Hilbert, Göttingen and the Development of Modern Mathematics 9781527523319, 1527523314

David Hilbert is one of the outstanding mathematicians of the twentieth century and probably the most influential. This

740 84 8MB

English Pages xii+272 [285] Year 2019

Report DMCA / Copyright

DOWNLOAD FILE

Polecaj historie

Hilbert, Göttingen and the Development of Modern Mathematics
 9781527523319, 1527523314

Citation preview

Hilbert, Göttingen and the Development of Modern Mathematics

Hilbert, Göttingen and the Development of Modern Mathematics By

Joan Roselló

Hilbert, Göttingen and the Development of Modern Mathematics By Joan Roselló This book first published 2019 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2019 by Joan Roselló All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-5275-2331-4 ISBN (13): 978-1-5275-2331-9

TABLE OF CONTENTS

List of Illustrations ..................................................................................... ix Introduction ................................................................................................. 1 Chapter One ................................................................................................. 7 Hilbert’s Early Career Chapter Two .............................................................................................. 15 The Theory of Algebraic Invariants Chapter Three ............................................................................................ 27 The Königsberg Lectures on Geometry Chapter Four .............................................................................................. 39 The Algebraic Theory of Numbers Chapter Five .............................................................................................. 53 Klein and the Mathematical Tradition of Göttingen Chapter Six ................................................................................................ 63 Hilbert’s First Years at Göttingen Chapter Seven............................................................................................ 71 The Foundations of Geometry Chapter Eight ............................................................................................. 79 The Axiomatization of Geometry Chapter Nine.............................................................................................. 87 Hilbert and American Postulational Analysis Chapter Ten ............................................................................................... 97 The Axiomatization of Analysis

vi

Table of Contents

Chapter Eleven ........................................................................................ 103 The Hilbert Problems Chapter Twelve ....................................................................................... 113 From Integral Equations to Hilbert Spaces Chapter Thirteen ...................................................................................... 123 Paradoxes in Göttingen Chapter Fourteen ..................................................................................... 131 The Consistency of Analysis Chapter Fifteen ........................................................................................ 139 Minkowski and the Early Reception of Relativity Theory at Göttingen Chapter Sixteen ....................................................................................... 149 Hilbert’s Foundations of Physics Chapter Seventeen ................................................................................... 159 Beyond Principia Mathematica Chapter Eighteen ..................................................................................... 171 The Great Debate on the Foundations of Mathematics Chapter Nineteen ..................................................................................... 181 The Foundations of Quantum Mechanics Chapter Twenty ....................................................................................... 193 Hilbert’s Program and Gödel’s Response Chapter Twenty-One ............................................................................... 205 The Noether School and the Rise of Modern Algebra Chapter Twenty-Two............................................................................... 215 Mathematics in Göttingen under the Nazis Chapter Twenty-Three............................................................................. 225 Hilbert, Bourbaki and the Structural Image of Mathematics Epilogue................................................................................................... 233 From Göttingen to Princeton and Paris

Hilbert, Göttingen and the Development of Modern Mathematics

vii

Appendix ................................................................................................. 239 The Hilbert Problems revisited Reference List.......................................................................................... 259 Author Index............................................................................................ 267

LIST OF ILLUSTRATIONS

Fig. 1-1 Pregel river with the castle of Konigsberg in the background. By Unknown௅Adam Kraft, Rudolf Naujok: Ostpreußen. Ein Bildwerk mit 220 Fotos. Würzburg 2002, ISBN 3-88189-444-6, Abb. 7, Public Domain, https://commons.wikimedia.org/w/index.php?curid=49851038. Fig. 1-2 Albertine University of Königsberg. By Alte Postkarte, Public Domain, https://commons.wikimedia.org/w/index.php?curid=3946063. Fig. 1-3 Hermann Minkowski, 1883. By Unknown௅Reid, Constance (1970) Hilbert, Berlin, Heidelberg: Springer Berlin Heidelberg Imprint Springer, p. 224 ISBN: 978-3-66227132-2., Public Domain, https://commons.wikimedia.org/w/index.php?curid=56283063. Fig. 1-4 David Hilbert, 1886. By http://www-history.mcs.st-andrews.ac.uk/history/PictDisplay/Hilbert. htmlReid, Constance (1970) Hilbert, Berlin, Heidelberg: Springer Berlin Heidelberg Imprint Springer, p. 226 ISBN: 978-3-662-27132-2., Public Domain, https://commons.wikimedia.org/w/index.php?curid=91396. Fig. 2-1 The founders of the DMV in Bremen in 1890. Author: Ortgies, J. jr. (photos provided by Ortgies, J. jr.) Source: Bernd Fischer, München. The Oberwolfach Photo Collection. Photo ID: 10606. Fig. 3-1 Adolf Hurwitz as an Extraordinarius in Königsberg. By Unknown௅Reid, Constance (1970) Hilbert, Berlin, Heidelberg: Springer Berlin Heidelberg Imprint Springer, p. 225 ISBN: 978-3-66227132-2., Public Domain, https://commons.wikimedia.org/w/index.php?curid=56283319.

x

List of Illustrations

Fig. 3-2 Projection of a figure in a plane. By Torretti, Roberto, “Nineteenth Century Geometry,” The Stanford Encyclopedia of Philosophy (Winter 2016 Edition), Edward N. Zalta (ed.), https://plato.stanford.edu/archives/win2016/entries/geometry-19th/. Fig. 3-3 Desargues’ theorem. By Weisstein, Eric W. “Desargues’ Theorem.” From MathWorld௅A Wolfram Web Resource. http://mathworld.wolfram.com/DesarguesTheorem.html/. Fig. 5-1 Felix Klein as a professor at Leipzig. By Unknown. Public Domain, https://commons.wikimedia.org/w/index.php?curid=38617. Fig. 6-1 The University of Göttingen at the beginning of the 20th century. By Vereinigung Göttinger Papierhändler, Göttingen௅ImageZeno.org, ID number 20000606855, Public Domain, https://commons.wikimedia.org/w/index.php?curid=64897305. Fig. 9-1 Eliakim Hasting Moore. By Unknown௅Historical photo, Public Domain, https://commons.wikimedia.org/w/index.php?curid=6090343. Fig. 11-1 David Hilbert, ca. 1900. By Unknown௅Reid, Constance (1970) Hilbert, Berlin, Heidelberg: Springer Berlin Heidelberg Imprint Springer, p. 227 ISBN: 978-3-66227132-2., Public Domain, https://commons.wikimedia.org/w/index.php?curid=56288719. Fig. 13-1 Ernst Zermelo in the 1900s. By Unknown (Mondadori Publishers), http://www.gettyimages.co.uk/detail/news-photo/portrait-of-the-germanmathematician-ernst-zermelo-1900s-news-photo/141551246, Public Domain, https://commons.wikimedia.org/w/index.php?curid=41238428. Fig 14-1. Logische Principien des mathematischen Denkens. Lecture notes by Max Born. Unpublished manuscript. Georg-August-Universität Göttingen. Niedersächsische Staats- und Universitätsbibliothek. Fig. 15-1 Poincaré, Mittag-Leffler, Runge and Landau. Author: Jacobs, Konrad (photos provided by Jacobs, Konrad).

Hilbert, Göttingen and the Development of Modern Mathematics

xi

Source: Konrad Jacobs, Erlangen. Oberwolfach Photo Collection. Photo ID: 17115 Fig. 16-1 David Hilbert, 1912. By Unknown௅Possibly Reid, Constance (1970) Hilbert, Berlin, Heidelberg: Springer Berlin Heidelberg Imprint Springer, p. 230 ISBN: 978-3-662-27132-2., Public Domain, https://commons.wikimedia.org/w/index.php?curid=36302. Fig. 16-2 Albert Einstein, 1916. By Paul Ehrenfest௅Museum Boerhaave, Public Domain, https://commons.wikimedia.org/w/index.php?curid=45353782. Fig. 17-1 Paul Bernays, ca. 1920. By Unknown. ETH-Bibliothek Zürich, Bildarchiv/Fotograf: Unbekannt/ Portr_10424/Public Domain Mark. Fig. 18-1 Brouwer and Bohr at the International Mathematical Congress, Zurich 1932. Public Domain, https://commons.wikimedia.org/w/index.php?curid=60221395. Fig. 19-1 The Mathematical Institute at Bunsenstrasse, 3-5. By Daniel Schwen - Own work, CC BY-SA 2.5, https://commons.wikimedia.org/w/index.php?curid=1933920. Fig. 19-2 Richard Courant in a lecture, 1932. Author: Piene, Kay (photos provided by Piene, Kay) Source: Piene, Ragni, Oslo. Copyright: MFO. The Oberwolfach Photo Collection, photo id=17002 Fig. 21-1 Emmy Noether with colleagues at Nikolausberg (near Göttingen). By Artin, Natascha. Source: archives of P. Roquette, Heidelberg and C. Kimberling, Evansville. The Oberwolfach Photo Collection. Photo ID: 9265. Fig. 22-1 David Hilbert in a lecture, 1932. Author: Piene, Kay (photos provided by Piene, Kay) Source: Piene, Ragni, Oslo. Copyright: MFO. The Oberwolfach Photo Collection, photo id=17004. Fig. 22-2 Courant, Landau and Weyl converse in Göttingen, ca. 1930. By Unknown. AIP Emilio Segrè Visual Archives, Nina Courant Collection.

xii

List of Illustrations

Fig. 22-3 Emmy Noether’s letter of dismissal, copy for the Rektorat. Niedersächsische Staats- und Universitätsbibliothek Göttingen. http://kulturerbe.niedersachsen.de/viewer/objekt/isil_DE7_gua_rek_3206_b_nr37/1/LOG_0000/. Fig. 23-1 Bourbaki founding congress in Besse-en-Chandesse, July 1935. By Unknown. Collection Privee/ Archives Charmet/Bridgeman images. Fig. 24-1 Kurt Gödel and Albert Einstein at Princeton, 1950. By Imagno. Hulton Collection. Getty Images. Fig. 24-2 Alexandre Grothendieck lecturing at the Seminar of Algebraic Geometry (1962-1964). Photo credit: IHES.

INTRODUCTION

David Hilbert is undoubtedly one of the most outstanding mathematicians of the twentieth century and probably the most influential of them all. His importance is explained not only by his capital contributions to the development of contemporary mathematics, but also by his prominent role in the conversion of the University of Göttingen into a mecca of mathematical research and, related to this, his personal character: his leadership, his lack of prejudice of all kinds, his internationalism, his ability to work in a team and his fraternization with students. Where does Hilbert’s mathematical talent lie? In his accurate Lebensgeschichte (Biography) on Hilbert, Otto Blumenthal, a disciple of Hilbert’s in Göttingen, says the following about: In the analysis of mathematical talent one has to differentiate between the ability to create new concepts and to generate new types of thought structures, and the gift for sensing deeper connections and the underlying unity. In Hilbert’s case, his greatness lies in an immensely powerful insight that penetrates into the depths of a question [...] Insofar as the creation of new ideas is concerned, I would place Minkowski higher, and from the classical great ones, Gauss, Galois and Riemann. But when it comes to penetrating insight, only a very few of the greatest were the equal to Hilbert.1

It is obviously not the case that Hilbert was not able “to create new concepts and generate new types of thought structures.” For example, Hilbert spaces, a key concept of modern functional analysis, and Hilbert class fields, a fundamental notion in algebraic number theory, are a clear demonstration of Hilbert’s talent regarding the introduction of key concepts in modern mathematics. It is also not the case that Hilbert was not able to demonstrate major new theorems of modern mathematics. For example, Hilbert’s Nullstellensatz (Hilbert’s zero theorem), Hilbert’s finite basis theorem and Hilbert-Waring theorem are first order contributions to modern mathematics. However, as noted by Blumenthal, it was probably Hilbert’s “penetrating insight” “for sensing deeper connections and the underlying 1

Hilbert 1965, vol. 3, 429.

2

Introduction

unity” in one or more branches of mathematics, the most characteristic feature of his contributions to mathematics. This is evident, for example, in the way in his famous work Die Theorie der algebraischen Zahlkörper (The Theory of Algebraic Number Fields) (1897), he was able to synthesize and reorganize the algebraic number theory of those days, to open new research perspectives and, finally, to establish connections of this subject with other branches of mathematics. But this ability of Hilbert “for sensing deeper connections and the underlying unity” is manifested mainly in the application of the axiomatic method to different branches of mathematics. Hilbert initially applied this method to geometry with the purpose of reconstructing it from a more solid basis and within the framework of a deductive system that would enable one to see the logical connections between the different geometries that emerged in the nineteenth century (basically non-Euclidean, projective and no Archimedean geometries) and Euclidean geometry. This is essentially the aim of Grundlagen der Geometrie (The Foundations of Geometry) (1899), a masterpiece in the history of modern mathematics which gave him a deserved worldwide reputation. Shortly after, Hilbert also applied the axiomatic method to analysis and later to physics and logic. In addition, his disciple Ernst Zermelo also applied the axiomatic method to the “naive” set theory initiated by Georg Cantor and developed by Richard Dedekind, and thus arose “axiomatic” set theory, a consolidated branch of mathematics today. Largely, the axiomatization and formalization of the different branches of mathematics has been considered throughout the twentieth century and so far this century their ideal state, since it was considered that rigor in a specific domain of mathematics could only be achieved if this became a formal axiomatic system, that is, an axiomatized theory. Not even the well-known Gödel’s theorems on the incompleteness of any mathematical theory capable of expressing a minimum arithmetical knowledge have put into question the viability and fertility of the axiomatic method (although certainly its limits). However, the genesis and development of the axiomatic method, as it is understood today, is inextricably linked to the name of David Hilbert. Here is a major source, if not the most important, of his influence on the development of mathematics in the XX and XXI centuries. Another of the facts that underlines Hilbert’s influence on contemporary mathematics and his talent “for sensing deeper connections and the underlying unity” in different branches of mathematics is his famous conference “Mathematische Probleme” (“Mathematical Problems”), given at the International Congress of Mathematicians held in

Hilbert, Göttingen and the Development of Modern Mathematics

3

Paris in 1900. In his lecture at the Sorbonne, Hilbert did not only present a simple list of important mathematical problems of his time, since the problems often pointed to theories that would provide light on these problems and to new problems that would arise from these theories. Indeed, some of the problems were not problems, but rather research programs. An example of this, but not the only one, would be the sixth problem in the list, which demanded an axiomatization of physics. In addition, the problems were grouped into four separate groups according to their topics, which showed their interconnection. In this sense, the success of the list of problems presented by Hilbert was not only that it was a list of well-chosen problems, but basically because it offered a coherent view of how different branches of mathematics would progress in the future and for the role deserved to rigor and axiomatization in the solution and the formulation of various problems. However, as we said before, the importance of the figure of Hilbert lies not exclusively in his mathematical talent, but also in his determining role in the gestation and consolidation of the Georg-August University of Göttingen as the most important mathematical research centre of the world. In this, as we will explain later, Hilbert was not alone, but he had the invaluable collaboration of the great Felix Klein, whose idea was from the beginning to turn Göttingen into a centre of mathematical excellence. And with this purpose in mind, he took all necessary steps to get Hilbert to occupy the chair of mathematics at that university in 1895. Actually, the life and academic career of Hilbert can be divided into two clearly differentiated phases: the period of Königsberg, which embraces from the moment of its birth to its consecration as one of the most brilliant mathematicians in Germany, and the Göttingen period, which goes from his arrival at this university until his death in the middle of the Second World War. Hilbert taught at the University of Königsberg from 1886 until 1895, the year in which he obtained a chair of mathematics at the University of Göttingen, an institution to which he continued to be bound to the rest of his life. During his stay at these universities, Hilbert taught numerous courses in algebra, number theory, geometry, analysis, logic, foundations of mathematics, physics, etc., with the resolution, as Constance Reid has pointed out in his magnificent book Hilbert, “to educate himself as well as his students through his choice of subjects and to not repeat lectures, as many docents did.”2 With the arrival of Hilbert, the University of Göttingen gradually became the paradigm of modern universities, the place where students and researchers of mathematics and related sciences, particularly physics, from 2

Reid 1970, 28.

4

Introduction

around the world, wanted to go to complete their training or progress in their research. As we will see throughout this essay, the list of first rate scientists working in Göttingen during the first three decades of the century is impressive and includes names such as Hilbert, Hermann Minkowski, Max Planck, Max Born, Werner Heisenberg, Emmy Noether, Hermann Weyl and John von Neumann, among many other leading scientists. However, what is most impressive for our history is that most of these scientists worked in direct collaboration and often under the direction of Hilbert. In this sense, Hilbert’s leadership and his influence on the scientific development of an epoch certainly has no parallel throughout history. Only the inexorable passage of time, in the case of Hilbert’s life, and the rise of Nazism, in the case of Göttingen, diminished his direct influence on the development of science in the first half of the 20th century. The arrival of the Nazis to power in 1934 meant the expulsion of Jewish scientists from all German universities, leading to a progressive impoverishment of the German scientific and cultural life. Reid explains a well-known anecdote, which tells Hilbert’s dislike about the situation. “Sitting next to the Nazis’ newly appointed minister of education at a banquet, he was asked, “And how is mathematics in Göttingen now that it has been freed of the Jewish influence?” “Mathematics in Göttingen?” Hilbert replied. “There is really none any more.””3 One of the peculiarities of Hilbert’s mathematical contributions lies in the fact that they can be divided more or less sharply in different periods during which Hilbert was dealing almost exclusively with a particular set of problems, whether they were relative to the theory of invariants, the algebraic theory of numbers or the integral equations. In this sense, Hermann Weyl, an outstanding disciple and successor of Hilbert in Göttingen, divided the contributions of his teacher into five periods: i. Theory of invariants (1885-1893). ii. Theory of algebraic number fields (1893-1898). iii. Foundations, (a) of geometry (1898-1902), (b) of mathematics in general (1922-1930). iv. Integral Equations (1902-1912). v. Physics (1910-1922). 4 Although, as noted by Weyl himself, these titles 3

Ibid., 205. The great set-theorist Abraham Fraenkel gives a slightly different version of the story. According to Fraenkel, Bernhard Rust, the Nazi Reich Minister for Science, Education, and Popular Culture, asked Hilbert: “Is it really true, Mr. Professor, that your institute suffered so much from the departure of the Jews and their friends?” to which Hilbert replied, in his characteristic East Prussian dialect: “Suffered? No, it hasn’t suffered, Mr. Minister. It simply doesn’t exist anymore!” (Fraenkel 2016, 135). 4 Weyl 1944, 617.

Hilbert, Göttingen and the Development of Modern Mathematics

5

are perhaps too specific and there are some overlaps,5 the truth is that the previous periodization of Hilbert’s work gives a fairly accurate general idea of the whole of his work and can serve as a guide the reader not to lose himself in the somewhat sinuous vital and intellectual itinerary that we will cover in this work. The great mathematical conquests achieved by Hilbert are undoubtedly a product of his formidable mathematical talent, but surely this talent would not have been developed with the force it did if Hilbert had not worked much of his academic life in Göttingen. Hilbert was one of the architects of the great mathematical tradition of Göttingen and of its leading position in mathematical research during the first three decades of the twentieth century, but one must not forget the leadership and achievements of other mathematicians such as Felix Klein, Hermann Weyl or Richard Courant in this direction. In this sense, the exhibition of Hilbert’s contributions to modern mathematics will always be based on the vital, geographical and historical context in which they originated and this will almost always suppose a reference to the intense intellectual and academic life of Göttingen. An exposition of this kind must necessarily follow a chronological approach, so throughout the book, the precise exposition of the mathematical concepts and problems addressed by Hilbert is combined with the description of the life of the author and his historical context.

5

Also, as noted by D. W. Lewis, “there were exceptions to the list above. For example, in 1909 Hilbert successfully solved Waring’s problem, a problem outstanding since 1770 about expressing a natural number as a sum of n-th powers […] Also in 1899 Hilbert managed to resuscitate Dirichlet’s Principle concerning the solution of boundary value problems, this being totally unrelated to the main research work he was pursuing at this period” (Lewis 1994, 44).

CHAPTER ONE HILBERT’S EARLY CAREER

David Hilbert was born in Wehlau,1 in the neighbourhood of Königsberg (now Kaliningrad, Russia), capital of East Prussia and hometown of the great philosopher Immanuel Kant, the 23rd of January in 1862. That same year, Otto von Bismarck was entrusted by King William I of Prussia with forming the next government, achieving, between 1864 and 1871, the political unification of Germany under Prussian rule. Shortly after the wars of unification, David’s father, Otto Hilbert, obtained a position as judge in the city of Königsberg, where he moved with his family. His mother, Maria Erdtmann was a peculiar woman, interested in astronomy and philosophy, and fascinated by prime numbers. David was raised and educated in the Prussian virtues of punctuality, thrift, discipline, duty and respect for the law. And also, like the rest of the children of Königsberg, in admiration and respect for Kant. In 1870 the Franco-Prussian war broke out and, in less than one year, it marked a definitive step towards the unification of Germany under Prussian leadership. In the autumn of 1872, when the Prussian army returned victorious from the war with France, a Jewish family called Minkowski arrived in Königsberg. His third son, Hermann, eight and a half years old, entered the Vorschule (preparatory school) of the Altstadt Gymnasium (Altstadt High-school) in Königsberg. Hermann Minkowski (1864-1909) soon showed an exceptional talent for mathematics and would become an inseparable friend and colleague of Hilbert during his youth. Later he would become a central figure in the gestation and consolidation of Gottingen as a global reference centre in mathematical research. The gymnasium chosen by the parents of David for the education of their son was the Friedrichskolleg. This school was considered the best of Königsberg and had had among its students Kant himself. But, unfortunately for Hilbert, it was a very traditional school, in which the emphasis was on the memorization of the contents and in its curriculum 1

See Reid (1970, 8).

8

Chapter One

the study of languages, particularly the classical ones, predominated. What Hilbert liked most were mathematics, as they were easy for him and did not require memorization. It should not be surprising, therefore, that the years at Friedrichskolleg were always remembered by Hilbert with disgust.

Fig. 1-1 Pregel river with the castle of Konigsberg in the background

In September of 1879, in his last years before entering to the university, Hilbert was transferred to the Wilhelm Gymnasium, a public gymnasium much more focused on mathematics. The period at Wilhelm Gymnasium was for Hilbert much more pleasant than the one at Friedrichskolleg and years later he would remember his stay there with gratitude and affection. After graduating from gymnasium and having passed the Abitur, the admission exam to the university, he entered the Albertina University of Königsberg. This university was then one of the most prestigious in Germany in the teaching of physics and mathematics, mainly due to the work carried out by the physicist Franz Neumann (1798-1895) and the leading authority on mathematical analysis Carl Jacobi (1804-1851). In his first semester at the university, Hilbert studied analytical geometry with Johann Georg Rosenhain (1816-1887) and differential calculus with Louis Saalschütz (1835-1913), two qualified mathematicians who had done their doctorates under the supervision of Jacobi and his

Hilbert’s Early Career

9

disciple Friedrich Richelot (1808-1875). In the second semester, Hilbert decided to go to Heidelberg, where he enrolled in courses with Lazarus Fuchs (1833-1902) on integral calculus and the theory of invariants. After Hilbert returned to Königsberg he remained there the following four years until he finished his studies. Among the professors he had at the University of Königsberg, it is worth mentioning the physicist Gustav Kirchoff (1824-1887) and the geometer Otto Hesse (1811-1874), disciples of Neumann and Jacobi. However, the professor who caused a more profound and lasting influence on Hilbert was Heinrich Weber (18421913), who held the chair of mathematics at the University of Königsberg from 1875 to 1883.

Fig. 1-2 Albertine University of Königsberg

Weber was a first-class mathematician who, at the time Hilbert met him for the first time, had just finished, with Richard Dedekind, one of the most important articles in the history of mathematics entitled “Theorie der algebraischen Funktionen einer Verfinderlichen” (“The theory of algebraic functions of one variable”), which was published in 1882 in the prestigious magazine Crelle’s Journal. Hilbert attended several courses given by Weber dealing with the theory of numbers and the theory of elliptic functions, as well as a seminar directed by him dealing with the theory of invariants. As we will explain later, the theory of algebraic invariants and the algebraic theory of numbers were the first fields to which Hilbert devoted his efforts and managed to make himself a name in the mathematical community.

10

Chapter One

When Weber left Königsberg in 1883 to go to the Technische Hochschule (Technical University) of Charlottenburg, his place was occupied by Ferdinand von Lindemann (1852-1939), a geometer who had studied with Alfred Clebsch (1833-1872) and his disciple, Felix Klein (1849-1925). Lindemann was one of the most famous mathematicians of the moment, as he had proved the transcendence of S, and would become Hilbert’s Doktorvater. Hilbert had shown some interesting results on continuous fractions and asked Lindemann if this topic could become the subject of his doctoral thesis. But Lindemann discovered that Hilbert’s results had been obtained before by Jacobi and proposed to Hilbert another subject, knowing he had previously studied the theory of invariants with Weber, namely the study of the invariant properties of harmonic spherical functions. Hilbert accepted the subject proposed by Lindemann and on December 11, 1884, he successfully defended his doctoral dissertation. Indeed, the theory of invariants would become the main topic of his research over the next eight years even to the point that in a letter to Minkowski he described himself jokingly as “an expert in the theory of invariants.”2 Apart from being his doctoral dissertation advisor, the truth is that Lindemann did not exert a great influence on Hilbert’s mathematical development. Regarding this, much more important were two young and brilliant mathematicians of the same generation as Hilbert: Hermann Minkowski, two years younger than Hilbert but who had enrolled half a year before him at the Albertine University, and Adolf Hurwitz (18591919), who was Lindemann’s associate but only three years older than Hilbert. Minkowski was a child prodigy: at the age of 15 he had already finished the curriculum of the Altstadt Gymnasium in Königsberg and two years later he won the Grand Prix des Sciences Mathématiques, announced by the Académie des Sciences in Paris, which would be awarded to anyone capable of solving the problem of representing a positive integer as the sum of five squares. Minkowski was shy and introverted, but Hilbert was soon interested in him and so they forged in the college an intense friendship that would accompany them throughout his life. On the other hand, Hurwitz had reached Königsberg on Easter in 1884 as a Professor Extraordinarius (ausserordentlicher), a kind of associated professor with a fixed salary. Hilbert repeatedly commented about him that “we, Minkowski and I, were totally astonished by his knowledge and we thought we could never get as far as him.”3 2 3

Hilbert 1965, vol. 3, 392. Ibid., 390.

Hilbert’s Early Career

11

After completing his doctorate, Hilbert stayed one more semester at Königsberg and attended a course given by Lindemann on the linear geometry of Plücker and the spherical geometry of Lie, and some conferences of Hurwitz on modular functions. In May 1885 Hilbert passed the Staatsexamen (state examination), which enabled him to work as a teacher at a gymnasium in case his plans to pursue a career in the university

Fig. 1-3 Hermann Minkowski, 1883

12

Chapter One

did not succeed. After the summer Hilbert went to Leipzig, where he was warmly welcomed by Felix Klein), who led a group of young researchers. Hilbert stayed the winter semester of 1885/86 working with Klein, who advised him to visit Paris to get in touch with French mathematicians in order to extend his mathematical horizons. Hilbert finally went to Paris with another mathematician from Klein’s group in Leipzig, Eduard Study (1862-1930), and there he met Henri Poincaré (1854-1912), Émile Picard (1856-1941), Paul Appell (18551930) and Charles Hermite (1822-1901). But linguistic and other kind of problems (the relations between France and Germany were not going through their best moment) did not allow him to take full advantage of his stay in Paris nor to establish a more fluid contact with Poincaré, who was maybe the most important mathematician of the moment (having succeeded Klein in this honour). Indeed, it was Hermite who, despite his advanced age (64), was more vividly interested in the two young visitors and explained to them his reciprocal theorem for binary invariants, encouraging them to extend it to ternary forms. He also told them about his correspondence with James Joseph Sylvester (1814-1897), who was trying to give a new proof of Gordan’s theorem, the most important theorem of invariant theory at the moment. Despite the difficulties of communication, Klein insisted that Hilbert take advantage of his stay and to make as many contacts as possible with French mathematicians. But Hilbert’s priority at that time was to finish his Habilitationschrift (habilitation thesis), which would enable him to become a Privatdozent in the university. So, he returned to Königsberg and presented in the summer of 1886 his Habilitationschrift, entitled Über invariante Eigenschaften specieller binärer Formen, insbesondere der Kugelfunctionen (About invariant properties of special binary forms, in particular the spherical functions). Hilbert worked as a Privatdozent, a kind of associated professor with no fixed salary and who was paid depending on the number of students he had, at Königsberg University from 1886 to 1892. The following summer Minkowski moved to Bonn where he also began to work as Privatdozent, so Hilbert was left with only the intellectual stimulus of his friend and previous professor Hurwitz. Hilbert himself would remember some years later the almost daily walks with Hurwitz, from 1886 to 1892, in which they discussed any subject of the mathematics at their time under the leadership of Hurwitz. Indeed, during the nine years that he stayed at Königsberg University, Hilbert taught courses on almost all the areas of higher mathematics of his time: theory

Hilbert’s Early Career

13

Fig. 1-4 David Hilbert, 1886

of invariants, theory of numbers, analytic, projective, algebraic and differential geometry, Galois theory, differential equations, theory of functions, potential theory (study of harmonic functions) and even hydrodynamics. On the other hand, his scientific production was mainly focused on the theory of invariants and number theory.



CHAPTER TWO THE THEORY OF ALGEBRAIC INVARIANTS

In the first semester as Privatdozent at the University of Königsberg, Hilbert prepared courses on the theory of invariants, determinants and hydrodynamics, but only the first course could be taught, since in the other two courses the number of students enrolled did not reach the minimum required by the university for the course to be taught.1 However, Hilbert was able to teach these courses in the second semester of the academic year of 1886/87, in which he also began to plan various courses on numerical equations and other subjects. That first year, Hilbert did not make any of the trips he had planned to break the isolation exerted by a place “so far from things” as Königsberg. Finally, in 1888, during Easter holidays, Hilbert made a trip to Leipzig, Göttingen and Berlin. In Leipzig, he met Paul A. Gordan (1837-1912), the “king of invariants” who had been the first to prove the existence of a finite basis for the invariant systems of a binary form. Still in Leipzig, he wrote to Klein in Göttingen: “With the stimulating help of Prof. Gordan an infinite series of thought vibrations has been generated within me, and in particular, so we believe, I have a wonderful short proof for the finiteness of binary systems of forms.”2 After spending a week with Gordan in Leipzig, Hilbert arrived in Göttingen, where he met Felix Klein and Hermann Schwarz (1843-1921). Before leaving Göttingen, during the month of March 1888, Hilbert wrote a new proof of Gordan’s theorem that synthesized and simplified it. From Göttingen, Hilbert went to Berlin where he visited Lazarus Fuchs, Hermann von Helmholtz (1821-1894), Karl Weierstrass (18151897), who had just retired, and Leopold Kronecker (1823-1891). Of all of them, the most important visit in Hilbert’s eyes was the one with Kronecker, with whom he talked about his investigations in the theory of

 1

J. J. Gray explains that “Königsberg attracted very few students in the late 1880s and 1890s, and Hilbert, who certainly became a very good lecturer, often had an audience of only 2 or 3” (Gray 2000, 22). 2 Hilbert to Klein, 21 March 1888, in Frei (1985, 39).



16

Chapter Two

invariants but also about the meaning of the word “to exist” in mathematics and Kronecker’s objections to the use of irrational numbers by Weierstrass. On his return trip to Königsberg, Hilbert still continued to think about Gordan’s problem. In the summer Hilbert went to Rauschen (nowadays Swedlogorsk), a seaside resort city on the Baltic sea, where he had spent the summer with his family ever since he was a child. There, on September 8, he sent a note entitled “Zur Theorie der algebraischen Gebilde” (“On the theory of algebraic figures”) to the journal Göttingen Nachrichten. In this note he solved the problem of proving the finiteness of the invariant systems in a trivial and complete way from a more abstract and less computational perspective. This proof would be refined and extended in several directions in a couple more notes sent to Göttingen Nachrichten the following year. All these results were summarized in the article “Zur Theorie der algebraischen Formen” (“On the theory of algebraic forms”), published in the prestigious journal Mathematische Annalen in 1890. In 1892 Hilbert generalized his proof to any n-ary forms in an article entitled “Über die vollen Invariantensysteme” (“On complete systems of invariants”), published the following year in Mathematische Annalen. The main results of the Annalen papers of 1890 and 1893 were presented some years later in the introductory course Theorie der algebraischen Invarianten (“Theory of algebraic invariants”), given by Hilbert in the summer semester of 1897 in the University of Göttingen. The theory of algebraic invariants was one of the most active research fields in the second half of the 19th century and, as we know, the field to which Hilbert dedicated his doctoral thesis and much of his efforts from 1885 to 1893. We could say, in a very summarized way, that the fundamental objectives of classical invariant theory were: first, the determination of the invariants of the forms (homogeneous polynomials) of any degree and with any number of variables, and second, the proof that there is a finite basis for all invariants of a given forms system (that is, that there is a finite set of independent invariants which generates all the invariants of the system). The origins of invariant theory can be found in the study of transformation of homogeneous polynomials by linear substitutions of the variables in the work of Joseph Louis Lagrange (1736-1813) Mécanique Analytique (1788). Some years later Carl Friedrich Gauss (1736-1813), in his Disquisitiones Arithmeticae (1801), established a new discipline, the theory of forms (binary and ternary), where he studied transformations, similarities, classifications and compositions of forms. In particular, in his study of binary quadratic forms, Gauss defined an equivalence relation



The Theory of Algebraic Invariants

17

between them and showed that the discriminant is an invariant under such relation. Although invariant theory was already hinted at in Gauss’s seminal work, the creation and development of this theory is mainly due to the so-called British school, particularly George Boole (1815-1864), Arthur Cayley (1821-1895) and James Joseph Sylvester. Forty years after the publication of Gauss’s work, Boole wrote two papers: “Exposition of a general theory of linear transformations” (Parts I, II), which are usually regarded as the beginning of Invariant Theory. However, as remarked by P. R. Wolfson, Boole’s principal aim in these papers “had been, first, to determine when two pairs of forms are equivalent, and second, if they are indeed equivalent, to determine those substitutions which take the first pair to the second.”3 The first who clearly stated the main objective of study of Invariant Theory was Arthur Cayley. This was, according to him, “to find all the derivatives of any number of functions, which have the property of preserving their form unaltered after any linear transformations of the variables.”4 So Cayley put in the forefront of the study of linear transformations the problem of finding all invariants of any number of forms, thereby initiating the principal line of research of classical invariant theory. In the decades of the 1840s and 1850s the English school of invariant theory, mainly led by Cayley and Sylvester, developed important tools still in use today to determine the invariants of a given system of binary forms, such as the Cayley ȳ-operator. In the meanwhile, Siegfried Aronhold (1819-1884) in Germany introduced a more flexible notation and a symbolic method that made it easier to calculate invariants and operate at a more abstract level. The notation and method were adopted by Alfred Clebsch and Paul Gordan, who worked extensively in the topic. One of the most important successes of the German school–and indeed of the theory of invariants in the late nineteenth century–was obtained by Gordan, who had published a proof of a finiteness theorem for binary forms in 1868. In particular, Gordan proved that given any system of binary forms of arbitrary degree, the set of simultaneous invariants of the system (possibly an infinite set) is finitely generated, i.e., there is a finite subset of invariants ‫ܫ‬ଵ ǡ ‫ܫ‬ଶ ǡ ǥ ǡ ‫ܫ‬௞ of the invariant set such that each element of this set is a polynomial in ‫ܫ‬ଵ ǡ ‫ܫ‬ଶ ǡ ǥ ǡ ‫ܫ‬௞ . As a result of this proof, which was obtained by cumbersome calculations and using the symbolic method, Gordan was crowned “King of Invariants.”

 3 4



Wolfson 2008, 45. Cayley 1846, 104.

Chapter Two

18

The Theory of algebraic invariants (1897) A homogeneous polynomial is a polynomial in which terms all have the same degree. For example,‫ݔ‬ଵଷ ൅ ʹ‫ݔ‬ଵଶ ‫ݔ‬ଶ ൅ ͹‫ݔ‬ଵ ‫ݔ‬ଶଶ is a homogenous polynomial of degree ͵, since the sum of the exponents in each term is always 3. Homogeneous polynomials are also often called algebraic forms or simply forms. A binary form ݂ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ሻ of degree ݊ in the variables ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ is a homogeneous polynomial of degree ݊ in the variables ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ . In the 1897 lectures, Hilbert only considers binary forms, but says that generalizing to ݊-ary forms poses in general no problems. He writes a ݊-ary binary form ݂ as ݊ ݂ ሺ௡ሻ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ሻ ൌ σ௡௜ୀ଴ ቀ ቁ ܽ௜ ‫ݔ‬ଵ௜ ‫ݔ‬ଶ௡ିଵ ݅ ݊ ݊ ൌ ܽ௜ ‫ݔ‬ଵ௡ ൅ ቀ ቁ ܽଵ ‫ݔ‬ଵ௡ିଵ ‫ݔ‬ଶ ൅ ቀ ቁ ܽଶ ‫ݔ‬ଵ௡ିଶ ‫ݔ‬ଶଶ ൅ ‫ ڮ‬൅ ܽ௡ ‫ݔ‬ଶ௡ Ǥ ͳ ʹ A linear change of variables ൫ߙ௜௝ ൯ is a transformation of the variables ‫ݔ‬ଵ and ‫ݔ‬ଶ given by ‫ݔ‬ଵ ൌ ߙଵଵ ‫ݔ‬ଵᇱ ൅ ߙଵଶ ‫ݔ‬ଶᇱ ‫ݔ‬ଶ ൌ ߙଶଵ ‫ݔ‬ଵᇱ ൅ ߙଶଶ ‫ݔ‬ଶᇱ such that the determinant ߜ of the entries, ߙଵଵ ߙଶଶ െ ߙଵଶ ߙଶଵ , is nonzero. Under a linear change of variables, the binary form ሺ೙ሻ ݂ ሺ௡ሻ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ሻis transformed into another binary form ݂ ᇱ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ሻ in the ᇱ ᇱ new variables ‫ݔ‬ଵ , ‫ݔ‬ଶ defined by ௡

݊ ሺ೙ሻ ݂ ᇱ ሺ‫ݔ‬ଵᇱ ǡ ‫ݔ‬ଶᇱ ሻ ൌ ෍ ቀ ቁ ܽ௜ ሺߙଵଵ ‫ݔ‬ଵᇱ ൅ ߙଵଶ ‫ݔ‬ଶᇱ ሻ௜ ሺߙଶଵ ‫ݔ‬ଵᇱ ൅ ߙଶଶ ‫ݔ‬ଶᇱ ሻ௡ି௜ Ǥ ݅ ௜ୀ଴

After expanding and rearranging this becomes ௡

݊ ሺ೙ሻ ሺ௜ሻ ሺ௡ି௜ሻ ݂ ᇱ ሺ‫ݔ‬ଵᇱ ǡ ‫ݔ‬ଶᇱ ሻ ൌ ෍ ቀ ቁ ܽ௜ᇱ ‫ݔ‬ଵᇱ ‫ݔ‬ଶᇱ Ǥ ݅ ௜ୀ଴

An invariant of the form ݂ is a polynomial function ‫ܫ‬ሺܽ଴ ǡ ܽଵ ǡ ǥ ǡ ܽ௡ ሻ of the coefficients of ݂ which changes only by a factor equal to a power of the transformation determinant ߜ when one makes a linear change of variables, i.e., ሺܽ଴ᇱ ǡ ܽଵᇱ ǡ ǥ ǡ ܽ௡ᇱ ሻ ൌ ߜ ௞ ‫ܫ‬ሺܽ଴ ǡ ܽଵ ǡ ǥ ǡ ܽ௡ ሻ for some ݇ ‫ א‬Գ. For example, let ݂ ൌ ܽ଴ ‫ݔ‬ଵଶ ൅ ʹܽଵ ‫ݔ‬ଵ ‫ݔ‬ଶ ൅ ܽଶ ‫ݔ‬ଶଶ (a binary form of degree 2, i.e., a quadratic form). Then ‫ ܫ‬ൌ ܽ଴ ܽଶ െ ܽଵଶ is మ an invariant of ݂ because ܽ଴ᇱ ܽଶᇱ െ ܽଵᇱ ൌ ߜ ଶ ሺܽ଴ ܽଶ െ ܽଵ ሻ (as Hilbert easily proves). cont. p. 19



The Theory of Algebraic Invariants

19

In lecture 24, Hilbert introduces the idea of simultaneous invariants. Here one begins with a system of algebraic forms, each with the same number of variables but not necessarily of the same degree. An invariant of the system is a polynomial in the set of coefficients of all the forms that changes only by a power of the transformation determinant when the same linear transformation is applied simultaneous to all the forms. In lectures 34-36, Hilbert proceeds to prove his general finiteness theorem, which asserts the existence of a finite basis for any set (possibly infinite) of simultaneous invariants of a system of algebraic forms of arbitrary degree. Hilbert proves this theorem following the lines of the proof he had already given in his 1890 paper, that is, he uses the well-known Hilbert Basis Theorem for polynomial ideals together with the so-called Cayley’s ȳprocess. The first is stated by Hilbert in the following terms: Hilbert Basis Theorem: Given any infinite series of forms with ݊ variables ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ǥ ǡ ‫ݔ‬௡ such as ‫ܨ‬ଵ ǡ ‫ܨ‬ଶ ǡ ‫ܨ‬ଷ ǡ ǥ , there is always a number ݉ such that any form [‫ ]ܨ‬of that series can be represented as ‫ ܨ‬ൌ ‫ܣ‬ଵ ‫ܨ‬ଵ ൅‫ܣ‬ଶ ‫ܨ‬ଶ ൅ ‫ ڮ‬൅ ‫ܣ‬௠ ‫ܨ‬௠ , where ‫ܣ‬ଵ ǡ ‫ܣ‬ଶ ǡ ǥ ǡ ‫ܣ‬௠ are suitable forms of the same ݊ variables. The ȳ-process is a differentiation process that produces an invariant when applied repeatedly to a polynomial. Hilbert’s basis theorem yields a set of invariants ‫ܫ‬ଵ ǡ ‫ܫ‬ଶ ǡ ǥ ǡ ‫ܫ‬௞ such that any ‫ܫ‬௜ ሺͳ ൑ ݅ ൑ ݇ሻ is of the form ‫ܫ‬௜ ൌ ‫ܨ‬ଵ ‫ܫ‬ଵ ൅‫ܨ‬ଶ ‫ܫ‬ଶ ൅ ‫ ڮ‬൅ ‫ܨ‬௞ ‫ܫ‬௞ for some forms ‫ܨ‬ଵ ǡ ‫ܨ‬ଶ ǡ ǥ ǡ ‫ܨ‬௞ . Applying the ȳ-process to each of the ‫ܨ‬௜ yields invariants ‫ܩ‬௜ such that ‫ܫ‬௜ ൌ ‫ܩ‬ଵ ‫ܫ‬ଵ ൅‫ܩ‬ଶ ‫ܫ‬ଶ ൅ ‫ ڮ‬൅ ‫ܩ‬௞ ‫ܫ‬௞ , where each ‫ܩ‬௜ has a degree less than the degree of ‫ܫ‬, since each ‫ܨ‬௜ has a degree of at least one. By expressing each ‫ܩ‬௜ in terms of the set ‫ܫ‬ଵ ǡ ‫ܫ‬ଶ ǡ ǥ ǡ ‫ܫ‬௞ and repeating the process as many times as needed, we can write ‫ ܫ‬as a polynomial in ‫ܫ‬ଵ ǡ ‫ܫ‬ଶ ǡ ǥ ǡ ‫ܫ‬௞ . Two other results in Hilbert’s course are worth mentioning. In Lecture 39, Hilbert introduces the famous Hilbert’s Nullstellensatz (Hilbert’s zeros theorem), an intriguing result on the zero set of families of forms (and by extension polynomials): Hilbert’s Nullstellensatz: Let ݂ଵ ǡ ǥ ǡ ݂௠ be ݉ rational homogeneous functions with ݊ variables ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ . Let ‫ ܨ‬ᇱ ǡ ‫ ܨ‬ᇱᇱ ǡ ‫ ܨ‬ᇱᇱ ǡ ǥ be any rational function with the property that they cont. p. 20



20

Chapter Two

vanish for all those systems of values of these variables for which the given ݉ functions ݂ଵ ǡ ǥ ǡ ݂௠ are all equal to zero. Then it is always possible to find an integer number ‫ ݎ‬such that every product ςሺ௥ሻ of any ‫ ݎ‬functions of the series ‫ ܨ‬ᇱ ǡ ‫ ܨ‬ᇱᇱ ǡ ‫ ܨ‬ᇱᇱ ǡ ǥ can be expressed in the form ςሺ௥ሻ ൌ ܽଵ ݂ଵ ൅ܽଶ ݂ଶ ൅ ‫ ڮ‬൅ ܽ௠ ݂௠ , where ܽଵ ǡ ǥ ǡ ܽ௠ are suitable selected rational homogeneous functions with the variables ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ . In Lecture 47 Hilbert states a third noteworthy result known today as Hilbert’s Syzygy Theorem. The set of invariants ‫ܫ‬ଵ ǡ ‫ܫ‬ଶ ǡ ǥ ǡ ‫ܫ‬௞ is not usually independent, i.e., there will be a set of relations between them. This set of relations forms an ideal and so must have a finite basis ܴଵ ǡ ܴଶ ǡ ǥ ǡ ܴ௝ by the finiteness theorem. Now, there may be also relations between these basic relations, i.e., expressions of the form ܵଵ ܴଵ ൅ܵଶ ܴଶ ൅ ‫ ڮ‬൅ ܵ௝ ܴ௝ ൌ Ͳ. These are the first-order syzygies. They again form an ideal to which the finiteness theorem applies so that they must have finite basis ܵଵ ǡ ܵଶ ǡ ǥ ǡ ܵ௟ . The relations between these relations are the second-order syzygies. It may seem that this process can be iterated ad infinitum, but Hilbert’s Syzygy Theorem asserts that the chain of syzygies terminates after finitely many steps. In Hilbert’s words: Hilbert’s Syzygy Theorem: The system of irreducible syzygies of first order, second order, etc. forms a chain of derived equation systems. This chain of syzygies breaks off in the end and there are no syzygies of an order higher than ݉ ൅ ͳ, if ݉ is the number of the invariants of the whole system.

Although the validity of Gordan’s result had already been proved for particular cases of higher-degree forms, nobody had been able to generalize Gordan’s result beyond binary forms. Twenty years after the publication of Gordan’s proof the problem was still open and had become the fundamental problem of the theory of invariants. Hilbert had discussed this problem with Hermite and Study in Paris and surely also with Klein and the circle of young mathematicians around him during his visit to Leipzig. Hilbert had been introduced to Gordan by Felix Klein and spent a week with him in Leipzig in the spring of 1888. In the following years, he wrote a series of articles which gave invariant theory a more abstract orientation.



The Theory of Algebraic Invariants

21

More specifically, in 1890 Hilbert introduced the idea of what would later be called a Noetherian polynomial algebra to prove Gordan’s theorem again and in 1893 he used the same idea to generalize Gordan’s theorem to ݊-ary forms, that is, to prove the finitude of any system of invariants of forms of any degree. Hilbert proved that there must be a finite basis by induction over the degree (number of variables) of the forms considered. The problem of Hilbert’s proof was that the argument used to demonstrate the existence of a finite basis was not constructive; it was rather an indirect proof (through reduction ad absurdum). That is to say, Hilbert did not give an effective procedure to calculate, for each system of forms of degree ݊, the number of the invariants of the basis, but showed that the negation of the statement asserting the existence of a basis for any system of forms and for all ݊ leads to a contradiction. This led Gordan, after having seen Hilbert’s proof, to affirm that “this is not mathematics, it is theology.” Later, Gordan modified his opinion, but Hilbert’s proof would open another front of controversy, this time with Leopold Kronecker, about what it means “to exist” in mathematics. In the article of 1893, read on his behalf at the International Mathematical Congress held in Chicago that year, Hilbert also reflected on the state of development of the theory of invariants and his contributions to it. According to him, in the development of mathematical theories one can distinguish three stages, which he characterizes as the naïve, formal and critic stages. In the case of the theory of invariants, Hilbert saw the work of Cayley and Sylvester as exponents of the naive stage, the work of Clebsch and Gordan as representative of the formal stage and, finally, his own contributions as the only exponent of the critical stage.5 In fact, in the last lines of the above-mentioned article, Hilbert expressed his conviction that “he had achieved the most important general objectives of a theory of functional fields of invariants.”6 More or less at the same time, Hilbert told Minkowski that “with the Annalen article I definitively leave the field of invariants and I now turn to the theory of numbers.”7 The well-known algebraist and historian of mathematics, Bartel Leendert van der Waerden (1903-1996), wrote in a review of Hilbert’s contributions to algebra that, although the articles of 1888 and 1893 constituted the conclusion of Hilbert’s research on the theory of invariants,

 5

We shall see later that Hilbert sees the axiomatic analysis of geometry and of any other branch of mathematics as the representative of the last stage of the development of the theory, that is, the critical stage. 6 Hilbert 1965, vol. 2, 344. 7 Ibid., vol. 3, 395.



Chapter Two

22

they would have “an overwhelming and revolutionary influence on algebraic thinking in the ensuing years.”8 According to van der Waerden “when Hilbert, in these articles, considers the fields of invariants as special cases of the fields of functions, he is at the edge of an historical development: before him the interest of the algebraists was mainly directed towards the possibility of explicitly representing all invariants in a basic given form; after him, the algebraists looked more towards the arithmetic and algebraic properties of the systems of rational and algebraic functions. From inside this circle of ideas, the general theories of abstract fields, rings and modules arose naturally.”9 As a matter of fact, in the years following the publication of Hilbert’s articles the theory of invariants would transform into something completely different from the theory established before their publication. The first step in that direction was given by Emmy Noether (1882-1935), a collaborator of Hilbert’s and van der Waerden’s teacher in Göttingen, when she generalized Hilbert’s finiteness theorem so that it was not connected to the theory of invariants but was a result of the modern algebra. To see this, note first that in modern terminology (as we have seen Hilbert doesn’t use the words ideal and ring and he speaks of forms instead of polynomials), Hilbert’s theorem can be stated and generalized as follows: Let ‫ܭ‬ሾ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ ሿ be the ring of polynomials with ݊ variables and coefficients belonging to any field. Then Hilbert basis Theorem: Every ideal in the polynomial ring ‫ܭ‬ሾ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ ሿ has a finite basis, i.e., every ideal is generated by a finite number of elements of the ring.

A ring in which every ideal is finitely generated (or equivalently, every ascending chain of ideals terminates) is called nowadays a Noetherian ring. Hence Hilbert’s theorem is usually stated in modern algebra textbooks in the following form: Hilbert basis Theorem: If ‫ ܭ‬is a Noetherian ring, then every polynomial ring ‫ܭ‬ሾ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ ሿ is also a Noetherian ring.

Thus, Noether’s work can be seen as a decisive step in the transformation of the theory of invariants and its consideration as a part within modern abstract algebra. The above is just an example of the generalizations and reinterpretations of Hilbert’s most significant theorems in invariant theory

 8 9



Ibid., vol. 2, 401. Ibid.

The Theory of Algebraic Invariants

23

by modern abstract algebra. The other two main theorems presented by Hilbert in the 1890 and 1893 paper can also be dealt in a similar way. Regarding the Nullstellensatz we have the following generalization: Let ‫ ܣ‬be an ideal of the polynomial ring ‫ܭ‬ሾ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ ሿ, where ‫ ܭ‬is an algebraic closed field. We can think of ‫ ܣ‬as generated by a finite number of polynomials ݂ଵ ǡ ݂ଶ ǡ ǥ ǡ ݂௡ . The zeros of ‫( ܣ‬literally Nullstellen means zero places) are the points ܽଵ ǡ ǥ ǡ ܽ௡ of the ݊-dimensional space ‫ ܭ‬௡ for which the identity ݂௜ ሺܽଵ ǡ ǥ ǡ ܽ௡ ሻ ൌ Ͳ, ሺͳ ൑ ݅ ൑ ‫ݎ‬ሻ, holds true. (In Geometry, the set of zeros of ‫ ܣ‬is called an algebraic variety. Hilbert called it an algebraischer Gebilde (algebraic configuration). We have then Hilbert’s Nullstellensatz: If the polynomial ݃ vanishes at all zeros of the ideal ‫ܣ‬, then there is a power ݃௠ belonging to ‫ ܣ‬such that ݃௠ ൌ σଵஸ௜ஸ௥ ‫ݑ‬௜ ݂௜ , ሺ‫ݑ‬௜ ‫ܭ א‬ሾ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ ሿሻ.

If ܴ is a commutative ring and ‫ ܣ‬and ideal of ܴ, then a Nullstelle of ‫ ܣ‬is a maximal ideal of ܴ which contains ‫ܣ‬. When an element ݃ ‫ ܴ א‬belongs to all these maximal ideals, then it is said that “݃ vanishes at all the zeros of ‫”ܣ‬. A ring in which Hilbert’s Nullstellensatz in this abstract sense is valid is called a Hilbert’s ring. We also have then the following generalization: Hilbert’s Nullstellensatz: If ‫ ܭ‬is a Hilbert’s ring, then every polynomial ring ‫ܭ‬ሾ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ ሿ is also a Hilbert’s ring.

