, vol. 10, pp. 127–130. Bernays, P. (1965), Betrachtungen zum Sequenzen-kalkul, in Contributions to Logic and Methodology in Honor of J. M. Bochenski, pp. 1–44, North-Holland, Amsterdam.

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Bernays, P. (1970), On the original Gentzen consistency proof for number theory, in J. Myhill et al., eds, Intuitionism and Proof Theory, pp. 409–417, North-Holland, Amsterdam. Church, A. (1940), A formulation of the simple theory of types, The Journal of Symbolic Logic, vol. 5, pp. 56–68. Church, A. (1941), The Calculi of Lambda-Conversion, Princeton University Press. Curry, H. (1963), Foundations of Mathematical Logic, as republished by Dover, New York, 1977. Curry, H. and R. Feys (1958), Combinatory Logic, vol. 1, North-Holland, Amsterdam. Dragalin, A. (1988), Mathematical Intuitionism: Introduction to Proof Theory, American Mathematical Society, Providence, Rhode Island. Russian original, 1979. Dyckhoff, R. (1992), Contraction-free sequent calculi for intuitionistic logic, The Journal of Symbolic Logic, vol. 57, pp. 795–807. Gentzen, G. (1932), Ueber die Existenz unabh¨ angiger Axiomensysteme zu unendlichen Satzsystemen, Mathematische Annalen, vol. 107, pp. 329–250. altnis zwischen intuitionistischer und klassischer ArithGentzen, G. (1933), Ueber das Verh¨ metik. Submitted for publication but withdrawn, first published in Archiv f¨ ur mathematische Logik, vol. 16 (1974), pp. 119–132. Gentzen, G. (1934-35), Untersuchungen u ¨ ber das logische Schliessen, Mathematische Zeitschrift, vol. 39, pp. 176–210 and 405–431. Gentzen, G. (1936), Die Widerspruchsfreiheit der reinen Zahlentheorie, Mathematische Annalen, vol. 112, pp. 493–565. Gentzen, G. (1938), Neue Fassung des Widerspruchsfreiheitsbeweises f¨ ur die reine Zahlentheorie, Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, vol. 4, pp. 19–44. Gentzen, G. (1943), Beweisbarkeit und Unbeweisbarkeit der Anfangsf¨ allen der transfiniten Induktion in der reinen Zahlentheorie, Mathematische Annalen, vol. 120, pp. 140–161. Gentzen, G. (1969), The Collected Papers of Gerhard Gentzen, ed. M. Szabo, NorthHolland, Amsterdam. ur die klassische Zahlentheorie, Gentzen, G. (1974), Der erste Widerspruchsfreiheitsbeweis f¨ Archiv f¨ ur mathematische Logik, vol. 16, pp. 97–118. G¨ odel, K. (1931), On formally undecidable propositions of Principia mathematica and related systems I (English translation of German original), in van Heijenoort (1967), pp. 596–617. G¨ odel, K. (1932), Zum intuitionistischen Aussagenkalk¨ ul, as reprinted in G¨ odel’s Collected Works, vol. 1, Oxford University Press, 1986. G¨ odel, K. (1933), Zur intuitionistischen Arithmetik und Zahlentheorie, as reprinted in G¨ odel (1986), pp. 286–295. G¨ odel, K. (1941), In what sense is intuitionistic logic constructive?, a lecture first published in G¨ odel’s Collected Works, vol. 3, pp. 189–200, Oxford University Press, 1995. Harrop, R. (1960), Concerning formulas of the type A → B ∨ C, A → (Ex)B(x) in intuitionistic formal systems, The Journal of Symbolic Logic, vol. 25, pp. 27–32. van Heijenoort, J., ed. (1967), From Frege to G¨ odel, A Source Book in Mathematical Logic, 1879–1931, Harvard University Press.

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Index

Bense, Max, 125, 192 Bernays, Ludwig, 89 Bernays, Paul, 23, 26–34, 38–42, 47, 52–55, 57–60, 62, 64–76, 78, 80–82, 84, 89–91, 93–97, 99–102, 108, 110, 111, 115, 119–121, 128–131, 134, 135, 137, 138, 161, 178, 192, 214, 227, 230, 255, 258–264, 266, 269, 270, 308, 309, 312, 313, 322, 323, 331, 334, 338, 339, 377, 387, 392, 395, 397, 400 Bernays-Brecher, Sara, 89 Bernstein, Felix, 23, 38, 108 Bertram, H., 248 Beth, Evert Willem, 81, 84, 85, 182, 184, 215 Beyerchen, Alan, 234 Bieberbach, Ludwig, 24, 46, 48, 56, 57, 86, 114, 134, 141, 142, 147, 150–162, 167, 168, 174, 180–184, 186–190, 192, 194–196, 199, 206, 208, 213–215, 220, 221, 223–226, 230, 231, 241, 242, 324, 331–334 Bieberbach, Ulrich, 225, 230, 231 Bilharz, Adele, 4, 6, 7 Bilharz, Alfons, 4–10, 13, 283 Bilharz, Bertha, 7, 13, 14 Bilharz, Elisa, 4 Bilharz, Joseph Anton, 4 Bilharz, Sophie, 1 Bilharz, Theodor, 4, 5, 7–10 Birkhoff, George D., 121 Blaschke, Wilhelm, 48, 50, 196 Bloch, Ernst, 173 Blome, Kurt, 200 Blume, W., 62 Blumenthal, Otto, 59, 65, 100, 305, 307, 324, 333 Bochenski, Joseph Maria, 229 Bochner, Salomon, 24, 205 B¨ odewadt, Uwe Timm, 117 Bohr, Harald, 50, 155 Bohr, Niels, 125, 170 Bollnow, Otto Friedrich, 83, 170 Bolyai, Janos, 297