As an immediate consequence of the above generalizations we have that every polynomial ring over a field is a Noetherian and Hilbertian (also called Jacobian) ring. Hilbert’s Nullstellensatz is nowadays usually regarded as a foundational theorem of algebraic geometry, since it yields a correspondence between geometric objects (varieties) and algebraic objects (prime ideals of polynomial rings). Finally, regarding the Syzygy Theorem we also have an intriguing generalization and reinterpretation in terms of modern abstract algebra. Let ‫ ܯ‬be a finitely generated ܴ-module (ܴ a commutative ring) and ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ its sets of generators. A syzygy of ‫ ܯ‬is an element ሺܽଵ ǡ ǥ ǡ ܽ௡ ሻ ‫ܴ א‬௡ for which ܽଵ ‫ݔ‬ଵ ൅ܽଶ ‫ݔ‬ଶ ൅ ‫ ڮ‬൅ ܽ௡ ‫ݔ‬௡ ൌ Ͳ. The set of all syzygies ܵሺ‫ܯ‬ሻ of the module ‫ ܯ‬is a submodule of ܴ௡ called the module of syzygies. Inductively we define the ݅-th module of syzygies by ܵ଴ ሺ‫ܯ‬ሻ ൌ ‫ ܯ‬and ܵ௜ ሺ‫ܯ‬ሻ ൌ ܵ൫ܵ௜ିଵ ሺ‫ܯ‬ሻ൯. In modern terms, Hilbert’s Syzygy theorem then asserts that every finitely generated graded module ܵሺ‫ܯ‬ሻ ൌ ۩ஶ ௜ୀ଴ ܵ௜ ሺ‫ܯ‬ሻ over the polynomial ring ԧሾ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ ሿ has a free resolution of length at most ݊, that is, its ݊௧௛ syzygy is free. Now, it turns out that this is valid for every finitely generated module (not necessarily graded) over a



Chapter Two

24

polynomial ring. Thus, we usually find the following generalization of Hilbert’s theorem in modern algebra textbooks: Hilbert’s syzygy theorem: Let ܴ ൌ ‫ܭ‬ሾ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ ሿ be a polynomial ring. Every finitely generated ܴ-module ‫ ܯ‬has a finite free resolution of length at most ݊.

Hilbert’s theorem is nowadays considered as an early result in homological algebra and the point of departure of homological methods in commutative algebra and algebraic geometry. The next step and, to a certain extent the definitive one, in the entry of invariant theory in modern abstract algebra was the work of a collaborator and disciple of Hilbert, Hermann Weyl (1885-1955). In his book The Classical Groups. Their Invariants and Representations, published in 1939 and widely read among mathematicians around the world, Weyl observed that one of the main aims of the book was “to give a modern introduction to the theory of invariants. It is high time for a rejuvenation of the classic theory of invariants, which has fallen into on almost petrified state.”10 In Weyl’s book, invariant theory is developed for all classical Lie groups. More specifically, one of the main themes of Weyl’s book is the study of the polynomial invariants for a standard classical group action, so that classical invariant theory is viewed as a part of the theory of linear representations of groups. In any case, after 1893 Hilbert left his investigation in the theory of invariants and concentrated his efforts in algebraic number theory. The reason for this was not only that Hilbert considered that his work in the theory of invariants had answered all the great open questions of this discipline, but also that he and his friend Minkowski had been commissioned by the Deutsche Mathematiker-Vereinigung (German Mathematicians Association) (DMV), which had been founded at the annual meeting of the Gesellschaft deutscher Naturforscher und Ärzte (Society of German Scientists and Doctors) in Bremen in 1890, to develop a study on the state of the art of number theory. 

 10



Weyl 1939, vii.

The Theory of Algebraic Invariants

Fig. 2-1 The founders of the DMV in Bremen in 1890



25





CHAPTER THREE THE KÖNIGSBERG LECTURES ON GEOMETRY

In 1892 Hurwitz, who had been working as an Extraordinarius at Königsberg for eight years, was appointed as the successor of Ferdinand Georg Frobenius (1849-1917) at the Eidgenössische Polytechnische Schule (Federal Polytechnical School) in Zürich.1 Hurwitz’s departure opened the possibility for Hilbert to take over his post, as finally happened in August of that year when the faculty unanimously voted him to succeed Hurwitz. Almost at the same time, Minkowski received an offer to work as an associated professor in Bonn, which he accepted at the request of Althoff. A few months earlier, Hilbert had become engaged to Käthe Jerosch, whom he would marry on October 12 of 1892. After ten months, on August 11, 1893, his first and only son, Franz Hilbert, was born. Although the best-known contributions in the three following years of Hilbert’s academic career correspond to the field of algebraic number theory, during his stay at Königsberg Hilbert devoted several courses to the study of geometry, in which projective geometry occupied a central place. This was undoubtedly one of the mainstream topics at that time, mainly due to the conception of the geometry exposed by Felix Klein in his famous Erlanger Programm (Erlangen Program) (1872) which placed projective geometry at the top of the different geometries known those days. Even though the origins of projective geometry can be traced back to ancient Greece and had an important impulse in the 17th century, especially thanks to the work of Girard Desargues (1591-1661), it was not until the 19th century that projective geometry received the definitive impulse and became a fundamental branch of geometry and, by extension, of mathematics. The work with which the golden age of projective geometry began was Geometrie Descriptive of Gaspard Monge (17461818). This work inspired the research on this topic of several of his students, particularly, Jean-Victor Poncelet (1788-1867), Lazare Carnot

 1

Renamed Eidgenössische Technische Hochschule (The Swiss Federal Institute of Technology) in 1911.



28

Chapter Three

Fig. 3-1 Adolf Hurwitz as an Extraordinarius in Königsberg



The Königsberg Lectures on Geometry

29

(1753-1823), Michel Chasles (1793-1880) and Charles Julien Brianchon (1785-1864). These French mathematicians obtained their geometric results without the use of coordinate systems or sets of equations, that is, by synthetic methods akin to those that were usual in Euclidean geometry until 17th century.2 In this way, the development of projective geometry went hand in hand with a revival of synthetic geometry. The most prominent figure in the development of projective geometry was undoubtedly Poncelet. The basic idea of Poncelet’s research program, as developed in the first part of his work Traité des propriétés projectives des figures (1822), was to renew the synthetic approach to geometry of the Ancients which tends, according to him, to study only problems attached to a single figure. On the contrary, the consideration of “indeterminate magnitudes” (instead of “absolute magnitudes”) and the use of algebraic formulas in analytical geometry makes it possible to consider at once a whole set of figures and to reason about this set by considering only one figure. Poncelet’s idea was then to endorse geometry with a method that allows a similar degree of generality, that is, a method by which, while applying the traditional methods of the Ancients to a particular case, would extend the results to all other analogous cases. To this end, Poncelet introduced the so-called principle of continuity or “permanence of the mathematical relations of the magnitudes considered,”3 which ensured, for a set of correlative figures,4 the validity of a property shown in a specific figure, beyond the changes in configuration, passages to the infinite or the loss of reality of certain elements. So, for example, what could be asserted of a figure that contains two intersecting lines must remain valid if the lines become parallel–and hence the need of points at infinity. And the same could be said of a figure in which a straight line and a curve meet or not in the real plane–and hence the need of imaginary points. In any case, the extension of the Euclidean space to a series of new points and straight lines presented several problems, the first of them being to justify the introduction of these extension elements. As we have seen, according to Poncelet, it was necessary to postulate them in accordance

 2

Analytic geometric emerged in the 17th century thanks to the contributions of René Descartes (1596-1650) and Pierre de Fermat (1601-1655). Throughout 18th century, the success of analytical geometry and differential geometry–which combined Cartesian geometry with differential and integral calculus–had pushed classical Euclidean methods to an aside. 3 Poncelet 1864, 319. 4 Two figures are correlatives if one is obtained from the other by modifying through a “progressive and continuous movement” the relative position of its component elements.



Chapter Three

30

with the principle of continuity in order to develop geometry from a unified and synthetic point of view. However, this principle suffers from a remarkable lack of rigor–as Augustin-Louis Cauchy (1789-1857), a contemporary of Poncelet, had already noted–and turns projective geometry into a kind of creative reorganization of Euclidean geometry. This led Karl von Staudt (1798-1867) to try to reformulate projective geometry and to present it more rigorously. In the preface to his work Geometrie der Lage (Geometry of position) (1847), von Staudt exposed clearly his intention of a purely synthetic development of geometry, without any appeal to metric considerations: I have tried in this work to obtain the geometry of position as an autonomous science, for which [the concept of] measure is not required.5

Thus, for example, he defined the notion of a projective mapping without appealing to the notion of measure of a segment or an angle by the property of preserving harmonicity–which is defined by incidence relations only. This approach led him to define the elements of extension on the basis of already known concepts and relations of Euclidean geometry, so that the projective space became a mere extension by definition of the Euclidean space. In a nutshell, the strategy of von Staudt consisted, instead of postulating new elements that explained the continuity of certain properties and geometrical relations à la Poncelet, of choosing these same forms or, rather, the abstract concepts derived from them, like the extension elements. Thus, for example, instead of postulating the existence of new points in infinity wherever parallel lines meet, von Staudt defined points at infinity as the concept-object that can be extracted from the form common to all parallel lines to a line L, namely the concept-object “the direction of L.” Similarly, instead of postulating the existence of new imaginary points corresponding to the intersection of a curve with a straight line when this intersection is given by an imaginary coordinate, he defined the imaginary points on the straight line as “involutions with one direction.” Despite the great advances that we can find in von Staudt’s work in the search of rigor, the first mathematician who provided a rigorous foundation for projective geometry was Moritz Pasch (1843-1930), who presented, in the work Vorlesungen über die neueren Geometrie (Lessons on modern geometry) (1882), the first serious example of axiomatization of a branch of mathematics since Euclid (flourished c. 300 BCE). Pasch’s Vorlesungen dealt with projective geometry, but what interests us here is

 5



Von Staudt 1847, III.

The Königsberg Lectures on Geometry

Projective geometry Plane Euclidean geometry studies the properties and the relationships between points and straight lines in the plane. These properties (length of the lines, size of angles, incidence, congruence, parallelism, etc.) are preserved when the transformations considered by Euclidean geometry are applied: the translations and rotations. But, apart from these, there are other types of transformations such as projections. These are the kind of transformations that, for example, we see in action when a painter draws a picture in perspective or when a light beam projects the shadow of an object on the wall or the floor of a room. Which are the properties of the figures that are preserved under projective transformations? Or, in other words, what are the projective properties? Consider, for example, two planes ī and H, and a point P out of them. Let ĭ be any figure in ī and draw straight lines from P crossing each point of ĭ. The figure formed by these points is the projection of ĭ in H from P.

Fig. 3-2 Projection of a figure in a plane As we can see in the figure above, the projection is usually different from the projected figure regarding its size and shape. Indeed, although some basic elements retain their properties after the projection (for example, the projection of a line is another line, and the point of intersection of two lines is projected at another point that is the intersection of the projections of the two original lines), measurements such as the length of the lines and the size of the angles, as well as the cont. p. 32



31

32

Chapter Three

forms of the figures, are not invariant. In addition, the concept of parallelism does not appear in projective geometry: two different lines in the same plane meet at one point, and if these lines are parallel in the sense of Euclidean geometry (because of being in different planes, as shown in the above figure), then their point of intersection is in the infinite. The plane that includes the line at the infinite containing all the ideal points (points in the infinite) where the parallel lines intersect is called the projective plane. Since the properties that are preserved under projective transformations are those of incidence–properties that are invariant under stretching, translation or rotation of the plane௅, the notions of distance and angle play no role in projective geometry. Hence in order to extend the Euclidean plane to the projective plane it suffices to begin with the affine plane and to complete it by adding certain “points at infinity”. Let A be an affine plane. For each line l in A, we will denote by [l] the pencil of lines parallel to l, and we will call [l] an ideal point, or point at infinity, in the direction of l. We write P* = [l]. We define the completion S of A as follows. The points of S are the points of A, plus all the ideal points of A. A line in S is either a) an ordinary line l of A, plus the ideal point P* = [l] of l, or b) the “line at infinity”, consisting of all the ideal points of A. It is then easy to prove that S is a projective plane in the sense of the following: Definition. A projective plane S is a set, whose elements are called points, and a set of subsets, called lines, satisfying the following four axioms. P1. Two distinct points P, Q of S are incident to one and only one line. P2. Any two lines meet in at least one point. P3. There exist three points not incident to the same line. P4. Every line is incident to at least three points. From these axioms the “duals” of P1, P3 and P4, which are obtained from the above statements by interchanging the words “point” and “line”, can be demonstrated. Now, from P1, P2, P3 and their dual statements the following can be proved: Theorem (Principle of duality): In any projective plane S consider an arbitrary statement, expressed in terms of points, cont. p. 33



The Königsberg Lectures on Geometry

lines and incidence, and which has been proven from the axioms. Then the dual statement, obtained when interchanging the words “point” and “line,” is a theorem as well. One of the more intriguing theorems of projective geometry is Desargues’ Theorem:

Fig. 3-3 Desargues’ theorem It was precisely the theorem of Desargues that motivated the development of projective geometry by Jean-Victor Poncelet. It asserts that if two triangles ‫ ܥܤܣ‬and ‫ܣ‬ᇱ ‫ܤ‬ᇱ ‫ ܥ‬ᇱ are in perspective from a point ܱ and the lines ‫ ܤܣ‬and ‫ܣ‬ᇱ ‫ܤ‬ᇱ meet at ܴ, the lines ‫ ܥܣ‬and ‫ܣ‬ᇱ ‫ ܥ‬ᇱ meet at ܵ, and the lines ‫ ܥܤ‬and ‫ܤ‬ᇱ ‫ ܥ‬ᇱ meet at ܶ, then the points ܴǡ ܵƒ†ܶ are incident to the same line. This is true in three-dimensional Euclidean space provided that no two of these lines are parallel. If this last case occurs, there will be only two points of intersection instead of three, and the theorem should be modified to include the result that these two points will be on a line parallel to the two lines. Instead of modifying the theorem to cover this special case, Poncelet modified the Euclidean space by postulating points at infinity. In this new projective space, each straight line has an added point at infinity and all parallel lines have an intersection point.



33

34

Chapter Three

his conception of the axiomatic method. For Pasch geometry is a natural science and so its purpose is to deduce from basic concepts (Grundbegriffe) and basic propositions (Grundsätze) obtained from ordinary spatial experience the laws of more complex phenomena. Once the basic concepts and propositions have been posed, the definitions of the derived concepts must be reduced to the basic concepts and the theorems should be exclusively deduced from the fundamental propositions and previously deduced theorems. However, although fundamental concepts and propositions should be derived from experience, deductiveness requires that meaning is left apart: Whenever geometry has to be really deductive, the process of inferring must be independent of the meaning of the geometrical concepts as well as of the figures. The only things that matter are the relations between geometrical concepts, as they are established in the theorems and definitions used.6

Thus, Pasch’s approach combined a formal stance regarding the validity of mathematical deductions (deductivist methodology) with a commitment to the empirical origin of the basic concepts of geometry (empiricist epistemology). As we shall later see this combination of formalism and empiricism will also be an important feature of Hilbert’s early approach to geometry. In any case, the decisive step in the consecration of projective geometry was the publication of Felix Klein’s Erlanger Programm. In this address to the University of Erlangen, Klein unified most of the existing geometries, including non-Euclidean geometry, by showing that they were special cases of projective geometry. According to Klein’s conception, geometry could be considered as the study of those properties of figures that remain invariant under the action of a specific group of transformations. Therefore, to classify the geometries it is only necessary to classify the groups of transformations. The larger the group of transformations (the fewer restrictions imposed on them), the more fundamental is geometry. Thus, from the point of view of Klein, Euclidean geometry was only a special case of affine geometry and the same occurred with affine geometry with regards to projective geometry. In short, according to Klein, projective geometry possessed conceptual and intuitive primacy over the other geometries since the projective transformations were the most general.

 6



Pasch 1882, 98; emphasis in original.

The Königsberg Lectures on Geometry

35

Projektive Geometrie (Projective Geometry) was precisely the title of Hilbert’s first course on geometry, taught during the summer semester of 1891. Hilbert’s lecture course was quite conventional and the presentation that he made in it of projective geometry was very similar to that one found in the work of Theodor Reye (1838-1919) Die Geometrie der Lage (1886), which was based on the homonymous work by von Staudt. Indeed, Hilbert’s lectures followed the point of view of von Staudt: projective geometry is developed in a synthetic way, that is, without following, as far as possible, metric considerations. Although the structure of Hilbert’s lectures was very similar to that of Reye’s book, Hilbert’s presentation of projective geometry was much more formal and rigorous than Reye’s. Hilbert, for example, introduced only the concepts, definitions and theorems strictly necessary for the development of projective geometry. And, in general, the deductive relationships between the different theorems were clearly exposed and discussed. A clear example of this is the proof that Hilbert offered of the Desargues’s theorem, much clearer and briefer than Reye’s. In the Introduction (Einleitung) to the lectures of 1891 we find one of the first references of Hilbert to the axiomatic method. According to Hilbert, there are three main branches or ways of approaching elemental geometry: intuitive geometry, which is based on the “simple facts of intuition;” axiomatic geometry, which investigates the axioms of geometry underlying the intuition and the axiomatic systems obtained when one or more axioms are left out; and analytical geometry, which reduces geometry to analysis in virtue of the correspondence between the points of a straight line and the real numbers. According to Hilbert, the value of intuitive geometry is merely aesthetic or pedagogical, while axiomatic geometry is especially important from the epistemological point of view. Finally, Hilbert argues, analytical geometry is the most important from the mathematical and scientific perspective. In any case, although Hilbert lectures of 1891 dealt with the first branch, all its subsequent courses would be inscribed in the second. Another point of interest in the Introduction of the 1891 course was its characterization of geometry as a natural science in which, unlike other branches of pure mathematics, the recourse to sensorial intuition (Anschauung) is essential. “Geometry, Hilbert says, is the theory of the properties of space,” which “is not a product of my reflection. Rather, it´s given to me through the senses. I need my senses to understand its properties.”7

 7



Hilbert 2004, 22.

36

Chapter Three

In September 1891, Hilbert attended a lecture given by Hermann Wiener (1857-1939) at a DMV meeting in Halle. The conference dealt with the foundations of geometry and in it Wiener defended the axiomatic approach to arithmetic and geometry. In particular, Wiener raised the possibility of an axiomatic development of projective geometry based on Desargues’s and Pascal’s theorems. According to Hilbert’s first doctoral student, Otto Blumenthal (1876-1944), the conference had a great impact on Hilbert. Actually, it seems to have been precisely after this conference that Hilbert made his famous statement that it should be possible to replace “point, line and plane” for “chair, table and beer mug” without affecting the validity of the theorems that are proved. As we shall see later, this statement, according to which the concepts of an axiomatic theory must not denote specific objects (and therefore do not need to have an intuitive content), would become a central aspect of Hilbert’s formalism in the following years. In April 1893, Hilbert wrote to Minkowski: “I am now getting familiarized with non-Euclidean geometry, because I want to give a course about it the next semester.”8 This would happen during the summer semester of 1893, but the lack of audience did not allow Hilbert to give the course announced and he had to postpone it until the summer semester of 1894. The lectures for this course were very adequately entitled Grundlagen der Geometrie (Foundations of Geometry), since this was the first course in which Hilbert seriously dealt with the foundations of some branch of mathematics and, specifically, geometry. The aim of the course was to produce a system of axioms as pure and accurate as possible of non-Euclidean and Euclidean geometry. Hilbert had described the axiomatic approach of geometry in the lectures corresponding to the 1891 course on projective geometry as the investigation of the “axioms underlying the facts presented by the geometry of intuition” and “the systematic investigation of the geometries that arise when one or more of these axioms is set aside.”9 Although in the lectures of 1894 Hilbert was not interested in this second level of research characteristic of the axiomatic method, he dealt with the first level of investigation mentioned above. Hence this lecture course not only represented an axiomatic presentation of geometry but was the first axiomatic analysis that Hilbert made of this science. This means that Hilbert was not only worried about axiomatizing Euclidean and nonEuclidean geometry, for which he closely follows the work of Pasch in

 8 9



Cited from Toepell (2000, 214). Hilbert 2004, 22.

The Königsberg Lectures on Geometry

37

1882 on this subject, but also worried about investigating these axioms to a certain extent. He analysed, for example, different points of view of the theory of parallelism and the independence of the Parallels Axiom, exhibiting a spherical model for hyperbolic geometry in which the other axioms are fulfilled, but in which “through a point there is an infinite number of straight lines parallel to a given straight line.”10 The lectures of 1894 opened again with the characterization of geometry as a natural science, but with a notable difference with respect to the lectures of 1891. Geometry is no longer defined now as “the theory of the properties of space,” otherwise the physical properties of space would be part of geometry, but rather as the science that deals with the “facts that determine the external form of things”, which he calls “geometric facts.” The task of geometry is then, like any other natural science, to arrange and describe the facts belonging to its field of study “by means of certain concepts, which are linked to each other by the laws of logic.”11 However, Hilbert considered that there was a significant difference between the degree of development of geometry and other natural sciences. In geometry, unlike other natural sciences, such as optics, the theory of electricity or mechanics, all (geometric) facts can be derived from the axioms, while in the rest of natural sciences they are still discovering new facts. In this sense, Hilbert says, geometry is the “most perfect, most complete” of natural sciences. Up to this point Hilbert remained at the same level as his colleague Pasch, who had also claimed that his axioms were derived from experience. But Hilbert will go further than Pasch and will now ask if the axioms are independent and complete: The problem of our colleague [Pasch] is this: what are the necessary and sufficient conditions, independent of each other, that one must put to a system of things, so that each property of these things corresponds to a geometric fact and vice versa, so that by means of this system of things a complete description and arrangement of all the geometrical facts is possible?12

Thus, the lectures of 1894 already contained the essential features of Hilbert’s point of view on how axiomatic analysis should proceed, namely: providing a scheme or network of concepts obtained by

 10

Ibid., 120. Ibid., 72. This Hilbertian characterization of geometry as a natural science was very influenced by the Heinrich Hertz (1857-1894) characterization of natural sciences from his Bildtheorie. 12 Ibid., 72-73. 11



38

Chapter Three

abstraction from the geometrical facts, corresponding to the axioms to the basic geometric facts. The problem was then to find a set of axioms that were mutually independent and complete, i.e., that were not redundant and that allowed the derivation of all the theorems (corresponding to geometrical facts) in the scheme of concepts (axiomatic system).





CHAPTER FOUR THE ALGEBRAIC THEORY OF NUMBERS

The same year, 1892, in which Hilbert had been named Extraordinarius professor at Königsberg, Lindemann received an offer from Munich. This meant that the position that he had occupied at Konigsberg for ten years remained vacant and opened the possibility for Hilbert to become his successor. The Faculty of Philosophy, to which the mathematics professors were ascribed, selected Hilbert and three other mathematicians as official candidates for the position and sent the list to Berlin. The following year, at the age of thirty-one, Hilbert became Ordinarius (ordentlicher) professor at Königsberg University. Friedrich Althoff, who had been the representative for the University of the Royal Prussian Ministry of Ecclesiastical, Educational and Medical Affairs since 1882 and would become, in 1897, the ministerial director of Universities and Higher Education, not only appointed Hilbert as Ordinarius under very favourable conditions, but also asked for advice about who could become his successor as Extraordinarius professor. After some complicated personal arrangements, Hilbert succeed in bringing his friend Minkowski, who was then a Privatdozent in Bonn, to Königsberg for Easter 1894. In this way, the two friends could work together again and resume their daily walks for one year, since at Easter 1895 Hilbert moved to Göttingen. During his years as an Extraordinarius and Ordinarius professor in Königsberg and his first years in Göttingen, Hilbert chiefly devoted his work to number theory. The first important result in this direction was the unification and simplification of the proofs by Hermite and Lindemann of the transcendence of ʌ and e respectively. This was during the winter of 1892 to 1893; after this Hilbert devoted all his efforts to the theory of number fields. The first contribution in this direction was the conference “Zwei neue Beweise für die Zerlegbarkeit der Zahlen eines Körpers in Primideale” (“Two new proofs of the divisibility of numbers of a field in a prime ideal”) held in September 1893 in the Congress of the DMV at Munich.



40

Chapter Four

It was precisely on occasion of this meeting that Hilbert and Minkowski were asked to write a Referat or report on the current development of number theory. It is worth noting the fact that Hilbert was requested to produce a report on the state of the art of a branch of mathematics as important as number theory, for this clearly shows that Hilbert was already considered an authority in this field, even though he had just begun publishing articles about it. The plan of the two friends and colleagues was to divide up the work, leaving to Minkowski the elementary parts of number theory like continued fractions, quadratic forms, and the geometry of numbers. Both started working on the report in 1894. Minkowski, however, abandoned the project very soon, because he was too busy writing his Geometrie der Zahlen (The Geometry of Numbers), which he had begun in 1890 and was partially published in 1896, so the report became exclusively the responsibility of Hilbert. Although in the end only Hilbert’s part was actually written, Minkowski did comment on Hilbert’s manuscript and read the galley proofs. As noted by J. J. Gray, number theory was a German subject: “From Gauss to Kummer and Kronecker, from Gauss to Dirichlet and then Dedekind, and thence to Hilbert and Minkowski, algebraic number theory had been largely German, largely indeed Prussian.”1 The point of departure of this German tradition was undoubtedly Gauss’ masterpiece Disquisitiones Arithmeticae (1801). In this work, Gauss raised different topics such as congruence, quadratic reciprocity and its generalization to higher powers, quadratic forms and cyclotomy, topics which would become the starting point for the research of authors of the calibre of Kronecker, Weber, Peter Gustav Lejeune Dirichlet (1805-1859), Ferdinand Gotthold Eisenstein (1823-1852), Ernst Kummer (1810-1893), Hilbert or Minkowski. As Felix Klein pointed out in his classic work Vorlesungen Über die Entwicklung der Mathematik im 19. Jahrhundert (Lectures on the development of mathematics in the nineteenth century), “in the Disquisitiones Arithmeticae, Gauss created modern number theory itself and fixed all its future development.”2 All the previous authors were certainly inspired by the issues raised by Gauss’ Disquisitiones Arithmeticae, although they reconsidered them in very different ways and extended them in often divergent directions. Indeed, as Gauss himself had predicted, there were numerous internal connections between these subjects and between them and other branches

 1 2



Gray 2000, 38. Klein 1926, 26.

The Algebraic Theory of Numbers

41

of mathematics, which were studied by the mathematicians mentioned above, often from different and even opposed points of view (this would be the case, for example, of Kronecker and Dedekind). One result of this was that the definition of some key concepts of algebraic number theory, such as class field (KlassenKörper) or ring (Ring), depended too much on the focus or interests of each author. So, it seemed logical that the DMV commissioned Hilbert and Minkowski to write a report on the state of the art of number theory. One of the more significant topics for the development of algebraic number theory dealt with by Gauss in his Disquisitiones was that of power residues, particularly quadratic residues, and the problem of generalizing the law of quadratic reciprocity to higher orders. It could be said, as Erich Hecke (1887-1947) put it in his Vorlesungen über die Theorie der algebraischen Zahlen (Lectures on the Theory of Algebraic Numbers), that “modern number theory dates from the discovery of the reciprocity law.”3 The origin of the research in number theory which led to the discovery of the reciprocity law can be found in the work of Pierre de Fermat. In his researches on number theory, Fermat was much inspired by the problems posed by Diophantus of Alexandria (flourished c. CE 250) in his famous work Arithmetica. For example, Problem 9 in Book V, asks for an odd number to be expressed as a sum of two squares, with several side conditions. And in most of the problems of Books IV and V, Diophantus supposes that every whole number is either a square, or a sum of two, three or four squares. In a letter from 1658 to Sir Kenelm Digby (an English adventurer and double agent), Fermat announces that he has proven Diophantus’s supposition and that he “can add a number of very celebrated propositions for which I also possess irrefutable proof. For example: Every prime number of the form Ͷ݊ ൅ ͳ is the sum of two squares, such as ͷǡ ͳ͵ǡ ͳ͹ǡ ʹͻǡ ͵͹ǡ Ͷͳǡ ݁‫ܿݐ‬. Every prime number of the form ͵݊ ൅ ͳ is the sum of a square and the triple of another square, for instance, ͹ǡ ͳ͵ǡ ͳͻǡ ͵ͳǡ ͵͹ǡ Ͷ͵ǡ ݁‫ܿݐ‬. Every prime number of the form ͺ݊ ൅ ͳ or ͺ݊ ൅ ͵ is the sum of a square and double another square, such as ͵ǡ ͳͳǡ ͳ͹ǡ ͳͻǡ Ͷͳǡ Ͷ͵ǡ ݁‫ܿݐ‬.”4

 3

Hecke 1923, 59. Cited from Knoebel et alia (2007, 233). Translated into the language of congruences, later introduced by Gauss, Fermat assertions become (for a prime ‫݌‬ and integers ‫ݔ‬ǡ ‫)ݕ‬: ‫ ݌‬ൌ ‫ ݔ‬ଶ ൅ ‫ ݕ‬ଶ if and only if ‫ͳ ؠ ݌‬ሺ‘†Ͷሻ (ˆ‘”‫ʹ ് ݌‬ሻ, ‫ ݌‬ൌ ‫ ݔ‬ଶ ൅ ͵‫ ݕ‬ଶ if and only if ‫ͳ ؠ ݌‬ሺ‘†͵ሻ (ˆ‘”‫ʹ ് ݌‬ǡ͵ǡ ‫ ݌‬ൌ ‫ ݔ‬ଶ ൅ ʹ‫ ݕ‬ଶ if and only if ‫ͳ ؠ ݌‬ǡ ͵ሺ‘†ͺሻ (ˆ‘”‫ʹ ് ݌‬ሻ. 4



42

Chapter Four

Unfortunately, no proofs of these results have been found in Fermat’s correspondence. Indeed, much of the work of proving Fermat’s conjectures was left to Leonard Euler (1707-1783). After discovering proofs for many of Fermat’s claims about sums of squares, Euler turned his attention to the following more general question: For a given nonzero integer ܽ, which prime numbers can be represented in the form ‫ ݔ‬ଶ ൅ ܽ‫ ݕ‬ଶ , with ‫ ݔ‬and ‫ ݕ‬positive integers?

(For the cases ܽ ൌ ͳǡ ʹ and ͵, a solution was essentially claimed by Fermat, as the letter to Digby quoted above shows). Another claim of Fermat’s, namely that a sum ‫ ݔ‬ଶ ൅ ܽ‫ ݕ‬ଶ , with ‫ ݔ‬and ‫ ݕ‬relatively prime positive integers, can never have a divisor of the form Ͷ݊ െ ͳ no matter how ‫ ݔ‬and ‫ ݕ‬are chosen, suggested a related problem to Euler: For a given nonzero integer ܽ, find all nontrivial prime divisors of numbers of the form ‫ ݔ‬ଶ ൅ ܽ‫ ݕ‬ଶ , with ‫ ݔ‬and ‫ ݕ‬positive integers.

It turns out that the nontrivial prime divisors p of numbers of the form ‫ ݔ‬ଶ ൅ ܽ‫ ݕ‬ଶ are precisely the odd primes p for which െܽ is a nonzero quadratic residue.5 Therefore, to solve the above problem Euler needed to find some procedure which allowed finding these prime numbers. Since for any odd prime there are ሺ‫ ݌‬െ ͳሻΤʹ quadratic residues and ሺ‫ ݌‬െ ͳሻΤʹ quadratic non-residues, and the quadratic residues are congruent modulo ‫݌‬ to the integers ͳଶ ,ʹଶ ǡ ͵ଶ ǡ ǥ ǡ ሺሺ‫ ݌‬െ ͳሻΤʹሻଶ , Euler suggested in 1748 the following criterion (in modern notation):6 Let ܽ be any integer and ‫ ݌‬an odd prime not dividing ܽ. Then ܽ is a quadratic residue of ‫ ݌‬if, and only if, ܽሺ௣ିଵሻΤଶ ‫ͳ ؠ‬ሺ‘†‫݌‬ሻ and is a quadratic non-residue of ‫ ݌‬if, and only if, ܽሺ௣ିଵሻΤଶ ‫ ؠ‬െͳሺ‘†‫݌‬ሻ.

This criterion is known today as Euler’s criterion and was Euler’s first important result for determining the quadratic residues of a prime ‫݌‬. Even though Euler’s criterion is very useful to determine whether a given integer is a quadratic residue or not, it is not so helpful when we wish to determine all the quadratic residues of a given prime. It was not until

 5

This is so because (using the congruence notation introduced later by Gauss) if ‫ ݔ‬ଶ ൅ ܽ‫ ݕ‬ଶ ൌ ݉‫݌‬, since ‫ ݕ‬is relatively prime to ‫݌‬, we can find an integer ‫ ݖ‬such that ‫ͳ ؠ ݖݕ‬ሺ‘†‫݌‬ሻ. Hence, multiplying ‫ ݔ‬ଶ ൅ ܽ‫ ݕ‬ଶ by ‫ ݖ‬ଶ , we obtain െܽ ‫ؠ‬ ሺ‫ݖݔ‬ሻଶ ሺ‘†‫݌‬ሻ. Conversely, if െܽ ‫݊ ؠ‬ଶ ሺ‘†‫݌‬ሻ with ݊ non-divisible by ‫݌‬, then െܽ ൌ ݊ଶ ൅ ݉‫ ݌‬for some integer ݉, and so ሺെ݉ሻ‫ ݌‬ൌ ݊ଶ ൅ ܽ ൉ ͳଶ . 6 Lemmermeyer (2000, 4) cites two papers, E134 and E262 in the Euler Archive.



The Algebraic Theory of Numbers

43

1773-75 that Euler stated a theorem equivalent to the quadratic reciprocity law, which provided the solution to the divisor problem, although he wasn’t able to prove it.7 Joseph Louis Lagrange, who was the successor of Euler at the Academy of Sciences in Berlin, made the next step in that he proposed to consider general quadratic forms, i.e., expressions as ܽ‫ ݔ‬ଶ ൅ ܾ‫ ݕݔ‬൅ ܿ‫ ݕ‬ଶ , rather than quadratic expressions of the form ‫ ݔ‬ଶ ൅ ܽ‫ ݕ‬ଶ , as the proper object of study in order to get a coherent theory of quadratic residues. As a result of this, Lagrange laid the foundations of the theory of quadratic forms, which would become one of the major topics of Gauss’s Disquisitiones. Lagrange was also able to prove a lot of theorems about the representability of primes in the form ‫ ݔ‬ଶ ൅ ܽ‫ ݕ‬ଶ and, most important, the claim that every positive integer can be written as a sum of four integer squares. However, the major advances in the study of the problem of representing prime numbers by quadratic forms were done by the French mathematician Adrien-Marie Legendre (1752-1833). To solve this problem, he presented a first version of the law of reciprocity in 1788. Ten years later he introduced the Legendre symbol: Since the analogous quantities ܰ ሺ௖ିଵሻΤଶ will occur often in our researches, ே we shall employ the abbreviation ቀ ቁ for expressing the residue that ௖

ܰ ሺ௖ିଵሻΤଶ gives upon division by ܿ, and which, according to what we just have seen, only assumes the values ൅ͳ or െͳ.8

This symbol enabled him to state the law of quadratic reciprocity in terms completely analogous to those in which it is stated today:

 7

In modern notation, Euler’s theorem says that: 1. If ‫ͳ ؠ ݌‬ሺ‘†Ͷሻ is prime and ‫ ݔ ؠ ݌‬ଶ ሺ‘†‫ݏ‬ሻ for some prime s, then േ‫ ݕ ؠ ݏ‬ଶ ሺ‘†‫݌‬ሻ. 2. If ‫͵ ؠ ݌‬ሺ‘†Ͷሻ is prime and െ‫ ݔ ؠ ݌‬ଶ ሺ‘†‫ݏ‬ሻ for some prime s, then ‫ ݕ ؠ ݏ‬ଶ ሺ‘†‫݌‬ሻ and െ‫ ݕ ء ݏ‬ଶ ሺ‘†‫݌‬ሻ. 3. If ‫͵ ؠ ݌‬ሺ‘†Ͷሻ is prime and െ‫ ݔ ء ݌‬ଶ ሺ‘†‫ݏ‬ሻ for some prime s, then െ‫ ݕ ؠ ݏ‬ଶ ሺ‘†‫݌‬ሻ and ‫ ݕ ء ݏ‬ଶ ሺ‘†‫݌‬ሻ. 4. If ‫ͳ ؠ ݌‬ሺ‘†Ͷሻ is prime and ‫ ݔ ء ݌‬ଶ ሺ‘†‫ݏ‬ሻ for some prime s, then േ‫ ݕ ؠ ݏ‬ଶ ሺ‘†‫݌‬ሻ. In fact, his theorem “is equivalent to the version of the quadratic reciprocity law that is best known today and that was formulated by Legendre and Gauss.” (Lemmermeyer 2000, 5). 8 Lemmermeyer 2000, 6.



Chapter Four

44

Whatever the prime numbers ݉ and ݊ are, if they are not both of the form ௡ ௠ Ͷ‫ ݔ‬െ ͳ, one always has ቀ௠ቁ ൌ ቀ ௡ ቁ ; and if both are of the form Ͷ‫ ݔ‬െ ͳ, ௡



௠ ௡



one has ቀ ቁ ൌ െ ቀ ቁ. These two general cases are contained in the ೙షభ ೘షభ Ǥ మ మ

formula ቀ ቁ ൌ ሺെͳሻ ௠



ቀ ቁ.9 ௡

Although Legendre tried several times to give a complete proof of this law, there always remained some gap to be fulfilled. The honour of giving a complete proof of the law of quadratic reciprocity was for Carl Friedrich Gauss, who proved it in several ways in his Disquisitiones Arithmeticae. After having introduced in the first three sections the fundamentals of the theory of congruences, in Section 4 Gauss develops the theory of quadratic residues. This section culminates in the “fundamental theorem” (art. 131) of this theory, from which “can be deduced almost everything that can be said about quadratic residues.”10 This is the law of quadratic reciprocity, which Gauss states in the following guise: If ‫ ݌‬is a prime number of the form Ͷ݊ ൅ ͳ, then so will be ൅‫݌‬, however if ‫ ݌‬is of the form Ͷ݊ ൅ ͵, then െ‫ ݌‬will be a [quadratic] residue or nonnonresidue of any prime number which is a residue or nonresidue of ‫݌‬.11

After 1824, Gauss also investigated cubic, biquadratic (quartic) and octic reciprocity and formulated statements for cubic and biquadratic reciprocity. He noticed that the statement of cubic and biquadratic laws requires the fields of cube or fourth roots of unity. For biquadratic reciprocity Gauss considered the ring Ժሾ݅ሿ ൌ ሼܽ ൅ ܾ݅ǣ ܽǡ ܾ ‫ א‬Ժሽ, which is now known as the Gaussian Integers and is regarded by many historians as the first appearance of algebraic numbers. Gauss provided a proof for cubic reciprocity and he also announced a proof of the biquadratic reciprocity law, but he never published them. This stimulated the work on reciprocity laws for small degrees, particularly cubic, quartic and quintic. The first public and complete proofs for the cubic and quartic residues were published in 1844 by Eisenstein, although Jacobi had already announced a reciprocity law for them in his 1836-37 Königsberg lectures. To generalize his results to higher residues Eisenstein was confronted with the problem of the absence of unique factorization, which was solved in 1845 by the introduction by Kummer of his “ideal numbers.” This enabled Eisenstein to prove a special case of what is now

 9

Ibid. Gauss 1801, 99. 11 Ibid. 10



The Algebraic Theory of Numbers

45

The law of quadratic reciprocity  If ܽ is an integer and p a prime number that does not divide ܽ, we say that ܽ is a quadratic residue of ‫ ݌‬if there is an integer ‫ ݔ‬such that ‫݌‬ divides ‫ ݔ‬ଶ െ ܽ. Equivalently, if we write ܽ ‫ܾ ؠ‬ሺ݉‫݌݀݋‬ሻ (read: ܽ is congruent with b modulo ‫ )݌‬to denote that ܽ െ ܾ is a multiple of ‫݌‬, then we have the following definition: Let ܽ be an integer and ‫ ݌‬a prime number that does not divide ܽ, then ܽ is a quadratic residue modulo ‫ ݌‬if the quadratic congruence ‫ ݔ‬ଶ ‫ܽ ؠ‬ሺ݉‫݌݀݋‬ሻ has a solution. Otherwise, ܽ is called a quadratic non-residue modulo ‫݌‬. If ‫ ݌‬is a prime odd number and a is an integer that is not divisible by ‫݌‬, ௔ then the Legendre´s symbol, denoted by ቀ ቁ indicates whether a is a ௣

quadratic residue modulo ‫ ݌‬or not. If it is, then Legendre’s symbol is equal to ͳ; if it is not, then it is equal to െͳ. The law of quadratic reciprocity answers the question: When is an integer a square modulo a prime ‫݌‬, i.e., a quadratic residue modulo ‫ ?݌‬In particular, this law states that for any odd primes ‫݌‬ǡ ‫ݍ‬: ௣ ௤ If ‫ͳ ؠ ݌‬ሺ‘†Ͷሻ or ‫ͳ ؠ ݍ‬ሺ‘†Ͷሻ, then ቀ ቁ ൌ ቀ ቁ, ௤











If ‫ ؠ ݌‬െͳሺ‘†Ͷሻ or ‫ ؠ ݍ‬െͳሺ‘†Ͷሻ, then ቀ ቁ ൌ െ ቀ ቁ. In the first case, ‫ ݌‬and ‫ ݍ‬are either both quadratic residues or else nonresidues each with respect to the other. In the second case, one of them is a quadratic residue and the other one a non-residue. In a more succinct way, we can reformulate these relationships as: Law of quadratic reciprocity: if ‫ ݌‬and ‫ ݍ‬are odd distinct prime numbers, then ௣







ሺ೛షభሻሺ೜షభሻ ర

ቀ ቁ ቀ ቁ ൌ ሺെͳሻ

.

Moreover, we have ିଵ

ቀ ቁ ൌ ሺെͳሻ ௣

೛షభ మ

and



ቀ ቁ ൌ ሺെͳሻ ௣

೛మ షభ ఴ

;

These are called the first and second supplementary law, respectively. ௔ It follows easily from the law of quadratic reciprocity that ቀ ቁ depends ௣



cont. p. 46

46

Chapter Four

on the residue class of ‫ ݌‬mod Ͷܽ. Generalizing the definition of quadratic residue, we will say that ܽ ‫ Ͳ ء‬is a residue of power ݇ modulo ‫ ݌‬if ‫ ݔ‬௞ ‫ܽ ؠ‬ሺ‘†‫݌‬ሻ has a solution, that is, if ܽ has a ݇-th root modulo ‫݌‬. In particular, we will say that ܽ is a cubical residue if ‫ ݔ‬ଷ ‫ܽ ؠ‬ሺ‘†‫݌‬ሻ has solution and that ܽ is a biquadratic residue if ‫ ݔ‬ସ ‫ܽ ؠ‬ሺ‘†‫݌‬ሻ has solution. In order to formulate the biquadratic or quartic of reciprocity, one needs an “infinite enlargement” of the integers, namely the ring Ժሾ݅ሿ ൌ ሼܽ ൅ ܾ݅ǣ ܽǡ ܾ ‫ א‬Ժሽ of Gaussian integers. For these integers, we have the following: Law of quartic reciprocity: Let ߨǡ ߣ ‫ א‬Ժሾ݅ሿ be primary primes, i.e., assume that ߨ ‫†‘ͳ ؠ ߣ ؠ‬ሺʹ ൅ ʹ݅ሻ; then ேగିଵ ேఒିଵ ߣ ߨ ቂ ቃ ൌ ሺെͳሻ ସ Ǥ ସ ൤ ൨Ǥ ߣ ߨ

(ܰߣ is the norm of ߣ, i.e. if ߣ ൌ ܽ ൅ ܾ݅, then ܰߣ ൌ ܽଶ ൅ ܾଶ ). There are also analogues of the first and second supplementary laws for quartic residues. A similar, although simpler, law holds for the cubic residue symbols and primary primes in Ժሾ‫݌‬ሿ, where ‫ ݌‬is a primitive cube root of unity. In all the above cases, the word “reciprocity” comes from the fact that these laws relate the solvability of the congruence ‫ ݔ‬௞ ‫݌ ؠ‬ሺ‘†‫ݍ‬ሻ to that of ‫ ݔ‬௞ ‫ݍ ؠ‬ሺ‘†‫݌‬ሻ.

called Eisenstein’ reciprocity law. The quintic case had to wait for Kummer who also stated a reciprocity law valid in all regular cyclotomic fields. Hilbert did the next step by reinterpreting the quadratic reciprocity law and generalizing it to arbitrary algebraic number fields in terms of the so-called “Hilbert symbol.” The fields of cyclotomic integers were central not only in the investigation of higher reciprocity laws but also in the study of Fermat’s Last Theorem. Both problems were of interest to Kummer and to make noteworthy progress it was necessary to establish unique factorization in the fields of cyclotomic integers. This problem was finally solved by him in the 1840s with the introduction of a new kind of complex number which he called “ideal complex numbers” (in analogy with “ideal” objects in



The Algebraic Theory of Numbers

47

geometry, such as points at infinity). Kummer’s major result was the proof that every element in the field of cyclotomic integers is a unique product of “ideal primes.” By the middle of the century considerable experience with number fields such as quadratic fields and cyclotomic fields had been acquired. This put the study of these algebraic number fields in the foreground. The first problem found while developing an arithmetic theory of an algebraic field of numbers ॶ was to choose a subset of ॶ, the algebraic integers of ॶ, which could play a role in ॶ similar to that played by the integers inԷ. In their study of biquadratic and cubic reciprocity Gauss, Jacobi, Eisenstein and Kummer employed several types of algebraic integers, but only Richard Dedekind (1831-1916) was able to define this concept in 1871–taking advantage of an important result proved by Eisenstein in 1850. The main problem in order to elaborate an arithmetic theory for algebraic number fields was to find a theory of divisibility for these fields. Kummer’s theory of divisibility was brilliant, but the fundamental concepts of ideal number and ideal prime were not intrinsically defined. Moreover, his theory only applied to cyclotomic integers. The immediate task was then to build a divisibility theory in which the fundamental concepts were clearly defined and would apply to more general domains of algebraic integers. This task was done independently by Dedekind and Kronecker.12 Dedekind formulated his theory of divisibility by means of ideals, an approach that was soon widely accepted. In Supplements X and XI to the second edition of Dirichlet’s Vorlesungen über Zahlentheorie (Lessons on Number Theory) (1863), Dedekind formulated his foundations for algebraic number theory, which incorporated the concepts that would become the core of modern presentations of the topic, such as, for example, the concepts of domain of rationality (a subset of complex numbers which is closed under the four arithmetic operations), algebraic field of numbers, algebraic integer, ideal, module, or lattice. It is generally accepted that the definitive transformation of algebraic number theory into a general study of algebraic fields of numbers, and so not limited to quadratic fields or to cyclotomic fields, took place in the work of Richard Dedekind. At the same time, Kronecker, a disciple of Kummer and doctoral student of Dirichlet, was making fundamental contributions to the

 12

A third approach was that of the Russian mathematician Yegor Zolotarev (18471878), who used the notion of exponential valuation for his theory. This was independently developed by Hensel, but it became accepted only in the 1920s due to the results of Hasse about quadratic forms.



48

Chapter Four

Algebraic numbers and algebraic fields of numbers  Algebraic numbers are the roots of polynomial equations with rational coefficients. More concretely, an algebraic number Į is a complex number that satisfies a polynomial equation ߙ௡ ‫ ݔ‬௡ ൅ ߙ௡ିଵ ‫ ݔ‬௡ିଵ ൅ ǥ ൅ ߙ଴ ൌ Ͳ with rational coefficients (or, equivalently by clearing denominators, integer coefficients) not all of them equal to 0. They are the essential elements of the so-called algebraic fields of numbers. An algebraic field of numbers ॶ is a subfield of the complex numbers ԧ that is a finite degree extension of the rational numbers Է, that is to say, ॶ has a finite dimension as a vector space over Է. Now, every algebraic field of numbers ॶ is of the form Էሺߙଵ ǡ ߙଶ ǡ ǥ ǡ ߙ௡ ሻ, where the ߙ௜ are algebraic numbers. Indeed, they are of the formॶ ൌ Էሺߙሻ , for an algebraic number Į; that is to say, every element ofॶ is a rational function of Į with rational coefficients (that is, a polynomial in Į). For example,Է itself is an algebraic field of numbers, but Թ or ԧ are not (since they have infinite dimension as vector spaces over Է). Some well-known examples of non-trivial algebraic number fields are the quadratic fields Է൫ξ݉൯, where ݉ ‫ א‬Ժ is not a perfect square, and the cyclotomic fields Էሺߞ௡ ሻ, where ߞ௡ is a primitive ݊–Š root of unity (i.e., ߞ௡ ൌ ‡š’ሺʹߨ݅Τ݊ሻ). The set of algebraic numbers is a field (this simply means that the sum, difference, product and quotient of two algebraic numbers is again an algebraic number), usually represented by ८. Indeed, ८ is a proper subfield of ԧ, given that there are transcendental (nonalgebraic) numbers such as ʌ and e. But note that ८ is not an algebraic field of numbers, since it is not a finite degree extension of Է (to see this, just think about the degrees of the solutions of‫ ݔ‬௡ െ ʹ ൌ Ͳ for every n). An algebraic integer is a complex number Į that satisfies a polynomial equation ‫ ݔ‬௡ ൅ ߙ௡ିଵ ‫ ݔ‬௡ିଵ ൅ ǥ ൅ ߙ଴ ൌ Ͳ, with rational coefficients not all equal to 0. Thus, the definition of algebraic integer results from the definition of algebraic number by restricting the polynomials to monic polynomials, that is, to those with leading coefficient 1. Some well-known examples of algebraic integers are the Gaussian integers ܽ ൅ ܾ݅, where ܽǡ ܾ ‫ א‬Ժ, and the cyclotomic integers ܽ଴ ൅ ܽଵ ߞ௡ଵ ൅ ‫ ڮ‬൅ ܽ௡ିଵ ߞ௡௡ିଵ , where ܽ௜ ‫ א‬Ժ and ߞ௡ is a primitive ݊–Š root of unity.



The Algebraic Theory of Numbers

49

theory. Kronecker used for his theory of divisibility the adjunction of variables, a method that almost disappeared in the 20th century despite the efforts of some prominent followers such as Weyl. In fact, Kronecker’s goal was much more ambitious than that of Dedekind, since he wanted to develop a theory that would include both algebraic number theory and algebraic geometry. But the articles of Kronecker were very difficult to understand and perhaps for that reason his work did not have such an immediate influence as that of Dedekind in the development of algebraic number theory. The main contributions of Dedekind and Kronecker to the subject took place during the 1860s and 1870s. In the following two decades, young and talented mathematicians such as Kurt Hensel (1861-1941), Kronecker’s disciple and editor of his five-volume collected works, Adolf Hurwitz and, needless to say, also David Hilbert were interested in this subject. Another important agent in the development of algebraic number theory was Heinrich Weber, who made important contributions to class field theory. Hilbert’s report, entitled Die Theorie der algebraischen Zahlkörper (Theory of algebraic fields of numbers), later known as Zahlbericht, was finally published in 1897. It was not only a comprehensive study of the state of development achieved by number theory at that time, but also (and mainly) a systematization of it based on Hilbert’s own vision of the development of this discipline over the last hundred years. In this sense one of Hilbert’s principal contributions was that “he rewrote the subject so that knowledge of its history and familiarity with the older texts was no longer required. New readers could start here.”13 As Hilbert explained in the preface of the Zahlbericht, Gauss, Dirichlet and Jacobi had expressed their surprise when seeing the close connection between the issues of number theory and algebraic problems (that is, the resolution of polynomial equations). The reason for this connection, Hilbert said, was now clear and diaphanous: both subdisciplines have their common roots in algebraic number theory. This theory had thus become, according to Hilbert, the essential component of modern number theory. Indeed, Hilbert’s Zahlbericht contributed decisively to the establishment of algebraic number theory as one of the most important branches of pure mathematics and it became the reference book used by most mathematicians of the following generation of number theorists such as Erich Hecke or Helmut Hasse (1898-1979), leaving its footprint on the textbooks on algebraic number theory that have been published since then

 13



Gray 2000, 45.

50

Chapter Four

to the present day. Moreover, in reformulating the results of its predecessors in Hilbert’s own terms, the Zahlbericht contained a handful of original results that opened new ways of research in algebraic number theory. The Zahlbericht is structured in a Preface and five parts.14 Part 1 introduces the theory of general number fields, i.e., the basic arithmetic theory of a general finite extension of the field of rational numbers: integers, ideals, discriminant, units, ideals classes, zeta-function, the ring of integers, etc. Part 2 deals with the Galois number field, that is, with the decomposition of primes in a Galois extension: decomposition group and inertia group, and the corresponding subfields. Part 3 deals with quadratic number fields. The greatest novelty here is the introduction of the local norm residue symbol, the so-called “Hilbert symbol” in terms of which he presents Gauss’s genus theory and also expresses the law of quadratic reciprocity (§69, Hilfsatz 14). Part 4 deals with cyclotomic fields and among his jewels there is the first complete proof of the Kronecker-Weber Theorem (§100, Satz 131) asserting that every abelian extension of the rational numbers is contained in a suitable cyclotomic field. The long and last fifth part deals with the arithmetic theory of ೙ Kummer’s fields, i.e., fields of the form ‫ܭ‬൫ ξߙ൯ , where ‡š’ሺʹߨ݅Τ݊ሻ ‫ܭ א‬. According to Hilbert, these are the fields “which Kummer took as a basis for his researches into higher reciprocity laws,” representing the theory of Kummer’s fields “the highest peak reached on the mountain of today’s knowledge of arithmetic.”15 In Hilbert’s own words, his main target in this part was to expose Kummer’s theory avoiding “Kummer’s elaborate computational machinery, so that here too Riemann’s principle may be realized and the proofs completed not by calculations but purely by ideas.”16 However, it should be noted that the critical evaluation of the scope of Hilbert achievements has been somewhat mixed. Thus, although some number theorists like Hasse or Emil Artin (1898-1962) have praised Hilbert’s exposition for his great simplification of Kummer’s theory, Hilbert’s approach has also been criticized by other mathematicians like André Weil (1906-1998), who has written that Hilbert’s exposition “is

 14

A detailed exposition of the topics dealt with by Hilbert in each part, and particularly of his most relevant theorems, can be found in (Schappacher 2005). 15 Hilbert 1998, X. The reason for this, as noted by Hilbert is that “almost all essential ideas and concepts of field theory, at least in a special setting, find an application in the proof of the higher reciprocity laws.” (Ibid.) 16 Ibid.



The Algebraic Theory of Numbers

51

little more than an account of Kummer’s number-theoretical work, with inessential improvements.”17 The advanced character of Hilbert’s report made it inaccessible to most readers, even of an academic type, so Hilbert gave a course in the winter semester of 1897/98 on quadratic fields of numbers and motivated one of the attendees to the course, Julius Sommer (1871-1943), to write a book on quadratic and cubic fields as an introduction to algebraic number theory. Similarly, Leigh Wilber Reid (1867-1961), who had done his Ph.D. under Hilbert’s supervision, published in 1910 a textbook, The Elements of the Theory of Algebraic Numbers, which dealt exclusively with quadratic extensions. In the Introduction of the book, written by Hilbert, he explained that “the theory of numbers is independent of the change of fashion and in it we do not see, as it is often the case in other departments of knowledge, one conception or method at one time given undue prominence, at another suffering undeserved neglect.”18 Despite Hilbert’s words of 1910, he indeed had to choose in the Zahlbericht between the different “conceptions or methods” that had been followed in number theory in recent years. Similar to what had happened with his attack to the problems of invariant theory, the way in which Hilbert addressed the problems of algebraic number theory in the Zahlbericht was closer to the conceptual approach followed by Dedekind than to the more computational one followed by Kronecker or Kummer. However, there is sometimes a mix of both approaches. So, for example, Hilbert defines ideals à la Dedekind as set of algebraic integers that are closed under linear combinations with algebraic integer coefficients. But he uses Kronecker’s theory of forms to prove the decomposition of ideals into prime ideals in the ring of integers of an arbitrary number field. These non-Dedekindian features of Hilbert text were probably the cause of later criticism by Noether and even Artin of Hilbert’s report, as documented by various historians.19 It must also be noted that Hilbert made less use than expected for a mathematician working at the end of 19th century of the unifying concepts of abstract algebra, for example, the concept of a group. In this sense, it is also worth mentioning that some key concepts in his work, such as the concepts of field, ring or ideal, referred to particular sets of numbers, rather than to general algebraic structures. Finally, we must not forget that the most original publications of Hilbert, in which he introduced for the first time the important notion of a field of classes of an algebraic number

 17

Kummer 1975, 1. Reid 1910, xvii. 19 See, for example, (Schappacher 2005, 703-4). 18



52

Chapter Four

field and he envisaged the idea of a general class field theory, only appeared between 1899 and 1902 and, therefore, after the publication of the Zahlbericht (see Chapter 6). Regardless of the criticism of Noether, Artin, Weil and others, it is clear that no other work has contributed so much to the consolidation of algebraic number theory as a well-established discipline within mathematics on which most of the 20th century’s research has been developed as Hilbert’s Zahlbericht. As Hermann Weyl observed, “while his work on invariants was an end, his work on algebraic numbers was a beginning. Most of the work of number theory specialists such as Furtwängler, Takagi, Hasse, Artin or Chevalley has been focused on proving the anticipated results of Hilbert.”20

 20



Weyl 1944, 627.

CHAPTER FIVE KLEIN AND THE MATHEMATICAL TRADITION OF GÖTTINGEN

Although from 1800 to 1830 approximately, France had held the European leadership in mathematical research, during the second half of the nineteenth century Germany took over from France. Under the influence of Dirichlet௅who had travelled to Paris in the decade of the twenties to study mathematics and went into close contact with Joseph Fourier (17681830) and Siméon Poisson (1781-1840) among others௅the University of Berlin became, from the thirties, the most important and influential mathematical centre of Germany. This influence became even more evident in the second half of the nineteenth century when, thanks to the teaching and research work of Weierstrass, Kummer and Kronecker, the University of Berlin became the most important mathematical research centre in the world. However, in the last quarter of the century, the University of Göttingen first challenged and then took over the supremacy of the University of Berlin regarding mathematical education and research. The great mathematical tradition of Göttingen had begun with Gauss௅ named Mathematicorum princeps௅, who was professionally linked to the University of Göttingen for more than 50 years. This tradition was continued by his immediate successors in the chair of mathematics at the University of Göttingen, Dirichlet, Bernhard Riemann (1826-1866) and Clebsch. However, Gauss never showed any special interest in the formation of future mathematicians, while Dirichlet, Riemann and Clebsch had very short careers in Göttingen, so they could influence only a few students. The result was that few young mathematicians chose Göttingen to begin their academic career; they were more inclined for Berlin or Königsberg. The true protagonist in the conversion of Göttingen in the first mathematical centre of Germany and in the establishment there of a community of world first class mathematicians was Felix Klein. Klein was born on April 25, 1849 in Düsseldorf, administrative capital of Prussian Rhineland and the main industrial centre of Prussia. After an elementary instruction in his home taught by his mother, when he was six

54

Chapter Five

years old he entered a private school in Düsseldorf and two and a half years later, in 1857, he continued his studies at the gymnasium in the same city. A traditional education, based predominantly on the cultivation of classical languages, has often been attributed to the German Gymnasien, but this ceased to be the case in most of the Prussian ones after the reforms undertaken in the 1820s and 1830s. Certainly, mathematics was well represented at the Düsseldorf gymnasium which Klein attended and it was taught by competent teachers. From the results achieved in the Abitur, we also know that Klein received good instruction in mathematics, science and humanities. After graduating, Klein entered the University of Bonn at the age of 16 and a half, where he studied not only mathematics but also natural sciences from 1865 to 1866. This was possible thanks to the existence at the Bonn University of the Seminar für die Gesamten Naturwissenschaften (Seminar on the entire Nature Sciences), which offered within its five sections devoted to physics, chemistry, geology, botany and zoology, a coherent and systematic program for the practical and theoretical study of the sciences. While studying at the University of Bonn, Klein was appointed as a laboratory assistant to Julius Plücker (1801-1868), a position he held from 1866 to 1868. Plücker held a chair of mathematics and experimental physics in Bonn, but when Klein became his assistant, Plücker’s interests had already been directed towards geometry. From the winter semester of the 1867/68 academic year, Klein concentrated on mathematics and gradually became influenced by Plücker and his analytical approach to geometry. Klein received his doctorate in 1868, after defending a thesis, supervised by Plücker, on linear geometry and its application to mechanics. That same year, Plücker died, leaving incomplete the second part of his work Neue Geometrie des Raumes (New Geometry of Space), in which he laid the foundations of the analytical approach to linear geometry. Given the circumstances, Klein was the most suitable person to complete the work of Plücker and so the editing of its work was entrusted to him. Klein’s editorial involvement in Plücker’s work put him in contact with Clebsch, who then occupied the chair of mathematics at the University of Göttingen and had formed around him an important group of researchers in algebraic geometry and the theory of invariants. Klein moved to Göttingen and studied there from January to August 1869 with Clebsch. He then continued his studies at the University of Berlin, where he stayed until April 1870 and attended the highly specialized Mathematisches Seminar (Seminar on Mathematics). We also know that he attended a conference done by Kronecker but did not want to

Klein and the Mathematical Tradition of Göttingen

55

attend a conference by Weierstrass, whose dominant position was not of his liking. In Berlin, Klein met Sophus Lie (1842-1899), whose cooperation with him would be very important in the following years. In April 1870, the two young mathematicians settled in Paris to learn firsthand the new advances in mathematics achieved by the French, but the outbreak of the Franco-German war in July 1870 put an end to his research stay in the French capital. Klein, who was very patriotic, volunteered for the army, where he served in the medical community. Infected by typhoid fever, he had to leave the army and took advantage of the recovery time to prepare his habilitation thesis. He submitted the habilitation thesis in January 1871 and during the following months he lectured as a Privatdozent at the University of Göttingen, where he frequented the Clebsch circle. It was precisely that year when Klein published the first of two writings entitled Über die sogennante nicht-euklidische Geometrie (About the so-called non-Euclidean Geometry), in which he showed that Euclidean and non-Euclidean geometries (hyperbolic and elliptical) could be considered special cases of a projective surface to which a specific conic section was attached. By furnishing the same projective model of Euclidean and non-Euclidean geometries, Klein also demonstrated that Euclidean geometry and non-Euclidean geometry were equiconsistent and, therefore, that they stood in an equal footing, ending all controversy about the consistency of non-Euclidean geometry. Despite some initial criticisms (Cayley, for example, never accepted Klein’s proof, considering it circular), Klein’s result on the status of non-Euclidean geometry was soon widely recognized as a first-order contribution to mathematics. This, together with Klein’s earlier contributions to linear geometry, gained him a great reputation. Clebsch had great regard for Klein’s talent and at his instance, he was made an ordentlicher Professor at the University of Erlangen. It was common at that time that new professors wrote a kind of program that served as a letter of introduction for their entrance to the university. Klein’s famous Anstrittrede (Inaugural Speech) was titled “Vergleichende Betrachtungen über neuere geometrische Forschungen” (“A Comparative Review of Recent Researches in Geometry”), which had a great international impact and was later known as the Erlanger Programm. Klein wrote this work as a response to a purely theoretical problem, but it was also a consequence of Klein’s dissatisfaction with the fragmentation and discredit, caused basically by the opposition between analytical and synthetic geometers, which characterized the geometry of that time in Germany.

56

Chapter Five

In the first section of the Erlanger Programm, Klein explained his research program in the following terms: “Given a manifold (Mannigfaltigkeit) and a group of transformations acting on it, investigate the properties of the figures (Gebilde) of that manifold that are invariant under all the transformations of that group.”1 This was for Klein, the proper object of study of every geometry. In other words, given a manifold and a group of transformations, a geometry is the invariant theory of the given transformation group. The above definition of geometry stresses the importance for geometry of two algebraic concepts, the concept of invariance and the concept of group. We have already studied the importance of the theory of invariants in the second half of the 19th century (see Chapter 2). Klein learned about this concept from his mentor Clebsch, who was together with Gordan, the most prominent figures in the study of algebraic invariants in Germany. Clebsch and Gordan had applied the concept of invariance to algebra and only to the linear group. Now Klein applied it to geometry and to the space transformation groups. However, the most significant contribution of Klein’s Erlanger Programm was the fundamental role he deserved to the concept of group in geometry. He was familiar with this concept because of his relationship with Sophus Lie and through the work of Camille Jordan (1838-1922). Klein did not discover the concept of group nor did he have the abstract concept of group (he only considered groups of transformations), but he was the first to realize its usefulness for the classification of different geometries: each geometry is the study of certain properties that do not change (the invariants) when certain geometric transformations are applied, which must have a group structure under the composition operation.2 The above definition served to define and classify all the existing geometries known at that time. Thus, for example, Euclidean geometry is the study of invariants through the isometry group, affine geometry is the study of invariants by the affine group, projective geometry is the study of invariants through the projective group, etc.

 1

Klein 1872, 7. Another mathematician who stressed and worked out the connections between groups of transformations and geometry was Poincaré. But his main contributions were in the decade of the eighties. 2

Klein and the Mathematical Tradition of Göttingen

57

Klein’s Erlanger Programm Euclidean plane geometry is the study of invariants under the set ‫ ܩ‬of all rigid movements: translations, rotations and reflections. Since the composition of any two such transformations and the inverse of any such transformation are also such transformations and the identity is also one such transformation, it follows that ‫ ܩ‬is a group, called the isometry group. The corresponding geometry is plane Euclidean geometry. Since geometric properties such as length, area, congruence and similarity of figures, perpendicularity, parallelism, collinearity of points and concurrence of lines are invariant under the group ‫ܩ‬, these properties are studied in plane Euclidean geometry. If the group ‫ ܩ‬is enlarged by including, together with translations, rotations and reflections, dilations and shears, all transformations composite from all above-mentioned transformations, we obtain the affine group. Under this enlarged group, properties such as length, area, perpendicularity and congruence of figures are no longer invariant and hence are no longer subjects of the study in the framework of plane affine geometry. However, parallelism, collinearity of points and concurrence of lines are still invariant and, consequently, constitute subject matter for the study of this geometry. Similarly, plane projective geometry is the study of those geometric properties which remain invariant under the group of the so-called projective transformations. Of the previously mentioned properties, only collinearity of points and concurrence of lines remain invariant. Another important invariant under this group of geometric transformations is the cross ratio of four collinear points. The groups of transformations stated above can be ordered by the inclusion relation in this way: •‘‡–”›‰”‘—’ ‫’—‘”‰‡˜‹–…‡Œ‘” ؿ ’—‘”‰‡‹ˆˆ ؿ‬ Now, since every transformation group defines a corresponding geometry, we conclude that ”‘Œ‡…–‹˜‡‰‡‘‡–”› ‫›”–‡‘‡‰ƒ‡†‹Ž…— ؿ ›”–‡‘‡‰‡‹ˆˆ ؿ‬ This gives sense to the privileged position that occupies Projective Geometry in Klein’s classification of geometry.

58

Chapter Five

Clebsch died of diphtheria in 1872 and that caused some young mathematicians to leave Göttingen for Erlangen, but the truth is that Klein never had more than seven students in any of his classes and could never form a circle of competent mathematicians around him. Due to his isolation, Klein did not hesitate to accept an offer of the Technische Hochschule (Higher Technical School) in Munich in 1875. Klein stayed in this faculty for five years and, in addition to having a large group of students in his classes, mathematical talents such as Adolf Hurwitz, Carl Runge (1856-1927), or Max Planck (1858-1947) attended his seminar and advanced classes. In 1880 Klein accepted a newly created chair of geometry in Leipzig. During his stay at this university, Klein renovated the university auditorium, set up a mathematical seminar, a mathematics library and a collection of mathematical models. In addition, in 1882 he instituted the first paid position of assistant in mathematics in all of Germany. Finally, as a professor, he directed more than half of the 36 doctoral theses that were presented during his stay in Leipzig and five habilitation theses (Habilitationschriften), which shows the impulse that Klein gave to Leipzig University with the objective of consolidating it as one of the most important research centres in Germany. In 1881, a young and brilliant French mathematician named Henri Poincaré began publishing a series of articles on the theory of so-called automorphic functions. Klein was interested immediately by the subject and initiated a scientific correspondence, not exempted of some rivalry, with Poincaré. One of Klein’s objectives was to prove a theorem for the uniformization of the automorphic functions that could serve as the basis for this theory. Finally, after a great intellectual effort, Klein managed to formulate this theorem and outlined the proof of it. But the effort left a mark on him and caused recurring periods of depression over the years 1883 and 1884, which led him to abandon the first line of mathematical research that was being done at that time. From the publication of his famous work Vorlesüngen über das Ikosaeder und die Auflösung der Gleichungen vom fünfte Grade (Lectures on the icosahedron and the solution of equations of fifth degree) in 1884, Klein also adopted a new role as a professor, devoting himself to sketching the broad lines of a theory and offering suggestions for its development, leaving his assistants and disciples the task of completing the work. Robert Fricke (1861-1930), who arrived in Leipzig in 1884, began his career as an assistant to Klein, but would soon become his most important contributor in the elaboration of the four volumes on automorphic and elliptical modular functions that both would write during the two following decades.

Klein and the Mathematical Tradition of Göttingen

59

Fig. 5-1 Felix Klein as a professor at Leipzig

But Klein’s greatest success in his new role as professor would coincide with his first years in Göttingen. In 1886 Klein accepted a chair at this university, where he had studied with Clebsch and where he had been

60

Chapter Five

qualified as a university professor. The representative of the Prussian culture ministry, Friedrich Althoff, travelled expressly to Leipzig to ask Klein to accept the offer. The good relations with Althoff, which would remain until his death in 1908, were essential for the realization of the plans that Klein had for the University of Göttingen. It was not until 1892 that Klein’s idea of turning Göttingen into a centre of excellence in exact sciences began to take shape. Apart from being a magnificent professor and lecturer, Klein brilliantly used an important innovation of German universities in the second half of the 19th century, the Seminar, in order to train new mathematical talents. In the Seminar, the most recent results of mathematical research were presented to students, who could discuss and present themselves these results or their own research works in turns. This made many students from all over the world, particularly the United States, go to German universities, attracted not only by the great minds that taught there, but also by a system that allowed them to know first-hand and openly discuss the most recent results. During his first ten years in Göttingen, Klein taught numerous courses, which were often published and circulated widely inside and outside of Göttingen. Klein’s courses focused on a wide variety of topics, such as mechanics, potential theory, and other topics amid mathematics and physics. Klein’s goal, in offering courses on these subjects, was to reclaim the spirit of the Göttingen tradition forged by mathematicians such as Gauss and Riemann, as well as breaking with his typecasting as a geometer. In 1893, Klein visited the North-Western University in Evanston, Illinois, where he lectured on the current state of mathematics for two weeks. In the last lecture, after explaining in the previous lectures the latest advances in mathematics obtained in Germany, Klein recommended to future American students who would like to go to Göttingen, to not exclusively attend their lectures due to “their encyclopaedic character” and that those who would like to do the doctorate, to come in contact with other professors to continue their specialization. As a Wissenschaftpolitiker, that is, as a political activist in defence of the sciences, Klein skillfully dealt with Althoff, the ministerial representative, to get the best teachers and researchers for the mathematics division at Göttingen. In this sense, Klein’s greatest goal was to bring Hilbert to Göttingen, although his work was not limited to that. In 1892 Klein founded, together with his colleague Heinrich Weber, the Göttingen Mathematischen Geselschaft (Mathematical Society of Göttingen), which would have a major importance in the development of mathematics in

Klein and the Mathematical Tradition of Göttingen

61

Göttingen. Originally, the members of the society met once a week to discuss recent research and news about the profession, but as the society grew, the opportunity was often used to invite visiting professors and mathematicians or physicists of recognized international prestige to explain their most recent discoveries. In addition, Klein revitalized the mathematics and physics seminar, gave a significant impulse to the mathematics library of the University of Göttingen, renewed the math curriculum, directed the task of archiving the posthumous writings of Gauss, offered summer courses for high school teachers, and supported and participated in the DMV activities. For several years, Klein also directed the journal Mathematische Annalen, founded in 1868 by Clebsch and Carl Neumann (1832-1925), which ended up equalling or overcoming in prestige the Journal fur die Reine und Angewandte Mathematiker௅better known as Crelle’s Journal, in honour of its founder August Leopold Crelle (1780-1855), who was also the editor until his death in 1855௅published by the University of Berlin. In 1896, Klein finally succeeded in offering the other chair of mathematics of the University of Göttingen to Hilbert and convincing him to accept the offer. Both Klein and Hilbert would cause Göttingen to become the world centre of mathematics during practically the first half of the twentieth century. The collaboration between both mathematicians would also be a key for the future of Mathematische Annalen, since in 1902 Klein presented Hilbert as editor-in-chief of the journal, a position he kept until 1930. In the 90’s, Klein continued his research on function theory, but became increasingly involved in his activities as Wissenschaftpolitiker. One of the main objectives in Klein’s reform agenda was to strengthen the relationship between mathematics and its applications in science and technology. In this sense, Klein promoted, with the help of the most powerful German industrialists of the moment, the creation in 1898 of the Göttinger Vereinigung zur Förderung der angewandten Physik (Göttingen association for the promotion of applied physics), which expanded in 1901 to also include mathematics. This association, which had such important firm members as Bayer, Krupp, Siemens & Halske, AEG and Norddeutschen Lloyd, among others, not only sponsored and co-financed research activities in Göttingen, but also contributed to the construction of new buildings, particularly the one of the new Institute of Physics of the University of Göttingen, inaugurated in 1904. Klein’s dream was to use the resources of the association to erect a building and a similar institute for mathematics. But the resistance of the Ministry of Finance first, the outbreak of World War I and the inflation of the early twenties, postponed

62

Chapter Five

its realization. It was only in 1929, after the death of Klein and thanks to the Rockefeller Foundation, when the first building of the Mathematical Institute finally opened.



CHAPTER SIX HILBERT’S FIRST YEARS AT GÖTTINGEN

Hilbert finished his report on the theory of numbers at the University of Göttingen, where he had incorporated during the Easter of 1895. As we already know, the efforts to bring Hilbert to Göttingen had been mainly taken by Felix Klein, who was not only a great mathematician, but also a first-class Wissenschaftpolitiker. Hilbert spent the rest of his academic career in the University of Göttingen, which became, under the leadership of Klein and Hilbert, the world’s most important mathematical research centre and a paradigm of modern universities. After a first failed attempt in 1892 led Heinrich Weber to Göttingen instead of Hilbert, Klein took advantage three years later of the fact that Weber had accepted a chair in Strasbourg to request the addition of Hilbert to the teaching staff of Göttingen. A letter from Klein to Hilbert shows Klein’s determination in his intentions and his conviction that Hilbert was the most appropriate person to carry them out, as well as his admiration for his mathematical talent: This evening the faculty will meet, and although I cannot know in advance what the commission will recommend, I want to inform you that I will make every effort to ensure that no one else but you is called here. You are the man I need as my scientific complement: due to the direction of your work, the strength of your mathematical talent and the fact that you are now in the middle of your productive career. I count on you to give a new impulse to the mathematical school here, which has grown and, apparently, will continue to grow much more [...] I cannot know if I will prevail in the faculty, and even less if the recommendation that I will make will ultimately be heard in Berlin. But one thing you should promise me today: that you will not decline the offer if it comes to you! 1

Hilbert finally arrived in 1895 and with his arrival Göttingen gradually became the centre of reference for mathematical research in Germany and throughout the world. In the mid-90s, matriculation in mathematics

 1



Frei 1985, 115.

64

Chapter Six

courses in German universities was constantly increasing. In Göttingen, the number of mathematics and science students had dropped from a maximum of 240 during the year 1882 to only 90 ten years later, but in 1900 it had risen again to 300, and since then it continued to grow, reaching almost 800 students enrolled in 1914. In addition, according to Klein’s own estimates, between 10 and 15 percent of students were at an advanced level and could benefit fully from the resources that offered the University of Göttingen.

Fig. 6-1 The University of Göttingen at the beginning of the 20th century

Among the facilities offered by the University two of them stood out, which had been promoted by Klein with the aim of making the University of Göttingen “more invincible than ever.” The first was a collection of mathematical models, which aimed at the use of physical models and experimental instruments in education and research. The second was the library of mathematics, popularly known as das Lesezimmer (the reading room), a library in which the books were placed on open shelves so that the students could freely consult them. As noted by van der Waerden, “today, every mathematics department has a library, in which every student can take the books and journals directly from the shelves, but in 1924, when I came to Gottingen as a student, this was an exception.”2 One of the advantages of the Lesezimmer, as van der Waerden

 2



Van der Waerden 1983, 1-2.

Hilbert’s First Years at Göttingen

65

remembered, was that sometimes you were going to look for a specific book and ended up consulting another one that was better. “In this way, I learned more in weeks or months in the Lesezimmer than many students learn in years and years.”3 Regarding the academic staff, the key date that precipitated events in the meteoric rise of Göttingen as a centre of mathematical excellence was 1902. That year, Hilbert was named successor of Lazarus Fuchs at the University of Berlin but declined the offer after Althoff had agreed to create an Ordinariat in Göttingen for Minkowski. To the arrival of Minkowski in 1902, was added two years later that of Carl Runge (18561927), who would occupy a newly created chair of applied mathematics. Other prominent figures who joined the University of Göttingen in those years were the astronomer Karl Schwarzschild (1873-1916), Ludwig Prandtl (1875-1953), a precursor of aerodynamic and hydrodynamic research, and the geophysicist Emil Wiechert (1861-1928). Even before Klein officially retired in 1912, Hilbert began to preside the weekly meetings of the Mathematical Society of Göttingen. After these meetings, many of the participants would meet at the Rohn’s Cafe, located in the Hainberg on the outskirts of the city, for a follow up session (Nachsitzung). There the young mathematicians could expose their ideas in a much more relaxed and kind environment than the meetings of the Mathematical Society, in which the Prussian sense of order used to prevail. Norbert Wiener (1894-1964), the famous American mathematician and father of cybernetics, who arrived in Göttingen in 1914 to study with Hilbert and Landau, would say a few years later that “the combination of science and social life in the Nachsitzungen at Rohn’s café up the hill was particularly attractive to me. The meetings had a certain resemblance to those of the Harvard Mathematical Society, but the older mathematicians were greater, the younger men were abler and more enthusiastic, and the contacts were freer. The Harvard Mathematical Society meetings were to the Göttingen meetings as near beer is to a deep draft of Münchener.”4 Another type of meeting took place on Thursday afternoons, the socalled Bonzenspaziergang (the walk of the mandarins) another tradition instituted by Klein. This walk, also going up the mountain until reaching Rohn’s café, gave Göttingen professors an opportunity to discuss academic affairs distantly. In these walks, in which the mandarin mathematicians (Klein, Hilbert, Minkowski and Runge) were always

 3 4



As reported by Reid (1970, 162). Wiener 1953, p. 211.

66

Chapter Six

involved, they discussed “current events in science and in the life of the University”5 (for example, the positions that were later taken in the formal meetings of the Faculty of Philosophy of Göttingen). Despite the diversity of courses taught by Klein during the first years in Göttingen, his plan to gradually abandon the first line of mathematical research and the new role he carried out as a professor would have been disastrous for mathematics at Göttingen if Hilbert had not occupied his place in both tasks: research and academic. But Hilbert responded fully to the demands of the moment and quickly became the reference as a professor and researcher in Germany and the entire world in the field of mathematics. Some examples of this are not only his contributions to number theory and geometry through masterpieces of modern mathematics such as the Zahlbericht and Grundlagen der Geometrie respectively, or his ability to glimpse the future of mathematics in his famous list of Mathematical Problems presented in Paris in 1900, but also the more than sixty doctoral theses directed by him in Göttingen between 1895 and 1914 and his leadership in the task of converting Göttingen in the Mecca of modern mathematics at the turn of the last century. In his excellent Lebensgeschichte, Otto Blumenthal, Hilbert’s first doctoral student in Göttingen, has left us a magnificent portrait of Hilbert’s first years as a teacher in this university. Blumenthal explains that Hilbert’s lecture courses were characterized by their austerity and lack of ornaments, by their sometimes-vacillating tone and a certain tendency to repeat the most important theorems for everyone to understand it, but that the diaphanous clarity and the wealth of the content in his lectures made them forget any defect in their form. In his lectures he always incorporated both new and his own results, but without emphasizing them expressly. Hilbert, says Blumenthal, always “tried to be clear, understandable to everyone. He lectured for students, not for himself.”6 During the eight and a half years he had been in Königsberg, Hilbert had never taught two courses on the same subject, except for a weekly one-hour course on determinants. This allowed him to adjust easily to Klein’s wishes and to teach courses on the most diverse subjects in Göttingen. The first semester he gave a course on determinants and elliptical functions and led, along with Klein, a seminar on real functions every Wednesday morning. In the seminars, Hilbert was always very attentive, he was generally friendly and appreciated the work of others, but he could also lose patience and cut off a student who made an inadequate

 5 6



Born 1978, 98. Hilbert 1965, vol. 3, 400.

Hilbert’s First Years at Göttingen

67

presentation (“This is chalk, chalk, nothing more than chalk!”, he used to exclaim with his strong Königsberg’s accent) or criticize it sharply if he considered it necessary (“But that’s completely trivial!”). After each seminar, Hilbert used to meet with the participants and walk to a rural establishment (Waldwirtschaft) on the hills overlooking Göttingen, where he talked about mathematics. With the most prominent participants in the seminar, his doctorates, with whom he had a more intimate contact, he used to take a longer weekly walk through the mountains around Göttingen. In these walks doctoral students could take the opportunity to ask questions about their research, but it was mainly Hilbert who spoke to them of algebraic number theory, which was the research topic that occupied him until 1899. At the beginning of 1896, Hilbert’s part of the Zahlbericht was practically finished, but Minkowski’s was not. In the month of February, Hilbert proposed that either the Minkowski’s part be published as it was with his part, or that it be published separately the following year. Minkowski accepted the second proposal with gratitude. Just a little later, Hilbert had already finished his report on algebraic number fields and sent it to press. As the galleys arrived they were sent to Minkowski for correction. In the autumn of 1896, Minkowski accepted, a bit displeased, a place in Zürich, where he met Hurwitz again, so that the two old friends corrected the galleys of Hilbert’s report that were still to be reviewed. In the month of April 1897, Hilbert finally sent the Zahlbericht to be printed. This allowed him to take care of some more specific investigations that he had had to leave aside during the writing of the report on number theory. More concretely, Hilbert concentrated on trying to generalize the law of reciprocity to the algebraic fields of numbers (quadratic, cyclotomic and Kummer’s fields) that he had investigated in the Zahlbericht. This allowed him to enunciate the law of quadratic reciprocity in a simple and elegant way that also made it applicable to these algebraic number fields. This result was the main target of the articles “Über die Theorie des relativquadratischen Zahlkörper” (1899) and “Über die Theorie der relative-Abelschen Zahlkörper” (1902),7 which constitute the culmination of Hilbert’s work in algebraic number theory. Furthermore, in this last article Hilbert sketched the basic features of what would later be called class field theory. More concretely, Hilbert introduced the notion of a non-ramified class field, although his idea of

 7

In Hilbert (1965, vol 1, 370-482 and 483-509, respectively). The last article is a reimpression with minor changes of a paper previously published in Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, mathematischphysikalische Klasse (1898).



68

Chapter Six

class field theory was quite a bit more general, not limited to the theory of non-ramified class fields. The origins of class field theory can be traced back to the work of Kronecker. In examining the work of Niels Henrik Abel (1802-1829), he observed that certain abelian extensions of imaginary quadratic number fields are generated by the adjunction of special values of automorphic functions arising from elliptic curves. Furthermore, he also observed that each abelian extension of Է is a subfield of a field obtained by adjoining to Է a special value of an automorphic function. This led him to state in the article “Über die algebraisch auflösbaren Gleichungen” (“On Algebraically Solvable Equations”) (1853) the following: (Kronecker-Weber Theorem). Let ‫ ܭ‬be a finite abelian extension of Է. Then there exists a positive integer ݊ such that ‫ ك ܭ‬Է൫݁ ଶగ௜ Τ௡ ൯.8

(The numbers ݁ ଶగ௜Τ௡ are the roots of unity which together with the rational numbers constitute the field of cyclotomic numbers. Hence the KroneckerWeber theorem can also be stated as saying that every abelian extension of Է is contained in a cyclotomic extension of Է). Kronecker wondered whether all abelian extensions of imaginary quadratic fields are obtained in this way, that is, they are subfields of fields obtained by adjoining to them special values of automorphic functions. This was Kronecker’s Jugendtraum (dream of his youth). So Kronecker posed the main question of class field theory, namely, finding all abelian extensions of given algebraic number fields. In the preface of the first paper above mentioned, Hilbert wrote: The methods which I have used in the following for the investigation of relative quadratic fields are, by a suitable generalization, also useful with the same results in the theory of relative abelian fields of arbitrary relative degree, and then lead specially to the general reciprocity law for arbitrary higher power residues in an arbitrary algebraic number domain.9

 8

Kronecker did not manage to prove the theorem for field extensions of degree a power of ʹ. In the article “Theorie der Abel’schen Zahlkörper” (“Theory of Abelian Number Fields”) (1886), Weber supplied a proof that was also incomplete. The first complete proof of the theorem was published in Hilbert’s paper “Ein neuer Beweis des Kronecker’schen Fundementalsatzes über Abel’sche Zahlkörper” (“A New Proof of Kronecker’s Fundamental Theorem for Abelian Number Fields”) (1896). 9 Hilbert 1965, vol 1, 370.



Hilbert’s First Years at Göttingen

69

The subject of the 1899 paper was in effect to generalize the law of quadratic reciprocity, so that it becomes valid not only for cyclotomic fields but also for any arbitrary algebraic number field. He conjectured the following (Satz 60): Hilbert’s reciprocity law: Let ݇ be an algebraic number field containing ఓǡఔ the ݉-th roots of unity; then for all ߤǡ ߥ ‫ ܭ א‬ൈ , we have ς௣ ቀ ቁ ൌ ͳ, ௣

where ቀ

ǤǡǤ ௣

ቁ is Hilbert’s ݉-th power norm residue symbol mod ‫݌‬, and the

product is extended over all prime places of ‫ܭ‬.

As we will explain in a moment, Hilbert’s conjectured reciprocity law was part and parcel of one of the problems listed in his famous address at the International Congress of Mathematicians (ICM) in Paris (1900). Hilbert’s approach in the 1899 article was based on the genus theory in a relative quadratic number field and so he had to impose on the base field ‫ ܭ‬a couple of restrictions that converted the relative quadratic field over ‫ܭ‬ into a relatively ramified field. However, in the paper of 1902 Hilbert banished the two conditions imposed on the base field so that the resultant relative quadratic number field was non-ramified. In this way, Hilbert obtained for the first time the notion of a non-ramified class field (nowadays we usually call the maximal unramified abelian extension of ‫ܭ‬ the Hilbert class field of ‫)ܭ‬. The introduction of non-ramified class fields in the context of relative abelian number fields is usually the point of departure of class field theory. After 1899 Hilbert no longer wrote about class field theory, although nine of the doctoral theses directed by him addressed issues related to this topic. It must also be noted that two of the problems stated by Hilbert in his famous address to the ICM in Paris in 1900 were on class field theory, namely problems 9 and 12. Problem 9 asks for a proof of the most general law of reciprocity in any number field. A first step in this direction had been made by Hilbert himself when he conjectured a reciprocity law for arbitrary algebraic number fields. Philipp Furtwängler (1869-1940), who proved in 1907 and 1930 most of Hilbert’s conjectures on class field theory, Teiji Takagi (1875-1960), Hasse and Artin succeeded in finding reciprocity laws in algebraic number fields of an increasing generality. Artin’s reciprocity law (1927) is usually considered as the final response to Hilbert’s quest for the “most general reciprocity law” for abelian extensions; for non-abelian extensions the problem is still open. Problem 12 asks for a generalization of Kronecker’s Jugendtraum, that is, the extension of Kronecker-Weber Theorem on abelian fields to any algebraic realm of rationality. Kronecker’s Jugendtraum was fulfilled later by



70

Chapter Six

Weber in 1908 and Rudolf Fueter (1880-1950) in 1914, who partially proved Kronecker’s conjecture, and finally by Takagi in 1920, who gave a complete proof of it within class field theory. Problem 12 is still a major open problem although Goro Shimura (1930-) has made a good deal of progress on this problem.



CHAPTER SEVEN THE FOUNDATIONS OF GEOMETRY

Hilbert’s announcement that he would give the course Elemente der Euklidischen Geometrie (Elements of Euclidean Geometry) in the winter semester of 1898/99 was a surprise for his students, even the more veterans such as Blumenthal who regularly participated in the long walks with Hilbert, since they “had never noticed that Hilbert was occupied with geometrical issues; he only spoke about number fields.”1 However, Hilbert’s interest in geometry was not new, given that he had already taught several courses in Königsberg on geometry. Moreover, Hilbert had also given two summer courses (Ferienkurse) in Göttingen aimed at Oberlehrer (professors in Gymnasien or in similar institutions) during the school holidays of 1896 and 1898. The lectures of 1898/99 offered a description of geometry very similar to that of the lectures on the foundations of geometry of 1894 (see Chapter 3). In both cases, geometry was characterized as “the most complete natural science” whose objective is “the logical derivation of all facts belonging to its domain from well-known basic propositions.”2 Both lecture courses presented the same point of view on how the axiomatic analysis of geometry proceeds, by providing a scheme or network of concepts obtained by abstraction from the geometrical facts, corresponding the axioms to the basic geometrical facts. Finally, in both courses the demand for completeness was understood as the claim that there can be no geometrical facts that have no corresponding theorem in the axiomatic system, that is, as a form of deductive completeness of the axiom system relative to the geometrical facts.3 The 1898/99 lecture notes presented, however, an important novelty with respect to those of 1894. This was the formalistic standpoint about

 1

Hilbert 1965, vol. 3, 402. Hilbert 2004, 302. 3 This must be taken with some caution, since no logical system is specified or presupposed. So “deductive completeness” cannot be understood exactly in the same sense that we usually found in logical textbooks. 2

72

Chapter Seven

the geometrical concepts or terms that Hilbert had learned at Wiener’s conference in 1891 and here was clearly stated for the first time. Thus, Hilbert now writes, to set up Euclidean geometry, we must think about three systems of things, points, lines and planes. However: Despite the chosen names, we should not allow ourselves to be tempted to attribute to these things certain geometrical properties, which we usually associate with these names. So far, we only know that each thing in a system is different from each thing of the other two systems. All other properties of these things will be given only through the axioms.4

This would also be the point of view of what is perhaps the best-known work of Hilbert and one of the most celebrated works in the history of mathematics: Grundlagen der Geometrie (The Foundations of Geometry). It is popularly known with the nickname of Festschrift, since it was part of a volume edited by the University of Göttingen to celebrate the inauguration of a monument dedicated to two of its most illustrious professors: Gauss and Weber. The volume in question consisted of two works, Hilbert’s work and a work by Emil Wiechert entitled Grundlagen der Elektrodynamik (Foundations of Electrodynamics), but nowadays nobody remembers this second work and the name of Festschrift has remained inextricably linked to Hilbert’s work. It is a well-known fact that Hilbert’s contribution to the Festschrift was composed quickly just after the lecture course of 1898/99 ended and that it was Klein who suggested to Hilbert that he present the material of these lectures to celebrate the inauguration of the Gauss-Weber monument, which was finally carried out on June 17, 1899.5 As is well known, Euclid had systematized elementary plane and solid geometry in his Elements, taking as reference Aristotle’s theory of science. Thus, we can find in the first book of the Elements a list of propositions, stated without proof, classified into three groups: definitions (ex: “A point is that of which there is no part”), common notions (ex: “The whole is greater than the part”) and postulates (ex: the parallel axiom, also called Euclid’s Postulate). From these propositions are proved, by means of logic, the most important theorems of plane geometry (Euclid’s preferred method of proof is the indirect proof or reductio ad absurdum). Nonetheless, since the seventeenth century the mathematical community had been aware that several proofs in the Elements used

 4

Ibid., 304. The following paragraphs are taken almost verbatim from Roselló (2012, 10910). 5

The Foundations of Geometry

73

assumptions or hypotheses which were not spelled out in its postulates, common notions and definitions. Hence a thorough reworking of Euclidean geometry was required in a way that would avoid these “gaps” in the deductive structure of the Elements. Moreover, the development during the XIX century of hyperbolic geometry by Nikolai Lobachevski (1792-1856) and János Bolyai (1802-1860) and elliptic geometry by Riemann (these are the so-called non-Euclidean geometries), and also of projective geometry by Jean-Victor Poncelet and non-Archimedean geometry by Giuseppe Veronese (1854-1917), raised the need for a deductive organization of all geometrical knowledge that made visible the logical connections between the new geometries and Euclidean geometry. For example, in hyperbolic geometry, straight lines can be extended indefinitely and there are infinite lines parallel to a given line passing through a point outside it, whereas in elliptic geometry, straight lines are finite and there are no parallel lines. Thus, in hyperbolic and elliptic geometry the first four postulates of Euclid are valid, but the fifth (the axiom of parallels) is not. Therefore, if we denote by Ȉ the set of the first four axioms of Euclid and by Į the axiom of parallels, then it follows that in Euclidean geometry 6 ‰ D is valid, whereas in non-Euclidean geometry 6 ‰ ™D is valid. Similarly, if we denote by E the axiom of Archimedes (an axiom of continuity, not explicitly stated by Euclid, but necessary for the deductions carried out in the Elements) and by Ȉ the other axioms of Euclidean geometry, then in Euclidean geometry 6 ‰ E is valid, while in non-Archimedean geometry 6 ‰ ™E is valid. However, since all these geometries arise in denying a postulate of Euclidean geometry, whose axioms express supposedly evident and necessary truths about space that are the foundation of modern physics and astronomy, we cannot exclude the possibility that all these geometries are inconsistent, i.e., contradictory. And in the case they were all consistent, which of them was true? Hilbert responded to all these questions in Grundlagen der Geometrie by applying the axiomatic method and a new conception of it he called formal axiomatics. The first objective of this method was to look for a system of axioms through which it would be possible to characterize the basic facts (Grundtatsachen) of Euclidean geometry and to prove its deductive completeness with respect to all the other geometric facts (Thatsachen), that is, a system of axioms for Euclidean geometry had to be found that would be complete in the sense it constituted a sufficient basis for a rigorous derivation of all the true statements in this domain of mathematics.

74

Chapter Seven

The parallel axiom and non-Euclidean geometries Euclid defined the notion of parallelism in Elements as follows: Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction. The fifth postulate, the famous parallel axiom or Euclid´s postulate is enunciated in the following terms: That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. To see the intuitive connection between the definition of parallelism and the parallel axiom it is sufficient to observe the following drawing:

In this figure, the angles ‫ ܥܣܤס‬i ‫ ܦܥܣס‬together are smaller than two straight angles and, in agreement to Euclid´s postulate the lines AB and CD will meet each other when being indefinitely extended, whereas if the lines AE and CD are parallel, then Euclid´s postulate guarantees that the angles ‫ ܥܣܧס‬i ‫ ܦܥܣס‬together are equal to two straight angles. Euclid’s fifth postulate often appears in modern mathematical texts in the following terms: In a plane, given a straight line and a point not on it, at most one line parallel to the given line can be drawn through the point. cont. p. 75

The Foundations of Geometry

75

This statement is known as Playfair’s axiom, in honour of the Scottish mathematician John Playfair (1748-1819). In the presence of the remaining postulates of Euclid, each of these axioms can be used to prove the other, so they are equivalent in the context of absolute geometry (this is the geometry obtained from Euclidean geometry when the parallel postulate is removed and none of its alternatives is used in place of it). Traditional non-Euclidean geometries (hyperbolic and elliptic geometries) arise when the parallel axiom is replaced with an alternative one. As we have seen, the parallel axiom is equivalent to the axiom asserting that, given a straight line and a point outside this line, there is exactly one line parallel to the given line passing through that point. Now, in hyperbolic geometry, by contrast, there are infinitely many lines through that point not intersecting the line, whereas in elliptic geometry, any line through that point intersects the line and so there are no parallel lines. The first treatises on hyperbolic geometry were published independently around 1830 by the Hungarian mathematician János Bolyai and the Russian mathematician Nikolai Lobachevski. Because of this, hyperbolic geometry is sometimes called Bolyai-Lobachevski geometry. In a famous lecture of 1845, the German mathematician Bernhard Riemann founded the field of Riemannian geometry. He constructed a family of geometries which are non-Euclidean by giving a formula for a family of Riemannian metrics on the unit ball of Euclidean space. The simplest of these geometries is elliptic geometry.

Secondly, the consistency of Euclidean geometry had to be proven, namely, the impossibility to derive from the axioms of geometry a contradiction (a statement and its negation). Actually, the proof of the consistency of Euclidean geometry was an urgent matter at the time of Hilbert, since the proofs of the consistency of non-Euclidean geometries assumed this fact. For example, Eugenio Beltrami (1835-1900) had proved in 1868 that if the hyperbolic geometry contained a contradiction, then a contradiction would also arise in Euclidean geometry, thereby that if Euclidean geometry was consistent, then hyperbolic geometry was also. Finally, it was necessary to prove the independence of the axioms of parallels and Archimedes, that is, it had to be proven that these axioms were not a logical consequence of the other axioms of Euclidean geometry. For only in this way it was possible to ensure that Euclidean

76

Chapter Seven

The imaginary or hyperbolic geometry of Lobachevski  In Euclidean geometry, the sum of the angles of a rectangle triangle is equal to ߨሺൌ ͳͺͲιሻ. In the article “On the principles of geometry” (1928), Lobachevski considered the possibility of a geometry, which he called imaginary, in which the sum of angles in a rectangle triangle was less than ߨ and the lines depend on the angles. Lobachevski defined the angle of parallelism as follows: Given a line l, a perpendicular a respect to l and a point P on a outside l, the angle of parallelism is the angle between the perpendicular p and the parallel l’ to l, that is, the first line through P that does not meet with l:

In Euclidean geometry, this angle is always equal to ߨΤʹ, but in Lobachevski geometry it is acute and is a function of a. In the abovementioned article, Lobachevski denotes this function by ‫ܨ‬ሺܽሻ, but in the following articles he will denote it by ȫሺܽሻ. It is clear that: Ž‹௔՜଴ ȫሺܽሻ ൌ ߨΤʹ and Ž‹௔՜ஶ ȫሺܽሻ ൌ Ͳ. Lobachevski extends this function to all the real values of a putting ȫሺͲሻ ൌ ߨΤʹ and Ž‹ ȫሺെܽሻ ൌ ߨ െ ȫሺܽሻ and shows that for every angle A, acute or obtuse, there is a value a (ܽ ൐ Ͳ if A is acute and ܽ ൏ Ͳ if A is obtuse) such that ‫ ܣ‬ൌ ȫሺܽሻ. Then Lobachevski finds the trigonometric formulas for rectilinear and spherical triangles in its space.

geometry and non-Euclidean geometry (non-Archimedean geometry) could not both be inconsistent, because if the parallel axiom (Archimedes axiom) was a logical consequence of the other axioms of Euclidean geometry, then non-Euclidean geometry (non-Archimedean geometry) would be inconsistent, for the parallel axiom would be valid in it (the Archimedean axiom) and its denial.

The Foundations of Geometry

77

Regarding the importance of the demonstration of the independence of the axioms of the parallels and of Archimedes in relation to the genesis of the axiomatization of geometry carried out by Hilbert in his work of 1899, Hilbert himself recognized in a letter to Gottlob Frege (1848-1925) that: I was necessarily led to set up my axiomatic system, since I wanted to be able to understand those geometrical propositions that I regard as the most important results of geometrical inquiries: that the parallel axiom is not a consequence of the other axioms, and similarly for the Archimedes axiom, etc. 6

In fact, the axiomatic method constituted for Hilbert the ideal method to demonstrate the logical connections between the different geometries that had been developed throughout the nineteenth century. For this reason, as Hilbert’s most illustrious contemporary mathematicians emphasized, his work constituted such an important turning point in the research on the foundations of geometry and, in general, of mathematics. Thus, for example, the American mathematician Oswald Veblen (1880-1960) wrote in an article on “Hilbert’s Foundations of Geometry” (1903), just three years after the appearance of Hilbert’s work, that: Since its appearance in 1899 Hilbert’s work on The Foundations of Geometry has had a wider circulation than any other modern essay in the realms of pure mathematics. 7

That same year, the great French mathematician Henri Poincaré affirmed in a review of Hilbert’s Grundlagen for the Bulletin of the American Mathematical Society, that: [Hilbert’s work] made the philosophy of mathematics take a long step forward, comparable to those which were due to Lobachevsky, to Riemann, to Helmholtz, and to Lie.8

We could say, in short, that Grundlagen der Geometrie introduced a new conception of the axiomatic method that changed the way of thinking and doing mathematics throughout the 20th century, becoming a reference textbook in geometry for future mathematicians and professors of mathematics. In addition, it was an authentic mathematical best-seller, with 14 editions in the twentieth century (the last one, as far as we know,

 6

Frege 1976, 65. Cited by Ehrlich (1994, xxiv). 8 Ibid. 7

Chapter Seven

78

in 1999 to celebrate its centenary), which gave a well-deserved worldwide reputation to his author, David Hilbert. 



CHAPTER EIGHT THE AXIOMATIZATION OF GEOMETRY

Hilbert described, in the Introduction of the Festschrift, that the purpose of this work was an attempt to find “a simple and complete system of mutually independent axioms and to derive from them the most important geometrical propositions.”1 We have previously mentioned the meaning and importance of the requirements of independence and completeness in the context of Hilbert’s geometrical researches. The requirement of simplicity means roughly that an axiom should contain “no more than a single idea.” This is a requirement that Hilbert had also formulated in his lectures of 1894 and 1898/99 and that he had taken again from Hertz. However, neither Hilbert nor the members of his circle of collaborators were ever capable of giving an explicit or formal definition of this feature. It is remarkable that consistency is not explicitly mentioned as one of the requirements to be fulfilled by the axiom system set out in the Festschrift. This is somewhat surprising, because consistency would become the main metamathematical question in Hilbert’s future research and, as said before, proofs of the consistency of non-Euclidean geometry presupposed the consistency of Euclidean geometry. However, Hilbert addressed this question right after introducing all the groups of axioms and discussing their immediate consequences. In the Festschrift Hilbert defined the Euclidean space as a domain of elements of an arbitrary nature subdivided into three different systems, the elements of which are subject to certain relationships that are specified by the axioms. The work began, indeed, with the following definition or explanation (Erklärung): Let us consider three distinct systems of things. The things composing the first system, we will call points and designate them by the letters ‫ܣ‬ǡ ‫ܤ‬ǡ ‫ܥ‬ǡ Ǥ Ǥ Ǥ Ǣ those of the second, we will call straight lines and designate them by the letters ܽǡ ܾǡ ܿǡ ǥ Ǣ and those of the third system, we will call planes and designate them by the Greek letters ߙǡ ߚǡ ߛǡ ǥ […] We think of these points, straight lines, and planes as having certain mutual relations,

 1



Hallet & Majer 2004, 436.

80

Chapter Eight which we indicate by means of such words as “lie,” “between,” “parallel,” “congruent,” “continuous,” etc. The complete and exact description of these relations follows as a consequence of the axioms of geometry.2

These axioms are formulated for three systems of things named “points,” “lines” and “planes” in order to describe or specify certain fundamental relations between these things which he indicates by the words “lie,” “between,” “parallel,” “congruent” and “continuous.” Euclid had defined “point” as “that which has no parts,” “line” as a “breadthless length” and “surface” as “that which has length and breadth only.” However, he hadn’t defined the meaning of “part,” “length” and “breadth.” Thus, only the spatial intuition of these undefined terms could help the understanding of the previous concepts. So Euclid’s definitions did not contribute at all to the rigor of geometry. Hilbert, however, made no attempt to define “point,” “line” or “plane,” but simply postulated the existence of three systems of arbitrary elements that he called “points,” “lines” and “planes,” but could have also called “chairs,” “tables” and “beer mugs”–the example, as reported by Blumenthal, is from Hilbert!–because what really matters is not the nature of the elements, but the fact that these elements satisfy the axioms. If we do not think of the above elements as objects of our spatial intuition, we do not have to think of the axioms as truths relating to real space. Actually, according to Hilbert, beyond expressing “certain associated fundamental facts of intuition,” what the axioms do is to determine or implicitly define the fundamental relationships between the elements of the different systems by enunciating their basic properties.3 Thus, in Hilbert’s axiomatization of geometry “the bond with reality is cut.”4 This bond of geometry with real space constituted not only for Euclid, but also for Pasch, Federigo Enriques (1871-1946), Veronese, Mario Pieri (1860-1913) or Klein, the source from which geometric intuition sprouts. From Hilbert’s Festschrift onwards “geometry has become pure mathematics. The questions of whether and how to apply to reality is the same in geometry as it is in other branches of mathematics.”5

 2

Hilbert 2004, 437. Hilbert doesn’t use the term implicit definition, introduced by Joseph Gergonne (1771-1859), but when he introduces the second and third group of axioms, he says that these groups of axioms define (definieren) respectively the relations of order and congruence. 4 Freudenthal 1962, 618. 5 Ibid. 3



The Axiomatization of Geometry

81

Since there are five basic relations between the elements of the different systems–“lie,” “between,” “congruent,” “parallel” and “continuous”–, there are also five groups of axioms that define these relations: I. Axioms of incidence, II. Axioms of order, III. Axioms of congruence, IV. Parallel Axiom and V. Axioms of continuity. The first group of axioms defines the relation of incidence or “lies on”: “a point lies on a line,” “a line lies on a plane” and “a point lies on a plane.” The second group of axioms defines the order relation or “lies between”: “a point lies on a line between two points.” This relation was not defined in the work of Euclid, despite being fundamental for proving most of the properties of plane geometry. Thus, for example, it is necessary to define the concepts of segment and angle and to distinguish between the inner and outer points of a triangle. The third group of axioms is constituted by a single axiom, the famous Euclid’s fifth postulate or parallel axiom. It would seem perhaps more logical that Hilbert had considered the relation of parallelism as a primitive relation, as he had done with incidence, order and congruence relations. But Hilbert preferred to define the relation of parallelism in terms of the relation of incidence and, indeed, what is more common today is to define first the relation of parallelism in the manner indicated and then to state the parallel axiom. Actually, the fact that the parallel axiom is considered apart from the rest of the axioms is due only to its historical importance, especially in relation to the development of non-Euclidean geometries. The fourth group of axioms defines the relation of congruence between segments and angles, which is basic to the introduction of the concept of measure, being the measure a real number (length of a segment or measurement of an angle) associated with a segment or angle (the concept of measure is introduced from the relation of congruence by stipulating that two segments or angles have the same measure if, and only if, they are congruent). The last group of axioms includes the Axiom of Archimedes (V.1) and an Axiom of completeness (Vollständigkeitsaxiom) (V.2), added for the first time in the French translation of 1900 and present from the second edition of the Festschrift on. These axioms allow to establish a bijective correspondence between the set of points in a straight line and the system of real numbers and, therefore, to use the real numbers to introduce the metric ideas of length and measure of angles from the relation of congruence. These ideas are at the same time essential for the study of the similarity of figures and areas. The first continuity axiom is the axiom of Archimedes, which is nothing more than a geometric statement of the



82

Chapter Eight

well-known principle of Archimedes. If we assume indeed the idea of length and ܽ is the length‫ܣܣ‬ଵ and ܾ the length AB (see the formulation of the axiom in the textbox below), then the axiom can be formulated in its most common analytic form as follows: (Arquimedes principle): Let ܽ ൐ Ͳ and ܾ ൐ Ͳ be arbitrary real numbers, then there is a positive integer n such that ݊ܽ ൐ ܾ.