Abderhalden, Emil, 210 Abraham, Max, 107 Ackermann, Wilhelm, 24, 28, 29, 31, 33, 41, 43, 58, 63, 64, 67, 69, 72–75, 78, 84–86, 89, 91, 92, 95–97, 99, 100, 102, 120, 122, 124, 128–131, 135, 137, 138, 170, 173, 180, 181, 194, 203, 226, 228, 229, 266–268, 307, 315, 318, 323, 331, 334, 335, 338, 355, 358, 376, 377 Ahrends, E., 174, 175 Ajdukiewicz, Kazimierz, 182 Alexander the Great, 334 Alexandrov, P.S., 30, 328 Althoff, Fritz, 189 Am´ ery, Jean, 83 Aquinas, St. Thomas, 161 Arai, Toshiyasu, 396 Aristotle, 368, 384 Armsen, Hella, 253 Armsen, Paul, 246–248, 253–255 Arrhenius, Svante, 15 Artin, Emil, 41, 50 Aumann, 124 Azevedo do Amaral, Ignacio M., 121 Bachelard, Gaston, 82, 83 Bachmann, Friedrich, 63, 66, 78, 79, 84, 86, 113, 137, 180, 182, 213, 214, 228 Bæumler, Alfred, 77, 174 Baldus, Richard, 35 Bandelier, Emil, 6 Barrau, J., 35 Bartels, Traugott, 18 Barth, Karl, 176, 177, 185 Bauer, F.L., 235, 239, 249 Bavink, Bernhard, 157, 158, 190 Becker, Albrecht, 182, 201 Becker, Oskar, 29, 105, 125, 141, 154, 198, 216–218, 265 Becker, R., 131 Behmann, Heinrich, 37, 38, 80, 113, 124, 137 Behnke, Heinrich, 83, 89, 98, 177–179, 214 Benes, Edvard, 256, 260, 261 433

434

Bolzano, Bernard, 177, 186, 229, 328 Bonaparte, Napoleon, 283, 288 Boole, George, 204, 229 Borel, Armand, 78 Borel, Emile, 302, 379 Born, Max, 225 Boseck, Karl-Heinz, 153, 154 Bowditch, Henry Pickering, 6 Brahe, Tycho de, 237 Braun, H., 117 Brecher, Gustav, 89 Brouwer, L.E.J., 25, 26, 34–38, 40, 45, 72, 75, 78, 79, 87, 92, 95, 115, 141, 147, 160, 167, 169, 186, 203, 213, 218, 225, 226, 295–297, 301–314, 320, 322–325, 327–334, 336, 337, 341, 344, 346, 351–354, 361–364, 379–381 Brugsch, Theodor, 10 Bruno, Baron von, 7, 8 Bruno, Giordano, 270 Brunschvicq, L., 83 Burian, Maria, 246 Cantor, Georg, 24, 27, 28, 34, 37, 64, 65, 67, 69–72, 82, 93, 99, 115, 127, 136, 157, 198, 242, 291, 292, 295, 299, 304, 308, 313, 326, 327, 348, 351, 359, 372, 373, 399 Carath´eodory, Constantin, 21, 23, 24, 83, 95, 196, 210, 248, 324, 333 Carlyle, Thomas, 307 Carnap, Rudolf, 37, 77, 79, 80, 94, 131, 137, 147, 174, 175, 182, 185, 190, 191, 208, 212, 229, 335 Carroll, Lewis, 158 Cartan, Henri, 83 Cassirer, Ernst, 83, 147 Castelhun, C.F., 6 Cauchy, Augustin Louis, 34, 104, 157, 158 Cavaill` es, Jean, 27, 38, 52, 64, 74, 81–84, 111, 112, 270 Cavalieri, Bonaventura, 283, 284, 287 Cayley, Arthur, 297 Chevalley, Claude, 111 Christian, Dean, 171 Church, Alonzo, 37, 64, 66, 72, 76, 87, 91, 92, 121, 124, 128, 356, 357, 361, 377, 378, 399 Chwistek, Leon, 91, 230 Clauß, Ludwig Ferdinand, 216, 217 Cohen, Jonas, 205 Collatz, Lothar, 21, 22, 24, 26, 31 Comte, Auguste, 338 Copernicus, 208, 224 Courant, Richard, 21–23, 28–30, 42, 43, 46–48, 50, 52, 83, 108, 109, 111, 138, 168, 178, 217, 219, 283, 290, 312, 328, 333, 338

INDEX

Couturat, Louis, 137, 147, 175, 204, 229 Curry, Haskell, 23, 86, 87, 97, 118, 128, 161, 397–399 Dannegger, Portus, 7, 8 Darboux, Gaston, 25 Dedekind, Richard, 23–25, 27, 34, 40, 64, 82, 113, 114, 121, 179, 184, 193, 228, 242, 299, 301, 311, 313, 317, 335 Dehn, Max, 205, 214 Denk, Viktor, 255–257, 259, 262 Descartes, Rene, 213, 229 Deussen, Paul, 177 Diefenbach, Karl Wilhelm, 3, 4 Diller, Justus, 251 Dilthey, Wilhelm, 200 Dinghas, Alexander, 153, 154 Dingler, Hugo, 38, 56, 62, 86, 103, 105, 119, 124, 134, 141, 142, 144, 145, 147–150, 152–154, 156, 167, 169–175, 186, 190, 191, 196–206, 208, 209, 211–215, 224, 227, 230–232, 267 Dirichlet, P.G.L., 134 Dirlmeier, Johann Peter Gustav, 200 D¨ orrie, Heinrich, 162, 167, 187 Dræger, Max, 206 Dragalin, Albert, 398 Driesch, Hans, 200 Dubislav, Walter, 155, 174, 185 DuBois-Reymond, Emil, 5, 6, 10, 295 DuBois-Reymond, Paul, 295 D¨ uhring, Eugen, 6, 7 D¨ urer, Albrecht, 191, 208, 214 Dyckhoff, Roy, 397 Ebert, Hans, 241 Eddington, Arthur Stanley, 198 Eichler, Martin, 117 Einstein, Albert, 143, 144, 171, 191, 197–199, 225 Engler, Gunther, 239 Enriques, Federigo, 80 Epple, Moritz, 114, 115 Erxleben, W., 215 Essler, Wilhelm, 267 Euclid, 134, 205, 207, 210, 212, 268, 296, 297, 367 Eudoxus, 193 Euler, Leonhard, 144, 176 Ewald, Dr., 245 Ewald, P.P., 169 Faber, Gustav, 164 Fasnacht, Jakob, 6 Fasnacht, Rosalie, 6 Feferman, Solomon, 395 Fehr, Salomon, 4 Feickert, Andreas, 164 Fenstad, Jens Erik, 31

INDEX

Fermat, Pierre de, 283, 288, 295, 296, 329, 337, 357, 370 Feyerabend, Paul, 173 Feynman, Richard, 237 Fichte, Johann Gottlieb, 176, 208 Finsler, Paul, 37, 95 Fleischmann, Wolfgang, 246 Folkerts, Menso, 230 Frænkel, Adolf Abraham, 37, 62, 66, 82, 85 Frank, Hans, 214 Frank, Orrin, Jr., 96 Frank, Philipp, 80, 191, 212 Frege, Alfred, 194, 241 Frege, Gottlob, 24, 34, 43–45, 63, 77, 113, 114, 150, 158, 162, 177, 181–183, 186, 192–195, 203, 204, 214, 215, 228–230, 241, 271, 297–301, 308, 313, 368, 369, 371, 376, 383, 384 Freisler, Roland, 62, 226 Frend, William, 297 Frese, Harald, 18 Freud, Sigmund, 271 Freudenthal, Hans, 373 Freyer, Hans, 217 Freytag L¨ oringhoff, 7, 8 Friedrichs, Kurt, 23, 101 Friedrichs, Otto, 83 Fueter, R., 210 F¨ uhrer, Dr., 171 F¨ uhrer, Wilhelm, 214 Gadamer, Hans-Georg, 170, 216, 217 Galileo, 283 Gandy, Robin, 71 Garibaldi, Giuseppe, 6 Garkisch, Edith, 246, 254 Gauss, Carl Friedrich, 34, 106, 134, 144, 157, 297, 328 Gehlen, Arnold, 170 Gehrcke, Ernst, 171 Gentzen, Agnes Alexandrine Alwine, 2, 12 Gentzen, Carl Friedrich Wilhelm, 3 Gentzen, Erich Karl Hermann, 2 Gentzen, Gerhard, Dipl.-Ing., 2, 16, 234, 240, 241, 243, 244, 247, 249, 250, 253, 254, 260, 262 Gentzen, Hans, 1–4, 11, 12, 236 Gentzen, Hans August Karl, 3 Gentzen, Hans Karl, 2 Gentzen, Iga, Aunt, 22, 242, 243 Gentzen, Max Wilhelm Julius, 2 Gentzen, Melanie, 1, 8, 12, 18, 117, 236, 260, 261, 263–265 Gentzen, Waltraut, see Student, Waltraut Gentzen, Wilhelm, 3 Gentzen, Wilhelm Johann Carl, 2, 12 Gentzen, Willi, Uncle, 21, 22, 242, 243 Georgiadou, Maria, 248

435

Geppert, Harald, 223, 225 Gissel, Dr., 171 Gistl, Dean, 211 Glaser, Walter, 237 Glaus, Beat, 54 Glivenko, Valery, 37, 92, 95, 381, 382 G¨ odel, Kurt, 27–29, 37–41, 45, 58, 60, 64, 66–71, 73–76, 78, 81, 82, 87, 88, 90–95, 99–101, 110, 114–116, 119–122, 126, 127, 129–131, 133, 135–139, 146, 161, 169, 176, 184, 215, 227, 228, 265, 294, 298, 335–341, 348, 350, 352, 353, 355–359, 377, 378, 381–383, 390–393, 395, 400 Goebbels, Joseph, 106, 168, 217, 238 Goeppert, Marie, 111 Goethe, Johann Wolfgang von, 158, 178, 222, 307 Gonseth, F., 89, 91, 97 Gordan, Paul, 292, 293, 309, 310, 316 G¨ oring, Hermann, 234 G¨ ortler, Helmut, 110, 132 Goudsmit, Samuel, 234 Graf, Ulrich, 162 Grattan-Guinness, Ivor, 198 Greenwood, Thomas, 121 Grell, H., 248 Grelling, Kurt, 38, 169, 170, 177, 267 Grimm, Friedrich, 132 Groos, Helmut, 183 Grossinger, Wilhelm, 9 Grunsky, Helmut, 153, 249 Gudden, 247, 254, 255 Gumbel, E.J., 160, 224 G¨ unther, Karl Friedrich, 158, 188 Hæring, Theodor, 183, 191 Hahn, Hans, 191, 212 Hamel, Georg, 154, 155, 161, 162, 220, 227, 249 Hamilton, William Rowan, 297 H¨ ansel, Justizrat, 12 Hardy, Godfrey Harold, 156, 242, 325 Harms, J¨ org, 250 Harrop, Ron, 397 Hartmann, Hans, 220 Hartmann, Nicolai, 185, 190, 197, 200 Hartner, Willy, 182 Hartogs, Friedrich, 24 Hasenjæger, Gisbert, 125, 182, 184, 249 Hasenjæger, Lord Mayor, 184 Hasse, Helmut, 28, 46–50, 53, 60–62, 83, 97, 98, 117, 127, 128, 131–134, 159, 162, 168, 174, 176, 177, 226, 249 Heckmann, Otto, 41 Heiber, Hans, 217 Heidegger, Martin, 83, 170, 177, 188, 216, 224