Archimedes’ axiom converts the set of points of a straight line into an Archimedean ordered field–assuming the order relation is defined in the usual form–, which ensures that all the points of a straight line can be injected into the real numbers–since it can be demonstrated that any Archimedean ordered field is included in the system of real numbers. However, it might happen that the above set of points, despite satisfying the previous axioms (the axiom of Archimedes included), could be extended with the addition of new points, with which the set of points of a straight line would be no more in one-to-one correspondence with the system of real numbers and then one could not use the real numbers to measure segments and angles. The immediate consequence of this is that the system of axioms presented by Hilbert in the first edition of Grundlagen der Geometrie was not complete. For example, it was incapable of proving that a straight line with points inside and outside a circle must intersect this circle. For this reason, from the second edition on, Hilbert added to the previous axiom system an axiom of completeness, which affirmed basically that it is not possible to add new points to the points in this system, so that the system that is obtained by composition satisfies all the axioms. In more technical terms, we could say that the axiom of completeness states that the Euclidean space characterized by the previous axioms, including the completeness Axiom, is a maximal (not extensible) model of the axioms I-V-1. To prove the consistency and independence of the axioms above, particularly the independence of the Euclidean postulate and Archimedes’ axiom, Hilbert developed and perfected a method whose basic idea was to find a model for the axioms of the theory in question (e.g., Euclidean geometry or hyperbolic geometry), that is, a well-defined set in which to interpret the primitive terms of the theory, so that the axioms turn out to be true and thereby, if the rules of inference preserve truth, the theorems inferred from them will also be true. If all the axioms of the theory turn out to be true in the model, then the theory is consistent relative to this model. If all axioms but one are true in the model, then this axiom is independent from the other ones.



The Axiomatization of Geometry

83

The axiomatization of Euclidean geometry (Grundlagen der Geometrie, 1899) Primitive terms: point, straight line, plane, lying on, lying between, being congruent. Group of axioms I: Axioms of connection (or incidence) I 1. Two distinct points ‫ ܣ‬and ‫ ܤ‬always completely determine a straight line a. We write ‫ ܤܣ‬ൌ ܽ or ‫ ܣܤ‬ൌ ܽ. I 2. Any two distinct points of a straight line completely determine that line; that is, if ‫ ܤܣ‬ൌ ܽ and ‫ ܥܣ‬ൌ ܽ, where ‫ܥ ് ܤ‬, then is also ‫ ܥܤ‬ൌ ܽ. I 3. Three points ‫ܣ‬ǡ ‫ܤ‬ǡ ‫ ܥ‬not lying in the same straight line always completely determine a plane ߙ. We write ‫ ܥܤܣ‬ൌ ߙ. I 4. Any three points A, B, C of a plane ߙ, which do not lie in the same straight line, completely determine that plane. I 5. If two points A, B of a straight line ܽ lie in a plane ߙ, then every point of ܽ lies in ߙ. I 6. If two planes ߙǡ ߚ have a point ‫ ܣ‬in common, then they have at least another point B in common I 7. Upon every straight line there exist at least two points, in every plane at least three points not lying in the same straight line, and in space there exist at least four points not lying in a plane. Group of axioms II: Axioms of order II 1. If ‫ܣ‬ǡ ‫ܤ‬ǡ ‫ ܥ‬are points of a straight line and ‫ ܤ‬lies between ‫ ܣ‬and ‫ܥ‬, then ‫ ܤ‬lies also between ‫ ܥ‬and ‫ܣ‬. II 2. If ‫ ܣ‬and ‫ ܥ‬are two points in a straight line, then there is always at least one point ‫ ܤ‬which lies between ‫ ܣ‬and ‫ܥ‬, and at least one point ‫ܦ‬ such that ‫ ܥ‬lies between ‫ ܣ‬and ‫ܦ‬. II 3. Of any three points of a straight line, there is always one, and only one, which lies between the other two. II 4. Any four points any ‫ܣ‬ǡ ‫ܤ‬ǡ ‫ܥ‬ǡ ‫ ܦ‬of a straight line can always be ordered in such a way that ‫ ܤ‬lies between ‫ ܣ‬and ‫ ܥ‬and also between ‫ܣ‬ and ‫ ܦ‬and, furthermore, that ‫ ܥ‬lies between ‫ ܣ‬and ‫ ܦ‬and also between ‫ ܤ‬and ‫ܦ‬. Definition. We call the system of two points ‫ ܣ‬and ‫ܤ‬, which lie on a straight-line ܽ, a segment and denote it by ‫ ܤܣ‬or ‫ܣܤ‬. The points between ‫ ܣ‬and ‫ ܤ‬are called the points of the segment ‫ ܤܣ‬or also the points lying within the segment ‫ܤܣ‬. All other points in the straight line ܽ are called the points lying outside the ‫ ܤܣ‬segment. The points ‫ܣ‬ǡ ‫ܤ‬ are called the final points of the segment ‫ܤܣ‬. II 5. Let ‫ܣ‬ǡ ‫ܤ‬ǡ ‫ ܥ‬be three points not lying in the same straight line and cont. p. 84



84

Chapter Eight

let ܽ be a straight line on the ‫ ܥܤܣ‬plane that does not meet with any of the points ‫ܣ‬ǡ ‫ܤ‬ǡ ‫ܥ‬. If, then, the straight line ܽ passes through a point of the segment ‫ܤܣ‬, it will always pass through either a point of the segment ‫ ܥܤ‬or through a point of the segment ‫ܥܣ‬. Group of axioms III: The axiom of parallels (Euclid’s postulate) III. In a plane ߙ, one can always trace through a point ‫ܣ‬, lying outside of a straight-line ܽ, one and only one straight line which does not intersect the line ܽ. This straight line is called the parallel to ܽ through the point ‫ܣ‬. Group of axioms IV: Axioms of congruence IV 1. If ‫ܣ‬ǡ ‫ ܤ‬are two points on a straight line ܽ and, furthermore, ‫ܣ‬ᇱ is a point on the same or another straight line ܽᇱ , then, on a determinate side of ‫ܣ‬ᇱ on the straight line ܽᇱ , one can always find one and only one point ‫ܤ‬ᇱ , so that the segment ‫( ܤܣ‬or ‫ )ܣܤ‬is congruent to the segment ‫ܣ‬ᇱ ‫ܤ‬ᇱ (or ‫ܤ‬ᇱ ‫ܣ‬ᇱ ); in symbols ‫ܣ ؠ ܤܣ‬ᇱ ‫ܤ‬ᇱ . Every segment is congruent to itself, i.e., we always have ‫ܤܣ ؠ ܤܣ‬. IV 2. If a segment ‫ ܤܣ‬is congruent to the segment ‫ܣ‬ᇱ ‫ܤ‬ᇱ and also to the segment ‫ܣ‬ᇱᇱ ‫ܤ‬ᇱᇱ , then the segment ‫ܣ‬ᇱ ‫ܤ‬ᇱ is also congruent to the segment ‫ܣ‬ᇱᇱ ‫ܤ‬ᇱᇱ ; that is, if ‫ܣ ؠ ܤܣ‬ᇱ ‫ܤ‬ᇱ and ‫ܣ ؠ ܤܣ‬ᇱᇱ ‫ܤ‬ᇱᇱ , then also ‫ܣ‬ᇱ ‫ܤ‬ᇱ ‫ܣ ؠ‬ᇱᇱ ‫ܤ‬ᇱᇱ . IV 3. Let ‫ ܤܣ‬and ‫ ܥܤ‬be two segments on the straight-line ܽ which have no points in common and, furthermore, let ‫ܣ‬ᇱ ‫ܤ‬ᇱ and ‫ܤ‬ᇱ ‫ ܥ‬ᇱ be two segments on the same or of another straight line ܽᇱ also without any point in common. Then, if ‫ܣ ؠ ܤܣ‬ᇱ ‫ܤ‬ᇱ and ‫ܤ ؠ ܥܤ‬ᇱ ‫ ܥ‬ᇱ , we always have that ‫ܣ ؠ ܥܣ‬ᇱ ‫ ܥ‬ᇱ . Definition. Let ߙ be an arbitrary plane and h, k any two different halfrays in ߙ that emerge from the point O which are part of two different straight lines. We call this system of two different half-rays h, k an angle and represent it with the symbol ‫ס‬ሺ݄ǡ ݇ሻ or ‫ס‬ሺ݇ǡ ݄ሻ. IV 4. Let ‫ס‬ሺ݄ǡ ݇ሻ be an angle in a plane ߙ and let ܽᇱ be a straight line in a plane ߙ ᇱ . Suppose also, that in the plane ߙ ᇱ , a definite side of ܽᇱ is assigned. We denote by ݄ᇱ a half-ray of the straight line ܽᇱ that has its origin in the point ܱᇱ of this line. Then in the plane ߙ ᇱ there is one and only one half-ray ݇ ᇱ such that the angle ሺ݄ǡ ݇ሻ, or ሺ݇ǡ ݄ሻ, is congruent to the angleሺ݄ᇱ ǡ ݇ ᇱ ሻ and, at the same time, all the interior points of the angle ሺ݄ᇱ ǡ ݇ ᇱ ሻ lie on the given side of ܽᇱ . In symbols: ‫ס‬ሺ݄ǡ ݇ሻ ‫ؠ‬ ‫ס‬ሺ݄ᇱ ǡ ݇ ᇱ ሻ. Every angle is congruent to itself, that is, we always have ‫ס‬ሺ݄ǡ ݇ሻ ‫ס ؠ‬ሺ݄ǡ ݇ሻ. IV 5. If the angleሺ݄ǡ ݇ሻ is congruent to the angle ሺ݄ᇱ ǡ ݇ ᇱ ሻ and also to the cont. p. 85



The Axiomatization of Geometry

85

angle ሺ݄ᇱᇱ ǡ ݇ ᇱᇱ ሻ, then the angle ሺ݄ᇱ ǡ ݇ ᇱ ሻ is congruent to ሺ݄ᇱᇱ ǡ ݇ ᇱᇱ ሻ, that is to say, if ‫ס‬ሺ݄ǡ ݇ሻ ‫ס ؠ‬ሺ݄ᇱ ǡ ݇ ᇱ ሻ and ‫ס‬ሺ݄ǡ ݇ሻ ‫ס ؠ‬ሺ݄ᇱᇱ ǡ ݇ ᇱᇱ ሻ, then we always also have ‫ס‬ሺ݄ᇱ ǡ ݇ ᇱ ሻ ‫ס ؠ‬ሺ݄ᇱᇱ ǡ ݇ ᇱᇱ ሻ. IV 6. If for the two triangles ‫ ܥܤܣ‬and ‫ܣ‬ᇱ ‫ܤ‬ᇱ ‫ ܥ‬ᇱ the congruences ‫ؠ ܤܣ‬ ‫ܣ‬ᇱ ‫ܤ‬ᇱ , ‫ܣ ؠ ܥܣ‬ᇱ ‫ ܥ‬ᇱ and ‫ܤס ؠ ܥܣܤס‬ᇱ ‫ܣ‬ᇱ ‫ ܥ‬ᇱ are valid, then the congruences ‫ܣס ؠ ܥܤܣס‬ᇱ ‫ܤ‬ᇱ ‫ ܥ‬ᇱ and ‫ܣס ؠ ܤܥܣס‬ᇱ ‫ ܥ‬ᇱ ‫ܤ‬ᇱ also hold. Group of axioms V: Axioms of continuity (Archimedes’ and completeness axioms) V 1. Let ‫ܣ‬ଵ be any point on a straight line between two arbitrary given points, A and B. Take the points ‫ܣ‬ଶ ǡ ‫ܣ‬ଷ ǡ ‫ܣ‬ସ ǡ ǥ in such a way that ‫ܣ‬ଵ lies between A and ‫ܣ‬ଶ , ‫ܣ‬ଶ between ‫ܣ‬ଵ and ‫ܣ‬ଷ , ‫ܣ‬ଷ between ‫ܣ‬ଶ and ‫ܣ‬ସ , and so on. Moreover, let the segments ‫ܣܣ‬ଵ , ‫ܣ‬ଵ ‫ܣ‬ଶ , ‫ܣ‬ଶ ‫ܣ‬ଷ , ‫ܣ‬ଷ ‫ܣ‬ସ ,... be equal to one another. Then there is always in the series of points ‫ܣ‬ଶ ǡ ‫ܣ‬ଷ ǡ ‫ܣ‬ସ ǡ ǥ a point ‫ܣ‬௡ such that B lies between A and ‫ܣ‬௡ . V 2. The elements of the geometry (points, straight lines and planes) constitute a system of objects that does not admit any extension when one accepts the previous axioms. In other words: To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that, in the system so obtained, the axioms I-V, VI are still valid.

Thus, for example, to prove the non-contradiction of the axioms of plane geometry, Hilbert appeals to a simple model provided by analytic geometry. In the first edition of Grundlagen, Hilbert used the system of algebraic numbers ȳ obtained from 1 for the four elementary operations and the operation หξͳ ൅ ݊ଶ ห ,where n is a previously given number. In this model, a point is represented by an ordered pair ‫ݔۃ‬ǡ ‫ ۄݕ‬of real numbers, a straight line by an ordered triple of numbers ‫ݑۃ‬ǡ ‫ݒ‬ǡ ‫ۄݓ‬. Analytic geometry then translates the fundamental relations and proves that the axioms are satisfied: The existence of the equation ‫ ݔݑ‬൅ ‫ ݕݒ‬൅ ‫ ݓ‬ൌ Ͳ expresses the fact that the point ‫ݔۃ‬ǡ ‫ ۄݕ‬lies on the line ‫ݑۃ‬ǡ ‫ݒ‬ǡ ‫ۄݓ‬, i.e., the relation of incidence; regarding the relation of order, ‫ݑۃ‬ǡ ‫ ۄݒ‬is said to be between ‫ݓۃ‬ǡ ‫ ۄݔ‬and ‫ݕۃ‬ǡ ‫ ۄݖ‬if there are real numbers ‫ݎ‬ǡ ‫ݏ‬ǡ ‫ ݐ‬such that ‫ ݑݎ‬൅ ‫ ݒݏ‬ൌ ‫ ݓݎ‬൅ ‫ ݔݏ‬ൌ ‫ ݕݎ‬൅ ‫ ݖݏ‬ൌ ‫ ݐ‬and either ‫ ݓ‬൏ ‫ ݑ‬൏ ‫ ݕ‬or ‫ ݓ‬൐ ‫ ݑ‬൐ ‫ ݕ‬or ‫ ݔ‬൏ ‫ ݒ‬൏ ‫ ݖ‬or ‫ ݔ‬൐ ‫ ݒ‬൐ ‫ݖ‬. Finally, the relation of congruence between segments and angles is defined by groups of translations and rotations in the plane (hence the need of the operation หξͳ ൅ ݊ଶ ห). In the domain ȳ, the axioms



86

Chapter Eight

of the I-IV groups and the axiom of Archimedes are then satisfied. However, the axiom of completeness is not satisfied, because ȳ is extensible to other domains in which all the previous axioms are satisfied. One of these domains is the field of real numbers, which is an Archimedean ordered field that is not extensible to any other ordered Archimedean field containing it properly. Therefore, if instead of ȳ we consider the field of real numbers, plane Cartesian geometry will provide the model of Euclidean geometry Hilbert was looking for, since under this interpretation, the axioms of groups I-IV and the two axioms of continuity (the Archimedean and completeness axioms) become truths of real number theory. Extending the above argument to spatial geometry is not a problem. Regarding the mutual independence of the different axioms, it is worth mentioning the proof of the independence of the parallel axiom of the parallel and Archimedes’ axiom, from which follows respectively the consistency of the hyperbolic geometry of Lobachevski and Bolyai and the non-Archimedean geometry of Veronese. Regarding the independence of the parallel postulate, Hilbert appealed to the abstract geometric model that Cayley had constructed inside a conic, where the points and straightlines are defined from the Euclidean points and straight-lines and a group of collineations allows fixing the relationships of angles and lengths. Felix Klein had indeed shown that Cayley’s geometric model satisfied all the axioms of Euclid, except for the axiom of the parallel. More exactly, that the geometry realized inside a conic is the geometry of Lobachevski. To prove the independence of Archimedes’ axiom from the other axioms, Hilbert considers the same set of algebraic numbers ȳ previously introduced and the set of algebraic functions in an indeterminate ‫ ݐ‬on this field௅with the same five operations as before and where ݊ is now an arbitrary function generated by the five operations. The order is obtained by defining ܽ ൏ ܾ for two functions ܽ and ܾ of ‫ ݐ‬if ܿ ൌ ܽ െ ܾ is always positive or negative for a ‫ ݐ‬large enough. For any positive rational number, ‫ ݍ‬െ ‫ ݐ‬is then always negative for a t large enough, so that ‫ ݍ‬൏ ‫ ݐ‬for every ‫ݍ‬. In other words, the axiom of Archimedes is not valid.



CHAPTER NINE HILBERT AND AMERICAN POSTULATIONAL ANALYSIS

One of the places where the impact of Hilbert’s Grundlagen was stronger was in the young American mathematical community. The last quarter of the 19th century was a time of great prosperity and growth in the United States, which had a decisive influence on the creation of new and important centres of mathematical research and, in general, on the formation of many of the institutions of higher education that are today leaders in the whole world. As great fortunes were made on the railroads, the telegraphs, and industrial expansion in general, individuals like Johns Hopkins and John D. Rockefeller endowed universities through their private philanthropy. The presidents of these new schools, well aware of the educational scenes abroad and especially in Germany, France, and Great Britain, crafted their new institutional philosophies informed by the examples of those foreign systems. In particular, many of them adopted the production of research and of future researchers as explicit missions for their faculties and schools.1

For example, Daniel Coit Gilman (1831-1908), the first president of Johns Hopkins University, asked himself in the opening lesson of this university in 1876: What are we aiming at? And he responded: “the encouragement of research . . . and the advancement of individual scholars, who by their excellence will advance the sciences they pursue, and the society where they dwell.”2 Gilman believed that teaching and research were interdependent, that success in one depends on success in the other, and that a modern university like Johns Hopkins must do both well.

 1

Parshall 1996, 292. Inaugural Address of Gilman as first president of the John Hopkins University (http://webapps.jhu.edu/jhuniverse/information_about_hopkins/about_jhu/daniel_c oit_gilman/).

2

88

Chapter Nine

At the University of Chicago, a university financed by the philanthropy of John D. Rockefeller and opened in 1892, a strong emphasis was placed on securing a research faculty of the highest quality. William Rainey Harper, the University’s first president, envisioned “a modern research university that would combine an English-style undergraduate college and a German-style graduate research institute.”3 This emphasis on research of such universities as Johns Hopkins and Chicago revolutionized higher education in America. The three figures who played a decisive role in the formation of a powerful community of researchers in mathematics in the United States were James Joseph Sylvester, Felix Klein and Eliakim Hastings Moore (1862-1932). Sylvester was a British mathematician who, with Arthur Cayley, was a cofounder of invariant theory. He went to the United States in 1876 where he became the first professor of Mathematics at Johns Hopkins and founded the American Journal of Mathematics in 1878. As remarked by E. T. Bell: Sylvester’s enthusiasm for algebra during his professorship at the Johns Hopkins University in 1877-1883 was without doubt the first significant influence the United States had experienced in its attempt to lift itself out of the mathematical barbarism it appears to have enjoyed prior to 1878 […] Under Sylvester’s personal inspiration, several of his pupils did creditable and even brilliant work; but when they left the warmth of his enthusiastic personality, they either abandoned mathematical research, or rapidly chilled in a deadening round of pedagogical drudgery in colleges and universities administered by mediocrities for the perpetuation of mediocrity.4

When Sylvester returned to Britain in 1883, his role as a mentor for the young American mathematicians desirous of an advanced mathematical research was taken over by Klein, who trained many PhD students in Leipzig and Göttingen (and indirectly in other German universities). In fact, during the 80s and 90s, Klein had a profound influence on many American young mathematicians who would be responsible for laying the foundations for mathematical research and its teaching in American universities in the beginnings of the 20th century. For example, during the winter semester of the 1887/88 academic year, Klein had six American students enrolled in their courses: M. W. Haskel and W. F. Osgood of Harvard, H. S. White of Wesleyan University, H. D. Thompson of Princeton, B. W. Snow of Cornell University and H. W. Tyler of the

 3 4

Information excerpted from https://www.uchicago.edu/about/history/. Bell 1938, 2.

Hilbert and American Postulational Analysis

89

Massachusetts Institute of Technology, which represented a substantial part of Klein’s advanced students. Another significant example of Klein’s enormous influence on the development of the United States mathematical community is the fact that for a brief period most of the American Mathematical Society officials, including six presidents and 13 Vicepresidents, had been his students. Among the most notable events that gave a definitive boost to mathematical research in the United States, it is worth highlighting the International Mathematical Congress, organized by Moore and his colleagues from the University of Chicago in conjunction with the World’s Columbian Exposition, held in Chicago in 1893. Klein attended the Congress commissioned by the Prussian Ministry of Education Friedrich Althoff and gave a very influential course of lectures on mathematics at the nearby Northwestern University in Evanston. The aim of these lectures was to “pass in review some of the principal phases of the most recent development of mathematical thought in Germany.”5 Actually, the mathematical spectrum reviewed in these lectures was broad: algebraic curves and surfaces, ideal numbers, higher algebraic equations, hyperelliptic and Abelian functions, non-Euclidean geometries, etc. But there were also lectures devoted to the work of his teacher Clebsch and of his colleague Sophus Lie, and to general topics such as “the relation of pure mathematics to the applied sciences,” “the study of mathematics at Göttingen” or “the development of mathematics at the German universities.” Among the attendees to the Evanston Colloquium–this was indeed the official name of the event–were Moore, professor of Mathematics at the University of Chicago, Oskar Bolza (1857-1942) and Heinrich Maschke (1853-1908), assistant professors at the same university. It was probably his attendance to the Evanston Colloquium that inspired Moore to create a first-rate mathematical research centre in Chicago that could compete with the monopoly of universities on the East Coast of the United States, particularly Harvard and Johns Hopkins. Moore had spent one year abroad studying mathematics in Germany after graduating at Yale University. “He went first to Gottingen, in the summer of 1885, where he studied the German language and prepared himself for the winter of 1885-86 in Berlin. The professors of mathematics most prominent in Gottingen at that time were Weber, Schwarz, and Klein. At Berlin, Weierstrass and Kronecker were lecturing.”6 After

 5 6

Klein 1894, 1. Bliss and Dickson 1935, 85.

90

Chapter Nine

returning to the USA in 1886 he took different posts at Northwestern University and Yale University. “When the University [of Chicago] opened in the autumn of 1892 he was appointed professor and acting head of the department of mathematics. In 1896, after four years of unusual success in organizing the new department, he was made its permanent head, and he held this position until his partial retirement from active service in 1931.”7

Fig. 9-1 Eliakim Hasting Moore

Before its creation, most American graduates who wanted to continue their training as mathematicians travelled to Germany, especially to Göttingen. But Moore, with the collaboration of Bolza and Maschke, two German mathematicians who had studied with Klein in Göttingen before immigrating to the United States, set up a department that was able to attract some of the most brilliant young American mathematicians who wanted to devote themselves to research. Thanks to the efforts of Moore, Bolza and Maschke toward strengthening several institutional and

 7

Ibid., 86.

Hilbert and American Postulational Analysis

91

organizational structures inspired in that of the German universities, particularly Göttingen, the University of Chicago quickly became the most important mathematical research centre in the United States: In addition to the regular lecture courses that they offered in the established areas of late nineteenth-century mathematics௅invariant theory, the theory of substitutions, elliptic function theory, among others௅the Chicago mathematicians also incorporated the seminar into their overall pedagogical approach. As especially Bolza and Maschke knew from firsthand experience, the seminar served as a fertile seedbed for the germination of new mathematical ideas along more specialized lines. The Chicagoans further augmented this learning device with what they called the “Mathematical Club,” a series of biweekly meetings throughout the academic year in which speakers, both faculty and students, presented expositions of the recently published results of other mathematicians or of their own evolving ideas. The atmosphere fostered by this faculty and through these means produced in short order a number of first-rate mathematicians, notably Leonard E. Dickson, Oswald Veblen, Robert L. Moore, and George D. Birkhoff.8

Considering the above facts, it comes as no surprise that one of the places where the publication of Grundlagen der Geometrie had an immediate influence was the United States and, specifically, the University of Chicago. In the fall of 1901, Moore conducted a seminar on the foundations of geometry and analysis, which constituted the basis of his paper “On the projective axioms of Geometry” (1902).9 He introduced his students to and discussed with them the latest literature on the topic, particularly Hilbert’s Festschrift and a recent article of Friedrich Schur (1856-1932), “Über die Grundlagen der Geometrie” (1901), where he contested Hilbert’s claim of the independence of the axioms of connection (incidence) and order. Regarding the axioms of connection, order and congruence, Hilbert had stated in Grundlagen that “it is easy to show that the axioms of these groups are each independent of the others of the same group.”10 But this, as Schur observed, did not grant that the axioms were, in fact, mutually independent. More specifically, Schur argued that three of Hilbert’s axioms of connection followed from the other four axioms of connection together with the five axioms of order. Moore agreed with

 8

Parshall 1996, 292. According to Moore “this paper has been prepared in connection with my current Chicago seminar-course on the foundations of geometry and analysis, and queries and remarks of members of this course, in particular, of Mr. O. Veblen, have been a source of much stimulus” (Moore 1902, 143, footnote *). 10 Hilbert 2004, 456. 9

92

Chapter Nine

Schur in that Hilbert’s system was redundant but noticed that he had not correctly identified the actual redundancy. Moore proved that the redundancy in Hilbert’s system involved only one axiom of connection and one axiom of order. To understand why Hilbert just worried about the mutual independence of axioms inside each group of axioms, it must be noticed that Hilbert’s groups of axioms correspond to geometric relations that have an intuitive independent content and which he therefore considers the fundamental relations. For example, in contrast to his predecessors, he sharply separates the connection and ordering axioms, which is not strictly necessary from a logical point of view. Hence the issues of independence between axioms of distinct groups were secondary for Hilbert, since the axiom system as a whole was built from geometrical intuitive groups of axioms (corresponding to each of the fundamental relations), not from the standpoint of logical economy. Thus, when Schur and Moore analysed the logical dependence between axioms of distinct groups they made a step forward that Hilbert had not foreseen. Moore’s analysis moved still further away from the original spirit of Hilbert’s axiomatic analysis. When analysing the different groups of axioms, Hilbert was implicitly granting these groups of axioms a genuine mathematical interest, as they expressed facts of our spatial intuition about relations between different systems of things, which were the object of study of geometry–at least in a first intuitive, pre-critical stage.11 For Moore, however, the axiom systems as such became the subject of study, regardless of the mathematical interest that these systems might have. The problem addressed by Moore was whether these systems of axioms could be formulated more conveniently from the deductive point of view, regardless of whether they had any intuitive geometrical meaning. Obviously, this type of analysis was also available to Hilbert, given his formalistic conception of the axiomatic method, but neither Hilbert nor his collaborators had any interest in developing a logical analysis of the axioms per se. For Hilbert, the axiomatic analysis was the final and ideal stage of any mathematical discipline. For Moore, the systems of axioms or postulates themselves were the starting point, thus creating a new

 11

Although from Hilbert’s formal standpoint, “the basic elements can be thought of in any way one likes,” for example, as chairs, tables and beer-mugs, and so the axioms can also be thought as defining the different kind of “relations between these things” (see Chapter 8), it is just insofar they express facts of our spatial intuition about the relations between points, lines and planes that they have a genuine mathematical interest, i.e. they can be called properly geometrical axioms.

Hilbert and American Postulational Analysis

93

scientific discipline, the analysis of systems of postulates or postulational analysis. This new perspective would be developed by Moore and other American mathematicians in the forthcoming years. Particularly important in this direction was the figure of Edward Huntington (1874-1952). Just a few months after the publication of Moore’s article, he published two papers “A complete set of postulates for the theory of absolute continuous magnitude” (1902) and “Simplified definition of a group” (1902), which are usually considered the first published contributions of American postulational analysts after Moore’s paper. In the first paper, Huntington proposed a set of six postulates or primitive propositions “from which the mathematical theory of absolute continuous magnitude can be deduced.”12 An important feature of Huntington’s presentation of these postulates is that they were considered independent of any interpretation, that is, as uninterpreted conditions that could be themselves object of a metatheorical study. The object of the paper was indeed To show that these six postulates form a complete set; that is, they are (I) consistent, (II) sufficient, (III) independent (or irreducible). By these three terms we mean: (I) there is at least one assemblage in which the chosen rule of combination satisfies all the six requirements; (II) there is essentially only one such assemblage possible; (III) none of the six postulates is a consequence of the other five.13

Consistency and independence were two of the three properties that Hilbert required to axiomatic systems (the other one was completeness). Sufficiency is what is nowadays called categoricity, as can be seen from Huntington’s proof that all models of his set of postulates are isomorphic. As we will see later, the term “categoricity” was introduced by Veblen in 1904. “But the notion of categoricity and its relevance for proving that an axiom system is semantically complete is due to Huntington, as Veblen acknowledges.”14 In the second paper, Huntington analysed the definition of group given by Heinrich Weber in his famous Lehrbuch der Algebra (1896). As observed by Huntington, Weber’s system of postulates contained many redundancies, so the purpose of his analysis was precisely to discover and

 12

Huntington 1902a, 264. Ibid. 14 Scanlan 1991, 986. A theory is semantically complete if any two models of it satisfy all the same sentences. It is clear that any categorical theory is semantically complete, as there is then only one isomorphism class of models, and these all satisfy the same sentences. 13

94

Chapter Nine

eliminate them. More specifically, Huntington reduced Weber’s postulates to just three postulates, which he then proved to be independent. In a subsequent paper Huntington gave a second set of four postulates for group theory. This paper was presented in a meeting of the American Mathematical Society (April 1902), where Moore also presented a paper, published in the Transactions the same year, in which he gave a set of six postulates. Moore analysed his own postulates and those of Huntington in relation to them, adding to Huntington’s first set the explicit statement of the closure postulate. In this way, Moore took a step further comparing different systems of postulates for the same discipline. These papers were only the beginning of a new trend, as confirmed by numerous articles on postulational analysis appearing in major American mathematical journals, particularly in the Transactions of the American Mathematical Society, but also in the Bulletin of the same society. The analysis of the systems of axioms or postulates was almost always very similar to and followed the guidelines marked by Huntington’s seminal papers, which were at the same time inspired by Hilbert’s requirements on the axiomatic method. As we have seen, the aim was to demonstrate that the sets of postulates were complete, that is, that the postulates were consistent, sufficient (categorical) and mutually independent. If different postulate systems for the same branch of mathematics were analysed, then the objective was to show that these systems were equivalent, i.e., that they amount to derive the same theorems. For example, Leonard E. Dickson (1874-1954), Moore’s first doctoral student, published between 1903 and 1905 several articles analysing different systems of postulates defining the concepts of field, associative linear algebra and group. Also, in 1904 Huntington published the influential paper “Sets of independent postulates for the algebra of logic” in which he compared distinct sets of postulates for Boolean algebra. But perhaps Moore’s most outstanding student in this direction was Oswald Veblen. His dissertation discussed a new system of axioms for geometry, in which the notions of point and order were used as basic notions, instead of Hilbert’s primitive concepts (point, line and plane) and relations (connection, order and congruence). These results were published in the paper “A system of axioms for geometry,” published in the Transactions of the American Mathematical Society (1904), a journal in which he published important articles in the same direction during the following years. In Veblen own terms: The propositions brought forward as axioms in this paper are stated in terms of a class of elements called “points” and a relation among points called “order;” they thus follow the trend of development inaugurated by

Hilbert and American Postulational Analysis

95

Pasch and continued by Peano rather than that of Hilbert or Pieri. All other geometrical concepts, such as line, plane, space, motion, are defined in terms of point and order. In particular, the congruence relations are made the subject of definitions rather than of axioms. This is accomplished by the aid of projective geometry according to the method first given analytically by Cayley and Klein. The terms “point” and “order” accordingly differ from the other terms of geometry in that they are undefined.15

Veblen gave concretely a set of twelve axioms which he demonstrated to be independent and constituted a categorical system. But in contrast to Hilbert and Huntington he didn’t worry about their consistency. According to Veblen, a system of axioms is categorical if there is essentially only one set of elements that satisfy these axioms, that is, if all their models are essentially identical or, as we would say today, isomorphic. Otherwise it is said that the system of axioms is disjunctive. He writes that “the categorical property of a system of propositions is referred to by Hilbert in his ‫ލ‬Axiom der Vollständigkeit.‫”ތ‬16 But to be more precise it should be said that categoricity is a consequence of Hilbert’s completeness axiom. The reason for this is that the associated system of axioms for the real number system is itself categorical and the completeness axiom, together with the Archimedean axiom, assures a biunivocal correspondence between the set of points of the Euclidean plane and the system of real numbers. And in an Archimedean ordered field, the Vollständigkeitaxiom for fields imposes categoricity. Over the years, the experience accumulated in postulational analysis led to a greater understanding of the essence and common features of axiomatic systems. As we have seen, postulational analysts not only showed how different mathematical disciplines could be explicitly axiomatized in a formalized, uninterpreted language, but they also showed how these axiomatized systems could themselves be the object of a (meta)mathematical study. In this sense, postulational analysis had a great influence on the development of mathematical logic in North America and, ultimately, on the creation of model theory, one of the most active research areas of logic nowadays. More immediately, postulational analysis also provided a wealthy collection of standardized axiomatic systems of different mathematical disciplines that were adopted by researchers in these disciplines or in the foundation of mathematics. An example of this is Alfred Tarski’s (1901-1983) famous paper “Der

 15 16

Veblen 1904, 344. Ibid., 346.

96

Chapter Nine

Wahrheitsbegriff in den formalisierten Sprachen” (“The concept of truth in formalized languages”) (1935), where the two only American mathematicians cited are Huntington and Veblen. This, at the same time, provided the natural framework from which the abstract research of structural type would come in the future. This was the case, for example, of the definition and structural analysis of the abstract notion of ring by Abraham Fraenkel (1891-1965), who was influenced by the work of postulational analysis through his uncle, the mathematician Alfred Loewy (1873-1935).17 Although from 1898 to 1938 numerous papers appeared on postulational analysis in the American mathematical journals, particularly in the period from 1928 to 1938,18 the most promising research on the formalization of mathematical theories and the scope of the axiomatic method in the same period was led by Hilbert himself and his circle of collaborators: Ernst Zermelo (1871-1953), Paul Bernays (1888-1977) and Wilhelm Ackermann (1896-1962), among others. Also, by the early 1930’s the most promising research on these topics was led by European logicians and mathematicians like Kurt Gödel (1906-1978), Alfred Tarski or Rudolf Carnap (1891-1970) whose research programs were directly inspired (at least to a great extent) by the work of Hilbert. It was in the work of Gödel, as we will see later, where the influence of Hilbert was most fruitful (though his results were mostly unexpected for Hilbert).



17 As remarked by L. Corry, Fraenkel “took Loewy’s and Hensel’s trains of ideas into a new direction, leading to the definition and early research of abstract rings” (Corry 1996, 201). 18 According to E. T. Bell’s statistics, 8,46% of the mathematical papers publishes in the USA between 1888 and 1938 were devoted to postulational analysis. 23.3% of the papers on postulational analysis were published between 1898 and 1907, 11.1% between 1908 and 1917, 23.5% between 1918 and 1927, and 41.1% between 1928 and 1937భమ (Bell 1938, 7).

CHAPTER TEN THE AXIOMATIZATION OF ANALYSIS

At the end of the summer of 1899, between the 17th and 23rd of September, the annual meeting of the DMV took place in Munich, together with that of the Society of German Scientists and Doctors. At this meeting those present discussed the possibility of hosting the International Congress of Mathematicians in Germany in 1904, for which the official request would be made at the next Congress that was to be held in Paris the following year. Hilbert was one of the 80 participants in the DMV Congress and presented two articles, one on Dirichlet’s principle and another on the concept of number. In the summer of 1899, just after the publication of Grundlagen der Geometrie, Hilbert had concentrated all his energies in trying to “revive” the so-called Dirichlet principle, which had focused the attention of the most prestigious mathematicians of the 19th century, due to its importance both theoretical௅to prove the existence and unicity of solutions for problems related to limit values௅and practical௅to solve numerous types of problems that arise in physics. In a fairly generic way, Dirichlet’s principle is a method to solve boundary value problems of elliptic partial differential equations which consists in reducing them to the (variational) problem of finding the minimum value of an integral (named the Dirichlet or energy integral) in a determinate class of functions subject to the condition that they take on prescribed boundary values (basically, that the values of the integral do not become infinite and they allow the problem initially raised to be solved). Bernhard Riemann had widely used Dirichlet’s principle in his famous dissertation of 1851 and was, in fact, the first to call it by this name in honour to his master Dirichlet. But the principle had been heavily criticized by Weierstrass in the late 1860’s, when he proved by means of an example that the differential boundary problem may have a solution, while the corresponding variational problem could not have a solution because the value of the Dirichlet integral when solving the problem becomes infinite in such cases. In other words, the Dirichlet principle was not valid in all cases. But this principle had too many applications in

Chapter Ten

98

physics to completely discard it, so several mathematicians such as Carl Neumann and Hilbert himself proposed to save it. In a few pages of the article read at the DMV Congress, Hilbert demonstrated how, by putting certain limitations on the nature of the differential equation in question and its boundary values, he could avoid Weierstrass’s objections and safeguard the vast majority of mathematical and physical applications of the Dirichlet principle. The other article read by Hilbert at the DMV Congress of September 1899 was entitled “Über den Zahlbegriff” (“On the concept of number”). Encouraged by the success achieved in Grundlagen der Geometrie by the application of the axiomatic method to geometry, in this paper Hilbert proposed applying this method to analysis instead of the genetic method. Hilbert described the genetic method, commonly used in the Weierstrass school and also by Kronecker or Dedekind, dominant figures of analysis in the second half of the nineteenth century, to introduce the concept of number in the following terms: Starting from the concept of the number 1, one usually imagines the further rational positive integers 2, 3, 4, ... as arising through the process of counting, and one develops their laws of calculation; then, by requiring that subtraction be universally applicable, one attains the negative numbers; next one defines fractions, say as a pair of numbers–so that every linear function possesses a zero; and finally one defines the real number as a cut or a fundamental sequence, thereby achieving the result that every entire rational indefinite (and indeed every continuous indefinite) function possesses a zero.1

Hilbert recalled, however, that, to erect geometry, one proceeds in an essentially different way, applying the axiomatic method: Here one customarily starts by assuming the existence of all the elements, i.e., one postulates at the outset three systems of things (namely the points, lines, and planes), and then–essentially following the pattern of Euclidbrings these elements into relationship with one another by means of certain axioms–namely the axioms of connection, of ordering, of congruence, and of continuity. The necessary task then arises of showing the consistency and the completeness of these axioms, i.e., it must be proved that the application of the given axioms can never lead to contradictions, and, further, that the system of axioms is adequate to prove all geometrical propositions.2

 1 2

Hilbert 1900, 180. Ibid., 181.

The Axiomatization of Analysis

99

Certainly, Hilbert had not only axiomatized Euclidean geometry in his work Grundlagen der Geometrie, but also had proved the consistency and independence of its axioms௅the completeness followed by the axiom of completeness. Now, Hilbert had proved the consistency of Euclidean geometry by furnishing a model in which all the axioms of plane geometry were satisfied. This model was that of analytical geometry, whose consistency was taken for granted. Thus, the pending task was first to axiomatize analysis and then to prove its consistency. In fact, the great Italian mathematician Giuseppe Peano (1858-1932) had axiomatized arithmetic at the end of the nineteenth century, characterizing the system of natural numbers through the set of axioms usually known as Peano axioms. Now, in the paper “Über den Zahlbegriff,” Hilbert axiomatized analysis, characterizing the system of real numbers as a maximal Archimedean ordered field, that is, an Archimedean ordered field not extensible to another field of the same kind that contains it. Unlike the authors who followed what Hilbert called the genetic method, Hilbert did not define the real numbers from previously given objects, such as rational numbers, and then prove that the system of real numbers satisfied the aforementioned properties. On the contrary, Hilbert asked his readers to imagine a “system of things” that already satisfied the properties enunciated in the axioms. The system of real numbers was precisely this system of things. This way of proceeding raised a couple of problems. The first was that it did not guarantee that there was any “system of things” that satisfied the previous properties, that is to say, that the system of real numbers really existed. This was indeed a basic problem in Hilbert’s eyes since he knew, from his correspondence with Georg Cantor (1845-1918), that Cantor, in a letter addressed to Dedekind on July, 1899, had noted that the existence of the Gedankenwelt (world of thought), in which Dedekind had based his proof of the existence of an infinite set, was equivalent to the hypothesis of the existence of the set of all sets, which led to a logical contradiction analogous to that which the consideration of all the ordinals or of all the cardinals led to. Cantor thought to solve these contradictions with the distinction between absolutely infinite or inconsistent pluralities (like the previous ones) and consistent pluralities or sets. However, this solution could not satisfy Hilbert, who considered the proof of the consistency of the system of axioms through which he had characterized the real numbers as the most appropriate mean of proving its existence. Thus, for example, the observation made by Hilbert in “Über den Zahlbegriff” that he saw in the proof of the consistency of the system of axioms presented in this paper “the proof of the existence of the totality of all real numbers or–in

Chapter Ten

100

the terminology of Cantor–the proof that the system of real numbers is a consistent (finished) set”3 was clearly an implicit critique of the solution adopted by Cantor. Generally speaking, the problem of proving the existence of a system of things was solved by proving that the axioms for this system of things were consistent, since for him consistency was the only criterion of existence. As Hilbert put it in a letter to Frege of December 29, 1899: If the arbitrarily posited axioms together with all their consequences do not contradict one another, then they are true and the things defined by the axioms exist. This is for me the criterion of truth and existence.4

The second problem raised by the application of the axiomatic method for introducing the real number system was that this procedure did not assure either, at least explicitly, that this system of things (in case of existing) was unique. This issue was solved thanks to the presence of the completeness axiom: if the real numbers are defined as a system of things satisfying the axioms that cannot be extended with new elements (and keep satisfying the axioms), then all the models of the axioms (all systems of things that satisfy the axioms) will be essentially the same, and therefore the system of real numbers will be univocally determined (up to isomorphism) by the axioms. The axiom of completeness was the most novel axiom of the four axiom groups proposed by Hilbert to axiomatize analysis. As we have already seen, Hilbert added an axiom analogous to that in the second edition of Grundlagen der Geometrie to the list of axioms present in the first edition. It was thanks to the completeness axiom of Grundlagen that it was possible to introduce the concept of measurement and to “prove all the theorems of geometry.” Similarly, Hilbert thought that a proposition about the real numbers was true if, and only if, it followed from the axioms that he had postulated for these numbers. Hilbert considered that the axiom of completeness guaranteed this, since, as we have already explained, it guaranteed what Veblen called the categoricity of the system of axioms and, therefore, that all true sentences in a model of the axioms were true in any other model of them. It followed from this (assuming that the underlying logic was semantically complete) that this system of axioms was complete in the sense that it allowed to prove or refute any proposition about the real numbers.

 3 4

Ewald 1996, vol. 2, 1095. Frege 1976, 68.

The Axiomatization of Analysis

101

The axiomatization of analysis (“Über den Zahlbericht,” 1900) Let us consider a system of things; we call these things numbers and we assign them names ܽǡ ܾǡ ܿǡ ǥ we suppose that these numbers have among them certain relations, the precise and complete description of which is given by the following axioms: I. Axioms of linking I 1. From the number ܽ and the number ܾ, there arises through “addition” a determined number ܿ, in symbols: ܽ ൅ ܾ ൌ ܿ or ܿ ൌ ܽ ൅ ܾ. I 2. If ܽ and ܾ are given numbers, then there always exists one and only one number ‫ ݔ‬and also one and only one number ‫ ݕ‬such that ܽ ൅ ‫ ݔ‬ൌ ܾ and ‫ ݕ‬൅ ܽ ൌ ܾ respectively. I 3. There is a determinate number –it is called Ͳ– such that for every ܽ, ܽ ൅ Ͳ ൌ ܽ and Ͳ ൅ ܽ ൌ ܽ. I 4. From the number ܽ and the number ܾ there arises in yet another way, through “multiplication”, a determinate number ܿ; in symbols: ܾܽ ൌ ܿ or ܿ ൌ ܾܽ. I 5. If ܽ and ܾ are arbitrary given numbers and ܽ is not Ͳ, then there always exists one and only one number ‫ݔ‬, and also one and only one number ‫ ݕ‬such that ܽ‫ ݔ‬ൌ ܾ and ‫ ܽݕ‬ൌ ܾ. I 6. There is a determinate number –it is called ͳ– such that for every ܽ, ܽ ή ͳ ൌ ܽ and ͳ ή ܽ ൌ ܽ. II. Axioms of calculation If ܽǡ ܾǡ ܿǡ ǥ are arbitrary numbers, then the following formulas always hold: II 1. ܽ ൅ ሺܾ ൅ ܿሻ ൌ ሺܽ ൅ ܾሻ ൅ ܿ II 2. ܽ ൅ ܾ ൌ ܾ ൅ ܽ II 3. ܽሺܾܿሻ ൌ ሺܾܽሻܿ II 4. ܽሺܾ ൅ ܿሻ ൌ ܾܽ ൅ ܽܿ II 5. ሺܽ ൅ ܾሻܿ ൌ ܽܿ ൅ ܾܿ II 6. ܾܽ ൌ ܾܽ III. Axioms of order III 1. If ܽǡ ܾ are any two different numbers, then a determinate one of them (say ܽ) is always greater (>) than the other; the latter is then called the smaller. In symbols: ܽ ൐ ܾ and ܾ ൏ ܽ . III 2. If ܽ ൐ ܾ and ܾ ൐ ܿ, then ܽ ൐ ܿ. III 3. If ܽ ൐ ܾ, then always ܽ ൅ ܿ ൐ ܾ ൅ ܿ and ܿ ൅ ܽ ൐ ܿ ൅ ܾ. III 4. If ܽ ൐ ܾ and ܿ ൐ Ͳ, then always ܽܿ ൐ ܾܿ and ܿܽ ൐ ܾܿ. IV. Axioms of continuity IV 1. (Archimedean axiom) If ܽ ൐ Ͳ and ܾ ൐ Ͳ are two arbitrary cont. p. 102

102

Chapter Ten

numbers, then it is always possible to add ܽ to itself so often that the resulting sum has the property that ܽ ൅ ܽ ൅ ‫ ڮ‬൅ ܽ ൐ ܾ. IV 2. (Axiom of completeness) It is not possible to add to the system of numbers another system of objects so that the axioms I, II, III and IV are also all satisfied in the combined system; in short: the numbers form a system of things which is incapable of being extended while continuing to satisfy all the axioms.

It was surely the success achieved with the safeguarding of the Dirichlet principle which impelled Hilbert to lecture, for the first time in his career, on the calculus of variations, during the summer semester of 1899. This is a branch of analysis that deals with problems in which (as in the Dirichlet principle) we search for maxima and minima of defined continuous functionals on some space of functions, or said in another way, functions ݂ሺ‫ݔ‬ሻ, belonging to a space of functions, for which a functional ‫ܫ‬ሺ݂ሻ reaches an extreme value, where ‫ܫ‬ሺ݂ሻ is composed of an integral that depends on x, the function ݂ሺ‫ݔ‬ሻ and some of its derivatives. Apart from this course, Hilbert also taught courses on differential calculus and group theory during the same semester. In the winter semester of the course 1899/1900, Hilbert taught courses on integral calculus, on the concept of number and the squaring of the circle, and on the curvature of surfaces. Amid this broad range of interests and academic activity, Hilbert received the invitation to give a plenary lecture at the Second International Congress of Mathematicians, which was to be held in Paris the following summer.

CHAPTER ELEVEN THE HILBERT PROBLEMS

Hilbert was aware of the opportunity to give a plenary conference in Paris represented for him and wanted to be up to the circumstances. In his New Year congratulations letter to Minkowski, then in Zürich, Hilbert took the opportunity to comment to him the good news of the invitation to Paris and asked him his opinion about the topic of his lecture. He doubted between answering, in his lecture, another one given by Poincaré at the First Congress of Mathematicians, held in Zurich in 1897, on the relationship between analysis and physics, or discussing the direction of mathematics in the next century posing some important problems that had to be answered by mathematicians in one way or another. Minkowski responded him in a letter of 5 January 1900, that Most alluring would be the attempt to look into the future, in other words, a characterisation of the problems to which the mathematicians should turn in the future. With this, you might conceivably have people talking about your speech even decades from now.1

Hilbert finally decided to follow the advice of Minkowski. However, in the month of June Hilbert had not yet sent the text of his lecture to the organizers, and the program of the Congress was sent to the participants without the name of Hilbert. Minkowski, very disappointed, explained to Hilbert by letter that “my desire to travel to the Congress has now practically disappeared.”2 But Minkowski soon recovered his enthusiasm for travelling to Paris since, in the middle of July, Hilbert sent him a new letter with a preliminary version of the proposed address. Minkowski’s answer to Hilbert’s letter expresses very well the expectations that this author had deposited in Hilbert’s lecture and the profound admiration for his friend:

 1

Rüdenberg and Zassenhaus 1973, 119. In March, he also sought the opinion of another close friend, Hurwitz (Reported by Grattan-Guinness 2000, 252). 2 Ibid., 127.

104

Chapter Eleven Certainly [your conference] will be the event of the congress and its success will be very lasting. I believe that this speech, which probably every mathematician without exception will read, will make you have even more power of attraction for young mathematicians [...] Now you have really delimited the mathematics for the twentieth century and in most circles you will be recognized gladly as its general director.3

These words became prophetic and, as Minkowski had foreseen, the 23 mathematical problems presented by Hilbert at the Sorbonne became the vanguard of mathematical research in the 20th century. The fact that Hilbert was invited to give a plenary conference in the Second Congress of Mathematicians was already indicative of the prestige achieved by Hilbert outside of Germany, but it was undoubtedly the Paris conference that confirmed the leadership of Hilbert in the International mathematical community. As remarked in the chapter of Landmark Writings in Western Mathematics 1640-1940 dedicated to Hilbert’s mathematical problems: That lecture and even more the written version of it has been of great influence on the development of mathematics in the 20th century, or so it would seem. It stems partly because of the stature of the lecturer, which was still to grow considerably in the decades to come; partly because the problems were well chosen; partly because they breathed a coherent view of what mathematics is all about; and perhaps most of all because of the incurable optimism in it all, a flat denial of Emil Du Bois-Reymond’s claim “Ignoramus et ignorabimus.”4

Hilbert’s speech was initially scheduled for the opening session of the Congress, but due to the delay in the arrival of the text, it was relegated to the joint session of the two general sections, one which dealt with Bibliography and History and the other with Teaching and Methods. These sections were considered of inferior rank to the sections on pure mathematics (Arithmetic, Algebra, Geometry, Analysis) or applied mathematics (Mechanics), but seen in perspective, they became, thanks to Hilbert’s talk, the true protagonists of the Congress. Hilbert’s lecture, entitled “Mathematische Probleme” (“Mathematical Problems”) consisted of a preamble and a list of 23 problems, of which Hilbert only read ten (Hilbert spoke in German, but had the good sense to

 3

Ibid, 129-30. Hazewinkel 2005, 733. The lecture was published for the first time in Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, mathematisch-physikalische Klasse, 1901, 253–297. The first French and English translations were published in 1902.

4

The Hilbert Problems

105

provide the attendees with the full text of the conference in French). In the preamble of the conference, Hilbert emphasized the importance of the existence of problems for the development of any science and, in particular, mathematics, as well as his conviction about the solubility of any mathematical problem, expressed through the slogan “in mathematics there is no ignorabimus”: This 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.5

As could be expected, the list of problems presented by Hilbert included problems related to invariant theory and algebraic number theory, two fields in which Hilbert had already established himself as a first rank mathematician. But, broadly speaking, the list contained a wide range of problems that corresponded essentially with the mathematical interests Hilbert had at that time: the foundations of mathematics and physics, algebraic number theory, algebra and geometry, analysis (particularly, calculus of variations) and its application to physics. The modernity in the approach to some problems–especially in the case of the problems related to the foundations of mathematics and physics–resided not only in the fact that there were unresolved problems that would determine an important part of future mathematical research in these fields, but also in the importance that Hilbert gave to axiomatization in order to solve or, even, to formulate them. The reason is that, as Hilbert explained, the general requirement that must be established for the solution of a mathematical problem is rigor in reasoning, which Hilbert described as follows: It must 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 in the statement of the problem and which must always be exactly formulated.6

However, Hilbert was opposed to the view that only the concepts of arithmetic and analysis are susceptible of a rigorous treatment, as this would lead to rejecting the concepts that arise from geometry, mechanics and physics. Thus, for example, this interpretation of the requirement of rigor

 5 6

Hilbert 1965, vol. 3, 298. Ibid., 293.

106

Chapter Eleven

would lead us to reject concepts of a geometrical origin, such as the continuum or the concept of irrational number, whose definition had become one of the central problems of analysis in the second half of the nineteenth century. Ultimately, according to Hilbert, whatever the origin of a concept or an idea was, the task of mathematics is “to investigate the principles underlying these ideas and so to establish them upon a simple and complete system of axioms”7 in order to achieve the aforementioned rigorous requirement. Thus, Hilbert considered the axiomatic method the most reliable mean to achieve the rigor necessary to solve mathematical problems, particularly those related to the foundations of mathematics and physics.

Fig. 11-1 David Hilbert, ca. 1900

Hilbert grouped together problems with a similar content, thus obtaining four separate groups. The first of them was constituted by six problems on

 7

Ibid., 295.

The Hilbert Problems

107

the foundations of mathematics and physics. In the second group were six problems of algebraic number theory, while the third one was a mixture of algebraic and geometrical problems covering several topics. In the last group were five problems of analysis, which was where the interests of Hilbert would be directed in the immediate future. Of these 23 problems, those discussed by Hilbert at the Congress were the problems 1, 2, 6, 7, 8, 13, 16, 19, 21 and 22. The first three were precisely the problems on the foundations of mathematics and physics that would have a greater relevance in the future regarding the development of the axiomatic method, either by Hilbert himself or by other authors of the Göttingen circle, such as his collaborator Ernst Zermelo. The first foundational problem discussed by Hilbert and indeed the problem that occupied the first place in the list presented by Hilbert was the proof of the continuum hypothesis, i.e., the proof of Cantor’s conjecture that every infinite set of real numbers is either countable (that is, it has one-to-one correspondence with the set of natural numbers) or has the power of the continuum (that is, it has one-to-one correspondence with the set of real numbers) and, therefore, there is no intermediate power between the power of the set of natural numbers and that of the continuum. Hilbert’s second problem was to find a direct proof of the noncontradiction or consistency of the axioms that determine the structure of the real numbers, which he simply called arithmetic axioms, that is, “to prove that a finite number of logical deductions based upon them can never lead to contradictory results.”8 As we already know, Hilbert had proved in Grundlagen der Geometrie the consistency of Euclidean geometry by assuming the consistency of the system of real numbers, which he had axiomatized in the article “Über den Zahlbegriff” as a maximal Archimedean ordered field. Therefore, to demonstrate the absolute consistency of the axioms of geometry (and, by extension, of mathematics), it was now necessary to prove the consistency of the axioms through which he had defined the real numbers. Finally, Hilbert’s sixth problem asked to axiomatize “those physical sciences in which mathematics already play an important role; in the first rank are the theory of probabilities and mechanics.” 9 Hilbert devoted many efforts throughout his life to the axiomatization of physical theories: mechanics, thermodynamics, probability theory, kinetic theory of gases,

 8 9

Ibid., 300. Ibid., 306.