436

Heine, Eduard, 291, 292 Heisenberg, Werner, 125, 159, 171 Hellinger, E., 312 Herbrand, Jacques, 27, 28, 33, 37, 38, 41, 66, 67, 73, 76, 81–83, 97, 111, 112, 355, 390 Herbst, Dr., 15, 18, 242 Herglotz, Gustav, 23, 41, 109 Hermann, Grete, 50, 80 Hermes, Hans, 30, 38, 78, 79, 84, 86, 118, 124, 137, 179, 181, 186, 204, 213, 228, 263, 265 Herrigel, Eugen, 183 Hertz, Elizabeth, 107 Hertz, Hans, 107 Hertz, Helene, 107 Hertz, Paul, 23, 28, 31–34, 38, 41, 44, 53, 107–109, 387 Hertz, Rudolf, 107 Herzog, Roman, 231 Hesse, Hermann, 228 Hessenberg, Gerhard, 37 Hetpur, Wladyslaw, 184 Heydrich, Reinhard, 248, 258, 260 Heyse, Hans, 177 Heyting, Arend, 37–40, 43–45, 128, 137, 334–336, 354, 364 Hilbert, David, 21–34, 37–44, 47, 50, 53, 55–58, 60–67, 69–80, 83, 84, 86–90, 92–94, 96, 98–102, 108, 111, 112, 114, 115, 119, 120, 122, 125, 127, 129–131, 133, 134, 137–139, 141, 144–147, 149, 150, 158, 160–162, 167, 169, 172–174, 176–180, 182, 183, 186, 187, 190–196, 198, 202–205, 207, 211, 213–215, 219, 224–226, 228, 229, 266–269, 291–341, 346, 351–353, 355, 356, 363–365, 373–378, 380, 381, 383, 384, 399, 400 Hillers, W., 201 Himmler, Heinrich, 174, 184, 188, 197, 224 Hitler, Adolf, 26, 62, 119, 133, 152, 165, 167, 168, 170, 179, 197, 213, 216, 218, 220, 221, 226, 230, 234 Hjelmslev, Johannes, 363 Hlavaty, Vaclav, 264, 265 Hlawka, Edmund, 192 Hofer, Hans Karl, 172, 192, 230 Hofmann, J.E., 153, 154, 284, 289 Hohenemser, Paul, 41, 101 Holzkamp, Klaus, 212 H¨ onigswald, Richard, 177 Hopf, Heinz, 328 H¨ oppener, Hugo, 4 Horner, 255, 256, 260 Howard, William, 398, 399 Hudelmaier, J¨ org, 397 Hunger, Ulrich, 235 Huntington, E.V., 121

INDEX

Hurwitz, Adolf, 293 Husserl, Edmond, 77, 149, 216, 217 Hyrtl, Josef, 5 Itelson, 175 Jacobi, Carl Gustav Jacob, 157 Jacobi, Karl, 79 Jæger, Werner, 177, 178 Jænsch, Erich Rudolf, 152, 156–158, 170, 186–191, 221 Ja´skowski, Stanislaw, 38, 80, 109, 179, 386 Jaspers, Karl, 175, 176 Jervell, Herman, 52 Johnson, Samuel, 309 Jordan, Pascual, 170–172 Jorgensen, J., 192, 206 K´ alm´ ar, L´ aszlo, 73, 76, 89, 121, 124, 138 Kaluza, Theodor, 131, 132, 233, 235 Kant, Immanuel, 7, 176, 182, 188, 208, 380 Kapp, Wolfgang, 217 Kas¨ uske, J¨ urgen, 2 Kayser, Heinrich, 166 Kepler, Johannes, 15, 187, 191, 208, 236, 237 Kerr, Alfred, 228 Kersten, Ina, 263 Ketonen, Oiva, 52, 97, 396–398 Khintchine, Alexander, 380 Kirchhoff, Gustav Robert, 5 Kleene, Stephen Cole, 76, 91, 97, 119, 121, 128, 397 Klein, Felix, 28, 37, 46, 48, 50, 103–107, 109, 132, 134, 160, 163, 189, 190, 206, 221, 226, 292, 293, 302, 328 Kluge, Alexander, 217 Kneser, Hellmuth, 21, 24, 30, 31, 33, 55–57, 61, 70, 79, 85, 86, 137–139, 156, 157, 196, 210, 221, 225, 230, 233, 265 Kneser, Martin, 24, 55–57, 79, 85, 138, 140 Koch, Peter, 200 Kochend¨ orffer, R., 117 Koebe, P., 328 K¨ ohler, Otto, 162, 228 Kolmogoroff, A.N., see Kolmogorov, A.N. Kolmogorov, A.N., 30, 70, 76, 78, 95, 169, 262, 380, 398 Kommerell, Max, 213 K¨ onig, Julius, 138 Korteweg, D.J., 303, 305, 306 Koschmieder, L., 312 Kovalevskaya, Sonya, 291, 292 Kowalewski, Gerhard, 248, 258, 261 Krahelska, Hanna, 230 Krammer, Franz, 240, 241, 244–247, 249, 250, 254–256 Kratzer, Adolf, 78, 80, 84, 86, 114, 178–180, 199, 200, 213

INDEX

Kraus, Fritz, 255, 256, 258–261, 263 Kraus, Olga, 256 Kreisel, Georg, 39, 60, 81, 88, 111, 262, 263, 392 Krieck, Ernst, 174, 191, 201, 202, 205, 206 Kronecker, Leopold, 291–293, 299, 302, 307, 308, 313, 323, 328, 329, 334, 344, 360 Kr¨ uger, Dr., 18 Kubach, Fritz, 145, 171, 199, 215, 224 Kummer, E.E., 126 Kunstat, Miroslav, 235 Lagrange, Joseph Louis, 157 Lalande, 175 Lambert, Johann Heinrich, 191, 208, 210, 212 Lammel, Ernst, 239, 247, 254, 255 Landau, Edmund, 23, 41, 46, 108, 153, 155, 167, 168, 188, 205, 224, 225, 328 Lange, Heinze, 49 Lautman, Albert, 52, 81–84, 111, 112 Lebesgue, Henri, 305, 324 Legris, Javier, 32 Lehmann, Gerhard, 77 Leibniz, Gottfried Wilhelm von, 134, 137, 147, 182, 186, 191, 194, 204, 208, 213, 219, 220, 224, 229, 333 Leisegang, Hans, 201, 202 LeLionais, F., 111 Lenard, Philipp, 105, 143, 144, 156, 159, 166, 171, 178, 191, 201, 203, 222 Lensch, Paul, 224 Leonardo da Vinci, 204 Le´sniewski, Stanislaw, 170, 179, 180, 182, 184 Levinas, Emmanuel, 83 Liebmann, Heinrich, 145, 211 Linke, Paul A., 183 Litten, Freddy, 219 Lobachevskii, Nicolai, 297 Loebell, Frank Richard, 208, 211 L¨ offler, Eugen, 139, 206 Lorbeer, Gerhard, 255–257, 259 Lorenzen, Paul, 141, 216, 231, 243, 244, 265, 266 Lothing, Dr., 255, 256 Lowell, Percival, 15 L¨ owenheim, Leopold, 177, 267 L¨ owith, Karl, 216 L¨ owner, Karl, 168 L¨ ubbe, Hermann, 186 L  ukasiewicz, Jan, 80, 96, 109, 128, 180, 181, 183–185, 263 Lullus, Raymundus, 229 Lutman-Kokoszynska, 182 Mach, Ernst, 171 Mach, Ludwig, 171 Mac Lane, Saunders, 23, 27, 52, 118, 222

437

Magnus, Heinrich Gustav, 10 Mahnke, Dietrich, 170, 204 Malcev, A., 60 Manger, Eva, 156, 230 Mann, Thomas, 24 Mannoury, Gerrit, 334, 381 Marcuse, Herbert, 83 Markov, A.A., 76, 169, 262 Marquard, Odo, 186 Maruhn, Karl, 248, 250, 251, 254, 255 Maunz, Theodor, 231 May, Dr., 15 May, Eduard, 141, 145, 147, 150, 152, 154, 167, 169, 172–175, 190, 191, 197, 199–202, 207, 209–213, 215, 267 May, Karl, 15 Meckel von Hemsbach, Johann Heinrich, 10 Meggers, William, 166 Mehrtens, Herbert, 154, 221, 222, 224, 239 Mengele, Josef, 210 Menger, Karl, 37, 39, 128 Mentzel, Rudolf, 48, 80, 151, 181, 210 M´ eray, Hugues Charles Robert, 242 Mesmer, G., 110 Meyer, Konrad, 184, 217 Meyerhof, Otto, 80 Michælis, Helmut, 21 Michælis, Hertha, 13, 14, 16, 18, 21, 30, 52, 126, 242, 269 Michælis, Otto, 13 Minkowski, Hermann, 125, 196, 205 Mittag-Leffler, Charlotte, 292 Mittag-Leffler, G¨ osta, 292 Mittelstraß, J¨ urgen, 231 Mohr, Ernst, 53, 217, 239, 241, 248–250 Moisil, Gr. C., 128 Moltke, Helmuth, 176 Morris, Charles W., 77 Mostowski, Andrzej, 80, 82, 264 Moufang, Ruth, 214 M¨ uller, Gerhard, 118 M¨ uller, Johann, 10 M¨ uller, Wilhelm, 147, 156, 172–174, 196–199, 203, 210, 211, 214, 219, 224 Nahin, Paul J., 284 Nasse, O., 5 Negri, Sara, 378, 389, 390 Nelson, Leonard, 29, 30 Neugebauer, Otto, 22, 41, 50, 101, 216, 218 Neumann, Friedrich, 51 Neurath, Otto, 191, 212 Newton, Isaac, 236, 237, 306, 333 Nietzsche, Friedrich, 6 Nikuradse, Johann, 217 Noether, Emmy, 23, 27, 47, 49, 108, 126, 138, 328 Nohl, Hermann, 29