108

Chapter Eleven

The Continuum Hypothesis (Cantor, 1878) In 1874 Cantor had shown that the set of algebraic numbers is countable, but that the continuum is not: it is not possible to establish a one-to-one correspondence between the set Թ of real numbers and the setԳ of natural numbers. In a publication of 1878, Cantor demonstrated that there are one-to-one correspondences between all sets Թ௡ ǡ ݊ ൒ ͳ, and therefore, that they all have the same power. So, there were still only two infinite powers: that of the countable sets and that of the non-countable sets. At the end of the same paper, Cantor stated the continuum hypothesis: every infinite set of real numbers is either numerable or it has the power of the continuum. From 1879 to 1884 Cantor published a series of investigations that constitute the origin of modern set theory. In them he introduced the transfinite numbers and the property of well-ordering. In modern notation, the transfinite numbers are the numbers: ߱ǡ ߱ ൅ ͳǡ ǥ ǡ ߱ʹǡ ߱ʹ ൅ ͳǡ ǥ ǡ ߱͵ǡ ǥ ǡ ߱ଶ ǡ ǥ ǡ ߱ଷ ǡ ǥ ǡ ߱ఠ ǡ ǥ Cantor named the set of natural numbers Գ the first class of numbers (I), he named the set of numbers whose predecessors are in a bijective correspondence with (I) the second class of the numbers (II), and so on. All the transfinite numbers from the list above belong to the second class of numbers, which Cantor had shown to be not countable. In 1883, Cantor proposed the principle of well-ordering, under which all sets can be ordered and gauged by the transfinite numbers and their classes of numbers. In this way the hypothesis of the continuum had been transformed into the hypothesis that Թ and the second class of numbers (II) have the same power. Finally, in a series of articles published between 1895 and 1897, Cantor specified the notion of power of a set through the concept of cardinal number and introduced the notation Յ଴ ǡ Յଵ ǡ Յଶ ǡ Ǥ Ǥ Ǥ ǡ Յఈ to represent the cardinal numbers of the successive classes of numbers. With this notation,Յ଴ is the cardinal number of Գ, the first class of numbers, Յଵ is the cardinal of the second class of numbers, the cardinal of the continuum is ʹՅబ (given that Թ ൌ ࣪ሺԳሻ) and the continuum hypothesis can now be formulated as ʹՅబ ൌ Յଵ.

electrodynamics, theory of relativity, etc. This should not be surprising since, as we know, according to Hilbert, the difference between geometry and physics is only a difference in the degree of development, not an essential difference, so Hilbert considered that physics theories were

The Hilbert Problems

109

susceptible to an axiomatic treatment as soon as they reached the sufficient degree of development.10 The three following problems discussed by Hilbert in Paris were problems related to number theory. The first problem (problem no. 7) was related to the Hermite and Lindemann demonstrations of the transcendence of the numbers ݁ and ߨ (and the unified and more rigorous proof by Hilbert himself) and asked for the irrationality and transcendence of the numbers of the form ߙ ఉ , with ߙ algebraic and ߚ algebraic and irrational; that is to say, numbers such as ʹξଶ o ݁ గ ൌ ݅ ିଶ௜ . The second one (problem no. 8) was a problem related to prime numbers and included the proof of the famous Riemann hypothesis. This is the most important mathematical conjecture not yet solved today, since as Riemann himself had shown, some of the properties of the zeta function that appears in the formulation of its conjecture have important implications on the distribution of prime numbers. Finally, the third problem on number theory discussed in Paris (problem no. 13) asked for a proof of the impossibility of the solution of septic equations by means of functions of two variables. It must be said, however, that the problems on number theory not discussed by Hilbert at the Paris conference were at least as important as those proposed by Hilbert before his auditorium and would determine a large part of the research on number theory carried out in the 20th century. Thus, for example, problem no. 9 asked for a general law of reciprocity for any field of numbers, which would be formulated by Emil Artin in 1928 and would lead to the theory of Abelian class fields. This was precisely the topic of problem no. 12, which asked for the extension of Kronecker’s theorem about Abelian fields to any algebraic domain of rationality. A case apart is problem no. 10, which asks for the determination of the solvability of a given Diophantine equation with integer coefficients, since by the nature of the problem and methods employed by its solution it could also be placed between the foundational problems. The next problem discussed by Hilbert in Paris (problem no. 16) was extracted from the work of the French mathematician Maurice d’Ocagne (1862-1938) and asked for the form of curves and surfaces defined by polynomial equations. This was the only problem belonging to the miscellaneous block of algebraic and geometrical problems (problems 13 to 18) presented by Hilbert to his auditorium. As remarked by J. J. Gray,

 10

This is the topic of Corry 2004.

Chapter Eleven

110

The run of Problems from 13 to 18 is the least coherent block. Hilbert’s touch was less sure here, and the influence of these Problems on the later development of mathematics has been less substantial.11

This was not the case of the final group of problems, with which “Hilbert was on surer ground.”12 To this block belonged the last three problems presented by Hilbert to his auditorium, each of which referred to central questions of the mathematics of his time and of all the 20th century. The first problem, number 19, asked to determine if a concrete type of problems in the calculus of variations (the regular problems, which are those that usually arise when the calculus of variations is used to solve certain physical problems) always have solutions which are expressible in terms of analytic functions, namely, as functions given locally by a series of convergent powers. The second problem, number 21, was referred to a type of linear differential equations of ݊-th order that Lazarus Fuchs had studied first, and then Klein and Poincaré. As Hilbert explained, Klein had exploited the fact that from these differential equations a group can be obtained. The problem was then to show that, given a group, a differential equation can be found that has the given group as its corresponding group. The last problem discussed by Hilbert, number 22, was probably chosen as homage to Poincaré. As Hilbert pointed out, Poincaré was the first to demonstrate that it is possible to standardize (that is, to represent parametrically) any algebraic relation between two variables thanks to the automorphic functions of one variable. Poincaré himself generalized this fundamental theorem for any non-algebraic analytical relation between two variables. The problem raised by Hilbert was then to find out if the resolving functions could be determined in such a way that they fulfilled certain additional conditions. Hilbert’s lecture did not impress his audience, which triggered a brief and disperse discussion on some of the topics dealt with in the lecture. It was commented, for example, that more progress had been made on the problem extracted from Ocagne’s work than that suggested by Hilbert. One of the illustrious attendees at the conference, Giuseppe Peano, leader of the Italian formalist school, said that his colleague Alessandro Padoa (1868-1937) had solved the problem of axiomatizing arithmetic and would make a report on the subject in the Congress.

 11 12

Gray 2000, 72-73. Ibid.

The Hilbert Problems

Hilbert Problems (1900) Asterisks denote the ten problems presented during the Paris lecture 1.* Cantor’s problem of the cardinal number of the continuum (continuum hypothesis). 2.* The compatibility of the arithmetical axioms. 3. The equality of the volumes of two tetrahedra with equal bases and equal altitudes. 4. The problem of the straight line as the shortest distance between two points (alternative geometries). 5. Lie’s concept of a continuous group of transformations, without the assumption of the differentiability of the functions defining the group. 6.* A mathematical treatment of the axioms of physics. 7.* The irrationality and transcendence of certain numbers. 8. * Problems of prime numbers (including the Riemann hypothesis). 9. A proof of the most general law of reciprocity in any number field. 10. The determination of the solvability of a Diophantine equation. 11. Quadratic forms with any algebraic numerical coefficients. 12. The extension of Kronecker’s theorem on Abelian fields to any algebraic realm of rationality. 13.* A proof of the impossibility of solving any equation of the 7th degree by means of functions of only two arguments. 14. A proof of the finiteness of certain complete systems of functions. 15. A rigorous foundation for Schubert’s enumerative calculus. 16.* The problem of the topology of algebraic curves and surfaces. 17. The expression of definite forms by squares. 18. The building-up of space from congruent polyhedra (n-dimensional crystallography, groups, etc.). 19.* Determining whether the solutions of regular problems in the calculus of variations are necessarily analytic. 20. The general problem of boundary values (variational problems). 21.* A proof of the existence of linear differential equations with a prescribed monodromic group. 22.* The uniformisation of analytic relations by means of automorphic functions. 23. Further development of the methods of the calculus of variations.

111

112

Chapter Eleven

Despite Hilbert’s conference drawing a mixed reception from the attendees, it became a great success as Minkowski had predicted, being quickly published in Germany and translated into French and English. In fact, Hilbert’s problems did not take long to attract young mathematicians around the world to try to unveil the future of mathematics, solving the problems posed by Hilbert. Thus, for example, the Russian mathematician Sergei Bernstein (1880-1968) travelled from Paris to Göttingen in 1904 to present to Hilbert his proof that, under the conditions prescribed by Hilbert, elliptical differential equations have analytical solutions (problem no. 19).

CHAPTER TWELVE FROM INTEGRAL EQUATIONS TO HILBERT SPACES

After Paris, Hilbert resumed his usual academic activity at the University of Göttingen. A quick look at the titles of the courses offered by Hilbert between 1900 and 1904 shows us that they dealt with a wide range of topics, which included those that had focused his interest before Paris (algebra, geometry and number theory), those that occupied the centre of his research at that moment (differential and integral calculus, function theory, potential theory and calculus of variations), and those that would do so in the immediate future (particularly, mechanics). Among all these issues, there is no doubt that the one that focused most of his intellectual efforts during those years was that of integral equations (those equations in which in the integrand appears an unknown function to be determined). At the end of the 19th century there was an increasing interest in the study of integral equations, mainly due to their connections with some of the differential equations of mathematical physics. From these investigations, the four forms of integral equations, called today Volterra and Fredholm equations of first and second kind, arose. We owe the first rigorous treatment of what we might call a general theory of integral equations to the Swedish astronomer and mathematician Erik Ivar Fredholm (1866-1927), who published his research on the topic in a series of articles published between 1900 and 1903. The research of Fredholm had a profound impact and attracted the attention of Hilbert, who decided to dedicate the seminar of the winter semester of 1900/01 to the study of the integral equations. There, the Swedish mathematician Erik Holmgren (1872/3-1943) explained to Hilbert and the other participants in the seminar the research on integral equations carried out by his fellow Fredholm. In his work, Fredholm presented the solution of the Fredholm equations of the second kind, named for him, in an original and elegant way that revealed a certain analogy between the integral equations and the linear equations of algebra, which led him to develop a theory of determinants for

114

Chapter Twelve

Fredholm and Hilbert about integral equations An integral equation is an equation that contains a function ݂ሺ‫ݔ‬ሻ and integrals over this function that must be solved for ݂ሺ‫ݔ‬ሻǤ If the limits of the integral are fixed, then the integral equation is called a Fredholm integral equation. If a limit is variable, is called a Volterra integral equation. If the unknown function only appears under the sign of integration, the equation is called of the “first kind.” If the function is inside and outside the integration sign, the equation is called of the “second kind.” A Fredholm equation of the first kind is thus an integral equation of the form ௕

݃ሺ‫ݔ‬ሻ ൌ න ‫ܭ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ݂ሺ‫ݕ‬ሻ݀‫ݕ‬ǡ ௔

where ݂ is the function to be solved, and ܽǡ ܾǡ ݃ and ‫ ܭ‬are known (‫ܭ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ is called the kernel of the integral). A Fredholm equation of the second kind is an integral equation of the form ௕

݃ሺ‫ݔ‬ሻ ൌ ݂ሺ‫ݔ‬ሻ െ න ‫ܭ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ݂ሺ‫ݕ‬ሻ݀‫ݕ‬Ǥ ௔

A Volterra equation of the first kind is an integral equation of the form ௫

݃ሺ‫ݔ‬ሻ ൌ න ‫ܭ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ݂ሺ‫ݕ‬ሻ݀‫ݕ‬Ǥ ௔

A Volterra equation of the second kind is an integral equation of de form ௫

݃ሺ‫ݔ‬ሻ ൌ ݂ሺ‫ݔ‬ሻ െ න ‫ܭ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ݂ሺ‫ݕ‬ሻ݀‫ݕ‬Ǥ ௔

An integral equation is called homogeneous if ݃ሺ‫ݔ‬ሻ ൌ Ͳ. Clearly, not all the equations belong to one of the previous forms. For example, Dickman function ఈ ‫ݕ‬ ݀‫ݕ‬ ‫ܨ‬ሺߙሻ ൌ න ‫ ܨ‬൬ ൰ ǡ ͳെ‫ݕ ݕ‬ ଴ is very similar to a homogeneous Volterra equation of the second kind, but is not of this type, as the integrand is of the form ‫ܨ‬൫݂ሺ‫ݕ‬ሻ൯, not of the form ‫ܨ‬ሺ‫ݕ‬ሻ. In his articles Fredholm studied integral equations of the second kind with a complex parameter ߣ: ௕

݃ሺ‫ݔ‬ሻ ൌ ݂ሺ‫ݔ‬ሻ െ ߣ න ‫ܭ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ݂ሺ‫ݕ‬ሻ݀‫ݕ‬Ǥ ௔

Fredholm then defined a determinant ‫ܦ‬௞ ሺߣሻ associated with the kernel  cont. p. 115

From Integral Equations to Hilbert Spaces

115



ߣ‫ ܭ‬and showed that ‫ܦ‬௞ ሺߣሻ is an integer function (a function defined on the whole complex plane and holomorphic at each point) of ߣ. The roots of the equation ‫ܦ‬௞ ሺߣሻ = 0 are called eigenvalues, and the solutions corresponding to the homogeneous equation ݃ሺ‫ݔ‬ሻ ൌ Ͳ are called the eigenfunctions of the equation. Fredholm also showed that ifߣ is not an eigenvalue, then the integral equation can be solved or “inverted” by putting ௕

݂ሺ‫ݔ‬ሻ ൌ ݃ሺ‫ݔ‬ሻ െ ߣ න ܵሺ‫ݔ‬ǡ ‫ݕ‬ሻ݃ሺ‫ݕ‬ሻ݀‫ݕ‬ǡ ௔

where S is called the kernel or solving function and is given as the ratio of the determinants–as in Cramer´s rules. This shows how much Fredholm was inspired by the theory of linear equations. Following Fredholm, Hilbert considered a system of equations of finite dimension, but with the addition of a complex parameter Ȝ: ௡

݂௜ െ ߣ ෍ ݇௜ǡ௝ ݂௝ ൌ ݃௜ ሺ݅ ൌ ͳǡʹǡ ǥ ǡ ݊ሻǤ ௝ୀଵ

Instead of giving the solution and verifying it as Fredholm had done, Hilbert made the step to the limit rigorous. In doing so, he was able to demonstrate in his theory of integration a series of results analogous to those of the theory of linear equations. In this sense, we could say that Hilbert treated integral equations as if they were linear equations and, as it has been said sometimes, began the algebraization of analysis. Hilbert turned the previous system into a system of bilinear forms. Introducing the notation ௡

ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൌ ෍ ‫ݔ‬௜ ‫ݕ‬௜ ǡ ௜ୀଵ

for the internal product of two vectors x and y, the system of equations ݂ െ ߣ‫ ݂ܭ‬ൌ ݃ is written as ሺ‫ݑ‬ǡ ݂ሻ െ ߣሺ‫ݑ‬ǡ ‫݂ܭ‬ሻ ൌ ሺ‫ݑ‬ǡ ݃ሻǡ where the vector f is considered a solution if the above equation is satisfied for every vector u. Hilbert then solves the system ݂ െ ߣ‫ ݂ܭ‬ൌ ݃ in a similar way to what Fredholm had done but going a step further. In the case in which the kernel is symmetric, that is, when ‫ܭ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൌ ‫ܭ‬ሺ‫ݕ‬ǡ ‫ݔ‬ሻ, Hilbert was able to develop a more complete theory than Fredholm’s one. In particular, he established an analogy between the cont. p. 116

Chapter Twelve

116

bilinear form





‫ܭ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൌ ෍ ෍ ݇௜ǡ௝ ‫ݔ‬௜ ‫ݕ‬௝ ௜ୀଵ ௝ୀଵ

and the integral ௕



න න ‫ܭ‬ሺ‫ݏ‬ǡ ‫ݐ‬ሻ ‫ݔ‬ሺ‫ݏ‬ሻ‫ݕ‬ሺ‫ݐ‬ሻ݀‫ݐ݀ݏ‬Ǥ ௔



In this case, it is demonstrated that the eigenvalues of the integral equation are a succession of real numbers ሺߣ௡ ሻ and the eigenfunctions corresponding to the different eigenvalues are orthogonal. With this notation we can enunciate, as an example of the scope of the theory, one of the most important results achieved by Hilbert and his disciple and collaborator Erhard Schmidt (1876-1959): Theorem of Hilbert-Schmidt: If ݂ሺ‫ݔ‬ሻ satisfies ௕

݂ሺ‫ݔ‬ሻ ൌ න ‫ܭ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ݃ሺ‫ݕ‬ሻ݀‫ݕ‬ ௔

for a continuous function ݂ሺ‫ݔ‬ሻ, then ݂ ൌ σ௡ ܿ௡ ߮௡ , where ሺ߮௡ ሻ are the orthonormal eigenfunctions for K and ௕

ܿ௡ ൌ න ߮௡ ሺ‫ݔ‬ሻ ݂ሺ‫ݔ‬ሻ݀‫ݔ‬Ǥ ௔

(We can see here the connexion with Fourier series: a series written in terms of orthogonal eigenfunctions is called a Fourier series).

integral equations. Hilbert immediately saw the possibility to achieve his goal of a unitary approach in analysis more easily through this way than through the calculus of variations in which he was working. In this sense, the influence of Fredholm on Hilbert is evident, although Hilbert was not so worried about the resolution of the integral equations as in the construction of a general theory of integral equations. Although some results of Fredholm and Hilbert had been anticipated in some cases–particularly by Carl Neumann and Henri Poincaré–, they were the first to deal with the study of integral equations in all their generality, regardless of their applications. During the semester of winter of 1901/02, Hilbert gave a course on potential theory, using the first results obtained in his study of integral equations. From then on, Hilbert would only speak

From Integral Equations to Hilbert Spaces

117

about integral equations to his students. Although the topic of the discussions had changed, the weekly walk with the members of the seminar continued. Hilbert was already, at only forty years old, a famous mathematician; he had been elected a member of several foreign academies and the German government had awarded him the title of Geheimrat–knight. It was an acknowledgment of the lengthy list of impressive achievements by Hilbert: his important theorems in the theory of invariants, his contributions and systematization of number theory carried out in the Zahlbericht, his influential and widely published book on the foundations of geometry, the list of problems presented in Paris that unveiled the future of mathematics, his results on the calculus of variations, the restitution of the Dirichlet principle, etc. The years in which Hilbert taught his lessons on analytic functions for only one person, Professor Franklin, were far away. Then, hundreds of students attended his classes, some had to sit on the windowsills to be able to listen to the most famous mathematician in Germany and, together with Poincaré, the world. Then, shortly after his fortieth birthday, Hilbert again had the opportunity to leave Göttingen. Lazarus Fuchs had died and Hilbert was offered his chair of mathematics at the University of Berlin. But Hilbert not only did not leave Göttingen, but he kindly asked Althoff, with the approval and support of Klein and as a compensation for his remaining in Göttingen, to create a new chair of mathematics at Göttingen and that this position should be offered to his old friend Minkowski. And so it was: Minkowski arrived at Göttingen in the summer semester of 1902, thus beginning a period of six uninterrupted years of joint work with Hilbert. Now, instead of the seminar with Klein, Hilbert would run it with Minkowski. Two years later an 18-year-old student named Hermann Weyl, who would become one of Hilbert’s most brilliant disciples and collaborators, and his successor in Göttingen from 1930 to 1933, arrived in Göttingen. Weyl was particularly captivated by Hilbert’s Zahlberitcht and resolved to study everything he had written. He presented his doctoral dissertation in 1908 under Hilbert’s supervision, in which he explored singular integral equations with special consideration of Fourier integral theorems. After his habilitation in 1910, he became a Privatdozent and was thereby entitled to lecture at the University of Göttingen. Almost all of Weyl’s publications during his stay in Göttingen until 1913 dealt with integral equations and their applications. He left Göttingen in 1913, when he was offered a professorship at the Eidgenössische Technische Hochschule in Zürich, where he lectured until 1930.

118

Chapter Twelve

The same year of Weyl’s arrival, on the initiative of Klein, Carl Runge received a call from the University of Göttingen to occupy a chair of applied mathematics. He took up the post in October of that year and held it until his retirement in 1925. After Runge’s appointment, Göttingen became the only German university with four full professors of mathematics (Klein, Hilbert, Minkowski and Runge). The cooperation among all four manifested itself in the weekly walk, “every Thursday at three o’clock,” during which they talked a little of everything: mathematics (and the neighbouring sciences: astronomy, mechanics and physics), academic organizational tasks and the practice of sport! (see also Chapter 6). It was during these years that the “Hilbert School” (in Blumenthal’s expression) reached its maximum splendour. Between 1901 and 1914, Hilbert supervised more than 40 doctoral theses, some of them of lasting value and veritable landmarks in 20th century mathematics. Those included, but are not limited to, those of Earle Raymond Hedrick (1901), Georg Hamel (1901), Oliver Kellogg (1902). Rudolf Fueter (1903), Charles Mason (1903), Teiji Takagi (1903), Sergei Bernstein (1904), Erhard Schmidt (1905), Ernst Hellinger (1907), Hermann Weyl (1908), Andreas Speiser (1909), Alfred Haar (1909), Richard Courant (1910), Erich Hecke (1910), Kurt Grelling (1910), Hugo Steinhaus (1911), and Hans Bolza (1913). During those years, many students and researchers from inside and outside of Germany went to Göttingen to study or collaborate with Hilbert and to share their valuable contributions with the Göttingen scientific community. Many of them remained afterwards as professors in Göttingen or went to other universities, becoming in most cases reputed mathematicians or scientists. In March 1904 the first in a series of five articles by Hilbert, published between 1904 and 1906, on integral equations finally appeared; a sixth article appeared a little later, in 1910. These articles, which would be later included in the work Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen (Elements of a general theory of linear integral equations), published in 1912, are among the most influential papers published in modern times and constitute the starting point of modern or abstract functional analysis. Hilbert himself would explain in the first of the articles mentioned that: The systematic development of a general theory of linear integral equations is of utmost importance for analysis as a whole, in particular for the theory of definite integrals and the theory of the expansion of arbitrary functions in infinite series, furthermore for the theory of linear differential equations and analytic functions as well for potential theory and the calculus of variations. I intend to treat in this book the question of solving integral

From Integral Equations to Hilbert Spaces

119

equations, and, above all, to explore the interrelations and general properties of their solutions.1

That Hilbert considered his methods as a logical extension of finitedimensional techniques is clear, both for internal evidence and for his own affirmations. “The method,” said Hilbert, “is to begin with an algebraic problem, namely the problem of orthogonal transformations of quadratic forms in ݊ variables in a sum of squares, and by rigorous passage to the limit for ݊ ൌ λ, to successfully solve the transcendental problem considered.”2 This method led Hilbert and his collaborators to the consideration and abstract study of the space κଶ of square summable sequences of real or complex numbers, that is, of the space consisting of all the infinite sequences ሼǥ ǡ ܿିଶ ǡ ܿିଵ ǡ ܿ଴ ǡ ܿଵ ǡ ܿଶ ǡ ǥ ሽ, ܿ௞ ‫ॶ א‬, where ॶ ൌ Թ‘”ॶ ൌ ԧ, for which ஶ

෍ ȁܿ௞ ȁଶ ൏ λǤ ௞ୀିஶ

In 1906, the same year in which Hilbert’s influential papers were published, the doctoral thesis of Maurice Fréchet (1878-1973) Sur quelques points du calcul fonctionnel (On some points of the functional calculus), which had a tremendous influence, both for the development of Functional Analysis and for Topology, also appeared. In this thesis, Fréchet introduced the abstract notions of distance and metric space, although this term was coined later by Felix Hausdorff (1868-1942), which allowed him to extend the usual notions of neighbourhood, limits, continuity, etc. to abstract sets. Fréchet also introduced the notions of compactness, completeness and separability and studied them in different infinite-dimension functional spaces showing the importance of these properties for their characterization. Fréchet’s topological ideas spread quickly. It is not strange, then, that they were applied in the context of the work on integral equations developed by Fredholm and Hilbert. Thus, as early as 1907, Erhard Schmidt would simplify and extend the results of Fredholm and Hilbert, but from a completely different point of view. Schmidt was the first to define κଶ as a space of infinite dimension, thinking about the sequences (ܿ௞ ) as points in this space and studying the geometry of κଶ as a function space in the modern sense of the term–using to this effect the language of norms, linearity, subspaces and orthogonal projection. Schmidt’s conceptual simplifications were immediately incorporated by Ernst Hellinger

 1 2

Hilbert 1912, 2. Ibid., 3.

Chapter Twelve

120 

Functional Analysis and Spectral Theory Rather loosely we could say that the object of study of functional analysis is that of a “function space,” that is, a topological space, the points of which are functions. Many such spaces are vectors spaces which have a metric often defined in terms of a norm which yields a distance between any points in the spaces. Indeed, every normed vector space has a topological structure: the norm induces a metric and the metric induces a topology. Hence, paraphrasing Dieudonné (1981, 1), we can define functional analysis as “the study of topological vector spaces and of mappings defined between subsets of such spaces, these mapping being assumed to satisfy various algebraic and topological conditions”. Well known examples of topological vector spaces are Hilbert spaces and Banach spaces. The definition above is wide enough to include the main topics dealt with in functional analysis: Hilbert spaces and the spectral theory of operators, the theory of normed linear spaces, the theory of Banach algebras and operator algebras (C*-algebras and von Neumann algebras), the general theory of topological vector spaces, generalized functions (distributions), and the theory of partial differential equations. The fundamental concepts and methods of functional analysis arose from diverse sources: the calculus of variations, the theory of integral equations, set theory and topology, and linear and abstract algebra. Indeed, as remarked by Dieudonné, functional analysis is “a rather complex blend of Algebra and Topology, and it should therefore surprise no one that the development of these two branches of mathematics had a strong influence on its own evolution” (Ibid). Besides Hilbert spaces (and intimately related to it), the topic in Functional Analysis in which the influence of Hilbert has been greater is spectral theory. This can be defined as the study of the spectra of linear operators and related properties. The spectrum (pl. spectra) of a linear operator ܶ on a complex Banach space ܺ is the set ߪሺܶሻ of complex numbers ߣ for which the operator ߣ‫ ܫ‬െ ܶ is not invertible (i.e., it does not have a defined bounded inverse everywhere), where ‫ܫ‬ is the identity operator on ܺ. Hilbert’s pioneer contributions to spectral theory stem from his work on integral equations and are on the basis of the mathematical formulation of quantum mechanics.

(1883-1950) and Hermann Weyl in their 1907 and 1908 dissertations under Hilbert.

From Integral Equations to Hilbert Spaces

121

Fréchet’s and Schmidt topological and geometrical characterization of function spaces had a great influence in two young mathematicians Frigyes Riesz (1880-1956) and Ernst Fischer (1875-1954), who proved independently that the space ‫ܮ‬ଶ of square Lebesgue integrable functions is a complete metric space. In fact, the first to use the term Hilbert space (or, rather, espace de Hilbert) was Riesz in his book of 1913 on systems of equations in infinitely many unknowns. However, it was not until 1929 that John von Neumann (1903-1957), another disciple and collaborator of Hilbert in the 1920s, defined the abstract concept of Hilbert space rigorously and formulated the mathematical structure of quantum mechanics in the language of the theory of Hilbert spaces. Before von Neumann, the term Hilbert space had been applied principally to the space κଶ of square summable sequences or to the space ‫ܮ‬ଶ of Lebesgue square integrable functions which Riesz had proved to be isomorphic to κଶ . The essential properties of these spaces were those of a vector space with an inner product which was complete and separable (i.e., which had a countable dense subset). Von Neumann then defined an abstract Hilbert space axiomatically as any separable, complete inner product space. He also defined a general linear operator on a Hilbert space as a linear transformation defined on some of its subsets, which enabled him to express the transformations of quantum mechanics in terms of operators on a Hilbert space.

CHAPTER THIRTEEN PARADOXES IN GÖTTINGEN

In 1897 Ernst Zermelo arrived in Göttingen. He had studied at the universities of Berlin, Halle and Freiburg, and finished his doctorate in 1894 at the University of Berlin with a dissertation on the calculus of variations. He remained there until 1897, when he moved to the University of Göttingen. Two years later Zermelo completed his Habilitationschrift on hydrodynamics, which he had already began with Max Planck (18581947) in Berlin. He delivered his habilitation thesis at the University of Göttingen on 4 March, 1899, which allowed him to become a privatdozent at that University, where he remained until 1910. Despite his early underpinnings in the calculus of variations and hydrodynamics, the reading of the work of Cantor and the influence of Hilbert changed Zermelo’s research interests to set theory and the foundations of mathematics, two areas in which he became a close collaborator of Hilbert. The same year as the arrival of Zermelo in Göttingen the first work on mathematics of a young English aristocrat who had studied philosophy and mathematics at Cambridge, entitled An Essay on the Foundations of Geometry, appeared. Bertrand Arthur William Russell (1872-1970), third count of Russell and future Nobel prize of Literature, would play an outstanding role in the genesis of the so-called crisis of foundations (Grundlagenkrisis) of mathematics that would explode at the beginning of the 20th century. In 1901 Russell had begun writing about how it was possible to derive mathematics from the symbolic language of Peano, which he had extended to encompass the logic of relations that same year. This led Russell to examine Cantor’s proof that the cardinal of any set ܽ is smaller than the cardinal of its power set ࣪ሺܽሻ, or what amounts to the same, that there is no greater cardinal (Cantor’s theorem). The problem with this result was that it seemed to contradict the Russellian hypothesis of the existence of a universal class, which would have to be expected to have the greatest cardinal (this is the so-called Cantor’s paradox). It was precisely the application to the universal class of the diagonal argument, used by Cantor

124

Chapter Thirteen

in the proof of the theorem that bears his name, that led Russell to the discovery of the paradox of the class of all classes that do not belong to themselves, the so-called Russell’s paradox. Russell communicated his famous paradox to Frege in a letter dated July 16, 1902. It was first published in the work by Russell, The Principles of Mathematics, which appeared the following year. Almost at the same time, Frege referred to the paradox in an appendix to the second volume of his work Grundgesetze der Arithmetik (The Basic Laws of Arithmetic) (1903), recognizing that the logical system shown there, on which he wanted to base all mathematics, was inconsistent. Frege sent a copy of this volume to Hilbert, who replied that he already knew this paradox and that he believed that “Dr. Zermelo had discovered it three or four years before.”1 Hilbert also added that “I had already found more convincing contradictions four or five years earlier, which led me to the conviction that traditional logic is inadequate and that the theory of concept formation needs to be profiled and refined.”2 In fact, Hilbert’s first contact with paradoxes goes back to his correspondence with Cantor between 1897 and 1900. According to Hilbert’s opinion, Cantor’s paradox and other paradoxes (the paradox of Zermelo-Russell and another mathematical paradox discovered by Hilbert himself) could be solved by applying the axiomatic method, because in an axiomatic theory only those concepts that can be deduced from a finite base of axioms are accepted and so the problem is then to postulate a sufficient group of axioms that does not lead to contradictions. The Zermelo-Russell paradox showed that both Frege’s logical system and the naive set theory developed by Cantor and Dedekind were inconsistent and that, therefore, that it was necessary to find an axiomatization of logic and set theory that could prevent the emergence of paradoxes and to ensure once and for all the consistency of both disciplines. This was not a minor problem, since for many mathematicians of the time these two disciplines constituted the foundation of mathematics as a whole and, therefore, the consistency of all branches of mathematics (for example, of geometry or analysis) depended on the consistency of logic and set theory. The prominent place occupied by both disciplines in the late nineteenth century was to a large extent the logical consequence of the progressive rigorization of analysis that had taken place in the 18th and 19th centuries, to which mathematicians such as Cauchy, Bernard Bolzano

 1 2

For this reason Russell’s paradox is also known as the Zermelo-Russell paradox. Frege 1976, 79-80.

Paradoxes in Göttingen

125

(1781-1848), Riemann and Weierstrass, among others, had contributed decisively. This process had been inextricably linked to the progressive abandonment of the resource of spatial or temporal intuition that had dominated the infinitesimal calculus since its very creation by Isaac Newton (1642/43-1727) and Gottfried Wilhelm Leibniz (1646-1716). The discovery of non-Euclidean geometries, which made the Euclidean space lose the privileged place that had been granted until then as the ultimate source of mathematical intuition, contributed decisively to this abandonment. Because of this, the traditional definition of real numbers as continuous magnitudes of Euclidean geometry intuitively grasped was replaced by arithmetical definitions in which they were defined from sequences or infinite sets of rational numbers and, ultimately, from natural numbers. This process of arithmetization of analysis naturally left the door open to the question: to what are the natural numbers reducible? And the answer was not long in coming, since for Cantor and Dedekind the natural numbers (and from them, the rest of numbers) were reducible to sets, whereas for Frege they were reducible to concepts. Thus, with the publication by these mathematicians of the works on the foundations of mathematics, a new era was opened in the progressive rigorization of analysis that led, on the one hand, to the birth of set theory in the work of Cantor and Dedekind, and, on the other, to a radical refunding of logic in the work of Frege. Both in the correspondence with Frege and in his 1903 book, Russell had indeed formulated his paradox indistinctly in intensional terms–that is, as the paradox that arises when considering the predicates that cannot be predicated of themselves–, and in extensional terms–that is, as the paradox that emerges when considering the class of all classes that are not members of themselves. This fact, along with the conviction of Frege and Russell that to avoid the paradox it was necessary to reformulate the logical system–to which they intend to reduce the theory of classes and the entire mathematics–, put logic in the centre of attention. Thus, the publication of Russell’s paradox could not go unnoticed by Hilbert and his group of collaborators in Göttingen, because it made it clear that logic (and not just set theory) was inconsistent. In this sense, Russell’s paradox questioned Hilbert’s axiomatic program and caused an important revision of it, since its core was the proof of the consistency of the axioms of arithmetic by means of the usual methods of logic and mathematics, and so required logic to be free of contradictions. Thus, if before 1903 Hilbert believed it was possible to solve or avoid paradoxes by means of an axiomatic reformulation of set theory, from now on it

126

Chapter Thirteen

seemed also necessary an axiomatization of logic that avoided contradictions.

Fig 13-1 Ernst Zermelo in the 1900s

Hilbert entrusted the task of axiomatization of logic and set theory to Zermelo, who presented his famous axiomatization of set theory in the

Paradoxes in Göttingen

127

paper “Untersuchungen über die Grundlagen der Mengenlehre I” (“Investigations in the foundations of set theory I”) (1908). Besides the axioms of choice (Axiom VI) and of infinite (Axiom VII), the most intriguing axiom presented by Zermelo is the axiom of separation (Axiom III), since it is in virtue of it that it is possible to avoid Russell’s paradox and the paradoxes of transfinite numbers. As Zermelo himself explains, this axiom is a substitute for the Cantorian definition of set or, more precisely, for the principle of comprehension, which he sees as a specification of that definition. According to this principle, it is assumed that to every logically expressible property there corresponds a set. But according to the axiom of separation “no one ever has the right to define sets independently, but only as subsets obtained by separation from sets already given and so contradictory constructions such as “the set of all sets” or “the set of all ordinal numbers” [...] are eliminated.”3 For example, Russell’s paradox arises when considering the property “set that is not an element of itself.” Under the assumption of the principle of comprehension, the previous property determines the set ‫ ݓ‬of all sets ‫ݔ‬ such that ‫ݔ ב ݔ‬, from which Russell derived that ‫ ݓ א ݓ‬՞ ‫ݓ ב ݓ‬. One way to eliminate this contradiction, proposed by Russell in 1905, would be to limit the size of what is called a set, so preventing the formation of sets too large such as the set of all sets having any given property. This is indeed the basic idea of Zermelo’s axiom of separation, which limits the cases where a set can be associated to a property to those in which the elements that have this property already belong to a previously given set, so that you can only get a subset of this set and thus Russell’s paradox is blocked. Although Zermelo’s set theory blocks the appearance of Russell’s paradox and the transfinite paradoxes such as Burali-Forti’s or Cantor’s paradox, it cannot prove its own consistency as would be desirable for the completion of Hilbert’s program of proving the consistency of mathematics. Indeed, as we know today by Gödel’s second incompleteness theorem, this is impossible. On the other hand, the axiomatization of logic constituted a problem that presented inextricable difficulties for Zermelo and Hilbert. Hence the publication in 1910 of the first volume of the work of Alfred North Whitehead (1861-1947) and Bertrand Russell Principia Mathematica (1910-1913), in which an axiomatization of logic that avoided logical paradoxes appeared for the first time, could be described some years later by Hilbert as “the crowning achievement of the work of axiomatization as a

 3

Zermelo 1908, 263.

128

Chapter Thirteen

The axiomatization of set theory (“Untersuchungen über die Grundlagen der Mengenlehre,” 1908) Def. 1. Set theory is concerned with a domain ी of objects, which we will call simply things (Dinge) and among which the “sets” constitute a part. If two symbols ܽ and ܾ designate the same thing, we write ܽ ൌ ܾ, otherwise ܽ ് ܾ. We say that a thing ܽ “exists” if it belongs to the domain ी; likewise, we say of a class ॆ of things that “there exist things of the class ॆ,” if ी contains at least one individual of this class. Def. 2. There are, among the things of domain ी, certain “fundamental relations” of the form ܽ ‫ܾ א‬. If for two things ܽ and ܾ, the relation ܽ ‫ ܾ א‬holds, it is said that “ܽ is an element of the set ܾ” or that “ܾ contains ܽ as an element” or that “ܾ possesses the element ܽ.” A thing ܾ that contains another thing ܽ as an element, can always be called a set, and only in this case–with one exception (axiom II). Def. 3. If every element ‫ ݔ‬of a set ‫ ܯ‬is also an element of the set ܰ, so that from ‫ ܯ א ݔ‬it always follows that ‫ܰ א ݔ‬, then we say that “‫ ܯ‬is a subset of ܰ” and we write ‫ܰ ؿ ܯ‬. It is always assumed that ‫ܯ ؿ ܯ‬ and that from‫ ܰ ؿ ܯ‬and ܰ ‫ ܴ ؿ‬it always follows that‫ܴ ؿ ܯ‬. Def. 4. A question or statement ू is called definite if the fundamental relations of the domain, with the help of the axioms and the universally valid laws of logic, enable one to decide, without any arbitrariness, its validity or invalidity. Similarly, a class-statement (Klassenaussage) ूሺ‫ݔ‬ሻ, in which the variable ‫ ݔ‬ranges over all individuals of a class ॆ, will be called “definite” if it is definite for each single individual of the class ॆ. Ax. I. If every element of a set ‫ ܯ‬is also an element of ܰ and vice versa; if, therefore, ‫ ܰ ؿ ܯ‬and ܰ ‫ ܯ ؿ‬, then always ‫ ܯ‬ൌ ܰ. Or, more briefly: every set is determined by its elements. (Axiom of determination). Ax. II. There is an (improper) set, the null set Ͳ, that contains no element at all. If ܽ is any one thing of the domain, there exists a set ሼܽሽ containing ܽ and only ܽ as an element; if ܽ and ܾ are any two things in the domain, there exists always a set ሼܽǡ ܾሽ containing ܽ and ܾ as elements, and no other thing ‫ ݔ‬different from both. (Axiom of elementary sets). Ax. III. If the class-statementूሺ‫ݔ‬ሻ is definite for all the elements of a set ‫ܯ‬, ‫ ܯ‬always has a subset‫ ूܯ‬containing all the elements ‫ ݔ‬of M for which ूሺ‫ݔ‬ሻ is true, and only such elements. (Axiom of separation)

Paradoxes in Göttingen

129

Ax. IV. To every set ܶthere corresponds another set ॏܶ (the power set of T) that contains all the subsets of ܶ and only these. (Axiom of the power set). Ax. V. To every set ܶ there corresponds a set्ܶ (the union set of ܶ) that contains as elements all the elements of ܶ and only these. (Axiom of the union). Ax. VI. If ܶ is a set whose elements are all sets different from Ͳ and mutually disjoint, its unionՁܶ contains at least one subset ܵଵ having one and only one element in common with each element of ܶ. (Axiom of choice). Ax. VII. The domain contains at least one set ܼ that contains the null set as an element and is so constituted that to each of its elements ܽ there corresponds a further element of the form ሼܽሽǡ that is to say, that for each of its elements ܽ it also contains the corresponding set ሼܽሽ as an element. (Axiom of infinity). To this set of axioms, two more were added later: the replacement axiom scheme and the axiom of foundation. The replacement axiom scheme, proposed independently by Thoralf Skolem (1887-1963) and A. Fraenkel in 1922, allows the construction of the series of ordinals. The axiom of foundation, adopted by J. von Neumann in 1925 and by Zermelo in 1930, restricts the category of sets so that we can better capture the sets commonly used in mathematics. The theory obtained by adding these two axioms to axioms I-VII is commonly referred to in the literature as Zermelo-Fraenkel set-theory with the axiom of choice (ZFC).

whole.”4 Meanwhile, life continued in Göttingen. Between 1890 and 1914 the number of Privatdozenten of mathematics at the University of Göttingen was impressive and included such prominent names as Hermann Weyl, Arnold Sommerfeld (1868-1951), Constantin Carathéodory (1873-1950), Gustav Herglotz (1881-1953), Erich Hecke, Max Born (1882-1970), Richard Courant (1888-1972), Theodor von Kármán (1881-1963), Otto Blumenthal, Ernst Zermelo, Paul Koebe (18821945), Robert Fricke (1861-1930) and Otto Toeplitz (1881-1940).

 4

Hilbert 1965, vol. 3, 153 (Ewald 1996, vol. 2, 1113).

130

Chapter Thirteen

As the university grew in number of students, teaching staff and prestige, new research institutes were created in fields such as physics, applied mathematics, electronics and geophysics. All of them worked in close collaboration with the staff of mathematics professors, among other things because they were subsidized and equipped by the Göttingen Association for the Promotion of Applied Physics, a consortium of industrialists and scientists created by Klein in 1898. Thus, when Runge was appointed as a mathematics professor in the winter semester of 1904/05, the staff of science professors was also impressive. The Physics professors were Eduard Riecke (1845-1915) and Woldemar Voigt (18501919). Karl Schwarzchild was a professor of astronomy. Hermann Theodor Simon (1870-1918) was the head of the Institute of Applied Electronics; Ludwig Prandtl, of the Institute of Applied Mechanics; Emil Wiechert, of the Institute of Geophysics. Max Born (future Nobel Prize in physics for his contributions to quantum mechanics) was then Hilbert’s “private assistant”. This was, according to Born, a “rather vague job […] unpaid but precious beyond description by providing me the opportunity of seeing him and listening to him every day.”5 As reported by C. Reid, “in the morning Born came to Hilbert’s house, where he usually found Minkowski already present. Together, the three discussed the subject matter of Hilbert’s coming lecture, which was often taking place that same morning.”6 But not all the work of the assistant consisted of mathematical speculation and the preparation of the classes. Born was one of those in charge, for example, of taking notes for a course taught by Hilbert in the summer semester of 1905,7 which we will refer to in the next chapter, notes that were then deposited in the Lesezimmer, so that all students could consult and discuss them.

 5

Born 1965, 4. Reid 1970, 103. 7 The other was Ernst Hellinger, then a mathematics student in Göttingen. 6

CHAPTER FOURTEEN THE CONSISTENCY OF ANALYSIS

In Grundlagen der Geometrie, Hilbert had demonstrated the consistency of Euclidean geometry through the exhibition of a model in which all the axioms of plane geometry were satisfied. This model was that of analytic geometry, whose consistency was taken for granted. Thus, the remaining task was to axiomatize analysis and to prove the consistency of the resulting axiom system. The first task was accomplished by Hilbert in the article “Über den Zahlbegriff”, where he characterized the real numbers system as a complete Archimedean ordered field (see Chapter 10). Nonetheless, because analysis is also an axiomatic theory, the proof of the consistency of the axioms of geometry set out in Grundlagen der Geometrie was just a proof of the relative consistency of these axioms. To demonstrate the absolute consistency of the axioms of geometry it was necessary to prove the consistency of the axioms that define the real numbers as a complete ordered Archimedean field. Thus, in his famous list of problems presented in the Congress of Paris in 1900, Hilbert had put the question of whether it was possible to give a direct proof of the noncontradiction of the axioms that determine the structure of the real numbers system, which he called simply the axioms of arithmetic (see Chapter 11). Hilbert was convinced of the possibility to easily find a syntactic, purely logical proof of the consistency of these axioms, reformulating the methods of proof usually employed in Dedekind’s and Weierstrass’s theory of irrationals to this end. But Hilbert’s confidence was cut short by Zermelo and Russell’s discovery of the paradoxes of logic and set theory (see Chapter 13), for it was now obvious that the consistency proof of analysis could not be done using means just proved to be inconsistent. This led Hilbert to think that a proof of the consistency of the axioms of analysis, at least by the means of pure logic, as proposed by him in his 1900 lecture, was impossible. And this, in turn, led him to redefine his program for the foundations of mathematics and his understanding of the relationship between arithmetic and logic.

Chapter Fourteen

132

This rethinking is already evident in the lecture “Über die Grundlagen der Logik und der Arithmetik” (“On the foundations of logic and arithmetic”) (1904) delivered at the Third International Congress of Mathematicians, celebrated in Heidelberg, where Hilbert also outlined for the first time a solution to the second problem of his list of 1900. Thus, in the first lines of this lecture, Hilbert warns of the fundamental difference between arithmetic and geometry regarding the inquiry into the foundations of both disciplines: In examining the foundations of geometry, it was possible for us to leave aside certain difficulties of a purely arithmetic nature; but recourse to another fundamental discipline does not seem to be allowed when the foundations of arithmetic are at issue.1

It is true, says Hilbert, that “arithmetic is often considered to be a part of logic, and the traditional fundamental logical notions are usually presupposed when it is a question of establishing a foundation of arithmetic.”2 So one could perhaps consider the possibility of grounding arithmetic in logic, but Hilbert warns: If we observe attentively, however, we realize that in the traditional exposition of the laws of logic certain fundamental arithmetic notions are already used, for example, the notion of set and, to some extent, also that of number. Thus, we find ourselves turning in a circle, and that is why a partly simultaneous development of the laws of logic and of arithmetic is required if paradoxes are to be avoided.3

Hilbert merely outlined in this paper the joint development of logic and arithmetic mentioned in the text above and the programmatic ideas that should lead ultimately to prove the consistency of analysis. In the case of logic, he did not specify a logical system and just talked about “familiar forms of logical inference.” Regarding arithmetic, he attempted to demonstrate its consistency by first reformulating the Peano axioms. The Hilbert axioms are the following: 1. 2. 3. 4.

‫ݔ‬ൌ‫ݔ‬ ሼ‫ ݔ‬ൌ ‫ݓ†ƒݕ‬ሺ‫ݔ‬ሻሽȁ‫ݓ‬ሺ‫ݕ‬ሻ ‫ݏ‬ሺ‫ݔݑ‬ሻ ൌ ‫ݑ‬ሺ‫ ݏ‬ᇱ ‫ݔ‬ሻ ‫ݏ‬ሺ‫ݔݑ‬ሻ ൌ ‫ݏ‬ሺ‫ݕݑ‬ሻȁ‫ ݔݑ‬ൌ ‫ݕݑ‬

 1

Van Heijenoort 1967, 130. Ibid., 131. 3 Ibid. 2

The Consistency of Analysis

5.

133

തതതതതതതതതതതതതതത ‫ݏ‬ሺ‫ݔݑ‬ሻ ൌ ‫ͳݑ‬,

where ‫ ݑ‬represents an infinite set, ‫ ݔݑ‬an element of this infinite set, ‫ݏ‬ denotes “successor” and ‫ ݏ‬ᇱ the “accompanying operation.”4 All axioms are easy to read, except perhaps no. 3 which states that “each element ‫ ݔݑ‬is followed by a definite thought-object ‫ݏ‬ሺ‫ݔݑ‬ሻ, which is equal to an element of the set ‫ݑ‬, namely, the element ‫ݑ‬ሺ‫ ݏ‬ᇱ ‫ݔ‬ሻ, which likewise belongs to the set ‫ݑ‬.”5 So Hilbert ignores the Peano axioms “ͳ is a number” and the socalled “principle of complete induction,” which will nevertheless be stated later. The argument sketched by Hilbert to demonstrate the consistency of the above axioms seems to be the following. To prove the consistency of the axioms 1, 2, 3 and 4, Hilbert’s reasons as follows: (i) A contradiction is a statement of the form ‫ ܣ‬and not ‫;ܣ‬ (ii) None of the previous axioms contains a negation; (iii) Its conjunction can never, therefore, engender a contradiction. This reasoning cannot be applied, however, to axiom 5, which explicitly contains a negation. Hilbert’s idea to demonstrate the non-contradiction of this axiom and, ultimately, the non-contradiction of the axioms of arithmetic, is to attribute to the axioms one property (“to be a homogeneous equation”) that is preserved by the rules of inference, so that if ‫ ܨ‬is a formula derivable from the axioms, then ‫ ܨ‬will be a homogeneous equation.6 Furthermore, the denial of a homogeneous equation is not homogeneous and, therefore, the corresponding formula will not be derivable. In this way, Hilbert proves or intends to prove by induction on derivations that: (iv) Axioms 1, 2, 3 and 4 are homogeneous; (v) All the statements inferred from the previous axioms are homogeneous; (vi) The negation of axiom 5, which is not homogeneous, is not derivable, therefore, from the axioms 1, 2, 3 and 4; but that is the same as saying that the conjunction 5 with 1, 2, 3, 4 is not contradictory and, since these last axioms are not

 4

The distinction between “successor” and the “accompanying operation” seems unnecessary. 5 Ibid., 133. 6 An equation ܽ ൌ ܾ is called homogeneous if and only if ܽ and ܾ have the same number of symbols occurrences.

134

Chapter Fourteen

contradictory, that the axioms of arithmetic are not contradictory. Although Hilbert was aware that he had only hinted how to give a “complete proof” of the consistency of arithmetic, he was convinced that “the considerations just sketched constitute the first case in which a direct proof of consistency bas been successfully carried out for axioms.”7 Furthermore, Hilbert continues, “if we translate the well-known axioms for mathematical induction into the language I have chosen, we arrive in a similar way at the consistency of this larger number of axioms, that is, at the proof of the consistent existence of what we call the smallest infinite set (that is, of the ordinal type ͳǡ ʹǡ ͵ǡ ǥ).”8 However, Hilbert’s confidence was cut off by Henri Poincaré, who published soon after a lengthy article titled “Les mathématiques et la logique” (1905, 1906) in which he evidenced the constant circularity of the reasoning used by Hilbert in the proof of the consistency of the axioms of arithmetic. More specifically, Poincaré observed that the consistency proofs of the axioms that refer to an infinite set of statements require complete induction, so that “Hilbert’s reasoning not only assumes the principle of induction, but assumes that this principle is given to us, not as a mere definition, but as an a priori synthetic judgment.”9 This is an especially serious problem because one of the most important and well known axioms of arithmetic is precisely the principle of complete induction, so as Poincaré observed, in order to prove its consistency, Hilbert is forced to employ the principle of complete induction itself! In fact, at the end of the 1904 conference Hilbert himself seems to realize the circularity of his proof that the negation of axiom 5 does not follow from the rest of axioms and, anticipating Poincaré’s objections, sketches the way to avoid circularity in this and other consistency proofs: Whenever in the preceding we spoke of several thought-objects, of several combinations, of various kinds of combinations, or of several arbitrary objects, a bounded number of such objects was to be understood. Now that we have established the definition of the finite number we are in position to comprehend the general meaning of this way of speaking. The meaning of the “arbitrary” consequence and of the “differing” of one proposition from all proposition of a certain kind is also now, on the basis of the

 7

Ibid., 135. Ibid. In other words, the consistency of the principle of complete induction with the axioms 1 to 5 can be demonstrated in a way similar to that used in order to prove these axioms. 9 Ewald 1996, vol. 2, 1059. 8

The Consistency of Analysis

135

definition of the finite number (corresponding to the idea of mathematical induction), susceptible of an exact description by means of a recursive procedure. It is in this way that we can carry out completely the proof, sketched above, that the proposition ‫ݏ‬൫‫ ݔݑ‬ሺ଴ሻ ൯ ൌ ‫ ͳݑ‬differs from every proposition obtained as a consequence of Axioms 1-4 by a finite number of steps; we need only consider the proof itself to be a mathematical object, namely a finite set whose elements are connected by propositions stating that the proof leads from 1-4 to 6, and we must then show that such a proof contains a contradiction and therefore does not exist consistently in the sense defined by us.10

These ideas preclude the direction of the future work of Hilbert and his collaborators on the foundations of arithmetic in the decade of the twenties. In particular, for the first time, the idea of a proof theory (Beweistheorie) appears in the above text (“we need only consider the proof itself to be a mathematical object”), that is, of a mathematical theory which studies the mathematical proofs formalized in the language of symbolic logic. As Hilbert will explain later, a formalized proof is a finite sequence of formulae whose structural properties are accessible to intuitive and finite (metamathematical) reasoning. This makes possible the consistency proofs, which deal with the proofs used in mathematics and not with the objects or abstract concepts these proofs refer to. Ultimately, thanks to formalization, a mathematical proof of consistency can be reduced to a finite string of simple arithmetic statements. Therefore, two principles of complete induction must be distinguished: an intuitive and finite one and another properly mathematical one. In this way Hilbert will respond in the twenties to the objections of Poincaré. In the last paragraphs of the 1904 conference, Hilbert writes that the consistency of analysis can be proved in an analogous way: The existence of the totality of real numbers can be demonstrated in a way similar to that in which the existence of the smallest infinite set can be proved; in fact, the axioms for real numbers as I have set them up […] can be expressed by precisely such formulas as the axioms hitherto assumed […], and the axioms for the totality of real numbers do not differ qualitatively in any respect from, say, the axioms necessary for the definitions of the integers.11

However, Hilbert did not specify what this analogy of expression among the axioms of arithmetic and the axioms of analysis consists of, nor did he

 10

Van Heijenoort 1967, 137. Proposition number 6 is precisely ‫ݏ‬൫‫ ݔݑ‬ሺ଴ሻ ൯ ൌ ‫ͳݑ‬, which could be read as “At least for one ‫ݔݑ‬, the successor of ‫ ݔݑ‬is ‫ͳݑ‬.” 11 Ibid., 137-38.