438

Novikoff, P.S., see Novikov, P.S. Novikov, P.S., 128 Oberh¨ auser, Georg, 9 Ono, Hiroakira, 118 Ono, Katzui, 118, 119, 124 Orozco, Teresa, 217 Osenberg, Werner, 234, 235, 239–241, 249 Ostwald, Wilhelm, 171 Padoa, Alessandro, 122 Painlev´ e, Paul, 324 Parsons, Charles, 110 Pasch, Moritz, 373 Pascha, Abba, 9 Paulsen, Friedrich, 176 Peacock, George, 297 Peano, Giuseppe, 368, 371, 373, 384 Peckhaus, Volker, 29, 108 Pepis, J´ ozef, 96, 97 Perron, Oskar, 21, 23, 24, 154, 156, 196–199, 202, 211, 219, 225, 230, 328, 334 Physiologus, 340 Pieri, Mario, 371 Pinder, Wilhelm, 83 Pinl, Max, 247, 254, 255, 257, 258, 261, 264, 267 Planck, Max, 149, 183, 190, 199, 204, 210 Poincar´e, Henri, 24, 25, 37, 75, 103, 158, 218, 301–303, 308, 310, 314, 315, 332, 344, 360, 379 Popper, Karl, 174, 191, 197, 212, 213, 231, 232 Post, Emil, 76, 128 Prandtl, Ludwig, 109, 110, 132 Prawitz, Dag, 386, 389 Pringsheim, Alfred, 24 Proclus, 207, 210, 212 Pythagoras, 367 Quine, Willard Van Orman, 109, 121, 193 Ramsey, Frank Plumpton, 35 Rasiowa, Helena, 81 Rathjen, Michæl, 396 Reich, Max, 51 Reichenbach, Hans, 8, 80, 128, 141, 142, 144, 177, 182, 208, 212 Reid, Constance, 219, 307, 322, 323, 325 Reidemeister, Elisabeth, 219 Reidemeister, Kurt, 219, 220, 312, 337 Reinhardt, Benno, 10 Reisig, Gerhard H.R., 240 Rellich, Franz, 47, 52 Rembs, Eduard, 167 Requard, Friedrich, 141, 144, 147–149, 172, 206, 227 Rieger, Ladislav Svante, 264

INDEX

Riehl, Aloys, 176 Riemann, Bernhard, 34, 134, 213, 295–297, 324, 332 Riezler, Kurt, 83 Rinow, W., 248 Ritter, Joachim, 83, 170, 186 Robbel, Gerd, 8 Robbins, Herbert, 283, 290 Robespierre, Maximilien, 153 Robinson, J.A., 81 Rohrbach, Hans, 53, 70, 98, 234, 235, 237–251, 253, 254, 259, 261, 263, 265, 266 Rosenberg, Alfred, 97, 152, 159, 174, 215, 224 Rosenthal, A., 205, 211 Rosser, J. Barkley, 78, 91, 94, 109, 121, 128 Runge, Carl, 109 Russell, Bertrand, 23, 25, 37, 38, 42, 43, 63, 64, 75, 80, 87, 113, 144, 182, 185, 186, 203, 228–230, 299–301, 304, 308, 310, 336, 368, 369, 371, 376, 383, 384 Rust, Bernhard, 133, 159, 180, 182, 206, 224 Saint Just, Louis Antone de., 153 Salamucha, Jan, 185, 229 Sauerbruch, 83 Schellinx, Harold, 52 Schemm, Hans, 105 Schiaparelli, Giovanni, 15 Schiller, Friedrich, 62, 111 Schilling, Claus, 210 Schilling, Kurt, 66, 173, 174 Schirn, Matthais, 28 Schischkoff, Georgi, 199, 200 Schlick, Moritz, 107, 144, 177, 182, 191, 205, 208, 212 Schmidt, Arnold, 31, 44, 47, 60–64, 67, 68, 84, 85, 87, 88, 96, 102, 124, 137, 227, 262, 265, 388 Schmidt, Erhard, 26 Schmidt, F.K., 50 Schmidt, Helmut, 231 Schmidt, Johann L., 159 Schmitt, Carl, 267 Schneider, Theodor, 117 Schoch, D., 153 Scholz, Erna, 78 Scholz, Heinrich, 38, 45, 55–57, 66, 72, 76–80, 83, 84, 86, 96, 102, 113, 114, 119, 124, 125, 129–131, 137, 138, 141, 144–146, 162, 167, 169, 170, 173–187, 190–196, 202–204, 208, 210, 213–215, 218, 224–226, 228–231, 247, 249, 251, 259, 263–265 Schopenhauer, Arthur, 7 Sch¨ orner, Ferdinand, 247, 250

INDEX

Schr¨ oder, Ernst, 138, 204, 213 Schr¨ odinger, Erwin, 26 Schroeder-Heister, Peter, 32, 44 Schr¨ oter, Karl, 124, 181, 182, 186, 195, 204, 226, 230, 265 Schultze, Walther, 163 Schulze-S¨ olde, Walter, 159 Schur, Issai, 26, 29, 46, 152, 153, 167, 205, 225 Sch¨ utte, Kurt, 38, 42, 44, 62, 65, 66, 75, 262, 397 Schwarz, Laurent, 25 Schweikert, Ferdinand Karl, 297 Schweitzer, Hermann, 181 Schwerin, Burckhardt, 15 Schwichtenberg, Helmut, 398 Sheriff, R.C., 22 Shiryaev, Albert Nikolaevich, 262 Shoenfield, Joseph, 119 Sieg, Wilfried, 111 Siegel, Carl Ludwig, 126, 131, 295 Siegmund-Schultze, Reinhard, 167 Simpson, Thomas, 283 Skolem, Thoralf, 31, 32, 37, 41, 60, 87, 124, 317, 376–378, 397 Skvorecky, Josef, 261 Smory´ nski, Craig, 29, 56, 115, 283, 291 Soboci´ nski, Boleslaw, 184, 230 Sohn-Rethel, Alfred, 83 Solovay, Robert M., 127 Specht, Minna, 80 Speer, Albert, 234 Speiser, Andreas, 187, 195, 196, 208, 210, 215 Spengler, Oswald, 34, 53 Spinoza, Baruch, 229 Spranger, Eduard, 178, 183, 190, 197 Stachowiak, Herbert, 201, 202 Stark, Johannes, 159, 166, 172, 201, 203, 224 Steck, Max, 56, 86, 124, 134, 141, 143–145, 147, 149, 150, 152, 154, 160, 161, 169, 172–175, 181, 182, 185–187, 190–192, 194–199, 202–215, 218, 221, 225, 230, 267 Stegm¨ uller, Wolfgang, 227, 231 Steiner, Jacob, 283 Steinitz, 177 Stengel, Dr., 16 Stinnes, Hugo, 224 Strauss, Leo, 83 Streicher, Julius, 222 Strobel, R., 240 Student, Barbara, 18, 126 Student, Hans, 126 Student, Hans Lothar, 18, 126 Student, Waltraut, 8, 10, 12, 15, 18, 26, 94, 126, 235, 238, 242, 262, 265, 266

439

S¨ uss, Wilhelm, 83, 98, 178, 196, 210, 238, 240, 248, 265 Sylvester, J.J., 158 Szabo, Manfred, 39, 259, 261, 265, 266 Takeuti, Gaisi, 395 Tarski, Alfred, 44, 64, 70, 79–82, 96, 117, 118, 120, 182, 184, 185, 193, 195, 204, 213, 227, 229, 265, 268 Taussky-Todd, Olga, 339 Teichm¨ uller, Oswald, 49, 53, 162, 167, 217, 225 Thiel, Christian, 33, 70, 87, 107, 114, 117, 125, 138, 144, 222, 235, 259, 266 Thiele, R¨ udiger, 400 Thomæ, J., 169 Thullen, Peter, 178 Th¨ uring, Bruno, 141, 156, 167, 171–174, 191, 196, 199, 203, 208, 211, 214, 215 Tietze, Heinrich, 23, 24, 50, 196, 210 Tietze, Walter, 245–247 Tirala, Lothar Gottlieb, 141–144, 222, 223 Toeplitz, Otto, 177, 216 Tomascheck, Rudolf, 172, 199, 214 Tornier, Erhard, 49, 53, 151, 156, 162, 167, 205, 217, 225 Torricelli, Evangelista, 283 Troelstra, Anne, 398 Troll, Wilhelm, 197, 208, 210 Tugendhat, Ernst, 186 Turing, Alan, 71, 72, 87, 91, 249, 356, 357, 361 Uhland, Ludwig, 9 Ulm, Helmut, 47, 52 Urban, Hans, see Rohrbach, Hans Vahlen, Theodor, 47, 86, 134, 152, 154, 156, 162, 167, 172, 174, 181, 190, 198, 206, 208, 213, 219, 220, 223, 224, 231 Vaihinger, Hans, 205 Valier, Max, 15, 236 van Benthem, Johan, 383 van Dalen, Dirk, 225, 333 van Heijenoort, Jean, 326 Vasari, Giorgio, 271 Veblen, Oswald, 39 Vihan, Premysl, 247 Vlach, Milan, 235 von Amiens, Nikolaus, 229 von Braun, Wernher, 240, 270 von B¨ ulow, Bernhard, 195 von Clausewitz, Karl, 176 von Coburg-Gotha, Ernst, 10 von Cues, Nikolaus, 191, 208 von Greiff, Bodo, 173 von Harnack, Adolf, 176 von Helmholtz, Hermann, 107 von Humboldt, Alexander, 10

440

von von von von von von

K´ arm´ an, Theodor, 22, 109 Kempski, J¨ urgen, 125, 177, 213 Lindemann, Ferdinand, 198, 291, 292 Mises, Richard, 26 Mohl, Hugo, 9 Neumann, Johann, 28, 37, 41, 58, 60, 66, 72, 96, 100, 249, 331, 334–338, 355, 377, 381, 390, 392, 400 von Plato, Jan, 40, 52, 59, 97, 109, 110, 291, 367, 375, 378, 379, 389, 390, 393 von Virchow, Rudolf, 5, 6, 10 von Weizs¨ acker, C.F., 171, 172 Vonderau, Markus, 197, 201, 202, 222, 223 Voss, Aurel, 24, 198, 268 Wærden, B.L. van der, 25, 37, 61, 187, 195 Waismann, Friedrich, 76 Walther, Alwin, 168 Wang, Hao, 40, 126, 127 Weber, Max, 22, 33, 102, 271 Weber, Werner, 46–49, 53, 107, 151, 153, 217, 223, 225 Wegner, Udo, 46–48 Weierstrass, Karl, 134, 152, 157, 158, 292 Weigel, Erhard, 229 Weitzenb¨ ock, Richard, 50 Weizs¨ acker, Ernst Freiherr von, 83 Weyl, Hermann, 22, 23, 25, 26, 28, 34–38, 41, 42, 46, 47, 49, 50, 53–55, 57–60, 66, 72, 74, 75, 81, 82, 87, 89, 91–93, 101, 103, 106, 110, 115, 125, 141, 149, 160, 186, 213–215, 225, 241, 259, 296, 310–314, 320–330, 334, 340, 344, 351, 353, 354, 361, 364, 374, 380 Whitehead, Alfred North, 203, 229, 301, 304, 308 Witt, Ernst, 38, 50, 53, 262, 263 Wolf, Helmut, 246 Wolf, Karl Lothar, 197, 210 Wundt, Max, 183, 192 Zermelo, Ernst, 23–25, 85, 126, 299, 303, 308, 309 Zilsel, Edgar, 94, 110, 111 Zollikofer, Kaspar Tobias von, 9 Zollikofer, Sabine von, 4, 9 Zweig, Arnold, 271