136

Chapter Fourteen

attempt to demonstrate the non-contradiction of the latter axioms following the methodology employed to prove the non-contradiction of the former.

Fig 14-1. Logische Principien des mathematischen Denkens. Lecture notes by Max Born. Unpublished manuscript.

In the summer semester of 1905, just a few months after the Heidelberg conference, Hilbert taught a course entitled Logische Principien des mathematischen Denkens (Logical Principles of Mathematical

The Consistency of Analysis

137

Thinking).12 In these lectures, Hilbert attempted to develop the idea, formulated at the conference of 1904 on the foundations of logic and arithmetic, of a piecewise simultaneous development of these two disciplines and to clarify some of the programmatic ideas outlined in that conference. In the Heidelberg conference Hilbert had said almost nothing about the logical laws required for the formalization of arithmetic. Now in the 1905 lectures he presented a logical calculus, which he regarded as the main tool for the simultaneous development of logic and arithmetic. In fact, in these lectures he applied for the first time the axiomatic method to logic and, more precisely, to propositional calculus. This means that he not only axiomatized propositional calculus, but also raised such metalogical questions about it as the independence and non-contradiction of its axioms and the decidability of its theorems. However, Hilbert did not advance much in relation to his project of a joint development of logic and arithmetic to prove the consistency of analysis. Indeed, as pointed out by V. Peckhaus, “Hilbert did not further elaborate his thoughts on the logical foundations of mathematics at that time since he believed Ernst Zermelo capable of solving the problems in axiomatizing logic and set theory”13 and thus to resolve the paradoxes discovered independently by Russell, Zermelo and himself. We already know that in 1908 Zermelo expounded his famous axiomatization of set theory which allows avoiding the set-theoretic paradoxes. The same year Zermelo also addressed the problem of the axiomatization of logic in a lecture given at the University of Freiburg entitled Mathematische Logik, but he just introduced a set of axioms for propositional calculus and the calculus of classes. Hence, in the lectures entitled Elemente und Prinzipienfragen der Mathematik (1910), Hilbert could say that Russell’s paradox had been solved in its set theoretic formulation by Zermelo, but that “it has not yet been resolved in a satisfactory way as a logical antinomy.”14 The first solution to this antinomy appeared in Russell’s paper “Mathematical Logic as based on the Theory of Types” (1908) and in Russell’s and Whitehead’s book Principia Mathematica (1910-13). In both places the solution comes via the so-called theory of ramified types, which is also applied in the book to reduce the whole edifice of mathematics to logic. This explains the interest in Hilbert’s circle at

 12

This is the course, whose lecture notes were taken by Ernst Hellinger and Max Born, we referred to in the last chapter. 13 Peckhaus 1994, 317. 14 Hilbert 1910, 159.

138

Chapter Fourteen

Göttingen in knowing Russell’s mathematical logic and, to a certain extent, the growing interest in logic and the foundations of mathematics in Göttingen from 1914 on.15 Because of the influence of Principia Mathematica and of Poincaré’s criticism of his talk of 1904, Hilbert abandoned the project of a joint development of logic and arithmetic in this period to attack the proof of the consistency of analysis and called for a reduction of the axioms of arithmetic to logic.

 15

The central figure in the absorption of Principia in Göttingen was Heinrich Behmann (1891-1970), who gave four different talks between 1914 and 1917 on the logical achievements of that work in the Colloquium of the Göttingen Mathematical Society. In 1918, he obtained his doctorate with a thesis entitled Die Antinomie der trans¿niten Zahl und ihre AuÀösung durch die Theorie von Russell und Whitehead (The Antinomy of Trans¿nite Numbers and its Resolution by the Theory of Russell and Whitehead), written under the supervision of Hilbert. See Mancosu (1999, 304-305).

CHAPTER FIFTEEN MINKOWSKI AND THE EARLY RECEPTION OF RELATIVITY THEORY AT GÖTTINGEN

After the Heidelberg conference, Hilbert continued his research on integral equations and began, at the request of Minkowski, the study of classical physics. A year later, Hilbert and Minkowski decided to conduct a weekly seminar on physics, more concretely on the electrodynamics of moving bodies. The participants of the seminar, Emil Wiechert, Max von Laue (1879-1960), Max Born, Max Abraham (1875-1922), Arnold Sommerfeld and Gustav Herglotz among others, studied the papers of Hendrik Lorentz (1853-1928), Poincaré and others on the difficulties which the theories of the electrodynamics had run into because of the well-known MichelsonMorley experiment. However, the most recent publications of Lorentz and Poincaré, in which the principle of relativity and the Lorentz transformation were exploited more fully, were neglected. In the same year of 1905, an examiner of patents at the Patent Office at Bern named Albert Einstein (1879-1955) published four epoch-making scientific papers dealing with similar topics to those dealt with in the seminar. In the first paper, Einstein described a method for determining molecular dimensions. In the second, he explained the photo-electric effect, for which he won the Nobel Prize in 1921. In the third one, he presented a molecular kinetic theory of heat. The last paper, titled “Electrodynamics of Moving Bodies”, was the first presentation of what became known as the Special Relativity Theory. However, according to Born, “nothing was as yet known in Göttingen, and in the HilbertMinkowski seminar the name of Einstein was never mentioned.”1 The interest of Hilbert and Minkowski in physics was neither new nor spurious. The axiomatization of physics was one of the problems on foundations Hilbert had presented at the Paris Congress, although he had never published any paper on this or a similar topic. It not was until the academic year of 1902/03 that he taught his first lecture course on

 1

Reid 1970, 105.

140

Chapter Fifteen

mechanics. The next lecture courses on the same topic were taught in 1905/06 and 1906/07, which shows Hilbert’s renewed interest in this subject.2 However, from then and until 1910, he taught no additional courses on physical matters. The contact of Minkowski with physics was closer. He had already lectured on mechanics at the Federal Polytechnical School in Zürich, where Walter Ritz (1878-1909), Albert Einstein, and Marcel Grossmann (1878-1936) were among his students. From 1902, he also lectured on these subject and other related topics at the University of Göttingen. From 1907 and until his death in 1909, Minkowski dedicated much efforts to the study of electrodynamics and the principle of relativity. He was indeed the first mathematician in Germany to take an interest in relativity theory and the main protagonist in the mathematical reformulation of this theory. In 1905, the Hungarian Academy of Science instituted a prize of 10,000 gold crowns, a very generous amount for the time, for the mathematician who had contributed most significantly to the progress of mathematics in the last 25 years. The award, later known as the Bolyai Prize in honour of the Hungarian mathematician János Bolyai, one of the discoverers of non-Euclidean geometry, fell into the hands of Henri Poincaré. However, as a courtesy to Hilbert, the committee that granted the prize unanimously voted that in the report sent to the Academy to justify their election they would put Hilbert’s contributions to mathematics on an equal footing with those of Poincaré. Klein, who was one of the members of the committee, would recognize later that the prize was awarded to Poincaré for the generality and breadth of his contributions to mathematics and ventured that “Hilbert will yet encompass as comprehensive a field as Poincaré!”3 Klein was not wrong. In fact, Hilbert was doing at that time what would perhaps be his most important contribution to mathematics and, undoubtedly, the coronation of his contributions to mathematical analysis: his research on what would later be called Hilbert spaces. Following the seminar conducted with Hilbert, Minkowski turned his attention to the theory of heat radiation. In 1906 he gave a lecture on recent work in this area by Max Planck (1858-1947) and Walther Nernst (1864-1941) for the Göttingen Mathematical Society, and in the summer semester of 1907 he also offered a course on this subject. Minkowski’s

 2

Since the founding of physical institutes in the 1870s, theoretical physics had become the affair of specialists like Max Planck or Ludwig Boltzmann (18441906). However, mathematicians stayed up-to-date with the latest research in analytical mechanics and usually taught this subject in universities. 3 Ibid., 106.

Minkowski and the Early Reception of Relativity Theory at Göttingen

141

course notes indicate that he was familiar with Planck’s pioneering article on relativistic thermodynamics, in which he praised Einstein’s relativity paper. On October 9, 1907, Minkowski wrote to Einstein to request an offprint of his 1905 paper to discuss it in his seminar with Hilbert on the electrodynamics of moving bodies. On November 5, 1907, he gave a lecture about Space and Time entitled “The Relativity Principle” in the Göttingen Mathematical Society. Einstein’s achievements came to Minkowski as a great surprise, since he did not believe Einstein possessed the mathematical background necessary to create such a theory. Remembering the year in which he had Einstein as a student in Zürich, Minkowski once exclaimed: “Ach, der Einstein, der schwänzte immer die Vorlesungen௅dem hatte ich das gar nicht zugetraut!” (Oh, that Einstein, always missing lectures࣓I really would not have believed him capable of it!).4 He also once admitted to his student Max Born, that “for me it came as a tremendous surprise, for in his student days Einstein had been a real lazybones. He never bothered about mathematics at all.”5 In 1906, while continuing his abstract research on integral equations and function spaces, Hilbert taught a course on calculus for first year students and began his ski classes. In the spring, Hilbert bought a bicycle and at age 45 he started learning to ride a bicycle. Unlike skiing, which was nothing more than a momentary pastime, bicycle trips, hiking or gardening, often accompanied his creative part. Thus, Hilbert, who often worked in the garden of his house, writing on a large blackboard that was hanging on the wall of his neighbour, could stop his creative work for a while and go biking, prune a tree or dig a hole in the garden to plant something. Shortly afterwards, according to Courant, who often watched Hilbert working in the garden of his house from the balcony of his room, he would continue with the problem he had left in a “fantastic balance between intense concentration and complete relaxation.”6 Richard Courant had begun his studies at the University of Breslau where he had become friends with Otto Toeplitz and Ernst Hellinger, who were a bit older than him. When they left for Göttingen, they convinced Courant to join them, because the academic level of Göttingen was much higher than that of Breslau. When Courant arrived in Göttingen in 1907, he attended Minkowski and Hilbert’s courses and a year later he became Hilbert’s assistant, a post that he held during four semesters. In 1910 he obtained his doctorate with a thesis on the Dirichlet principle and in 1912

 4

Ibid., 105. Seelig 1956, 28. 6 Reid 1970, 109. 5

142

Chapter Fifteen

he got the venia legendi with a habilitation thesis about the existence proofs in mathematics. However, it was not until 1918, after the Great War had finished, that he started working as an associated professor in Göttingen. In the summer of 1908 Hilbert fell into a depression caused, according to Blumenthal, by his imprudent physical and mental excesses, in any case without any other reason or a known external cause. Because of this, he entered a sanatorium in the Harz mountains near Göttingen. There he rested for a few months and once he recovered he returned to classes in the fall of that same year. Meanwhile, Minkowski was at the top of his creative activity. On December 21, 1907, Minkowski had presented a talk to the Göttingen scientific society, which was published on 5 April 1908, in Göttinger Nachrichten under the title “Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern” (“The Basic Equations for Electromagnetic Processes in Moving Bodies”). On September 21, 1908, in the 80th annual general meeting of the Society of German Scientists and Doctors at Köln (Cologne), Minkowski presented his famous talk “Raum und Zeit” (“Space and Time”). In it he presented his famous mathematical model, the so-called Minkowski space-time. In this model it was possible to represent mathematically the physical properties of the universe described by Einstein’s theory of special relativity, according to which the notions of space and time cease to be absolute and become indissolubly united. Recovered from his illness, in the fall of 1908 Hilbert attacked a wellknown problem of number theory. The English mathematician Edward Waring (1734-1798) had conjectured in his Meditationes Algebraicae (1770) that every natural number could be expressed as a sum of at most four squares, nine cubes or nineteen fourth powers and, in general, as a finite number of ݊-powers. This assertion has been traditionally interpreted as saying that every positive integer can be expressed as a sum of at most ݃ሺ݇ሻ ݇–Š powers of positive integers, where ݃ሺ݇ሻ depends only on ݇, not on the number being represented.7 The case ݇ ൌ ʹ had been stated by Fermat in 1640 and, after long withstanding unsuccessful attacks by Euler, it was finally proved by Lagrange in 1770, who showed that each positive integer could be expressed as a sum of at most four squares

 7

So, for example, Waring speculated that ݃ሺʹሻ ൌ Ͷ, ݃ሺ͵ሻ ൌ ͻ and ݃ሺͶሻ ൌ ͳͻ, that is, it takes no more than 4 squares, 9 cubes, or 19 fourth powers to express any integer.

Minkowski and the Early Reception of Relativity Theory at Göttingen

143

of positive integers. After Lagrange success, particular cases of the problem were solved for ݇ ൌ ͵ǡ Ͷǡ ͷǡ ͸ǡ ͹ǡ ͺǡ ͳͲ. Although after 1770 little progress had been made in demonstrating Waring’s conjecture, interest had recently grown due to the possibility of applying certain analytical methods. Hurwitz had worked in this direction, but he stopped, seeing himself unable to solve the problem. Hilbert started just where Hurwitz had left off and at the end of 1908 he managed to prove the generalized Waring’s conjecture, nowadays known as the Hilbert-Waring theorem. The great English mathematician Godfrey Harold Hardy (1877-1947), who some years later would give another proof of the theorem along with John Edensor Littlewood (1885-1977), would show his great admiration for Hilbert’s solution to this historic problem and would affirm that “it is absolutely and triumphantly successful [...] one of the landmarks in the modern theory of numbers.”8 Hilbert’s mathematical conquests, his international recognition, his intense academic life, contact with nature, his natural optimism and friendship with Minkowski must offered Hilbert an intellectual stimulus that was hardly surmountable and probably something very similar to the happiness that some Greek philosophers like Pythagoras, Plato or Aristotle saw indissolubly associated with the bios theoretikos, the life dedicated to the scientific activity. But in this clear and radiant sky a black cloud would suddenly appear, a premonition that human life is fragile and vulnerable. On Thursday, January 7, 1909, the four mathematics professors did their weekly walk at three o’clock. Perhaps Minkowski explained his recent work on the mathematics of special relativity and Hilbert his extraordinary proof of Waring’s conjecture, which he would present at the next meeting of the mathematics seminar. On Friday, Minkowski and Hilbert continued their usual academic tasks. Then, on Sunday afternoon Minkowski suffered a serious appendicitis attack and at night he was admitted to the hospital. On Monday his condition worsened and on Tuesday at noon he asked to see his family and Hilbert for the last time. When Hilbert arrived at the hospital, Minkowski was already dead. On Wednesday, Hilbert explained in the classroom, with tears in his eyes, the death of his old and dear friend. On Thursday, at three o’clock, instead of the usual mathematical walk, the mathematics professors accompanied Minkowski’s body in his funeral procession. Hilbert would never fully recover from Minkowski’s loss. To a certain extent, Göttingen neither. After a certain debate among the mathematics professors, the position left vacant by Minkowski’s death was offered to

 8

Ibid., 114.

144

Chapter Fifteen

Edmund Landau (1877-1938), who had previously worked as Privatdozent in Berlin and was only 32 years old. Landau’s specialty was the application of analytical methods to number theory and, to a lesser extent, function theory. He had no interest in geometry or mathematical physics and, in general, he always showed an absolute contempt for applied mathematics, contrary to what had been the spirit of the Göttingen mathematics faculty, as it had been conceived by Klein. The year of the death of Minkowski, a mathematician from Darmstadt, Paul Wolfskehl, left in his will the amount of 100,000 marks for anyone who could prove Fermat’s last theorem with the proviso that as long as it was not possible to prove it, a committee of the Scientific Society of Göttingen could discretionally dispose of the interests produced by that amount of money. Hilbert became the chairman of the Wolfskehl-Stiftung Committee (Committee for the Wolfskehl Foundation) created for this purpose and thus, during the following years, money was mainly used to invite professors of recognized prestige in mathematics and physics to lecture in Göttingen. That same year the committee invited the man who had been the great rival of Klein and had snatched the first Bolyai prize from Hilbert, the great French mathematician Henri Poincaré, to give a series of lectures at the Philosophical Faculty. The topics chosen by Poincaré were that of the integral equations and that of the theory of relativity, subjects to which he himself had contributed in a very significant way and that he knew that they were of interest to the mathematicians in Göttingen. Hilbert offered a great reception in his house in honor of Poincaré and Klein, who turned sixty at that time. In 1909 the philosopher Leonard Nelson (1882-1927), who could often be seen walking and discussing with Hilbert about philosophy, logic and mathematics, obtained the qualification for teaching as a Privatdozent in the mathematical-scientific division of the Philosophical Faculty of Göttingen. Nelson’s first attempt for the habilitation in 1904 had been rejected in 1906 by the majority of the Philosophical Faculty, particularly by those belonging to the philological-historical division. However, Klein and Hilbert had supported Nelson’s habilitation on the grounds of his mediating position in the philosophical foundations of mathematics and of his knowledge of recent developments in mathematics. The fact is that Nelson’s direct and irreverent style had offended the philosopher Edmund Husserl (1859-1938), whose phenomenology was publicly ridiculed by Nelson. Although he was finally habilitated in 1909, he did not obtain a post as associate professor until 1919. In 1917, after Husserl’s retirement, the majority of the Philosophical Faculty proposed Georg Misch (18781965) for this position, but Klein, Hilbert, Peter Debye (1884-1966) and

Minkowski and the Early Reception of Relativity Theory at Göttingen

145

Fig. 15-1 From left to right: Poincaré, Gösta Mittag-Leffler (1846-1927), Runge and Landau at Hilbert’s home on the occasion of Klein’s sixtieth birthday.

146

Chapter Fifteen

Runge, among others, proposed Nelson instead. In the summer of 1918, Landau and Carathéodory also sided with Hilbert’s minority group. After long-withstanding accusations between the two divisions of the Philosophical Faculty and negotiations with the Prussian Ministry of Education௅in which Hilbert played a principal role promoting Nelson௅, Misch was appointed full professor as a successor to Heinrich Maier (1867-1933), who had held this position since 1911/12, and Nelson was appointed associate professor. In this way, a schism between the two divisions of the Philosophical Faculty was avoided. As remarked by Volker Peckhaus, Hilbert’s long-withstanding support for the promotion of Nelson, as well as his efforts on securing a better position for Ernst Zermelo, were part and parcel of his strategy aimed at securing institutional support for research in logic, set theory and the foundations of mathematics.9 Also Hermann Weyl, who would be one of the most brilliant mathematicians of Hilbert’s next generation, was appointed as Privatdozent in Göttingen in 1909, a post that he held until 1913. It was then that Weyl began a long and lasting friendship with Hilbert and Courant, who would become Hilbert’s assistant in 1910. That year Hilbert sent his latest communication on integral equations to the Göttingen Scientific Society. Weyl and Courant would play a key role in the future development of mathematics: Weyl as a successor to Hilbert’s chair in mathematics at Göttingen and as professor at the Institute of Advanced Studies at Princeton from 1933 and Courant as the first director of the Mathematical Institute of Göttingen and, from 1936 onwards, as professor at the University of New York and promoter of its Institute of Mathematical Sciences, which currently bears his name. In 1910, the Hungarian Academy of Science awarded the second Bolyai prize “to David Hilbert, who by the profundity of this thought, the originality of his methods, and the rigorous logic of his demonstrations has already exercised considerable influence on the progress of the mathematical sciences.”10 Poincaré, as the secretary of the committee responsible for deciding the winner of the prize, was in charge of summarizing the contributions of Hilbert to be presented to the Academy. He mentioned to this effect Hilbert’s proof of Gordan’s theorem, the new proof of the transcendence of the numbers ݁ and ߨ, his work on algebraic number fields, his research on the foundations of geometry, the salvaging

 9

See Peckhaus (1990, 4-22). Reid 1970, 125.

10

Minkowski and the Early Reception of Relativity Theory at Göttingen

147

of the Dirichlet principle, his proof of Waring’s conjecture and his contributions to the theory of integral equations. Two years later the book Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen (Elements of a general theory of linear integral equations) (1912) was published. It contained the articles published by Hilbert between 1904 and 1910 on integral equations. Hilbert’s research on this topic had led him to the territory where mathematics and physics came together. It should not be surprising, then, that from 1912, Hilbert was devoted almost exclusively to physics, a discipline that had attracted the attention of the best mathematicians of the moment due to the discoveries that had been made in the previous decade.

CHAPTER SIXTEEN HILBERT’S FOUNDATIONS OF PHYSICS

The first decade of the 20th century had been undoubtedly a golden age for physics: Heinrich Hertz had found the existence of the electromagnetic waves predicted by James Maxwell (1831-1879), Wilhelm Röntgen (18451923) had discovered X-rays, the Curies radioactivity, 1 Joseph John Thomson (1856-1940) electrons, Albert Einstein had formulated the special theory of relativity and Max Planck quantum theory. However, Hilbert and other mathematicians of the time thought it lacked a certain rigor in the development of theoretical physics. Now, according to Hilbert, this rigor could only be achieved through the axiomatic method. “Physics, Hilbert exclaimed on an occasion, is much too hard for physicists.” 2 Therefore, it was time for Hilbert to attack the problem, which he had put in the sixth place of his famous Paris list, to axiomatize physics. Although Hilbert’s mathematical efforts were concentrated until 1912 on mathematical problems and more specifically, on the theory of linear integral equations, after Minkowski’s death he returned to lecture on physical issues. Thus, after teaching mechanics and continuum mechanics in the winter and summer semester of the academic year 1910/11, Hilbert also taught statistical mechanics in the winter semester of 1911/12, and kinetic gas theory and radiation theory in the winter and summer semester of 1911/12. However, the decisive turn in Hilbert’s research interests occurred in 1912, once he had finished his monograph on the theory of linear integral equations. For now he considered the applications of this theory as a means of clarifying the foundations of kinetic theory and radiation theory. Hilbert’s idea was basically to apply the axiomatic method and his theory of integral equations to specific branches of physics as a concrete accomplishment of his axiomatization program of physics postulated in Paris. The two fields that Hilbert attacked were first the kinetic theory of gases and then the elemental theory of radiation, fields whose concepts necessarily led to integral equations as the only possible

 1 2

Marie Curie (1867-1934) and Pierre Curie (1859-1906). Reid 1970, 127.

150

Chapter Sixteen

expression of the data. Hilbert would devote his efforts during the next few years almost exclusively to mathematical physics.

Fig. 16-1 David Hilbert, 1912

Hilbert’s Foundations of Physics

151

From 1912 to 1914 Hilbert published a series of articles on kinetic theory and the theory of radiation, which had a considerable impact in the community of young researchers in Göttingen and elsewhere. After his first paper on kinetic theory, Hilbert conducted a seminar on this topic together with his former student Erich Hecke. The seminar was also attended by the Göttingen docents Max Born, Paul Hertz (1881-1940), Theodor von Kármán and Erwin Madelung (1881-1972). Also in 1912, to be aware of the latest developments in physics, Hilbert called his old friend and collaborator Arnold Sommerfeld to ask him to look for a young researcher to become his assistant in this field. Sommerfeld offered the post to Paul Ewald (1888-1985), who had studied with Hilbert at Göttingen and had recently finished his dissertation in Munich. Now, he would return to Göttingen as Hilbert’s assistant on physics. He was the first to hold this position, which Hilbert would maintain for many years to come. In 1913 Niels Bohr (1885-1962), who often visited Göttingen, proposed his well-known theory on the movement of electrons in atoms and Hilbert did not miss the opportunity to talk with him. In fact, in the last years Hilbert had been able to arrange the visit of prominent researchers in the field of physics through the Wolfskehl Foundation. Thus, in 1910 Lorentz was invited to Göttingen to talk about the theory of relativity and radiation and in 1912 Sommerfeld gave a series of lectures on the latest advances in physics. 3 In the spring of 1913, Hilbert organized, under the auspices of the Wolfskehl Foundation, an important conference in Göttingen on the kinetic theory of matter. The meeting was held at the Royal Academy of Sciences in Göttingen and the lecturers included some of the most prominent physicists of the time: Max Planck, Peter Debye, Walther Nernst, Marian von Smoluchowski (1872-1917), Arnold Sommerfeld, and Hendrik Lorentz. As remarked by Leo Corry, “by the year 1913 Hilbert’s interest in a wide variety of physical disciplines had become a truly central feature of his current research and teaching concerns.”4 So, for example, “the topics of his lectures on physics had expanded way beyond the more traditional ones of classical mechanics and continuum mechanics and now covered also statistical mechanics, radiation theory and the molecular theory of matter.”5 In the winter semester of 1913/14 Hilbert taught his last course on a physical issue before eventually turning to general relativity. It dealt

 3

In the year 1911 the financial aid of the Wolfskehl-Stiftung had gone to Zermelo for his contributions to set theory and in attention to his delicate health. 4 Corry 1999, 173. 5 Ibid., 174.

152

Chapter Sixteen

with electromagnetic oscillations. At the same time, the meetings of the Göttingen Mathematical Society discussed several topics that were crucial to Hilbert’s path in general relativity. So, for example, Max Born lectured on “Mie’s Theory of Matter” (November 25 and December 16, 1913), whereas Friedrich Böhm (1885-1965) lectured on the “Recent Work of Einstein and Grossmann on Gravitation” (December 9, 1913). Debye’s talk at the Royal Academy of Sciences, which dealt with the equation of state, the quantum hypothesis and heat conduction, had impressed Hilbert so much that the following year, 1914, he and the mathematician Alfréd Haar (1885-1933) became professors at Göttingen. That year the Great War began. The German government wrote an infamous “declaration to the cultural world” (Aufruf an die Kulturwelt) denying that Germany had provoked the war and refuting each one of “the lies and accusations of the enemy” in this direction. The Kaiser’s government asked the most prominent artists and intellectuals to sign the declaration and most of the scientists did so (Klein, Planck, Röntgen, Nernst, Paul Ehrlich (1854-1915), Emil Fischer (1852-1919), August von Wassermann (1866-1925), Wilhelm Wien (1864-1928), among others). Hilbert refused to sign the declaration on the pretext that he could not verify each one of the statements that were made in it. Despite the war, life continued in Göttingen. The mathematical walk on Thursday continued as usual, but now Landau, Prandtl and Carathéodory were added. However, most students and young professors were called up. The Lesezimmer, always crowded by students before the war, was now almost always empty. Hilbert, on the other hand, was absorbed with physics, although his research and docent activities on physical issues decreased considerably when compared with that of the previous years. In 1914 he published a third paper on the foundations of the theory of radiation and lectured once again on statistical mechanics. In the summer semester of 1915, Hilbert also gave a course on the structure of the matter. During that summer Albert Einstein visited Göttingen, invited by the Wolfskehl Foundation. In the week from June 28 to July 5, he gave a series of six lectures to the Mathematical Society of Göttingen in which he explained his latest research on gravitation and relativity as expounded in the paper “The Formal Foundation of the General Theory of Relativity” (1914). All that is left from these lectures is 11 pages of notes of his first lecture written by an unknown auditor of part of these lectures. There is also no documentation that certifies who attended the Einstein lectures. Apart from Klein and Hilbert, there were professors of a certain age such as Runge and Wiechert who were not called up for the military

Hilbert’s Foundations of Physics

153

service and who probably attended Einstein’s lectures. One of those attending the conferences given by Einstein was Emmy Noether, who had been appointed to the University of Göttingen that same year at the request of Klein and Hilbert. Little more is known about that visit, but Einstein would recognize in a letter addressed to Sommerfeld on July 15, just a week after his stay in Göttingen, that he “had the great pleasure of seeing that everything was understood to the details” and that he was “quite enchanted with Hilbert.”6 Needless to say, Hilbert was also enchanted with Einstein and really interested in Einstein’s latest approach to general relativity theory. Apart from the excitement caused by Einstein’s visit and the subsequent discussion about his ideas, Hilbert and Einstein enjoyed each other’s companionship and established a lasting personal bond. This was due not only to the mutual admiration they professed, but also to their open character, remote of any kind of pedantry, and their contempt for all kinds of social conventions. They also shared a great interest in politics and a firm belief in the international spirit of science. So, for example, Einstein also refused to sign the German “declaration to the cultural world,” although he had Swiss citizenship and, therefore, no one could accuse him of treason. Hilbert was not only German, but Prussian, so his refusal must be understood as a true act of political and vital courage. In the Wolfskehl lectures at Göttingen Einstein presented his so-called Entwurf theory, the precursor of the general theory of relativity, as expounded in the “Formal Foundation” paper of 1914. One of the main targets of this paper was the mathematical derivation of the field equations of the Entwurf theory from a variational principle. However, the equations presented in this paper were not generally covariant. Hilbert had studied this paper carefully and had understood its arguments and technical intricacies “down to the details.” So it is possible that he found problems in Einstein’s derivation of the field equations and sought his own way to derive fully covariant equations from variational principles. More concretely, as remarked by U. Majer and T. Sauer, “his heuristic idea was to try to connect Einstein’s theory to a generalized version of Maxwellian electrodynamics which had recently been proposed by Gustav Mie […] Hilbert’s central idea had been to combine both Mie’s and Einstein’s theories by means of an invariant variational integral governing the theory.”7

 6 7

Schulmann 1998, 147. Hilbert 2009, 10-11.

154

Chapter Sixteen

During the following months, Hilbert dedicated all his efforts to what he called Die Grundlagen der Physik (The Foundations of Physics), that is, the formulation of a unified theory of gravitational fields based on Gustav Mie’s (1869-1957) electromagnetic theory of matter. On November 20, 1915, Hilbert presented for publication in the Nachrichten of the Mathematical-Physical Class of the Göttingen Academy of Science his “First Communication” on “The Foundations of Physics.” Almost at the same time, Einstein presented to the Prussian Academy of Berlin a series of four papers (each separated from the other by a week: the 4th, 11th, 18th and 25th of November) in which he expounded his final version of the theory of general relativity. Hilbert’s communication and Einstein’s fourth note presented for the first time the generally covariant equations for the gravitational field. Since Hilbert’s paper was presented five days before Einstein’s definitive paper on the topic, some historians have suggested the priority of Hilbert over Einstein in the formulation of a generally covariant theory of gravitation, including field equations. However, Hilbert’s paper did not appear until 31 March 1916 and when Hilbert submitted his text on 20 November 1915 it did not contain these equations. Through a close analysis of Hilbert’s papers, L. Corry, J. Renn and J. Stachel have discovered a first set of proofs of the paper, bearing a printer’s stamp of December 6, which displays substantial differences from the published version. In particular, these historians emphasize the fact that the proofs do not yet contain the explicit form of the gravitational field equations in terms of the Ricci tensor and the Riemann curvature scalar.8 Only later, some time after 6 December, could Hilbert have added the key passage containing the gravitational field equations into the page proofs. Thus, it seems that Hilbert did not anticipate Einstein in the formulation of these equations. The intense epistolary exchange between Einstein and Hilbert on November 1915 and the shipment of their respective publications shows the engagement of Hilbert in the mathematical formulation of the theory of general relativity. This provoked a certain suspicion on the part of Einstein about the partaking or “nostrification” (Nostrifizierung) of his theory by Hilbert, but the fact is that Hilbert never claimed priority and often admitted privately and publicly that the great idea was Einstein’s. 9 On

 8

See Corry, Renn and Stachel (1997). On November 26, 1915 a day after Einstein presented the final version of the field equations he wrote his close friend Heinrich Zangger: “The general relativity problem is now finally dealt with. The perihelion motion of Mercury is explained wonderfully by the theory […] The theory has unique beauty. Only one colleague

9

Hilbert’s Foundations of Physics

155

December 20, 1915, Einstein wrote to Hilbert that “there has been certain resentment between us, the cause of which I do not want to analyse. I have fought against the associated feeling of bitterness with complete success. I think of you again with unmixed kindness, and I ask you to try to do the same with me. It is objectively a shame when two real guys that have emerged from this shabby world do not give each other a little pleasure.”10 On 4 December 1915, Hilbert had presented to the Göttingen Academy of Sciences a “Second Communication” on the “Foundations of Physics.” However, its processing was postponed and a second version of it was presented to the Academy on 26 February 1916, which was also withdrawn from print at the beginnings of March. By mid-February offprints of Hilbert’s First Communication were available and it is presumable that Hilbert sent a copy to Einstein together with an invitation to visit Göttingen again. Einstein’s response form 18 February did not mention Hilbert’s paper but did state his intention to visit Göttingen. He probably arrived there on 2 March 1916 and remained in Göttingen as Hilbert’s guest for a few days. In the March 1916 printed version of his November 20 paper, Hilbert added a reference to Einstein’s November 25 paper and wrote: “the differential equations of gravitation that result are, as it seems to me, in agreement with the magnificent theory of general relativity established by Einstein in his last papers.” 11 On May, 27, 1916 Hilbert invited again Einstein to visit Göttingen again and stay with him; but in spite of several invitations over the next few years, Einstein never came, although they continued to correspond over issues connected with Hilbert’s paper.

 has understood it really, but he tries in a tricky way to ‘nostrify’ it (an expression due to Abraham). In my personal experience I have not learnt any better the wretchedness of the human species as on occasion of this theory and everything related to it. However, that does not concern me in the slightest.” Einstein to Zangger, CPAE 8, Doc. 152 10 Einstein to Hilbert, CPAE 8, Doc. 167. 11 Cited from Corry, Renn and Stachel (1997, 344). It is worth mentioning, however, that Hilbert did not only cite Einstein’s fourth note in this passage, but he also pointed out that the derivation of the explicit form of the Einstein’s equations form the variational formulation is indeed trivial.

156

Chapter Sixteen

Fig. 16-2 Albert Einstein, 1916

When the summer semester began, Hilbert had a new assistant for physics, the mathematician Richard Bär (1892-1940). During this semester he lectured on partial differential equations in continuation of the lecture course on ordinary differential equations given in the previous semester.

Hilbert’s Foundations of Physics

157

He also conducted his weekly seminar on the “Structure of the matter” together with Debye and taught a course entitled “Introduction to the Principles of Physics.” This was followed by a course on “The Foundations of Physics” in the winter-semester of 1916/17, the Ausarbeitung of which was produced by Bär. In August 1917, Bär left Göttingen. Instead of him, Paul Bernays arrived in Göttingen as Hilbert’s assistant on logic and the foundations of mathematics. This demonstrates that Hilbert had turned his full attention to foundational topics in the fields of logic and mathematics. It does not mean, however, that Hilbert had abandoned physics or did not lecture on the foundations of physics anymore, but his lectures took a more comprehensive and philosophical approach. So, for example, he tried to connect the theory of relativity, his own search of a unified theory of the gravitational and electromagnetic fields and quantum theory. It is also worth mentioning that Hilbert would become the following years an enthusiastic advocate of Einstein’s theory of relativity, which he described in the closing passage of the 1922/23 lecture course entitled Wissen und mathematisches Denken (Knowledge and mathematical Thought) as “one of the most tremendous achievements of human spirit.”12

 12

Cited from Corry (2004, 430).

CHAPTER SEVENTEEN BEYOND PRINCIPIA MATHEMATICA

After the 1904 Heidelberg conference on the foundations of logic and arithmetic, Hilbert would not publish anything more about issues related to logic and the foundations of mathematics until 1917. However, during this period he taught several different courses on these issues, which shows that he continued to be engaged with logical and foundational issues, although these were not the main topic of his research.1 In the summer semester of 1917, Hilbert taught a course entitled Mengenlehre (Set theory), which supposed a remarkable shift in Hilbert’s main research interests. Near the end of these lectures, Hilbert observed that he hoped “to be able to go more deeply into the foundations of logic” in the next semester. Actually, during the following seven years, Hilbert gave a series of lecture courses௅most of them with the assistance of Bernays, who was also responsible for preparing the lecture notes௅, which reveal a profound revision of his views on logic and the foundations of mathematics.2 In September of 1917, Hilbert went to Zürich to present a conference at the Swiss Mathematical Society. It seems likely that was in the course of this stay in Zurich that Hilbert asked Bernays, a former student in Göttingen, to become his assistant for logic and the foundations of mathematics.3 Bernays accepted and began a fruitful collaboration 1 These are Logische Prinzipien des mathematischen Denkens (SS 1905), Prinzipien der Mathematik (SS 1908, WS 1908/09), Elemente und Prinzipienfragen der Mathematik (SS 1910, SS 1913), Logische Grundlagen der Mathematik (WS 1911/12) and Probleme und Prinzipienfragen der Mathematik (WS 1914/15). 2 These are Prinzipien der Mathematik (WS 1917/18), Logik-Kalkül (WS 1920), Probleme der mathematischen Logik (SS 1920); Grundlagen der Mathematik (WS 1921/22); and Logische Grundlagen der Mathematik (WS 1922/23 and WS 1923/24). 3 According to Constance Reid, Hilbert would have visited Zürich in the spring of 1917 and arranged an excursion with two young mathematicians from the circle of Hurwitz: George Pólya (1887-1985) and Paul Bernays. Surprisingly for both, in

160

Chapter Seventeen

between the two men in the field of logic and the foundations of mathematics that would culminate in one of the landmark works of the twentieth century in this field: Grundlagen der Mathematik (The Foundations of Mathematics), coauthored by Hilbert and Bernays and published in two volumes at the beginning of the thirties. The Zürich conference, entitled “Axiomatisches Denken” (“Axiomatic thought”), supposed Hilbert’s public return, after the Heidelberg conference in 1904, to the field of logic and the foundations of mathematics. As we already know, in this talk Hilbert praised the axiomatization of logic carried out by Whitehead and Russell at Principia Mathematica (1910-13), describing it as “the crowning achievement of the work of axiomatization as a whole,”4 but at the same time, he resumed the subject of proof theory, which he had set aside since the 1904 conference. In Principia, Russell and Whitehead expounded their well-known solution of the Russell paradox and other paradoxes through the theory of ramified types, a solution that Russell had previously exposed in the paper “Mathematical Logic as based on the theory of types” (1908). Moreover, in Principia the theory of ramified types is also applied to reduce the whole edifice of mathematics to logic. The profound impact that the Russell paradox had in Hilbert’s consistency program (see Chapter 13) and the fact that Russell’s logicism offered a solution to Hilbert’s quest for the consistency of mathematics explains the interest in Hilbert’s circle at Göttingen for knowing Russell’s mathematical logic and, to a certain extent, the growing interest in logic and the foundations of mathematics in Göttingen from 1914 on.5 Because of the influence of Principia Mathematica and of Poincaré’s criticism of his Heidelberg talk of 1904, Hilbert abandoned the project of a joint development of logic and arithmetic to attack the proof of the consistency of analysis and called for a reduction of the axioms of arithmetic to logic. As Hilbert writes in the lecture notes Mengenlehre:

the ride uphill to the Zürichberg there was no talk of mathematics, but of philosophy. Bernays, who had studied a bit of philosophy in Göttingen and was a friend of Leonard Nelson, had then much more to say than Pólya, so when they finished the excursion, Hilbert asked him to be his assistant at Göttingen (Reid 1970, 151-52). We have followed here the datation given in Hilbert (2013) of Hilbert’s visit to Zürich (See Hilbert 2013, 35-36, n. 9, for more information on this topic). 4 Hilbert 1965, vol. 3, 152. (Ewald 1996, vol. 2, 1113). 5 As we have already explained, the central figure in the absorption of Principia in Göttingen was Heinrich Behmann (see Chapter 14, n. 15).

Beyond Principia Mathematica

161

If we set up the axioms of arithmetic but forego their further reduction and take over uncritically the laws of logic, then we have to realize that we have not overcome the difficulties for a first philosophical-epistemological foundation; rather, we have just cut them off in this way.6

Hilbert then asks “to what we can further reduce the axioms” and responds himself “to the laws of logic!”7 All this certainly put logic in the forefront of Hilbert’s research on the foundations of mathematics. However, as Bernays would later remark in his article “Hilbert Untersuchungen über die Grundlagen der Arithmetik” (“Hilbert’s investigations on the foundations of arithmetic”) (1935), Principia Mathematica could only provide an “empirical confidence” of the consistency of the logical axioms, not “complete certainty” as Hilbert sought, because the only thing that can be done in the framework of Principia in this regard is to derive theorems and to see that no contradiction follows from them. Thus, the problem of proving the consistency of the axioms of logic and, with it, that of proving the consistency of arithmetic and set theory, still remained open. Moreover, as Hilbert writes in Axiomatisches Denken: When we consider the matter more closely we soon recognize that the question of the consistency of the integers and of sets is not one that stands alone, but that it belongs to a vast domain of difficult epistemological questions which have a specifically mathematical tint: for example (to characterize this domain of questions briefly) the problem of the solvability in principle of every mathematical question, the problem of the subsequent checkability of the results of a mathematical investigation, the question of a criterion of simplicity for mathematical proofs, the question of the relationship between content and formalism in mathematics and logic, and finally the problem of decidability of a mathematical question in a finite number of operations.8

Hence, Hilbert continued, “we cannot rest content with the axiomatization of logic until all questions of this sort and their interconnections have been understood and cleared up.”9 All these issues and, in particular, the decidability of a mathematical question in a finite number of steps, seem to Hilbert “to form an important new field of research which remain to be developed. To conquer this field we must, I am persuaded, make the concept of mathematical proof itself into a specific object of

6

Quoted in Sieg (2013, 100). Ibid. 8 Hilbert 1965, vol. 3, 152. (Ewald 1996, vol. 2, 1113). 9 Ibid. 7

162

Chapter Seventeen

Fig. 17-1 Paul Bernays, ca. 1920

investigation.”10 These are words with which Hilbert summarized again the central idea of his Beweistheorie, which he had already expressed at 10

Ibid., 155 (1115).

Beyond Principia Mathematica

163

the end of the Heidelberg conference of 1904 and will develop in a series of lecture courses, talks and papers along the twenties. In short, Hilbert’s insight was to apply the axiomatic method to the logic of Principia, which, as Russell and Whitehead have “experimentally” demonstrated, it can be used to provide a foundation for mathematics. However, Principia Mathematica could only provide an “empirical confidence” in the consistency of the logical axioms. So the first step in Hilbert’s renewed program for the foundation of mathematics was to present logic as a formal axiomatic system in such a way that the axiomatic method could be applied to it in order to assail a “complete certainty” about the consistency of its axioms. This was precisely the aim of Hilbert’s next lectures. After the Zürich conference, Hilbert announced for the winter semester of 1917/18 the intriguing lecture course Prinzipien der Mathematik, the protocols of which were prepared by Bernays. These lecture notes deserve to be studied not only because Hilbert and Ackermann relied on them to write his famous book Grundzüge der theoretischen Logik (Principles of theoretical logic) (1928), but also because they presented, for the first time in history, first-order logic as a separate and independent system from second and higher-order logics and raised for the former logic the same metalogical questions that had been posed for propositional logic in the lecture notes of 1905. Indeed, as remarked by W. Ewald and W. Sieg, they “are an historical milestone, fully comparable in importance to Frege’s Begriffschrift and to Whitehead and Russell’s Principia Mathematica.”11 In these lecture notes Hilbert calls restricted function calculus what we now call (many-sorted) first-order logic, that is, the logical calculus in which only quantification of individual variables (i.e., variables whose rank is a collection of individuals) is allowed. In order to present the restricted function calculus as a formal axiomatic system, Hilbert first enunciates all the primitive symbols: the variables and constants: individual, propositional and functional; the logical symbols: negation, disjunction and the universal quantifier (the rest of logical connectives and the existential quantifier are introduced as auxiliary symbols); and, finally, parentheses. Once this has been done, Hilbert defines the admissible expressions of the calculus, that is, the formulas of first-order logic recursively in metalanguage. The importance of defining the formulas of language recursively lies in the fact that it allows to treat these formulas as purely syntactic objects, that is, as strings of symbols devoid of any meaning, which made it possible to prove statements about all the first-order formulas. This 11

Hilbert 2013, 10.

164

Chapter Seventeen

recursive definition of the set of first-order formulas in these lectures is therefore a worthy fact to emphasize not only because it was an effective definition of this set, but also because it was an indispensable tool for most proofs regarding metalogical questions, so important for Hilbert. As Hilbert remarks, the basic function of restricted function calculus is the presentation of theories from an axiomatic standpoint. Now, according to him: The calculus is well suited for this purpose mainly for two reasons: one, because its application prevents that௅without being noticed௅assumptions are used that have not been introduced as axioms, and, furthermore, because the logical dependencies so crucial in axiomatic investigations are represented by the symbolism of the calculus in a particularly perspicuous way.12

Nevertheless, logic not only aims to formalize mathematical theories, but also to provide a foundation for mathematics and, therefore, the restricted function calculus is not enough; what Hilbert calls the extended function calculus (which correspond to what we nowadays call second-order logic) is also necessary. This calculus admits not only the quantification over individual objects, but also over functions and predicates. As Hilbert explains, this is necessary to carry out the reduction of number theory and set theory to logic, for example, to formulate the principle of complete induction or to define the relation of identity and the concept of number. However, Hilbert continues, non-restricted quantification over functions and predicates leads to contradictions. It is then necessary to introduce what he calls the Stufen-Kälkul (calculus of levels), which is just Hilbert’s version of Russell’s theory of ramified types. In either system, one takes for granted a domain of individuals and basic properties and relations between them, from which all further properties and relations (of any level) are defined constructively by the logical operations. Now, according to Hilbert, the reduction of mathematics conducted by Whitehead and Russell in Principia Mathematica requires introducing a further axiom, the axiom of reducibility, which presupposes the existence of certain predicates and relations at each level which have not been obtained from the basic properties and relations, “so that their multiplicity depends neither on the definitions actually given, nor in any way on our ability to give a definition.”13 As Hilbert admits, “such a procedure appears at first very strange, but it is unavoidable so long as our goal is to 12 13

Hilbert 1917/18, 187 (Hilbert 2013, 179). Ibid., 232 (206).

Beyond Principia Mathematica

165

develop the foundations of set theory and analysis from the system of functions of our calculus.”14 These remarks clearly indicate that Hilbert’s allegiance to logicism was not complete. The problem was that the introduction of the axiom of reducibility requires the expansion of the system of basic properties and relations in such a way that the axiom is satisfied. But this cannot be achieved in a purely logical constructive way. Thus, as remarked by Hilbert in the lecture course Grundlagen der Mathematik (1921/22): There remains only the possibility to assume that the system of predicates and relations of the first-level is an independently existing totality satisfying the axiom of reducibility. In this way we return to the axiomatic standpoint and give up the goal of a logical foundation of arithmetic and analysis. Because now a reduction to logic is given only nominally.15

With these words, Hilbert abandoned Russell’s logicist program and resumed his old idea of a proof theory. Indeed, it was in this lecture course that Hilbert formulated for the first time his new proof theory and the finitist position (finite Einstellung), which are the essential components of the so-called Hilbert’s program, his latest proposal to address the problem of the foundation of mathematics (see Chapter 20). The above argument would be later rehearsed in the talk “Die Grundlagen der Mathematik” (“The foundations of mathematics”) (1926) and extended to Russell’s axiom of infinity: Russell’s and Whitehead’s theory of foundations is a general logical investigation of wide scope. But the foundation that it provides for mathematics rests, first, upon the axiom of infinity and, then, upon what is called the axiom of reducibility, and both of these axioms are genuine contentual assumptions that are not supported by a consistency proof; they are assumptions whose validity in fact remains dubious and that, in any case, my theory does not require.16

By the end of 1918 the Great War was over and the effects on Göttingen were evident: The Lesezimmer presented many holes in its collection, some professors such as Debye or Carathéodory left Göttingen and the construction of the Mathematical Institute was postponed sine die. The conditions of life also worsened remarkably: food was scarce, the Papiermark (paper mark) lost a good part of its value and, in general, the

14

Ibid. Hilbert 1921/22, 232. 16 Van Heijenoort 1967, 473. 15

166

Chapter Seventeen

future did not look good.17 On November 18, 1919, Hurwitz died and for the second time, Hilbert adressed the Göttingen Scientific Society to pay a heartfelt tribute to the memory of a friend of his youth. The year of 1919 was also when Wilhelm Ackermann returned to Göttingen. Ackermann had entered the University of Göttingen in 1914 to study mathematics, physics and philosophy. But the outbreak of the First World War meant that he had to join the ranks in 1915 and remain in the army until 1919. That year he resumed his studies at Göttingen, obtaining his doctorate in 1925 with a thesis entitled Begründung des “tertium non datur” mittels der Hilbertschen Theorie der Widerspruchsfreiheit (Foundation of the “tertium non datur” by means of Hilbert’s theory of non-contradiction), written under the supervision of Hilbert. Although it contained significant mistakes, Ackermann’s thesis was of particular interest to Hilbert as it was the first non-trivial example of a finitist proof of a part of elementary analysis (more concretely, of arithmetic without induction). After successfully defending his thesis, Ackermann went to Cambridge with a scholarship from the Rockefeller Foundation. He remained there the first half of 1925 and returned later to Göttingen, where he became, together with Bernays, Hilbert’s main collaborator in questions related to logic and the foundations of mathematics. One of the fruits of this collaboration was the development of a logical system, called epsilon calculus, originally devised by Hilbert and that would play a fundamental role in Hilbert’s proof theory. Another fruit of the collaboration between Hilbert and Ackermann was the writing of the book Grundzüge der theoretischen Logik (Elements of theoretical logic), published in 1928. Although, as the authors acknowledge in the preface, Grundzüge der theoretischen Logik is based entirely on the lectures notes of 1917/18 and the first edition of the book appeared ten years after Hilbert’s lectures in Göttingen, the importance of Hilbert’s and Ackermann’s book in the history of contemporary logic should not be underestimated. Firstly, because the book allowed the dissemination outside the University of Göttingen of the direction that the logical investigations of Hilbert’s circle of collaborators (particularly Bernays and Ackermann) were taking. Secondly, because Grundzüge der theoretischen Logik soon occupied the

17

In 1914 the Goldmark was replaced by the Papiermark in order to finance the war effort. From 1918 the continuing loose money policy resulted in inflation, and in 1923, in hyperinflation.

%H\RQGPrincipia Mathematica 



The axiomatization of predicate calculus Grundzüge der theoretischen Logik  7KH/DQJXDJHRI6HQWHQWLDO&DOFXOXV:HVKDOOHPSOR\FDSLWDOLWDOLF OHWWHUV ܺǡ ܻǡ ܼǡ ܷǡ ǥ WR VWDQG IRU VHQWHQFHV 7R LQGLFDWH WKH ORJLFDO FRPELQDWLRQ RI VHQWHQFHV ZH VKDOO LQWURGXFH WKH IROORZLQJ ILYH V\PEROV  ܺത UHDG³QRWܺ´ VWDQGVIRUWKHRSSRVLWHRUFRQWUDGLFWRU\RIܺ WKDWLVIRUWKDWVHQWHQFHZKLFKLVWUXHLIܺLVIDOVHDQGZKLFKLV IDOVHLIܺLVWUXH  ܺƬܻ UHDG³ܻܺܽ݊݀´ VWDQGVIRUWKHVHQWHQFHZKLFKLVWUXH LIDQGRQO\LIERWKܺDQGܻDUHWUXH  ܺ ‫ ܻ ש‬UHDG³ܺ‫ ´ܻݎ݋‬VWDQGVIRUWKHVHQWHQFHZKLFKLVWUXHLI DQGRQO\LIDWOHDVWRQHRIWKHWZRVHQWHQFHVܺǡ ܻLVWUXH  ܺ ՜ ܻ UHDG ³݂݅ܺ‫  ´ܻ݄݊݁ݐ‬VWDQGV IRU WKH VHQWHQFH ZKLFK LV IDOVHLIDQGRQO\LIܺLVWUXHDQGܻLVIDOVH  ܺ ‫ ܻ ׽‬UHDG ³݂ܺ݅ܽ݊݀‫  ´ܻ݂݅ݕ݈݊݋‬DOVR ZULWWHQ ܺ ՞ ܻ R ܺ ֐ ܻ VWDQGV IRU WKH VHQWHQFH ZKLFK LV WUXH LI DQG RQO\ LI ܺƒ†ܻDUHERWKWUXHRUERWKIDOVH 7KH /DQJXDJH RI 3UHGLFDWH &DOFXOXV $PRQJ WKH V\PEROV ZKLFK RFFXULQWKHSUHGLFDWHFDOFXOXVWKHUHDUHILUVWDOOYDULDEOHVRIGLIIHUHQW NLQGV>@:HGLVWLQJXLVKDPRQJ 6HQWHQWLDOYDULDEOHVܺǡ ܻǡ ܼǡ ǥ ,QGLYLGXDOYDULDEOHV‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬ǡ ǥ 3UHGLFDWHYDULDEOHV‫ܣ‬ǡ ‫ܤ‬ǡ ‫ܥ‬ǡ ǥ ‫ܨ‬ǡ ‫ܩ‬ǡ ‫ܪ‬ǡ ǥ >Hilbert and Ackermann have previously introduced:@ 8QLYHUVDO4XDQWLILHUሺ‫ݔ‬ሻ‫ܣ‬ሺ‫ݔ‬ሻ UHDG)RUDOO‫ܣݔ‬ሺ‫ݔ‬ሻLVWUXH  ([LVWHQWLDO 4XDQWLILHU ሺ‫ݔܧ‬ሻ‫ܨ‬ሺ‫ݔ‬ሻ UHDG ³7KHUH H[LVWV DQ ‫ ݔ‬IRU ZKLFK ‫ܣ‬ሺ‫ݔ‬ሻLVWUXH´  5HFXUVLYH 'HILQLWLRQ RI )RUPXOD %\ WKH WHUP IRUPXOD ZH QRZ XQGHUVWDQG WKRVH DQG RQO\ WKRVH FRPELQDWLRQV RI V\PEROV RI RXU FDOFXOXV ZKLFK PD\ EH VKRZQ WR EH VXFK E\ D ILQLWH QXPEHU RI DSSOLFDWLRQVRIWKHIROORZLQJUXOHV  $VHQWHQWLDOYDULDEOHLVDIRUPXOD  3UHGLFDWH YDULDEOHV ZKRVH DUJXPHQW SODFHV DUH ILOOHG E\ LQGYLGXDOYDULDEOHVDUHIRUPXODV ഥ LVDOVRD  ,IDQ\FRPELQDWLRQिRIV\PEROVLVDIRUPXODWKHQॏ IRUPXOD  ,IिDQGीDUHIRUPXODVVXFKWKDWWKHVDPHLQGLYLGXDOYDULDEOH GRHVQRWRFFXUERXQGLQRQHRIWKHPDQGIUHHLQWKHRWKHUWKHQ cont. p. 168



168

Chapter Seventeen

िƬी, ि ‫ ש‬ी, ि ՜ ी, ि ‫ ׽‬ी are also formulas. 5. If िሺ‫ݔ‬ሻ is a formula in which the variable ‫ ݔ‬occurs as a free variable, then ሺ‫ݔ‬ሻिሺ‫ݔ‬ሻ and ሺ‫ݔܧ‬ሻिሺ‫ݔ‬ሻ are also formulas. The corresponding statement holds for other free variables. The Axioms of Predicate Calculus: We now state the system of axioms for the predicate calculus. As primitive logical formulas, we first have the axioms of the sentential calculus, which for the sake of simplicity we shall give in the same form as before. ƒሻ ܺ ‫ ܺ ש‬՜ ܺ. „ሻ ܺ ՜ ܺ ‫ܻ ש‬. …ሻ ܺ ‫ ܻ ש‬՜ ܻ ‫ܺ ש‬. †ሻ ሺܺ ՜ ܻሻ ՜ ሺܼ ‫ ܺ ש‬՜ ܼ ‫ܻ ש‬ሻ. ഥ ‫ ש‬ॐ). (ि ՜ ॐ is again to be understood as an abbreviation for ि In addition, we know have a group of two axioms for “all” and “there exists” (the universal and existential quantifiers). ‡ሻ ሺ‫ݔ‬ሻ‫ܨ‬ሺ‫ݔ‬ሻ ՜ ሺ‫ݕ‬ሻ ˆሻ ‫ܨ‬ሺ‫ݕ‬ሻ ՜ ሺ‫ݔܧ‬ሻ‫ܨ‬ሺ‫ݔ‬ሻ. The first of these axioms means, “If a predicate ‫ ܨ‬holds for all ‫ݔ‬, then it also holds for any arbitrary ‫ݕ‬.” The second formula means, “If a predicate ‫ ܨ‬holds for some particular ‫ݕ‬, then there is an ‫ ݔ‬for which ‫ܨ‬ holds.” Rules of Inference: We have the following rules for obtaining new formulas from the primitive logical formulas, as well as from formulas obtained therefrom. (Į) Rules of substitution [Hilbert and Ackermann set up here the rules for replacing the sentential variables, free individual variables and predicate variables occurring in a formula by other variables of the same type]. (ȕ) Rule of implication. From two formulas of the form ि i ि ՜ ी, the new formula ी is obtained. (Ȗ) Rules for the Universal and Existential Quantifiers. (Ȗ1) From a formula ሺिሻ ՜ ीሺ‫ݔ‬ሻ, in which the part after the ՜ contains the free variable x while x does not occur in ि, the formula ሺिሻ ՜ ሺ‫ݔ‬ሻीሺ‫ݔ‬ሻ is obtained. (Ȗ2) Under the same hypothesis concerning ि and ीሺ‫ݔ‬ሻ, a formula ीሺ‫ݔ‬ሻ ՜ ि yields a new formula ሺ‫ݔܧ‬ሻीሺ‫ݔ‬ሻ ՜ ि. (į) Rules for Rewriting Bound Variables [Hilbert and Ackermann explain how the replacement of a bound variable by a universal or existential quantifier must be made].

Beyond Principia Mathematica

169

place of Principia Mathematica as a reference book for the new generation of logicians, headed by Kurt Gödel. Finally, because it became the book that marked the consolidation of first-order as the logical system par excellence and served as a basis for future metatheorical research on it. In addition, the 1928 book presented some notable improvements over the 1917/18 lectures. For example, the axiomatization of propositional and first order logic carried out in the 1917/18 lectures presented some unnecessary complications, since some of the axioms and rules of inference presented there were redundant. Another of the fundamental contributions of the book of 1928 in relation to the lectures of 1917/18 was the exposition of the problem of the semantic completeness of function calculus. In the lectures of 1917/18, Hilbert had raised and solved only the problem of consistency and Post-completeness of the functional calculus. According to the 1917/18 lectures, a system of axioms was complete “if the addition of a formula, so far not proven, to the basic formula system [i.e., axiom system] always gives rise to a contradictory system.”18 But now, Hilbert and Ackermann also raise, for the first time in history, the problem of the semantic completeness of the calculus: If this system of axioms is complete, at least in the sense that all logical formulas that are correct in every domain of individuals can be derived, is a question that has not yet been answered.19

It is a well-known fact that Gödel would answer this question affirmatively in his doctoral thesis, defended in 1929 at the University of Vienna, which would be published a year later in the form of an article with the significant title “Die Vollständigkeit der Axiom des logischen Funktionenkälkuls” (“The Completeness of the Axioms of the Functional Calculus of Logic”).

18 19

Hilbert 1917/18, 152 (Hilbert 2013, 157). Hilbert and Ackermann 1928, 68.

CHAPTER EIGHTEEN THE GREAT DEBATE ON THE FOUNDATIONS OF MATHEMATICS

The year 1908 was not just the year when Russell and Zermelo published for the first time their answers to the problem of paradoxes, the theory of logical types and the axiomatic theory of sets, but also the year in which the Dutch mathematician Luitzen Egbertus Jan Brouwer (1881-1966) published his famous article: “Onbetrouwbaarheid der logische principes” (“On the Unreliability of the Logical Principles”), which can be considered the first manifesto of the intuitionist current, one of the most important and influential trends in the field of logic and the foundations of mathematics of the 20th century. As a program for the foundation of mathematics, Brouwerian intuitionism emerged in clear opposition to Frege and Russell’s logicism and to Hilbert’s formalism, so that the controversy with these two schools was somewhat inevitable. This was already apparent in Brouwer’s doctoral thesis entitled Over der Grondslagen der Wiskunden (On the Foundations of Mathematics) (1907), where he lashed out not only against the attempts of logicists and formalists to reduce mathematics to a mechanical manipulation of symbols, but also against those authors like Hilbert and Poincaré who identify the existence of a mathematical system with the absence of contradiction, that is, with its consistency. For Brouwer, to prove the existence of a mathematical object it was necessary to construct it in our intuition. Brouwer identified mathematics with the process or constructive activity of the human mind from elements of what he called the primordial intuition of time. By defining mathematics as a constructive thoughtactivity from intuition, Brouwer always insisted on the fact that this activity is non-linguistic. This radical distinction between mathematics and mathematical language also brought with it an equally and absolute distinction between mathematics and logic. Brouwer, in effect, defined theoretical logic as an application of mathematics to the language of logical reasoning, which is a particular case of mathematical reasoning,

172

Chapter Eighteen

namely, “that special kind of mathematical reasoning which remains if, considering mathematical structures, one restricts oneself to relations of whole and part.”1 The result of dealing mathematically with the linguistic record of mathematical activity proper is the realization of a certain regularity in the symbolic expression of this record. Logical principles, the laws of classical logic, such as the principles of syllogism, contradiction or excluded middle, are nothing more than the expression of these regularities observed in records of previously constructed mathematical systems. In his dissertation of 1907, Brouwer had considered the logical principles as mere tautological statements, without any informational content, so that they could not cause any error, provided they were applied to previously constructed mathematical systems, independently of whether these were finite or infinite. Later, in his famous and revolutionary article of 1908, Brouwer wonders about the confidence in the principles of syllogism, of contradiction and of the excluded middle even when these are applied to already constructed mathematical systems and his response is that “this confidence is well-founded for each of the first two principles, but not for the last.”2 More concretely, Brouwer affirms that “as long as only certain finite discrete systems are posited [...] the principium tertii exclusi is a reliable principle of reasoning,” but “in infinite systems the principium tertii exclusi is as yet not reliable.”3 Brouwer didn’t accept indeed the existence of complete or finished infinite sets. The only infinite sets that could be generated from intuition were the countable infinite sets, that is, the sets equipotent to the natural numbers, and the continuum. However, Brouwer regarded these sets as entities in a continuous process of generation in and from time, not as complete or finished totalities to which it might be assigned a particular cardinal number. As a result, Brouwer denied the existence of Cantor’s transfinite numbers and the mathematical meaning of the Continuum Hypothesis (in other words, Brouwer accepted, as Aristotle, the potential infinite, but not the actual infinite). The unreliability of the principle of excluded middle had the direct consequence of the lack of validity of the demonstrations by reductio ad absurdum since this kind of proofs is based on that principle. But these demonstrations had been used recurrently in the history of mathematics, especially in geometry. For instance, Hilbert had demonstrated his finiteness theorem by a reductio ad absurdum, that is, he had proved the existence of a finite basis for any system of invariants by deriving a 1

Brouwer 1975, 73. Ibid., 109. 3 Ibid., 110. 2

The Great Debate on the Foundations of Mathematics

173

contradiction from the supposition of its non-existence. But for Brouwer this was not enough: to prove the existence of a mathematical object (for example, a finite basis for any system of invariants) it was necessary to construct it in the intuition. In his inaugural lecture at the University of Amsterdam, titled “Intutionisme en Formalisme” (“Intuitionism and Formalism”) (1912), quickly translated to German and English, Brouwer renewed his attack on formalism. During the following years, Brouwer published a series of articles whose main objective was to reconstruct mathematics from an intuitionist point of view. To this end, Brouwer erected not only a brandnew intuitionist set theory based on the concepts of spread and species, the intuitionist equivalents of the classical concepts of set and property, but also a new and revolutionary analysis of the continuum, the mathematical substrate that underlies geometric considerations and which is the object of study of mathematical analysis, based on the notion of choice sequence. At the beginning of the 1920s Brouwer’s intuitionism was gaining support among young mathematicians. Between 1918 and 1919 Brouwer had published three intriguing papers on set theory: “Begrundung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossene Dritten. Erster Teil, Allgemeine Mengenlehre” (“Foundation of set theory independent of the law of excluded middle. First Part, General Set Theory”) (1918), which was followed by a second and a third part published respectively in 1919 and 1923, and “Intuitionistische Mengenlehre” (“Intuitionistic set theory”) (1919). At the same time, Hermann Weyl, a former student of Hilbert in Göttingen and one of the most brilliant mathematicians of the moment, had tried in his Das Kontinuum (1918) to ground predicatively the real number system in the natural numbers in order to avoid paradoxes. However, after talks he had had with Brouwer in the Swiss Alps in the summer of 1919 and the publication of Brouwer’s papers on set theory in 1918 and 1919, he converted to intuitionism and renounced his own program for the foundations of mathematics. Weyl’s public recognition of this conversion to the intuitionist side took place in the paper “Über die neue Grundlagenkrise der Mathematik” (“On the new foundational crisis of mathematics”) (1921), where he diagnosed “a new crisis in the foundations of mathematics” and hailed Brouwer’s approach to the continuum via the choice sequences and his rejection of the principle of excluded middle as the revolution:

174

Chapter Eighteen And Brouwer ௅that is the revolution! […] It is Brouwer to whom we owe the new solution of the continuum problem.4

As Weyl recognized in a letter to Brouwer from 06/05/1920, when he had already finished the paper to be published the next year, the article was not a scientific publication, but rather a “propaganda pamphlet” written in a bombastic style “suited to rouse the sleepers.” The publication of Weyl’s paper was indeed the starting point of the foundational crisis and the great debate between intuitionists and formalists that developed along the decade of the twenties, the two dominant themes of which were: 1. The meaning of “existence” and “constructivity” in mathematics 2. The status of the principle of excluded middle and logic.5 The first to react to Brouwer and Weyl’s challenge was not a sleeper, but the most prominent and active mathematician of the time, David Hilbert, who launched a counterattack the same year. Hilbert saw Brouwer and Weyl’s constraints on mathematics–rejection of the principle of excluded middle for infinite totalities, non-acceptance of pure existence proofs, anathematization of the actual infinite, artificial importation of the vicious circle principle in analysis, etc.–as a threat to the whole of mathematical heritage and to his own contributions to mathematics. It was then natural that Hilbert was the first to react to Weyl’s challenge and that he saw in Brouwer’s and Weyl’s intuitionist programme the return of Kronecker’s constraints on mathematical practice. Hilbert first reaction came in a series of lectures delivered in Copenhagen and Hamburg, which were published (at least partially) the following year in the article “Neubegrundung der Mathematik. Erste Mitteilung” (“New foundations of mathematics. First Communication”) (1922): What Weyl and Brouwer do amounts, in principle, to following the erstwhile path of Kronecker: they seek to ground mathematics by throwing overboard all phenomena that make them uneasy and by establishing a dictatorship of prohibitions à la Kronecker. But this means to dismember and mutilate our science, and if we follow such reformers, we run the danger of losing a large number of our most valuable treasures.6

Among the treasures to be lost following Brouwer and Weyl’s path Hilbert cites, among others, the general concept of irrational number, the transfinite numbers of Cantor and the principle of excluded middle. But as 4

Mancosu 1998, 99. See Hesseling 2003, Chapters IV and V. 6 Mancosu 1998, 200. 5

The Great Debate on the Foundations of Mathematics

175

said before, he could also have cited his finite basis theorem. Fortunately, continues Hilbert, “Weyl and Brouwer will be unable to push their programme through. No; Brouwer is not, as Weyl believes, the revolution, but only a repetition, with the old tools, of an attempted putsch.”7 Before Hilbert’s conference was published, Paul Bernays gave a lecture at the Mathematikertagung in Jena entitled “Über Hilbert’s Gedanken zur Grundlegung der Arithmetik” (“On Hilbert’s thoughts concerning the grounding of arithmetic”) (1922). Hilbert again spoke on the foundations of mathematics in 1922. This time in a lecture entitled “Die logischen Grundlagen der Mathematik” (“The logical foundations of mathematics”) (1923), delivered to the Society of German Scientist and Doctors and published the following year in Mathematische Annalen. Here, as in his 1921 lecture in Hamburg, Hilbert presented his own proposal for a definitive solution to the problem of the foundations of mathematics imbued with the philosophical standpoint necessary to answer Brouwer and Weyl’s criticisms: finitism. In 1923, the mathematician and set-theorist Abraham Fraenkel entered the scene, becoming the main commentator of the foundational debate during the twenties. Other early participants in the debate were Julius Wolff (18821945), Paul Finsler (1894-1970), Richard Baldus (1885-1945) and Oskar Becker (1889-1964). In the meanwhile, Brouwer developed his campaign for intuitionism following a double path. On the one hand, he espoused the basic parts of intuitionistic mathematics in the papers “Zur Begründung der intuitionistischen Mathematik, I, II, III” published in 1925, 1926 and 1927 in Mathematische Annalen. On the other hand, he furnished new constructive proofs of classical and intuitionist theorems of algebra and analysis. Of this kind are, for instance, the papers “Besitz jede reelle Zahl eine Dezimalbruchentwickelung hat?” (“Does every real number have a decimal expansion?”) (1921), “Intuitionistischer Beweis des Fundamentalsatzes der Algebra” (“Intuitionist proof of the fundamental theorem of algebra”) (1924),8 and “Beweis dass jede volle Funktion gleichmässig stetig is” (“Proof that every full function is uniformly 7

Ibid. Hilbert refers here to Kronecker’s “dictatorial prohibitions” of everything that was not for him an integer. 8 This paper was written together with his Ph. D. student B. de Loor. It presents an intuitionistic proof of the fundamental theorem of algebra, which states that every algebraic equation ‫ ݔ‬௡ ൅ ܽ௡ିଵ ‫ ݔ‬௡ିଵ ൅ ‫ ڮ‬൅ ܽଵ ‫ ݔ‬൅ ܽ଴ ൌ Ͳ, with complex coefficients, has a complex solution. The problem with the usual proofs of this theorem, beginning with Gauss, was that they proceed by reductio ad absurdum and so they were not admissible from the intuitionistic standpoint.

176

Chapter Eighteen

continuous” (1924), where he proves that every function mapping the closed unit interval ሼ‫ א ݎ‬Թǣ Ͳ ൑ ‫ ݎ‬൑ ͳሽ into Թ is uniformly continuous.9 Beside the above-mentioned papers, Brouwer published many other noteworthy papers aiming to reconstruct mathematics from the intuitionistic standpoint. After his conferences in Hamburg and Copenhagen in 1921, Hilbert lectured again on the foundations of mathematics in a meeting of the Society of German Scientist and Doctors that was held in Leipzig on September 1922. The conference, entitled “Die logischen Grundlagen der Mathematik” (“The Logical Foundations of Mathematics”), was published the following year in Mathematische Annalen. In June 1925, the same month in which Klein died, Hilbert took advantage of the occasion offered by the Weierstrass-Woche (Weierstrass-Week) in Münster to give a lecture titled “Über das Unendliche” (“On the Infinite”), which would also be published the following year in Mathematische Annalen. Finally, in July 1927, Hilbert delivered the talk “Die Grundlagen der Mathematik” (“The Foundations of Mathematics”) at the University of Hamburg. In these conferences, Hilbert continued to present his program for the foundation of mathematics. In addition to Hilbert and Bernays, his closest collaborator, other important mathematicians were involved in technical work on the program throughout the 1920s and 1930s. Weyl, in turn, submitted in 1923 and published the following year in the journal Mathematische Zeitschrift, a paper entitled “Randbemerkungen zu Hauptproblemen der Mathematik” (“Marginal notes on the main problems of mathematics”) (1924). In this paper, Weyl reacted to Hilbert’s paper of 1921, but also clarified some of his ideas regarding the intuitionist position on mathematical existence and substantially modified his initial position toward Brouwer’s Intuitionism. Apart from the papers of 1921 and 1923, Weyl contributed or referred to the debate on two more occasions. The first was in the paper “Die heutige Erkenntnislage in der Mathematik” (“The current epistemological situation in mathematics”) (1925) and his subsequent extension Philosophie der Mathematik und Naturwissenschaft (Philosophy of mathematics and natural science) (1927), where he espoused the state of the art of the controversy on foundational issues, discussing both intuitionism and formalism. The next occasion was in 1927, during the seminar talk in Hamburg in which Hilbert delivered his lecture on the foundations of mathematics mentioned above. In his lecture, Hilbert had attacked intuitionism, and Weyl took 9

This result is often called Brouwer’s Theorem and follows from it that (in intuitionistic mathematics) every function from Թ to Թ is continuous.

The Great Debate on the Foundations of Mathematics

177

upon himself the defence of Brouwer’s intuitionism and particularly of its role in connection with Hilbert’s program.10 From 1924 the debate extended beyond the initial group of directly involved: Brouwer, Weyl, Hilbert and Bernays and to a less extent, Fraenkel, Wolff, Finsler, Becker and Baldus. Not only did the number of persons involved in the controversy increase but also the languages and countries involved. In particular, the debate was brought to the English and French reading public by Arnold Dresden (1882-1954) and Rolin Wavre (1896-1949) respectively. For the new mathematicians and philosophers involved in the debate or presenting positive or negative results regarding the issues touched upon in the debate it is worth mentioning, among the most prominent ones, John von Neumann, Kurt Grelling (1886-1942), Andrey Kolmogorov (1903-1987), Arendt Heyting (1898-1980), Valerii Glivenko (1896-1940) and Thoralf Skolem. From 1928 onward, the foundational debate remitted significantly. In 1928, for the first time since 1922, the number of contributions to the debate dropped. That was also the year when the events that led to the definitive isolation of Brouwer and the loss of his influence were precipitated. After the war, the Conseil International de Recherches, created following the Treaty of Versailles, had instituted a boycott policy towards German scientists that prevented them from having any international contact. So, when in 1926 the Council abolished the paragraph that boycotted German scientists, mathematicians from Göttingen, headed by Hilbert, saw no disadvantage in accepting the invitation to participate in the Eighth International Congress of Mathematicians which was to be held in Bologna two years later. In contrast to this attitude, most mathematicians from the universities of Berlin and Munich opposed this participation, arguing that the congress was linked to the International Mathematical Union and the International Research Council, which were still hostile to German scientists. The leader of this group of nationalist mathematicians, Ludwig Bieberbach (1886-1982), a Berlin professor who would later be known for his persecution of Jewish mathematicians during Nazism, sent a letter to German high school teachers and university professors asking them to boycott the conference. Brouwer, who had a fluid contact with the University of Berlin (which had offered him a post in 1919 that he politely declined), supported Bieberbach in a circular published in Mathematische Annalen. 10

See “Diskussionsbemerkungen zu dem zweiten Hilbertschen Vortrag über die Grundlagen der Mathematik” (“Comments on Hilbert’s Second Lecture on the Foundations of Mathematics”) (1927).

Chapter Eighteen

178

Hilbert responded with another letter in which he criticized Bieberbach’s attitude and called for participation in the congress as an “act of the most elemental courtesy.” Finally, in August 1928, Hilbert headed the German delegation, the second most numerous with up to 67 mathematicians, in the International Congress of Mathematicians at Bologna. At the opening session of the Congress, Hilbert spoke eloquently to the attendees: It makes me very happy that after a long, hard time all the mathematicians of the world are represented here. That is as it should be and as it must be for the prosperity of our beloved science. Let us consider that we as mathematicians stand on the highest pinnacle of the cultivation of the exact sciences. We have no other choice than to assume this highest place, because all limits, especially national ones, are contrary to the nature of mathematics. It is a complete misunderstanding of our science to construct differences according to people and races, and the reasons for which this has been done are very shabby ones. Mathematics knows no races […] For mathematics the whole cultural world is a single country.11

This was not the only battle lost by Brouwer in 1928. That was also the year in which an unpleasant conflict, the so-called Annalenstreit, took place in the editorial board of the Mathematische Annalen. Shortly after his return from Bologna, Hilbert sent a short letter to Brouwer in which he informed him of his dismissal from the editorial board of the journal. The only reason given by Hilbert was that he could not continue to work with him because of the incompatibility of their points of view on fundamental issues. With this Hilbert referred to the hostile position towards foreign mathematicians that Brouwer had expressed in his editorial work in the journal (for example, he had opposed the participation of some French mathematicians in the special issue dedicated to Riemann) and to his opposition to the participation of the German mathematicians in the Bologna Congress. In the same letter, Hilbert also informed Brouwer that he had the authorization of Blumenthal and Carathéodory to retire his name from the next number of the journal, since its permanence in the editorial board was not appropriate for the reasons mentioned. At the time of Hilbert’s letter, the chief editors of the journal were Hilbert, Einstein, Blumenthal and Carathéodory, while the editorial board was formed by Bieberbach, Bohr, Brouwer, Courant, Walther von Dyck (1856-1934), Otto Hölder (1859-1937), von Kármán and Sommerfeld. Hilbert also wrote to Einstein, in order to ask for his support for the dismissal of Brouwer, but Einstein preferred to stay neutral in this “frog 11

Curbera 2009, 83.

The Great Debate on the Foundations of Mathematics

179

and mouse battle among mathematicians” (as he described it). Meanwhile, most members of the editorial board did not want to irritate Hilbert by objecting to his design. Finally, after an intense and bitter exchange of letters and meetings among the actors of this little tragedy (including the owner of the journal, Ferdinand Springer), the name of Brouwer was removed from the editorial board of the Mathematische Annalen. Actually, the Bologna affair and Brouwer’s attitude towards foreign mathematicians seems to have been just an excuse and that the true reason for Brouwer’s illegal dismissal was that Hilbert, believing that he was about to die, wanted to be sure that after his death Brouwer would not be too influential in the journal.12 Brouwer vehemently protested his dismissal and, in the end, the whole editorial board was dissolved and reassembled immediately in a very small format, with Hilbert as the sole editor in chief and Blumenthal and Hecke as members of the editorial board. In particular, Einstein and Carathéodory resigned their positions as chief editors of the journal. Brouwer’s intellectual and personal feud with Hilbert, together with his incapability of creating a group of disciples that continued his work and the lack of recognition of his contributions, left Brouwer bitter and isolated, and put an end to a very creative decade in his work. He abandoned his Intuitionist Programme and withdrew into silence. The paper “Intuitionistische Betrachtungen über den Formalismus” (“Intuitionist Reflections on Formalism”) (1928) and the Vienna lectures, “Mathematik, Wissenschaft und Sprache” (“Mathematics, Science and Language”) (1928) and Die Struktur des Kontinuums (The structure of the continuum) (1930), mark the end of Brouwer’s creative work and of his intuitionist campaign. Although from 1928 the foundational crisis was over, there were still important events related to the debate and contributions to some of the issues touched on in the debate which are worth mentioning. In September of 1930 the second Tagung für Erkenntnislehre der exakten Wissenschaften (Conference on the epistemology of the exact sciences) took place in Königsberg. The main theme of the conference was the foundations of mathematics and for the first time the three big currents in the philosophy of mathematics were represented in one and the same meeting: formalism was represented by John von Neumann, intuitionism by Arendt Heyting, and logicism by Rudolf Carnap. Regarding the later contributions to the debate, they were usually concrete realizations of the intuitionist or formalist program, looking sometimes for a middle way 12 Hilbert was indeed seriously ill at that time, but recovered and did not die until 1943.

180

Chapter Eighteen

between them, rather than philosophical discussions about the topics dealt with in the debate. Among such contributions, are worth mentioning the formalization of intuitionist logic (Heyting), its relationship with classical logic (Gödel) and the relation among consistency, satisfiability and existence (again Gödel).

Fig. 18-1 Brouwer (right) and Bohr at the International Mathematical Congress, Zürich 1932.

CHAPTER NINETEEN THE FOUNDATIONS OF QUANTUM MECHANICS

On November 9, 1918, just before First World War ended, Kaiser Wilhelm II abdicated and fled to the Netherlands where he was protected by Queen Wilhelmina. A new democratic government of Germany was declared in February 1919 in the small town of Weimar. The so-called Weimar Republic was faced from its beginnings with outrageous difficulties: the enormous reparation payments to the Allied powers, a high rate of unemployment and inflation completely out of control generated political instability and a generalized sensation of confusion. However, beginning in 1924, the overall situation began to improve and the period until 1929, when the great depression began, became known as the Golden Age. The feelings of the Göttingen academic staff and, particularly Hilbert, were not that different from that of the rest of German academics: The shock of defeat at the end of the First World War left many German academics dumbfounded and numb. Even Hilbert, an outspoken internationalist, was deeply disillusioned by the chaos and instability that plagued the early Weimar years.1

Indeed, during the Weimar era Göttingen was “one of the strongest outposts of support of the National Socialist Worker Party” and “in national elections, the Nazis always fared far better in Göttingen than in the state of Prussia at large.”2 Sympathy for the Nazis was also the majority feeling among the student organizations and the politically active professors. The fraternal organizations that dominated student life in Gottingen had a long history of anti-Semitism and were known for their reactionary politics. When the Prussian Minister of Culture, Carl Heinrich Becker, brought out a constitution for a National Student Union that made discrimination by race and religion illegal, 86 percent of the Gottingen student body voted against it. Rather than accept a Student Union in which 1 2

Rowe 2018, 315. Ibid., 323.

Chapter Nineteen

182

Jews and other undesirables would be granted free access, they evidently preferred to have none at all. Five years later the Nazis founded their own student organization in Gottingen in 1926; they had attained an absolute majority in the student congress. This shift to the right largely met with the approval of the Gottingen faculty, as its members too had strong leanings in this direction. Although they were not as radical as the students, most of those who were politically active belonged to the two traditional parties of the right: the German National Peoples’ Party (DNVP) and the German Peoples’ Party (DVP).3

The Göttingen Philosophical Faculty was composed of two Sparten (divisions)௅the mathematical-scientific division and the philologicalhistorical division௅which until 1922 voted together on matters that affected the faculty as a whole. In most cases this system functioned quite well, since the concerns of one group were usually a matter of indifference to the other group. But even before the war, controversies between the two Sparten arose that ultimately made this arrangement untenable. These conflicts were mainly due to the different ideological and political biases dominating each of the Sparten: conservative and right-wing in the humanist section, liberal and internationalist in the mathematics and natural science division. Two instances of such conflicts concerned the habilitation of Leonard Nelson and Emmy Noether that are dealt with elsewhere (see Chapter 15 and Chapter 21 respectively). Another instance was the appointment of the physicist Johannes Stark (1874-1957) to the chair in experimental physics at Göttingen. Stark was a Göttingen fellow: In 1900 he habilitated in physics at Göttingen and over the following years he taught and researched at the Göttingen Institute of Physics. It was also here where in 1905 he discovered the optical Doppler effect in canal rays for which he received the Nobel Prize in 1919. Although his scientific credentials were superior to those of his competitors and he had the support of the rightwing physicists Wilhelm Wien and Philipp Lenard (1862-1947), Hilbert opposed the appointment because he was a volkisch nationalist and declared anti-Semite and this made him unacceptable for the position. After Debye came to accept Hilbert’s view, nobody was in a position to oppose Hilbert’s refusal and Stark was never called to occupy the chair in experimental physics at the Georgia Augusta. The overall liberal and internationalist atmosphere of the Göttingen mathematical community was no accident:

3

Rowe 1986, 445-46.

The Foundations of Quantum Mechanics

183

As one of the world’s leading centers for mathematics and physics, Gottingen attracted students and scholars, many of whom became worldrenowned; many among them were women, foreigners, or Jews. Certainly there was no deliberate policy behind this௅much less an international Jewish conspiracy, as Hugo Dingler would have it. But then neither was it entirely an accident. It was, to a large extent, the indirect result of a principle that Felix Klein had followed from the beginning of his career, a principle that knew only one criterion for evaluating a mathematician’s worth௅talent. These are only some of the events and circumstances that can be cited in support of the thesis that the Gottingen scientific community was a true phenomenon of Weimar culture.4

At the beginnings of the twenties, Hilbert continued to offer lectures on the theory of relativity but his interests on physical issues shifted over to quantum mechanics. In 1913 Niels Bohr had presented his theory of the atomic structure and set the basis of what would later be called old quantum theory. Since its beginnings in 1913, Bohr’s atomic theory had been followed in Göttingen, particularly by Courant and Hilbert, who invited Bohr to lecture at the seminar for mathematics and physics under the auspices of the Wolksfehl Foundation. The lectures were held between 11 and 22 June 1922 and they become a momentous event for science. About one hundred people attended the lectures and participated in the event, which soon came to be called the Bohr Festspiele (Bohr Festival). Among the attendees were the most prominent physicists and mathematicians of the moment: Courant, Hilbert, Runge, Born, Sommerfeld, Wolfgang Pauli (1900-1958), Werner Heisenberg (19011976), Pascual Jordan (1902-1980), Paul Ehrenfest (1880-1933), James Franck (1882-1964), Hans Kramers (1894-1952) and Alfred Landé (18881976), among others. The Bohr lectures showed the limitations of the Bohr-Sommerfeld old quantum theory and had a decisive influence on the development of quantum theory in Göttingen, particularly in the future research of young physicists like Heisenberg and Pauli, but also for Born and his group of collaborators and even for Hilbert. After Debye left Göttingen in 1920, the three chairs in physics had to be refilled. The vacancies were filled by the theoretician Max Born and the experimental physicists James Franck and Robert Pohl (1884-1976). Early in the 1920’s, the institutions related to physics (mainly the astronomic observatory, the geophysical institute and the institute for applied mathematics and mechanics) were reorganized into four institutes: two experimental, a theoretical one, and an upgraded Institute for Applied Electricity under Max Reich’s (1874-1941) direction. Also, in 1925, a new 4

Ibid., 448.

184

Chapter Nineteen

Institute for Fluid Dynamics was established under the direction of Ludwig Prandtl. The reorganization and establishment of new institutes was made possible thanks to the financial support of funding organizations that promoted scientific research through Germany. This was the case of the Helmholtz Foundation, the Notgemeinschaft (which later became the DFG, the German Research Foundation), the Kaiser Wilhelm Foundation and the Rockefeller Foundation, which financed the new Mathematical Institute. Thus, at the end of the twenties Göttingen had acquired a much greater size and became a world-wide reference centre for the study and research of applied and theoretical physics. Although most of Hilbert’s efforts at the beginnings of the twenties were taken up by the realization of his program for the foundations of mathematics, he was still seriously engaged with the foundations of physics and continued lecturing on this topic. However, as explained in Chapter 16, these lectures and talks usually followed a more comprehensive and philosophical approach. Of this kind were, for instance, the talk given in Copenhagen (“Nature and Mathematical knowledge”) in 1921 and a series of three lectures (entitled “Epistemological questions of Modern Physics”) given in the University of Hamburg in 1923. Beginning in 1922, Hilbert’s assistant for physics was Lothar Nordheim (1899-1985), who like the other assistants for physics was chosen for Hilbert by Sommerfeld. When he arrived in Göttingen, he hadn’t yet finished his degree on physics in Munich, so he had to combine his work as Hilbert’s assistant with the preparation of his dissertation under the supervision of Max Born. For the winter semester of 1922/23, surely stimulated by the Bohr Wolksfehl lectures of that summer, Hilbert announced a lecture course entitled “Mathematical Foundations of Quantum Theory.” The lecture notes presented old quantum theory and were worked out by Nordheim and Gustav Heckmann (1898-1996), a physicist graduated from Göttingen who also wrote his dissertation under the supervision of Born.5 On January 23, 1922, when the great debate on the foundations of mathematics between intuitionists and formalists that dominated the 5

After attending a conference by Leonard Nelson at the Pedagogical Society in Göttingen on the Socratic method in December 1922, he became progressively interested in philosophy and pedagogy. He is also known for being the instigator of the so called “Urgent Call for Unity” from 1932, one of the last attempts by intellectuals of the time (e.g., Albert Einstein, Käthe Kollwitz and others) to prevent the takeover of power by Hitler and the Nazis.

The Foundations of Quantum Mechanics

185

mathematical scene of the 20s had just begun, Hilbert turned 60. To celebrate the event, the journal Die Naturwissenschaften dedicated one of its issues to him, and several of his former students wrote about his life and his contributions to the different fields of mathematics in which he had worked: algebra, geometry, analysis, mathematical physics and philosophy of mathematics. Likewise, a party in his honour was held in Göttingen, where Klein, who was 73 years old and wheelchair-bound, and much of the circle of former students and collaborators of Hilbert were present. The party was largely the passing from the old generation of mathematicians, headed by Klein and Hilbert, to a new batch in which Richard Courant would play a very important role as the successor of Klein. That year, the conflicts between the two divisions of the Faculty of Philosophy led to its final breakup and the separation of the MathematicalScientific Faculty from the Philosophical Faculty. In the wake of these events the Mathematical Institute of Göttingen, the great dream of Klein, was founded. The foundation was however more nominal than anything else, given that its construction did not begin until 1927 and it was not completed until 1929. Courant, Hilbert and Landau became directors of the Institute, but it was Courant who took over the main workload of management until 1933. Another mathematician whom Klein and Hilbert wanted in Göttingen was Hermann Weyl, who was then in his thirties like Courant. Weyl, however, despite the estimation and reverence he felt for Klein and Hilbert, refused the invitation because he could not afford “to exchange the tranquillity of life in Zürich for the uncertainties of post-war Germany.”6

Fig. 19-1 The Mathematical Institute at Bunsenstrasse, 3-5, inaugurated on 3.12.1929 6

Reid 1970, 161.

186

Chapter Nineteen

Despite the problems that Germany was experiencing in the early 1920s, particularly the galloping inflation, the University of Göttingen was slowly recovering much of its splendour. Major mathematicians such as Harald Bohr and Godfrey Hardy often visited Göttingen and new talented mathematicians, such as Carl Ludwig Siegel (1896-1981), arrived. Mathematical research was more alive than ever and important groups of researchers were formed around Courant, Landau and Noether.

Fig. 19-2 Richard Courant in a lecture, 1932

Another important research group was also setting up around Max Born, who was appointed to the chair of theoretical physics in 1921. The first assistants of Born were none less than Wolfgang Pauli and Werner Heisenberg. At the same time, Born’s best friend, James Franck, joined Göttingen as a professor of experimental physics. The seminar on the structure of matter, which during the war had been directed by Hilbert and

The Foundations of Quantum Mechanics

187

Debye, was now carried by Born and Franck, and among the attendees over the 1920s we can find some of the most prominent physicists of the last century; these include Heisenberg and Pauli, Robert Oppenheimer (1904-1967), Karl Taylor Compton (1887-1954), Pascual Jordan, Paul Dirac (1902-1984), Linus Pauling (1901-1994), Fritz Houtermans (19031966) and Patrick Blackett (1897-1974), among others. Lothar Nordheim worked with Hilbert at his home. During the period he was Hilbert’s assistant he helped him prepare his lecture notes on physics and explained the latest developments in physics, particularly in quantum mechanics, to him. He was indeed not very happy with this appointment and when he left his position as Hilbert’s assistant he felt somewhat relieved: First I must say, that during the time I was his assistant, he was very sick. And not the genius he had been. He lived very much in the past in a way. His mathematical interest was logic, which was not terribly appealing to me. But he had the conviction that the best thing for a young man was to work with him. That was a reward in itself. And everything else, financial and family considerations, would be way down in importance.7

Although Hilbert barely participated in the seminar and discussions about the latest advances in physics, his influence was still felt greatly due to the publication of two important books. The first was Max von Laue’s book Das Relativitätsprinzip of 1921.8 This was the first detailed exposition of the general theory of relativity and, as recognized by the author, it made ample use of Hilbert’s lecture notes on general relativity of 1916/17. The second was the 1924 book Methoden der mathematischen Physik (Methods of Mathematical Physics) by Hilbert and Courant, a classic work and the most referenced of modern physics. The book was written entirely by Courant, but, as he explained in the preface, Hilbert’s name on the cover of the book was fully justified by the fact that much of the material of the book came from lecture courses and talks given by Hilbert. The book represented a formidable tool for classical theoretical physicists, who from now on could find in a single book all the mathematics necessary for their investigations, however advanced they might be. But the book by Courant and Hilbert and, ultimately, Hilbert’s contributions to the theory of integral equations, still had a broader scope of application that would 7

Interview of Lothar Nordheim by Bruce Wheaton on 1977 July 24, Niels Bohr Library & Archives, American Institute of Physics, College Park, MD USA, www.aip.org/history-programs/niels-bohr-library/oral-histories/5074. 8 Das Relativitätsprinzip. Zweiter Band: Die allgemeine Relativitätstheorie und Einsteins Lehre von der Schwerkraft. Braunschweig: Vieweg.

188

Chapter Nineteen

make it an indispensable tool for the future development of quantum mechanics. As remarked by Constance Reid: The Courant-Hilbert book on mathematical methods of physics, which had appeared at the end of 1924, before both Heisenberg’s and Schrödinger work, instead of being outdated by the new discoveries, seemed to have been written expressly for the physicists who now had to deal with them. Hilbert’s own work at the beginning of the century on integral equations, the theory of eigenfunctions and eigenvalues of 1903-04 and the theory of infinitely many variables of 1905-06, turned out to be the appropriate mathematics for quantum mechanics (as was first established by Born in a joint paper with Heisenberg and Jordan).9

Apart from Nordheim, who often visited Hilbert at home, was a young mathematician of Hungarian origin, John von Neumann, who had studied in Berlin with Erhard Schmidt, another of Hilbert’s students and a key actor in the theory of integral equations. In 1924, von Neumann was only 21 years old and was deeply interested in Hilbert’s axiomatic approach to physics and his proof theory. Between Hilbert and von Neumann there was an age difference of more than forty years, but they spent long hours discussing physics or the foundations of mathematics at Hilbert’s home. However, the most intimate assistant and collaborator of Hilbert during those years was still Bernays, who according to Nordheim “was completely taken up by him.”10 In 1925 Felix Klein died, and with his death a golden age for mathematics at Göttingen was also beginning to languish. That same year Runge retired and his place was occupied by Gustav Herglotz. Hilbert, who was 63 years, was diagnosed with pernicious anaemia, an illness that was considered generally mortal at that time. But Hilbert was lucky that a treatment for pernicious anaemia was just beginning to be experimentally tested in the United States. So, through the many contacts of all kinds that Hilbert had at that time, it was possible to get the miraculous drug in Göttingen and, after a brief time, Hilbert’s general state improved ostensibly. In 1925 it was also the year in which the events that would bring in a few years to the definitive mathematical formulation of quantum mechanics were precipitated. That year Werner Heisenberg, who was then the assistant of Max Born, brought him an article in which he eliminated the electron orbits with defined radii and periods of rotation because they were not observable and insisted that the theory be elaborated from tables 9

Reid 1970, 182. Interview of Lothar Nordheim. See note 7 for a complete reference.

10

The Foundations of Quantum Mechanics

189

or matrices of a certain type. Heisenberg thought that this mathematical formulation of quantum mechanics was one of the things that had to be improved, but in fact it was his greatest discovery. Born quickly identified Heisenberg’s mathematics with matrix algebra and called Pascual Jordan, who had been one of Courant’s assistants in the preparation of Hilbert’s book on mathematical physics, to help him. Just sixty days after Heisenberg had brought Born his paper, an important article signed by Born and Jordan appeared, in which they gave a rigorous mathematical formulation of the matrix mechanics of Heisenberg and established the most important principles of quantum mechanics and its application to thermodynamics. A sequel of this article, written by Born, Heisenberg and Jordan, was published next year (1926). That same year, Born’s studies on the statistical interpretation of quantum mechanics, for which he would receive the coveted Nobel Prize in Physics, were published. The first of four papers on wave mechanics that Erwin Schrödinger (1887-1961) would write and publish in the first-half of 1926 appeared a few months after the publication of Heisenberg’s paper on matrix mechanics. Although the matrix and wave mechanics afforded the same results, they started from different assumptions and used different mathematical methods, so that the physicists were astonished. The first to raise the question of the relationship between matrix and wave mechanics was Schrödinger himself, who published an article in the spring of 1926 (which appeared between the second and third articles of the aforementioned four) which contained an outline of a proof of the mathematical equivalence of both theories, but not a rigorous one. Hilbert was immediately interested in the subject and asked Nordheim to instruct him about the new developments in quantum mechanics. In the spring of 1926 Hilbert announced his second course on quantum mechanics, which would take the title “Mathematical methods of quantum theory.” As explained by Tilman Sauer in the Introduction to the recent edition of Hilbert’s lecture course by Springer: The course of 1926/27 was held immediately after the development of the “new quantum mechanics” by Werner Heisenberg, Max Born, Pascual Jordan, Erwin Schrödinger, Paul Dirac, and others. It contains all the known new results, e.g. Heisenberg’s original approach leading to “matrix mechanics,” Schrödinger “wave mechanics” and their application to the harmonic oscillator, the rigid rotation and the hydrogen atom. Further topics are perturbation theory, Fermi-Dirac vs. Bose-Einstein statistics, the ideal quantum gas, the Klein-Gordon equations, the probability interpretation

190

Chapter Nineteen of Schrödinger wave function by Born, and Pascual Jordan’s axiomatization of the new theory.11

Thus, Hilbert’s course offered a comprehensive and fully updated vision of the new quantum theory. When Hilbert began lecturing, in October 1926, important results had already been achieved in this field using both the matrix and the wave method, but the mathematical equivalence between the two methods was not yet established. Hilbert would declare in the inaugural session of the course that the discipline was still in an immature state, but that he was convinced that this situation would soon improve. And that really happened. Shortly afterwards, Dirac and Jordan independently provided a unification of the two mechanics, although their approach made an essential use of the delta functions, which were not yet considered a rigorous mathematical concept. In his 1926 paper “Über eine neue Begründung der Quantenmechanik” (“On a new Foundation of Quantum Mechanics”), Jordan had proposed an axiomatic foundation of quantum mechanics which was aimed at showing the underlying common thread of the various approaches to quantum theory. At the time Jordan was a researcher in the Institut fur theoretische Physik, where he obtained his Ph.D. in 1924 and became Privatdozent in 1926. Hilbert adopted the axioms proposed by Jordan for quantum mechanics at the end of the course and, despite explicitly attributing merit to Jordan, he added that this had been done “according to the principles and following the approach which I applied a generation ago to the foundations of geometry.”12 Hilbert’s latest publication in the field of physics was an article signed by Nordheim, von Neumann and Hilbert entitled “Über die Grundlagen der Quantenmechanik” (“On the Foundations of Quantum Mechanics”), which was published in Mathematische Annalen in 1927. The article was largely based on Hilbert’s lectures of 1926/27 and adopted, with small modifications, the axiomatic approach of Dirac and Jordan based on the Delta functions. At the end of the article the authors acknowledged that their presentation of the new quantum theory was temporary and unsatisfactory and that more work was needed to complete it. The person in charge of carrying out this work was precisely von Neumann, who published, when he was only twenty-three years old, three papers in which he provided his famous axiomatization of quantum mechanics and a completely rigorous mathematical proof of the equivalence between matrix and wave mechanics. 11 12

Hilbert 2009, 505. Hilbert 1926-27, 204 (cited by Corry 2004, 417).

The Foundations of Quantum Mechanics

191

The papers that von Neumann published between 1927 and 1929 culminated in his great work Mathematische Grundlagen der Quantenmechanik (Mathematical Foundations of Quantum Mechanics) of 1932, which is still a reference work for the study of the mathematical structure of elemental quantum mechanics. For the description of this structure, von Neumann defined the abstract concept of Hilbert space, of which only a few specific examples were previously known (see Chapter 12). Instead of formulating the mathematical structure of quantum mechanics from the Delta functions of Dirac, von Neumann did it from the axiomatic theory of Hilbert spaces and the linear operators over them. The previous example shows once again the influence of Hilbert in the formulation of a revolutionary physical theory such as quantum mechanics. This influence was given not so much by his participation, more or less directly, in the research that would lead to the definitive mathematical formulation of quantum mechanics, as by his mastery through the seminars, talks and courses in Göttingen where he provided the basic concepts and tools that would make this formulation possible: his theory of integral equations, his work in what would later be called Hilbert spaces and, last but not least, his axiomatic approach to the foundations of physics.

CHAPTER TWENTY HILBERT’S PROGRAM AND GÖDEL’S RESPONSE

Although throughout the decade of the twenties, Hilbert closely followed the advances in the foundation of physics, the fact is that as of 1922 his most relevant contributions took place in the field of the foundations of mathematics. As we have already explained, Hilbert’s new proposal for the foundation of mathematics, the so-called Hilbert program, was developed in several lecture courses, conferences and articles throughout the 1920s by Hilbert and his collaborators, particularly Bernays. So let us take stock. From 1917 to 1924 Hilbert taught a series of lecture courses which reveal a profound revision of his views on logic and the foundations of mathematics.1 In particular, the lecture courses of 1921/22, 1922/23 and 1923/24 show the development of Hilbert’s ideas concerning proof theory and the finitist consistency program. The basic ideas of proof theory were already outlined in the conferences held at Copenhagen and Hamburg in the spring and summer of 1921. These were partially published in the paper “Neubegründung der Mathematik” (“New Grounding of Mathematics”) in 1922. However, it was in the lecture course “Grundlagen der Mathematik” (“Foundations of Mathematics”), taught in the winter semester of 1921/22, that the main tenets of Hilbert’s program were assembled, namely, proof theory and the finitist consistency program. It was, indeed, in these lectures that the terms “finite Mathematik,” “transfinite Schlussweisen” and “Hilbertschen Beweistheorie” appeared for the first time. Hilbert’s program was presented for the first time outside Göttingen in a conference held in Leipzig in 1922, published the next year with the title “Die logische Grundlagen der Mathematik.” The main objective of Hilbert’s program was to offer a direct proof of the consistency of analysis, since this would solve once and for all the problem of the foundation of classical mathematics. The problem of the 1

See Chapter 17, n. 2 for the list of these lecture courses.

194

Chapter Twenty

consistency of analysis was an issue that Hilbert had been considering since the Heidelberg conference of 1904 and to which a fully satisfactory solution should be found, especially after the devastating criticism addressed by Poincaré to Hilbert’s proof of the consistency of the axioms of arithmetic given in that talk (see Chapter 14). The problem for Hilbert was that now he had to answer not only the charges of circularity launched by Poincaré, but also Weyl and Brouwer’s objections to the use of the actual infinite in set theory and classical analysis and, in particular, to the use of the law of excluded middle in infinite domains. As remarked by Hilbert in “Neubegründung der Mathematik,” to answer the above objections and to “do full justice to the constructive tendencies, to the extent that they are natural,”2 he adopted a new philosophical perspective, later called by him the finitist position (finite Einstellung). This finitist stance basically consisted of restricting mathematical thinking only to “certain extra-logical objects, which exist intuitively as an immediate experience before all thought”3 and to those operations and methods of reasoning about these objects that do not require the introduction of abstract concepts, especially the complete infinite totalities. More precisely stated, according to Hilbert, the objects that constitute the domain of finitist mathematics are the signs (Zeichen). In the case of arithmetic these signs are the numerals (Zahlzeichen) ͳǡ ͳͳǡ ͳͳͳǡ ǥ which are usually represented by the signs ͳǡ ʹǡ ͵ǡ ǥ or also by the letters ܽǡ ܾǡ ܿǡ ǥ Arithmetic operations acceptable from the finitist point of view are those that are recursively defined, i.e., the sum or concatenation, multiplication and exponentiation (from which subtraction and division are defined as usual). The arithmetical statements acceptable from the finitist standpoint are the equalities and inequalities between numerals (for example: ʹ ് ͵, ܽ ൏ ܾ,ʹ ൅ ͵ ൌ ͷ, etc.), and the statements in which basic decidable properties are attributed to the numerals (e.g.: “͵ is prime”). Obviously, these atomic statements can be combined using the standard logical operations: conjunction, disjunction, negation and conditional, obtaining again finitist acceptable statements. Hilbert also considered acceptable from the finitist point of view the so-called bounded formulas, that is, statements of the type ‫ ݔ׊‬൏ ‫߮ݐ‬ሺ‫ݔ‬ሻ or ‫ ݔ׌‬൏ ‫߮ݐ‬ሺ‫ݔ‬ሻ, where ‫ ݐ‬is a numeral and ߮ is a formula obtained by any of the procedures described above. Thus, in the article “Über das Unendliche” (“On the infinite”) (1925), Hilbert gave as an example of such quantificational 2 3

Ewald 1996, vol. 2, 1119. Ibid., 1121.

Hilbert’s Program and Gödel’s Response

195

statements, the statement “there is a prime number between ‫ ݌‬and ‫݌‬Ǩ ൅ ͳ,” where ‫ ݌‬represents the largest known prime number so far. This statement, wrote Hilbert, “serves merely to abbreviate the proposition: Certainly ‫ ݌‬൅ ͳ or ‫ ݌‬൅ ʹ or ‫ ݌‬൅ ͵ or …‫݌‬Ǩ ൅ ͳ is a prime number,”4 which consists of a finite number of disjunctions and is therefore decidable in a finite number of steps. The problematic statements from the finitist standpoint are the unbounded quantificational statements, i.e., statements of the type ‫߮ݔ׊‬ሺ‫ݔ‬ሻ or ‫߮ݔ׌‬ሺ‫ݔ‬ሻ whose range of quantification is all the numerals. For example, the statement “there is a prime number larger than ‫݌‬,” where ‫ ݌‬is as before, is not acceptable from the finitist standpoint. The reason is that this statement says that there is a number greater than ‫݌‬, belonging to an infinite totality of numbers, that is a prime number, but it is not sure that we can find in a finite number of steps a number of the form ‫ ݌‬൅ ݊ that is prime and obviously we cannot test all numbers greater than ‫ ݌‬to check whether any of them is a prime number. The universal statement “for every number ݊, ݊ ൅ ͳ ൌ ͳ ൅ ݊” is also a problematic statement, because its negation, the statement “there exists a number ݊ such that ݊ ൅ ͳ ് ͳ ൅ ݊,” is an existential statement whose range is an infinite totality of numbers and, therefore, is not acceptable from the finitist point of view. However, Hilbert considered universal statements as the one above acceptable from the finitist standpoint, provided that such statements would not be interpreted as an infinite conjunction, “but only as a hypothetical judgment that comes to assert something when a numeral is given.”5 Indeed, today all universal statements whose particular instances express decidable properties of the numerals are often regarded as acceptable from the finitist standpoint. These statements can be expressed as formulas of the form ‫߰ݔ׊‬ሺ‫ݔ‬ሻ, where ߰ሺ‫ݔ‬ሻ is a bounded formula, which are characterized moreover by the fact that their negation is also decidable (and thus, if they are false are refutable in a finite number of steps). In general, all quantificational statements over finite domains of objects are acceptable from the finitist standpoint, while quantificational statements over infinite domain of objects are problematic. The result is that while the application of certain laws of classical logic is safe when applied to finite totalities, its application to infinite totalities needs to be justified. This is the case, for example, of the De Morgan laws for quantificational statements, the equivalences ൓‫߮ݔ׊‬ሺ‫ݔ‬ሻ ‫ݔ׌ ؠ‬൓߮ሺ‫ݔ‬ሻ and 4 5

Van Heijenoort 1967, 377. Ibid., 378.