INDEX

Titles in This Series 33 Eckart Menzler-Trott, Logic’s lost genius: The life of Gerhard Gentzen, 2007 32 Karen Hunger Parshall and Jeremy J. Gray, Editors, Episodes in the history of modern algebra (1800–1950), 2007 31 Judith R. Goodstein, Editor, The Volterra chronicles: The life and times of an extraordinary mathematician 1860–1940, 2007 30 Michael Rosen, Editor, Exposition by Emil Artin: A selection, 2006 29 J. L. Berggren and R. S. D. Thomas, Editors, Euclid’s Phaenomena: A translation and study of a Hellenistic treatise in spherical astronomy, 2006 28 Simon Altmann and Eduardo L. Ortiz, Editors, Mathematics and social utopias in France: Olinde Rodrigues and his times, 2005 27 Mikl´ os R´ edei, Editor, John von Neumann: Selected letters, 2005 26 B. N. Delone, The St. Petersburg school of number theory, 2005 25 J. M. Plotkin, Editor, Hausdorff on ordered sets, 2005 24 Hans Niels Jahnke, Editor, A history of analysis, 2003 23 Karen Hunger Parshall and Adrain C. Rice, Editors, Mathematics unbound: The evolution of an international mathematical research community, 1800–1945, 2002 22 Bruce C. Berndt and Robert A. Rankin, Editors, Ramanujan: Essays and surveys, 2001 21 Armand Borel, Essays in the history of Lie groups and algebraic groups, 2001 20 Kolmogorov in perspective, 2000 19 Hermann Grassmann, Extension theory, 2000 18 Joe Albree, David C. Arney, and V. Frederick Rickey, A station favorable to the pursuits of science: Primary materials in the history of mathematics at the United States Military Academy, 2000 17 Jacques Hadamard (Jeremy J. Gray and Abe Shenitzer, Editors), Non-Euclidean geometry in the theory of automorphic functions, 1999 16 P. G. L. Dirichlet (with Supplements by R. Dedekind), Lectures on number theory, 1999 15 Charles W. Curtis, Pioneers of representation theory: Frobenius, Burnside, Schur, and Brauer, 1999 14 Vladimir Mazya and Tatyana Shaposhnikova, Jacques Hadamard, a universal mathematician, 1998 13 Lars G˚ arding, Mathematics and mathematicians: Mathematics in Sweden before 1950, 1998 12 Walter Rudin, The way I remember it, 1997 11 June Barrow-Green, Poincar´e and the three body problem, 1997 10 John Stillwell, Sources of hyperbolic geometry, 1996 9 Bruce C. Berndt and Robert A. Rankin, Ramanujan: Letters and commentary, 1995 8 Karen Hunger Parshall and David E. Rowe, The emergence of the American mathematical research community, 1876–1900: J. J. Sylvester, Felix Klein, and E. H. Moore, 1994 7 Henk J. M. Bos, Lectures in the history of mathematics, 1993 6 Smilka Zdravkovska and Peter L. Duren, Editors, Golden years of Moscow mathematics, 1993 5 George W. Mackey, The scope and history of commutative and noncommutative harmonic analysis, 1992 4 Charles W. McArthur, Operations analysis in the U.S. Army Eighth Air Force in World War II, 1990

TITLES IN THIS SERIES

3 Peter L. Duren et al., Editors, A century of mathematics in America, part III, 1989 2 Peter L. Duren et al., Editors, A century of mathematics in America, part II, 1989 1 Peter L. Duren et al., Editors, A century of mathematics in America, part I, 1988

Gerhard Gentzen (1909–1945) is the founder of modern structural proof theory. His lasting methods, rules, and structures resulted not only in the technical mathematical discipline called “proof theory” but also in verification programs that are essential in computer science. The appearance, clarity, and elegance of Gentzen’s work on natural deduction, the sequent calculus, and ordinal proof theory continue to be impressive even today. The present book gives the first comprehensive, detailed, accurate scientific biography expounding the life and work of Gerhard Gentzen, one of our greatest logicians, until his arrest and death in Prague in 1945. Particular emphasis in the book is put on the conditions of scientific research, in this case mathematical logic, in National Socialist Germany, the ideological fight for “German logic”, and their mutual protagonists. Numerous hitherto unpublished sources, family documents, archival material, interviews, and letters, as well as Gentzen’s lectures for the mathematical public, make this book an indispensable source of information on this important mathematician, his work, and his time. The volume is completed by two deep substantial essays by Jan von Plato and Craig Smoryn´ski on Gentzen’s proof theory; its relation to the ideas of Hilbert, Brouwer, Weyl, and Gödel; and its development up to the present day. Smoryn´ski explains the Hilbert program in more than the usual slogan form and shows why consistency is important. Von Plato shows in detail the benefits of Gentzen’s program. This important book is a self-contained starting point for any work on Gentzen and his logic. The book is accessible to a wide audience with different backgrounds and is suitable for general readers, researchers, students, and teachers.

Eckart Menzler-Trott

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