196

Chapter Twenty

൓‫߮ݔ׌‬ሺ‫ݔ‬ሻ ‫ݔ׊ ؠ‬൓߮ሺ‫ݔ‬ሻ, since both the negation of the universal statement ‫߮ݔ׊‬ሺ‫ݔ‬ሻ and that of the existential statement ‫߮ݔ׌‬ሺ‫ݔ‬ሻ are not acceptable from the finitist standpoint and, therefore, the above equivalences are not even intelligible from this point of view. The same applies to the principle of excluded middle, which states (in the version given by Hilbert) that either all objects have a specific property or there is an object that does not have this property, i.e., ‫߮ݔ׊‬ሺ‫ݔ‬ሻ ‫ݔ׌ ש‬൓߮ሺ‫ݔ‬ሻ. Obviously, the principle of excluded middle is trivial when applied to finite totalities of objects, since we can always test all objects in the domain and see if they all have the property in question or, conversely, if they do not have it. Nonetheless, this is impossible when the domain is infinite, because we cannot test all objects in the domain and, therefore, neither can we verify that they all possess the property in question nor can we be sure to find something that doesn’t possess it. In any case, the previous transfinite modes of inference are constantly used in analysis and set theory. Therefore, as stated by Hilbert in the article “Die logische Grundlagen der Mathematik” (“The logical Foundations of Mathematics”) (1923), “if we wish to give a rigorous grounding of mathematics, we are not entitled to adopt as logically unproblematic the usual modes of inference that we find in analysis.”6 Hilbert’s goal was not, however, to abandon these laws of classical logic as did the intuitionist, for their use in analysis and set theory has always led to correct results, but rather to justify their use in accordance with the finitist stance adopted. As Hilbert wrote, “the free use and the full mastery of the transfinite is to be achieved on the territory of the finite!”7 But how is this possible if, as we have seen, Hilbert considered these transfinite modes of inference as the precise point at which classical logic goes beyond the limits acceptable from the finitist standpoint? Hilbert’s answer was essentially to consider these transfinite modes of inference as ideal statements, that is, as statements that are meaningless from the finitist standpoint, but nevertheless should be added to the logical theory in order to preserve the laws of classical logic. Hilbert’s basic idea was that in the same way that imaginary numbers are introduced in analysis to preserve the laws of algebra, for example, those concerning the existence and number of roots of an equation, it was necessary to “adjoin the ideal propositions to the finitary ones in order to maintain the formally simple laws of ordinary Aristotelian logic.”8 This method of ideal elements was subject to only one condition, namely, the proof of consistency, since 6

Ewald 1996, vol. 2, 1140. Ibid. 8 Van Heijenoort 1967, 379. 7

Hilbert’s Program and Gödel’s Response

Sketch of a finitist consistency proof (“Neubegründung der Mathematik,” 1922) The system considered by Hilbert includes the following five axioms: 1. ܽൌܽ 2. ܽ ൌܾ ՜ܽ൅ͳൌܾ൅ͳ 3. ܽ൅ͳൌܾ൅ͳ՜ܽ ൌܾ 4. ܽ ൌ ܿ ՜ ሺܾ ൌ ܿ ՜ ܽ ൌ ܾሻ 5. ܽ൅ͳ്ͳ and modus ponens as the only deduction rule (which allows us to deduce the formula ‫ ܤ‬from the formulas ‫ ܣ‬and ‫ ܣ‬՜ ‫)ܤ‬. A proof from the previous axioms is defined as a sequence of finite length such that for each formula of the sequence, either this formula is an axiom or is obtained by replacing certain variables of a previous formula by other variables or numerals or, finally, is obtained by modus ponens from two previous formulas in the sequence. The last formula in the proof is the proven formula or theorem. The axiom system just described is consistent if an equation ߙ ൌ ߚ and its negation ߙ ് ߚ cannot be simultaneously proved. To demonstrate this fact, Hilbert proves the following two claims: 6. Every provable formula can contain up to two occurrences of the symbol ՜. 7. An equation ߙ ൌ ߚ is provable if, and only if, ߙ and ߚ have the same sign. From these two lemmas, whose proofs do not pose too many problems, Hilbert proves the consistency of the axioms 1-5 as follows: By Lemma 7 all provable equations are of the form ߙ ൌ ߚ. Therefore, to prove that the previous system is consistent we must show that no such formula as ߙ ് ߚ is provable. To see this, let us remark first that no such formula can be obtained by direct substitution from the axioms, since only the axiom 5 allows us to infer inequalities by direct substitution, but these inequalities are of the type ܽ ൅ ͳ ് ͳ. Moreover, if we had derived a formula of this type by modus ponens, it would have been necessary to have previously inferred a formula of the form ‫ ܥ‬՜ ߙ ് ߚ. But since this formula could not have been obtained by substitution from the axioms, it would in turn have required that we had previously deduced a formula of type ‫ ܤ‬՜ ሺ‫ ܥ‬՜ ߙ ് ߚሻ, and this formula, for the same reasons, would have required that we had previously inferred a formula of the type ‫ ܣ‬՜ ൫‫ ܤ‬՜ ሺ‫ ܥ‬՜ ߙ ് ߚሻ൯. But, by Lemma 6, such a formula is not provable because it contains more than two occurrences of the sign ՜. It follows then that no such formula as ߙ ് ߚ can be proven, as Hilbert wanted to prove.

197

198

Chapter Twenty

any domain’s extension with the addition of ideal elements is legitimate only if it generates no contradiction in the smaller original domain. Therefore, it must be demonstrated that in adding ideal propositions to the finitist ones it is not possible to derive any contradiction in the original domain of finite mathematics. Moreover, for this proof to be acceptable for the intuitionist too, it must be carried out exclusively from the statements and methods of reasoning acceptable from the finitist standpoint. This was exactly the aim of Hilbert’s proof theory (Hilbertsche Beweistheorie). The basic idea of Hilbert’s proof theory was, first, to rigorously formalize the whole edifice of mathematics, so that “mathematics proper or mathematics in the strict sense becomes a stock of formulae”9 and, secondly, to demonstrate the consistency of the formal system obtained in the previous step using only the statements and methods of reasoning acceptable from the finitist standpoint. In a formal system, the formulas become a mere succession of signs, subject to certain rules of formation and devoid of any meaning, while mathematical proofs become rows (sequences of finite length) of formulas of the formal language, where each formula of the sequence follows from the above according to certain rules of transformation or inference. The advantage of working with a formal system is that it allows, in principle, to demonstrate the consistency of an axiomatic system from a purely syntactic perspective, i.e., using only certain syntactic properties of the formulas and deductions of the formal system in question. As we have seen in the above sketch of a finitist consistency proof carried out in “Begründung der Mathematik” (see the Textbox above), Hilbert proved the consistency of axiomatic systems using only the syntactic properties of formulas (e.g., the fact that the sign ՜ appears in a provable formula more than twice) and proofs (e.g., the fact that if a formula ‫ ܤ‬is obtained by modus ponens, then a formula of the type ‫ ܣ‬՜ ‫ܤ‬ must have been deduced previously) of the axiomatic systems. Naturally, this is only possible if the concepts of formula and proof have been precisely formulated, that is, if the axiomatic system has been rigorously formalized. Moreover, reasoning over a formal language presupposes considering this language as the object language of our research, while the language through which we carry out this reasoning becomes a metalanguage. For example, in the proof of consistency we have just seen, Hilbert uses the letters ‫ܣ‬ǡ ‫ܤ‬ǡ ‫ܥ‬ǡ ǥ as meta-variables for the formulas of the object language and ߙǡ ߚǡ ߛǡ ǥ as meta-variables for the terms. 9

Ewald 1996, vol. 2, 1137.

Hilbert’s Program and Gödel’s Response

199

The distinction between language and metalanguage, or, as Hilbert put it, between mathematics and metamathematics, allowed him to answer the charges of circularity expressed by Poincaré and Brouwer about his proof of consistency in the Heidelberg conference of 1904 by distinguishing between contentual metamathematical induction and formal mathematical induction. The first applies to formalized proofs, which are finite objects concretely given, whereas the second applies to the infinite objects or concepts to which these proofs refer. So there is no risk of circularity. The above sketch of how the consistency proof of an axiom system could proceed is very simple. However Hilbert outlined in the article “Die logischen Grundlagen der Mathematik” the proof of the consistency of a subsystem, without quantifiers, of elementary arithmetic from which, as he himself says, “the elementary theory of numbers can also been obtained […] by means of “finite” logic and purely intuitive thought (which includes recursion and intuitive induction for finite existing totalities).”10 The problem obviously arises when Hilbert begins to extend his proof theory to set theory and analysis (the consistency proof of which is the ultimate goal of the proof theory) because in this case the use of transfinite modes of inference is inevitable. To handle these transfinite modes of inference from a finitist point of view, Hilbert introduced a special operator߬ (later substituted by its dual ߝ), a choice function that associates to each predicate ‫ܣ‬ሺܽሻ with a variable ܽ, a particular object ߬ሺ‫ܣ‬ሻ. The ߬ operator chooses for each predicate a counterexample, that is, a negative representative of the property ‫ܣ‬. For example, as Hilbert observed, if ‫ ܣ‬represents the predicate “corruptible,” then ߬ሺ‫ܣ‬ሻ will be a man of such integrity that if he becomes corruptible, then all men would be corruptible. So, this operator is governed by the following transfinite axiom: ‫ܣ‬൫߬ሺ‫ܣ‬ሻ൯ ՜ ‫ܣ‬ሺܽሻ, that is, if ߬ሺ‫ܣ‬ሻ has the property ‫ܣ‬, then every object has it too. Hilbert can now define the quantifiers in terms of the ߬ operator, for it is clear that (i) if ߬ሺ‫ܣ‬ሻ has the property ‫ܣ‬, then every object has the property ‫ ܣ‬and vice versa, that is, ‫ܣݔ׊‬ሺ‫ݔ‬ሻ ‫ܣ ؠ‬൫߬ሺ‫ܣ‬ሻ൯; and (ii) if ߬ሺ൓‫ܣ‬ሻ (in the example above: the man with less integrity exists) verifies the property ‫( ܣ‬is corruptible), then there is an object which satisfies ‫( ܣ‬a corruptible man) and vice versa, i.e., ‫ܣݔ׌‬ሺ‫ݔ‬ሻ ‫ܣ ؠ‬൫߬ሺ൓‫ܣ‬ሻ൯. From the transfinite axiom and the previous definitions of the universal and existential quantifiers follow almost immediately the De Morgan laws for quantificational statements and the principle of excluded middle for infinite totalities. Thus, the transfinite 10

Ibid., 1139.

200

Chapter Twenty

axiom allows Hilbert to eliminate quantifiers and, assuming that the system obtained by adding this axiom is consistent, to extend the validity of the laws of classical logic to infinite totalities and, therefore, justify the common resource in classical mathematics to these transfinite modes of inference (particularly the principle of excluded middle). Indeed, this axiom greatly simplifies proof theory, for now a formalized proof shall consist only of substitutions and applications of propositional calculus, which in turn allows that the consistency proof of the formal system can be carried out using only reasoning of a finite kind. Hilbert’s strategy was, in effect, to replace the finitely many ߬-terms (or ߝ-terms) in any proof by successively assigning numerical values to them by an effective procedure. The hope was that this procedure could be shown to be completed in a finite number of steps, so that only quantifier-free formulas were left behind. Hilbert reviewed the development of his finitist consistency program in the address “Probleme der Grundlegund der Mathematik” (“Problems in the Foundations of Mathematics”) at the inaugural session of the International Congress of Mathematicians held in Bologna in September of 1928. According to Hilbert, there were four problems that remained to be solved to complete his ambitious program for the foundations of mathematics: 1. Finitist proof of the consistency of analysis (second-order function calculus). According to Hilbert, Ackermann had already carried out the main part of the proof and only “an elementary theorem of finiteness which is purely arithmetical” was left to prove. 2. Finitist proof of the consistency of set theory (higher-order function calculi). Hilbert mentions only those parts of set theory necessary for the development of the classical mathematics of this time. 3. Proof of the completeness of the axiom systems for number theory and analysis, where completeness is to be understood here in a sense analogue to that of the Post-completeness of the restricted function calculus. 4. Proof of the semantic completeness of the system of axioms for the restricted function calculus, that is, proof that all universally valid sentences are provable from the axioms and logical rules of inference. The problems posed by Hilbert aroused the interest of a young Austrian logician, named Kurt Gödel, who was then only 22 years old. Gödel had studied at the University of Vienna, where he had such distinguished professors as the philosophers Heinrich Gomperz (18731942), Moritz Schlick (1882-1936) and Rudolf Carnap, and the mathematicians Philipp Furtwängler, Karl Menger (1902-1985) and Hans Hahn (1879-1934). The latter would be his Doktorvater and would introduce him to the group of philosophers and mathematicians formed

Hilbert’s Program and Gödel’s Response

201

around Schlick, later known as the Wiener Kreis (Vienna Circle). Between 1926 and 1928 Gödel regularly participated in the meetings of the Circle and it was there that he became acquainted with Russell’s logicist program for the foundation of mathematics. In 1928 Gödel’s interest in logic and the foundation of mathematics was further stimulated by the two conferences offered by Brouwer in Vienna that year (“Mathematik, Wissenchaft und Sprache” (“Mathematics, science and language”) and “Die Struktur des Kontinums” (“The structure of the continuum”), as well as his attendance at the course “The Philosophical Foundations of Arithmetic” taught by Rudolf Carnap at the University of Vienna in the winter semester of 1928/29. Given the lively interest of the Vienna Circle in the problems of foundations of mathematics, it is likely that Gödel knew the contents of Hilbert’s conference very soon after it took place. In any case, it is sure that Gödel had knowledge of it before 1931, since it is explicitly mentioned in his famous article on the incompleteness of arithmetic of that year.11 But what is worth mentioning is the fact that Gödel was able to solve the four problems posed by Hilbert in the course of the two years following the conclusion of the conference. During the summer of 1929, Gödel demonstrated the semantic completeness of first-order logic, thus solving Hilbert’s fourth problem positively. In the summer of 1930, Gödel attacked the first problem posed by Hilbert at the Bologna conference: the consistency of analysis. Gödel’s idea was to first prove the (relative) consistency of analysis with respect to number theory and then offer a direct proof of the (absolute) consistency of the theory of numbers. To demonstrate the relative consistency of analysis, Gödel considered the possibility of representing each real number by means of a formula ߮ሺ‫ݔ‬ሻ of the language of arithmetic. Gödel realized then that “߮ሺ‫ݔ‬ሻ is demonstrable” is definable in the language of arithmetic and, therefore, that the concept of demonstrability is definable in arithmetic. However, Gödel had known for some time now that the concept of truth was not definable in arithmetic and that this was the solution to the semantic paradoxes such as, for example, the liar paradox. Therefore, the concept of demonstrability could not coincide with that of truth and, in the event that all demonstrable formulas were true, then there should be some true formula that was not demonstrable. Gödel then 11

We refer to the paper “Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme” (“On formally undecidable propositions of Principia Mathematica and related systems”), received by the journal Monashefte für Mathematik und Physik on Novembre 17, 1930, and published the following year.

202

Chapter Twenty

constructed a sentence ‫ ݌‬of a formal system ܵ which says of itself that it is not demonstrable in ܵ. Intuitively, ‫ ݌‬must be indemonstrable in ܵ and, therefore, true. But if ܵ is not very strange, then ‫ ݌‬is not refutable in ܵ, since it is true. Thus, Gödel discovered the existence of a sentence ‫ ݌‬of the theory of numbers that is undecidable, thus demonstrating that this theory is incomplete, solving the third problem of Hilbert in a negative way. After announcing the previous result in the Second Congress on Epistemology of Exact Sciences held in Königsberg in September of that same year, Gödel improved and expanded the previous result, making ‫ ݌‬an elegant arithmetical proposition and noting that any proposition of a formal system ܵ that naturally expresses the consistency of this system can also be used as the undecidable proposition in ܵ. Thus, any proof of the consistency of ܵ can not be fully formalized in ܵ. In particular, the proposition that expresses the consistency of formal number theory or axiomatic set theory can be formalized in these systems and, therefore, it is not possible to find a proof of their consistency formalizable in them.12 Therefore, since Hilbert’s finitist methods should be formalizable in these systems, Gödel not only had demonstrated the impossibility of finding a finite demonstration of the consistency of analysis and set theory (problems 1 and 2 of Hilbert), but even of the theory of numbers (contrary to Hilbert’s belief that it had already been found by Ackermann). Gödel’s results were stunning and they annulled two basic convictions of Hilbert, his firm belief that in mathematics there are no irresolvable problems and his absolute confidence in the provability of the consistency of mathematics. However, when we look at the influence of Gödel’s results in mathematical practice in general, we find that although most mathematicians are aware of the incompleteness of any minimally interesting mathematical theory and, therefore, the theoretical possibility that any problem in which they are working is unsolvable in this theory, they do not stop working on this problem and trying to solve it. In other words, Gödel’s incompleteness theorems have not undermined the confidence of mathematicians in the resolution of mathematical problems in general. In fact, as Hilbert said in the 1900 Paris conference, “this 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.”13 12

These results constitute the core of the 1931 Gödel’s paper mentioned in the previous note 13 Hilbert 1965, vol. 3, 298.

Hilbert’s Program and Gödel’s Response

203

It is true, on the other hand, that Gödel’s incompleteness theorems, insofar as they claim that no formal system with a minimal arithmetic content can demonstrate its own consistency, clearly show the impossibility of carrying out Hilbert’s consistency program. Because, whatever be the system with which Hilbert could have identified finitist mathematics, it is clear from Gödel’s results that it could not demonstrate its own consistency and, much less, the consistency of whole mathematics. However, it does not follow from the fact that theories such as, for example, arithmetic, group theory or analysis cannot prove their consistency that they are inconsistent. In addition, the concern for the problem of consistency has now been substantially diminished and hardly anyone doubts today that, say, arithmetic, group theory or analysis are consistent. Moreover, in the past, all inconsistencies resulted from obvious defects in the formulation of the axiomatic theories, and these defects have been eliminated (this was the case, for instance, of set theory). Of course, there may still be hidden flaws which we are not aware of, but none has yet emerged, even after a huge amount of detailed research on these systems. In short, although it is true that Gödel’s theorems demolished the Hilbertian project of demonstrating the consistency of the whole mathematics and expressed certain inherent limitations of the axiomatic method,14 this is still the paradigm of mathematical rigor and one of the fundamental tools for the development of the different branches of mathematics. One proof of this is the importance of the axiomatic ideal in the development of modern algebra by Emmy Noether, Emil Artin and B. L. van der Waerden among others, or in the group of French mathematicians grouped under the pseudonym of Nicolas Bourbaki, topics which we will discuss in the following chapters.

14

This not excludes, however, the possibility of partial realizations of Hilbert’s program. Despite Gödel’s theorems, one can give a finitistic reduction for substantial portions of infinitistic mathematics. See, for example, the paper by Stephen Simpson (1945- ) “Partial realizations of Hilbert’s program,” in Journal of Symbolic Logic, 53, no. 2(1988): 349-363.

CHAPTER TWENTY-ONE THE NOETHER SCHOOL AND THE RISE OF MODERN ALGEBRA

There is no doubt that the central figure in the development of the modern conception of algebra௅the link between Hilbert’s name and modern algebra௅is Emmy Noether.1 Born in Erlangen (Bavaria, Germany), March 23, 1882, Emmy Noether was the daughter of the mathematician Max Noether (1844-1921). From 1889 to 1897 she attended the Städtische Höhere Töchterschule, the local public school for girls. In the spring of 1900, when she was eighteen, she sat and passed the examination to obtain the certificate to teach French and English at girls’ schools. She wanted to continue her studies at university level, but at that time German universities did not allow women to enroll as regular students. Female students were only admitted as Hospitanten (auditors), without acces to final examinations and only with the consent of the professors whose lectures they wished to attend. In the winter semester of 1900/01, Noether was registered as auditor for courses in language, history, and mathematics at Erlangen University. She was one of only two women studying with 984 men. The next two years she prepared for the Reifeprüfung, the national graduation exam that entitled the graduate to enter a university of his choice, which she passed in July 1903. She spent the winter semester of 1903/04 at the University of Göttingen, where she attended as auditor the lectures of the astronomer Karl Scwarzschild and the mathematicians Minkowski, Blumenthal, Klein and Hilbert. After the first semester she returned to Erlangen, since its university had changed its policies to admit women as regular students with the same rights as men. Noether enrolled in the Philosophical Faculty and in the next four years she only took mathematics courses. On 1

Most of the biographical information about Emmy Noether may be found in Dick (1981). In it we can also find the “Obituary of Emmy Noether” by B. L. van der Waerden (pp. 100-111), the memorial adress “Emmy Noether” by H. Weyl (pp. 112-152) and the adress “In Memory of Emmy Noether” by P. S. Alexandroff (pp. 153-179).

206

Chapter Twenty-One

December 13, 1907, she passed her doctoral oral examination and in July 2, 1908, her doctoral dissertation was registered. It was written under the supervision of Paul Gordan, who was an old friend of her father, and dealt with the theory of invariants. Once Noether had completed the doctorate it would have been natural for her to begin her academic career as Privatdozent at Erlangen. But this possibility was vetoed for women, so from 1908 to 1915 she had to settle for being an informal and unpaid member of the mathematics department, where she continued her research and replaced her father when he was ill in his classes at the university. In 1909 Noether presented the paper “Zur Invariantentheorie der Formen von ݊ Variabeln” (“On the theory of invariants for forms of ݊ variables”) at the DMV conference in Salzburg. A revised account of it appeared in a paper of the same title in 1911 in the Journal für die reine und angewandte Mathematik. In 1913 she presented another paper, entitled “Rational Funktionenkörper” (“Fields of rational functions”), at the DMV conference in Vienna. These two research papers along with her doctoral dissertation established Noether’s reputation in invariant theory. In 1910 Gordan retired from his position as Ordinarius at Erlangen. His immediate succesor, Erhard Schmidt, occupied his post for a short time and was followed by Ernst Fischer. According to Weyl, it was under the direction of Fischer that “the transition from Gordan’s formal standpoint to the Hilbert method of approach was accomplished.”2 In 1915, Hilbert and Klein called Noether in Göttingen. At this time, Hilbert was working on the mathematics of relativity theory and, as some problems he encountered required a good knowledge of invariant theory, he asked Noether to be his assistant. Thus, during the next four years, Noether published nine articles on the theory of invariants related to the theory of relativity. From 1920 onwards, she only wrote one paper on the theory of invariants. After Noether arrived in Göttingen, Hilbert and Klein tried to obtain a lectureship for her. On November 9, 1915, as part of the formal process of qualifying for Habilitation, Noether gave the lecture “Über ganze tranzendentte Zahlen” (“On transcendental integers”) before the Mathematical Society in Göttingen. The appointment was rejected, however, by the Senate of the Philosophical Faculty, because according to the rules concerning Privatdozenten of 1908, Habilitation could only be granted to male candidates. The better-known story is that, against the arguments contrary to the qualification of Noether expressed by the 2

Dick 1981, 123.

The Noether School and the Rise of Modern Algebra

207

philologists and historians of the Philosophical Faculty because of being a woman, Hilbert replied: “I do not see that the sex of the candidate is an argument against her admission as Privatdozent. After all, we are a university, and not a bathing establishment.”3 Despite not obtaining the Habilitation, Noether became an active member of the Division of Mathematics and Natural Sciences of the Philosophical Faculty and an intimate collaborator of Hilbert: In the catalogue of lectures for the winter semester 1916/17 of the GeorgAugust-University in Göttingen, we find the following listing: “Mathematical-physical seminar. Theory of invariants: Prof. Hilbert, with the assistance of Frl. Dr. E. Noether, Mondays 4-6 P.M., free of charge.” From then on, this was a standing addition to all listings of Hilbert’s seminars, problem sessions, and even main lectures, up to and including the summer semester of 1919.4

In 1919, a year after the First World War had ended, Germany became a Republic and the rights of women were improved (for example, they acquired the right to vote). This opened for Noether the possibility of Habilitation. On July 23, 1918, she had lectured before the Mathematical Society on “Invariante variationsprobleme” (“Invariant variational problems”). The final version of the lecture was published in 1918 in the Göttinger Nachrichten and was considered her Habilitation thesis. On May 21, 1919, the Habilitation request was approved and on June 4, 1919, she defended her thesis before a court formed by Klein, Hilbert, Courant, Landau and Debye. She was granted the venia legendi and became a Privatdozentin, a position that allowed her to teach the students who requested it. She was the first woman qualified for teaching mathematics by the University of Göttingen. In 1922, when she was 40 years old, she was appointed as Nichtbeamter ausserordentlicher Professor (a kind of non-tenured associated professor). As remarked by Weyl, “this was a mere title carrying no obligations and no salary. She was, however, entrusted with a Lehrauftrag for algebra, which carried a modest remuneration.”5 Noether’s 1918 paper on “Invariant variational problems” was indeed a milestone in invariant theory. As remarked by Noether in the curriculum vitae attached to the Habilitation act, the paper “deals with arbitrary finite or infinite continuous groups, in the sense of Lie, and discloses what consequences it has for a variational problem to be invariant with respect to such a group. The general results contain, as special cases, the theorems 3

Ibid., 125 (also in Reid 1970, 143). Ibid., 32. 5 Ibid., 125. 4

208

Chapter Twenty-One

on first integrals as they are known in mechanics; furthermore, the conservation theorems and the interdependences among the field equations in the theory of relativity.”6

Fig. 21-1 Emmy Noether with some of her students, colleagues and friends in a rural establishment in the mountains surrounding Göttingen. From left to right: Ernst Witt; Paul Bernays; Helene Weyl; Hermann Weyl; Joachim Weyl, Emil Artin; Emmy Noether; Ernst Knauf; ??; Chiuntze Tsen; Erna Bannow (who later became Witt’s wife).

Hermann Weyl, Noether’s colleague during his years in Göttingen, divided her scientific production in “three clearly distinct epochs: (1) the period of relative dependence, 1907-1919; (2) the investigations grouped around the general theory of ideals, 1920-26; (3) the study of the noncommutative algebras, their representations by linear transformations, and their applications to the study of commutative number fields and their arithmetics, from 1927 on.”7 As we already know, in the first period Noether’s work focused on invariant theory and its applications to the theory of relativity. Indeed, it was not until 1920 that, in the words of her great friend, the Russian mathematician Pavel Alexandroff (1896-1982), Noether became “the creator of a new direction in algebra and the leading, the most consistent and brilliant representative of a certain mathematical 6 7

Ibid., 36. Ibid., 141.

The Noether School and the Rise of Modern Algebra

209

doctrine௅all that which is characterized by the words begriffliche Mathematik (conceptual mathematics).”8 Noether’s figure is often identified with the axiomatic and conceptual approach in algebra. According to van der Waerden, the essence of Noether’s begriffliche Mathematik was contained in the following maxim: Any relationships between numbers, functions and operations only become transparent, generally applicable, and fully productive after they have been isolated from their particular objects and been formulated as universally valid concepts.9

Alexandroff linked this conceptual approach to what can be called structural turn in algebra: It was she [Noether] who taught us to think in terms of simple and general algebraic concepts: homeomorphic mappings, groups and rings with operators, ideals–and not in terms of cumbersome algebraic computations; and thereby opened up the path to finding algebraic principles in places where such principles had been obscured by some complicated special situation which was not at all suited for the accustomed approach of the classical algebraists.10

These ideas sound commonplace today, but they were not in Noether’s time. Algebra in the nineteenth century dealt with quadratic forms, cyclotomy, finite groups and fields, field extensions, ideals in rings and invariant theory. But all these topics were connected in one way or another with the study of different kinds of numbers: rational, real and complex numbers. Even in the late 19th and early 20th century algebraic structures such as groups, rings, fields or ideals were still regarded as means to an end, namely the study of concrete problems of number theory, polynomial algebra or geometry. This perception began to change with a paper of 1910 by the German mathematician Ernst Steinitz (1871-1928) titled “Algebraische Theorie der Körper” (“The algebraic theory of fields”). Steinitz’ paper marks the moment when fields themselves became objects of interest and were not merely means to other ends. This was also the main idea behind Noether’s conceptual approach to the study of the

8

Ibid., 156. We have replaced “abstract mathematics” (in the original text) by “conceptual mathematics,” which is a more exact rendering of the words begriffliche Mathematik. 9 Ibid., 101. 10 Ibid., 158.

210

Chapter Twenty-One

various algebraic structures beginning in 1920.11 The generalization beyond the realm of number of the core concepts of algebra (groups, rings, fields, homomorphisms, etc.) by Emmy Noether and Emil Artin marked the beginning of modern structural algebra. Noether’s research and courses in Göttingen in the period 1920-26 focused on the theory of ideals and of commutative rings, which she introduced in a couple of papers in 1921 and 1926 with a very elegant, abstract approach. In these papers she also demonstrated noteworthy results which until then had only been proven for particular cases. Indeed, the modern concepts of ring, ideal and module over a ring, appeared for the first time in history in the aforementioned 1921 article entitled “Idealtheorie in Ringbereichen” (“Theory of Ideals in Ring Domains”). On the other hand, the 1926 article, entitled “Abstrakter Aufbau del Idealtheorie in Algebraischen Zahl- und Funktionskörpern” (“Abstract Construction of the Theory of Ideals in Number Fields and Algebraic Functions”), is worth mentioning since it was the first time when the algebraic concepts were introduced axiomatically. Specifically, Noether formulated five axioms that ensured that every ideal is a product of prime ideals. The rings that satisfy these axioms are now called Dedekind rings, because Dedekind’s theory of ideals in number fields and algebraic functions of one variable is valid in them. Since 1927, Noether directed her interest towards non-commutative algebra, specifically to the study of ideals and modules over noncommutative rings, as she was aware of its usefulness to address arithmetic issues. One of the fundamental works of this last period was the paper of 1929 “Grosser und Hyperkomplexe Darstellungstheorie” (“Hypercomplex Quantities and Representation Theory”), which introduced important developments in the theory of group representations. Although the approach followed by Noether in the second (and third) period of her work is quite different from that followed until 1920, there is a common thread between the two periods that is worth noting and it goes back to Hilbert’s investigations on the theory of invariants. It was under the influence of Hilbert that Noether became interested in the axiomatization of different algebraic concepts (abstract fields, rings, 11 Besides the work of Steinitz (and Weber) in field theory, the main predecessors of Noether’s structural approach were Cayley and Frobenius in group theory, Dedekind in lattice theory, and Joseph Wedderburn (1882-1948) and Leonard Dickson in the theory of hypercomplex systems. But the main influence was the work of Dedekind. When discussing her own work, Noether used to say with modesty “Es steht schon bei Dedekind” (“It’s already in Dedekind”) and urged her students to read all of Dedekind’s work in the theory of ideals.

The Noether School and the Rise of Modern Algebra

211

modules, etc.) and perhaps the most important result achieved in this direction was the axiomatization of the theory of rings we have already mentioned. In the framework of this theory, Noether generalized Hilbert’s finiteness theorem, which thereafter became another result of modern algebra. In 1924, a young Dutchman named Bartel Leendert van der Waerden joined the research group formed around Noether in the early twenties. He had studied mathematics and physics at the University of Amsterdam from 1919 to 1924. That year he arrived in Göttingen where “a new world opened up before his eyes.” Just after his arrival, van der Waerden enrolled in Noether’s course “Gruppentheorie und hypercomplexe Zahlen” (“Group Theory and Hypercomplex Numbers”), given in the winter semester of 1924/25. There he learned that the questions about algebraic geometry and other issues that had worried him in Amsterdam could be addressed with the tools developed by Dedekind, Weber, Hilbert, Emanuel Lasker (1868-1941), Francis Macaulay (1862-1937), Steinitz and Noether herself. Consequently, he began to study abstract algebra and to work on his main problem: the foundations of algebraic geometry. In 1925 van der Waerden returned to The Netherlands where he presented his thesis on the latter issue. That same year he went to Hamburg with a grant from the Rockefeller Foundation to study with Erich Hecke, Emil Artin and Otto Schreier (1901-1929). Artin taught an algebra course in the summer semester and had promised a book on the subject to Ferdinand Springer. So he proposed to van der Waerden to take notes of his summer course and to write the book promised to Springer together. Artin was so pleased with the quality of van der Waerden’s notes that he suggested he write the book alone. Back in Göttingen, van der Waerden attended Noether’s course “Hypercomplexe Grösser and Darstellungstheorie” (“Hypercomplex Quantities and Representation Theory”) during the winter semester of 1927/28. Again, van der Waerden took notes of the course and Noether took advantage of them for the publication of the paper with the same title in Mathematische Annalen which we have mentioned earlier and which would have a profound influence on the development of algebra. The results of Artin’s and Noether’s research, presented in the courses which van der Waerden attended in Göttingen and Hamburg respectively, constitute the greater part of van der Waerden’s book Moderne Algebra, published in 1930 when he was only 27 years old. Moderne Algebra is probably the most influential treatise on algebra of the twentieth century

212

Chapter Twenty-One

and the first to offer an overview of the discipline from the conceptual and abstract point of view characteristic of modern structural algebra. In a 1975 article, van der Waerden explained the origins of some abstract notions and themes of his book Moderne Algebra, such as the concepts of group, field, ideal, group and algebra representations, etc. For example, he explained that the first person to give an abstract definition of field and make an extensive and unified study of the theory of fields was Steinitz in his paper of 1910, which he relied on for writing the fifth chapter of Moderne Algebra entitled Köpertheorie (Field Theory). Regarding group theory, van der Waerden explains that he learned it from the course Noether taught in Göttingen during the academic year of 1924/25 and from discussions with Artin and Schreier in Hamburg, as well as from the books by Andreas Speiser (1885-1927) and William Burnside (1852-1927) on group theory. Also, for the theory of ideals, van der Waerden mentions Noether’s articles of 1921 and 1926 on the theory of ideals, whereas regarding representation theory he mentions Noether’s paper of 1929. While the book by van der Waerden systematizes a series of new results that had been achieved in the field of algebra in recent years by Dedekind, Hilbert, Artin and Noether, among others, what caused a major impact was his way of conceiving and presenting this discipline, inherited largely from Noether. Moderne Algebra was indeed the first algebra textbook that presented systematically, relying on abstract set theory and applying the axiomatic method, the various areas of algebra as a homogeneous whole. Moreover, the approach followed by van der Waerden in his book was based on the recognition that groups, ideals, rings, fields, etc. are indeed realizations of the same idea, namely, the idea of an algebraic structure and that algebra is precisely the study of these structures. This structural conception of algebra would be resumed and exploited by Bourbaki, a group of French mathematicians who would largely lead the development of mathematics in the second half of the twentieth century (see Chapter 23). The abstract and conceptual approach to algebra has become so commonplace today that we easily forget that in the first half of the twentieth century this approach had to be gradually asserted. As we have seen, Noether’s role in this dissemination was authoritative, but of course she was not alone. She had illustrious predecessors such as Dedekind, Steinitz, Weber, Cayley, Frobenius, Wedderburn and Dickson, but also many graduate students and young colleagues (Grete Herman, Köthe, Krull, Deuring, Fitting, Witt, Tsen, Shoda, Levitski, van der Waerden) and associates (Schmidt, Artin, Hasse, Alexandroff, Pontrjagin, Hopf) that

The Noether School and the Rise of Modern Algebra

213

spread her work and made possible the abstract turn in algebra. When we speak of Noether’s school, this does not mean just the circle of her direct pupils, but those mathematicians from home and abroad who, in close exchange with Emmy Noether, but quite independently, developed abstract algebra and contributed to the dissemination of Noether’s conceptual approach to mathematics.

CHAPTER TWENTY-TWO MATHEMATICS IN GÖTTINGEN UNDER THE NAZIS

Hilbert retired from his professorship in Göttingen in 1930 at the age of 68. Upon his retirement, he was honoured by his home city of Königsberg. The presentation of the “honorary citizenship” was scheduled to be made at the 91st Convention of the Society of German Scientist and Doctors, which was being held in Königsberg in September of that year. The meeting was held in conjunction with the Sixth Conference of German Physicists and Mathematicians and the Second Congress on the Epistemology of Exact Sciences. It was at the joint inaugural conference that Hilbert was appointed honorary citizen of the city of Königsberg and gave the lecture “Naturerkennen und Logik” (“Natural Knowledge and Logic”). In his address, Hilbert outlined the basic tenets of his research program and repeated his conviction, already expressed in his famous 1900 address of Paris, that every mathematical problem is solvable. Arrangements were made for Hilbert to repeat the conclusion of his talk over the local radio station; and, breaking off the meeting, he was accompanied to the broadcasting studio. A record of Hilbert’s talk pronounced at the broadcasting studio exists. It ends with the same motto which is placed over his grave in Göttingen: Wir müssen wissen. Wir werden wissen. One of the attendees to the satellite Congress on Epistemology was Kurt Gödel. The last day a roundtable on the foundations of mathematics, directed by Hans Hahn, was scheduled. At the end of the session, Gödel spoke for the first time and, after an intervention by von Neumann, announced his result on the incompleteness of formal systems. As we have already explained, Gödel’s result annulled Hilbert’s firm belief that in mathematics there are no irresolvable problems and his absolute confidence in the provability of the consistency of mathematics (see Chapter 20). It is likely that Gödel attended the inaugural address of Hilbert. They never knew each other personally or kept any correspondence.

216

Chapter Twenty-Two

Fig. 22-1 David Hilbert in a lecture, 1932

According to Reid, when Hilbert first learned about Gödel’s work from Bernays, he was “somewhat angry”: At first he was only angry and frustrated, but then he began to try to deal constructively with the problem. Bernays found himself impressed that even now, at the very end of his career, Hilbert was able to make great changes in his program. It was not yet clear just what influence Gödel’s work would ultimately have. Gödel himself felt௅and expressed the thought in his paper௅that his work did not contradict Hilbert’s formalistic point of view. Broadened methods would permit the loosening of the requirements of formalizing. Hilbert himself now took a step in this direction. This was the replacing of the schema of complete induction by a looser rule called

Mathematics in Göttingen under the Nazis

217

“unendliche Induktion.” In 1931 two papers in the new direction appeared.1

Although in 1931 Hilbert had already retired as a professor, he still gave a weekly class entitled “Introduction to Philosophy based on Modern Science.” The four Ordinarii (full professors) of mathematics were then Hermann Weyl, Edmund Landau, Richard Courant and Gustav Herglotz. Weyl, who the previous year had changed the tranquillity of Zürich for Göttingen to occupy Hilbert’s chair, lectured on a wide range of topics that included differential geometry, algebraic topology and philosophy of mathematics, besides conducting a weekly seminar on the representation theory of groups. When Minkowski died in 1909 Landau, who occupied Minkowski’s post, lectured to an enormous number of students on number theory. Courant, who had occupied Klein’s chair upon his death in 1925, was the director of the Mathematical Institute. Herglotz, who followed Runge when he retired in 1925, also lectured on a wide variety of topics: Lie groups, mechanics, geometric optics, functions with real variables, etc. The Extraordinarii (associate professors) included Paul Bernays, Emmy Noether and Paul Hertz. Bernays was Hilbert’s assistant, now at the expense of Hilbert himself, and worked with him in the preparation of Grundlagen der Mathematik. He also taught Klein’s course Elementary Mathematics from a superior point of view, a summer course basically designed for high school teachers. Noether imparted courses on the subjects that at that time occupied her research, such as the representation of groups and algebras. Paul Hertz lectured on physics and the theory of causality, just next door to the Institute of Physics, where Max Born, Richard Pohl and James Franck, among others, also lectured. There were also many Privatdozenten and young Assistenten, such as Hans Lewy (1904-1988), Otto Neugebauer (1899-1990), Arnold Schmidt (1902-1967), Herbert Busemann (1905-1994), Werner Fenchel (19051988), Franz Rellich (1906-1955) and Wilhelm Magnus (1907-1990). Among the most prominent students there were Gerhard Gentzen (19091945), Fritz John (1910-1994), Peter Scherk (1910-1985), Olga TausskyTodd (1906-1995) and Ernst Witt (1911-1991). Visiting professors continued to come to Göttingen, attracted by the international prestige and wanting to present the fruits of their latest research. For example, during the course of 1931/32 Godfrey Hardy visited Göttingen invited by Landau, Pavel Alexandroff presented his latest formulation of algebraic topology, Emil Artin arrived from Hamburg to present his research on class field

 1

Reid 1970, 198-99.

218

Chapter Twenty-Two

theory and Richard von Misses (1883-1953) came from Berlin to discuss his foundations of the theory of probability. These lectures were attended by the impressive staff of members of the Mathematical Institute, Hilbert included, and after each one they discussed the different theories presented with a critical spirit and open eye. On the other hand, the social life continued being intense and included a weekly celebration in Weyl’s house and also some celebrations in Landau’s house, the best in Göttingen. Also, the rides and excursions to the mountains around Göttingen were frequented, and normally ended in a rural establishment to drink coffee and continue talking about mathematics and politics. In short, Göttingen was still a centre of excellence in mathematical research and a paradigm of modern universities in the early thirties.

Fig. 22-2 Courant, Landau and Weyl converse in Göttingen, ca. 1930

Unfortunately, the situation for mathematical and scientific research in Göttingen and the rest of the German universities would worsen dramatically in a few years. In 1930 Germany faced serious economic and political problems caused by the Great Depression of 1929: a deficit of 850 million marks, unemployment that affected six million people, the lack of foreign financing, fear of a new escalation of inflation like that which had occurred in the early 1920s, etc. To alleviate this economic

Mathematics in Göttingen under the Nazis

219

situation, the government proposed a series of economic measures that the Reichstag, the German parliament, refused. As a result of this, the Reichstag was dissolved and new elections were convened to confirm the rise of Adolf Hitler’s National Socialist German’s Workers Party (NSDAP). During 1932, politics and life in Germany worsened progressively, with constant street fights between Nazi brown shirts and communist groups. In January 1933, Hitler’s Nazi Party aligned itself with the German National Party, a conservative nationalist party led by Franz von Papen, to contest the German Chancellery elections. The joint vote of both formations allowed Hitler to be appointed by German President Paul von Hindenburg, Chancellor of Germany on January 30, 1933. Hitler himself, eager for more power, convened new general elections for March 5, 1933. On February 27, one week before, the building of the Reichstag was set on fire. This fact benefited Hitler’s party, which obtained, with the help of an intense campaign of propaganda based on the omnipresence of Hitler’s allegations and images, 43.9 % of the votes. On March 23, the Reichstag approved the so-called Enabling Act, which gave the cabinet the right to enact laws without the consent of the parliament and ended, de facto, with parliamentary government, turning Hitler into a constitutional dictator. On April 7, 1933, the German government passed a law prohibiting Jews from taking office in the state administration unless they had entered before 1914 or had served in the army during the First World War. Likewise, the suspension of work was contemplated for all those civil servants who were not sufficiently committed to the new order. Because German university professors, regardless of whether they were Ordinarii, Extraordinarii, Privatdozenten or Assistenten, were all state officials, the April 7 law meant the expulsion of all Dozenten of Jewish origin from the University of Göttingen and the rest of German universities. In addition, some professors were married to Jews, or had friends or relatives of Jewish origin, so many of them decided to leave although they were not required to by law. The consequences for the Mathematical Institute were terrible. Courant, Noether and Bernstein were dismissed immediately. Courant went first to Cambridge where he remained only one year, since in 1934 he joined New York University as a visiting professor. In 1935 he was invited to build up the Department of Mathematics at the Graduate School of Arts and Sciences, where he began to build the nucleus of a small research group following the style of the research groups at Göttingen. After the War the group grew and was reinforced with financial aid, becoming the Institute for Mathematics and Mechanics. After Courant’s retirement in 1958 as director, the institute was renamed Courant

220

Chapter Twenty-Two

Institute of Mathematical Sciences in honour of its organizer and first director.

Fig. 22.3 Emmy Noether’s letter of dismissal, copy for the Rektorat

Mathematics in Göttingen under the Nazis

221

The same year as the rise of Hitler to power, in the United States the Emergency Committee in Aid of Displaced Foreign Scholars had been created for assisting professors who were barred from teaching by the Nazis and helping them finding new appointments in the USA. This committee obtained a professorship for Noether at Bryn Mawr College, a women’s liberal arts college in Pennsylvania and with the aid of the Rockefeller Foundation, paid the salary of her first year there. In 1934, her appointment at Bryn Mawr was renewed for a couple of years more and she was invited to deliver weekly lectures at the Institute of Advanced Studies at Princeton by Abraham Flexner and Oswald Veblen.2 But fortune did not last long, since the following year an operation to eliminate a uterine tumour was complicated and she died shortly after, on April 14, 1935. When Courant left, Otto Neugebauer was appointed director of the Mathematical Institute, but he only lasted one day in this position. Neugebauer was required to sign a declaration of loyalty to the new government, but by refusing to sign it he was immediately suspended of salary and work. On April 27, Bernays, Paul Hertz and Hans Lewy were also dismissed, and Landau was warned that he could not lecture the next semester. Hermann Weyl was not a Jew, but his wife was, and therefore his two children were also considered Jews. So, in the summer semester of 1933, he accepted a post in the newly created Institute of Advanced Studies at Princeton and left Göttingen. Landau had been a civil servant before World War I and so he was not affected by the April 7 law. However, at the recommendation of the dean not to lecture the next semester, his calculus course in the summer semester was taught by his assistant Werner Weber (1906-1975), a fervent national-socialist. Landau insisted on preparing the course and during each class he stayed in his office. When he attempted to resume his calculus classes in the winter semester, the students staged a boycott with SA guards standing at the doors and forced Landau to retreat to his office. On November 2, Oswald Teichmüller (1913-1943) presented Landau with a letter formally explaining why they did not find him “fit for teaching.” He spent the winter semester at Groningen, in the Netherlands. On 7 February 1934, he was officially retired and moved to Berlin. After this he only lectured outside Germany, spending some time in Cambridge and in The Netherlands. Altogether, eighteen mathematicians left voluntarily or were expelled from the Mathematical Institute of Göttingen during 1933. At the

 2

As a woman she could not be appointed for a teaching position in Princeton.

Chapter Twenty-Two

222

University of Berlin things were also getting tough, and 23 professors of the faculty of mathematics, including Max Dehn (1878-1952), Hans Freudenthal (1905-1990) and Richard von Misses, had to leave their positions. And the same thing happened in the rest of German universities, although to a lesser extent in the case of mathematical faculties due to their smaller size. The number of German speaking mathematicians expulsed or persecuted during the Nazi period is presented in the following table:3 Aachen (Amsterdam) Berlin Bonn Brunswick Breslau Cologne Dresden Elsterwerda Essen Frankfurt Freiberg Freiburg Giessen

1/2 0/1 41/62 1/3 1/1 8/11 1/2 0/1 0/1 0/1 9/14 0/1 4/6 0/2

Göttingen Graz Greifswald Halle Hamburg Heidelberg Karlsruhe Kassel Kiel Königsberg Landsberg Leipzig Mansfeld Marburg

24/28 0/1 0/1 ½ 4/4 4/5 2/4 0/2 2/4 7/8 0/1 2/2 0/1 ¼

Munich Münster (Prague) Rostock Saarbrücken Schweidnitz (Stockholm) (Trieste) Tübingen Vacha Vienna (Warsaw) Würzburg (Zurich)

4/5 1/1 5/13 0/1 0/1 0/1 0/1 1/1 0/1 0/1 20/27 0/1 0/2 0/1

When Weyl resigned to his post, it was offered to Helmut Hasse. Hasse was a first-class mathematician who had been a disciple of Landau, Noether and Hecke in Göttingen and was acceptable in the eyes of the Nazi authorities for his strong nationalist and right-wing feelings, so he was also offered the direction of the Mathematical Institute. At the request of Weyl, Hasse accepted the position and from 1934 he became director of the Mathematical Institute of Göttingen, although he had to share this position with Erhard Tornier (1894-1982). Despite his ideological

 3

From Siegmund Schultze (2009, 66). As explained by the author, “the first figure denotes the number of emigrants, the number after the slash denotes the total number of those expelled and persecuted, including emigrants. Some differences compared to the total number of those persecuted result from double counting of certain persons or from uncertainty as to the place of expulsion/persecution. As is generally the case in this book, only German-speaking mathematicians are included, which is important to note for Amsterdam (persecution of Freudenthal), Trieste (Frucht), and Stockholm (threat to Müntz). Places outside Germany, where Germans were persecuted, are set in parentheses.” (Ibid., 65, n. 23).

Mathematics in Göttingen under the Nazis

223

proximity to Nazism, Hasse had numerous problems with his most fundamentalist nationalist colleagues such as Oswald Teichmüller, Werner Weber and especially Tornier, who on several occasions manoeuvred to take Hasse out as co-director of the Mathematical Institute. Tornier was also, together with Theodor Vahlen (1869-1945) and Ludwig Bieberbach, one of the greatest ideologues of Deutsche Mathematik, a movement that wanted to promote “German mathematics” and eliminate Jewish influence on it. However, The movement for a “Deutsche Mathematik” did not involve solely the expulsion of Jews, or the restriction to a few Nazi-promoted topics. Above all, the concept of a “German” mathematics not only involved a perverse and radical application of ideas that were commonplace in the scientific thinking of the day, but, at the same time, it also gave political meaning to various familiar currents within mathematics.4

Among these ideas there were those that assigned specific structures of personality (psychological characters, modes of thought, etc.) to different ethnic or “racial” types, thus enabling to discern “German” mathematics from other kind of mathematics, such as Oriental or Jewish mathematics.5 Such ideas led to the journal Deutsche Mathematik, edited by Vahlen and Bieberbach from 1936 to 1943. Besides them, other mathematicians that usually published in this journal were Teichmüller, a brilliant young mathematician and convinced Nazi, and Werner Weber, who has written his dissertation under Noether’s supervision and had been Landau’s assistant from 1928. Despite this, he was the leader of the student boycott against Landau and in 1934 he also attempted to prevent Hasse from assuming the direction of the Mathematical Institute. As of 1934, the chairs of Courant, Weyl and Landau were occupied by Theodor Kaluza (1885-1954), Hasse and Tornier respectively. Herglotz (the successor to Runge) continued until 1948, when he was granted the title of Professor Emeritus. Tornier’s chair was occupied by Rolf Nevanlinna (1895-1980) between 1936 and 1937 and by Car Ludwig Siegel between 1938 and 1940. Both Hasse and Siegel were first-rate mathematicians, so it cannot be said that Göttingen did not continue to be a centre of excellence in the research and teaching of mathematics during the Nazi period. However, the splendour of the golden age of Klein, Hilbert, Minkowski and Runge or of Courant, Weyl, Noether and Landau, among many others, would never be recovered.

 4 5

Segal 1986, 119. Jews were usually described as “Orientals” by German writers from time ago.

224

Chapter Twenty-Two

During the following years, the Nazis continued their policy of exclusion and humiliation towards non-Aryans and those who did not agree with their ideas. Thus, to the forty-five professors expelled from Göttingen in 1933 a total of seventy more docents expelled during the winter semester of 1935/36 and the summer of 1944 followed. Hilbert died in 1943, the year in which World War II began to change its sign and the Nazi leaders began to glimpse, even in the distance, their particular dusk. The decline of Germany, even its defeat, had begun many years ago.

CHAPTER TWENTY-THREE HILBERT, BOURBAKI AND THE STRUCTURAL IMAGE OF MATHEMATICS

The First World War had had more dramatic consequences for the development of mathematics in France than in Germany. The war had decimated the French population, including scientists and mathematicians. Thus, between 50% and 60% of the 1910, 1911 and 1912 graduates of the École Normale Supérieure in Paris, where the French scientific and intellectual elite was formed, had died in the Great War. One of the consequences of this was that at the end of the war and during the 1920s, the average age of mathematicians who held relevant positions at the Académie des sciences or at the Sorbonne was particularly high. After the War, French science was exhausted and with minimal international contacts or information about the latest developments abroad, especially about those coming from the most advanced German research centres (Göttingen, Hamburg, Berlin), as some young French mathematicians, such as Jacques Herbrand (1908-1931), Claude Chevalley (1909-1984), André Weil and Jean Leray (1906-1998), could verify during visits to those centres. This led to the mathematicians Émile Borel (18711956) in France and David Birkhoff (1884-1944) in the United States to persuade French patrons like Edmond de Rothschild and Americans such as John D. Rockefeller to contribute to the financing of an institution that supported courses and international exchanges in the field of mathematics and theoretical physics. As a result of this, the Henri Poincaré Institute was inaugurated in 1928, which would gradually become a research centre of international prestige in both mathematics and physics. In general, in the thirties the situation of mathematics in France had not improved much in relation to the previous decade. André Weil, then a young Maître de Conférences (the equivalent of assistant professor) at the University of Strasbourg still in his twenties, has described it in the following terms: At that time, scientific life in France was dominated by two or three coteries of academicians, some of whom were visibly driven more by their

Chapter Twenty-Three

226

appetite for power than by a devotion to science. This situation, along with the hecatomb of 1914-1918 which had slaughtered virtually an entire generation, had had a disastrous effect on the level of research in France.1

André Weil and Henri Cartan (1904-2008), another assistant professor at Strasbourg, were very concerned about the teaching of analysis at the French universities. One of their main duties at Strasbourg was the teaching of differential and integral calculus. The standard text was the Traité d’Analyse of Édouard Goursat (1858-1936), which they found wanting in many ways. Cartan frequently asked Weyl how to present this material, so to end with these problems once and for all, Weil proposed to Henri Cartan, to “write a new textbook for analysis.” Weil and Cartan used to visit Paris regularly, where they attended the Mathematics Seminar held every Monday at the Henri Poincaré Institute. This allowed them to meet with former normalien such as Claude Chevalley, Jean Delsarte (1903-1968), Jean Dieudonné (1906-1992) and René de Possel (1905-1974). They met for lunch at Café Capoulade, Boulevard de Saint-Michel, and at one of these meetings, Weil asked them what they thought of his idea of writing a new textbook of analysis and if there was someone who was interested in collaborating. All attendees at the meetings at Café Capoulade were aware of the situation of research and teaching of mathematics in France in the thirties, so the reaction to Weil’s proposal was enthusiastic. They agreed to meet on Monday, December 10, 1934 to formally discuss the idea of Weil, and a few months later, in July 1935, they celebrated the first Congress in Besse-enChandesse. During that summer, they also decided to sign their writings with the pseudonym Nicolas Bourbaki, borrowed from a French general who was active in the Franco-Prussian war. The 1935 Congress would be, then, the first of the Bourbaki Congresses that have continued to be held regularly to this day, usually three times a year for one or two weeks. In 1935 Bourbaki had already abandoned the project of writing a textbook on analysis and proposed a much more ambitious project: to write a work that, in Weil’s words, constituted “a sufficiently broad and solid basis to support the essential of modern mathematics.” According to the initial project, the work had to consist of six volumes (Theory of sets, Algebra, Topology, Functions of One Real Variable, Topological Vector Spaces and Integration), which would be published progressively, although without following any specific order. There is no doubt that Éléments de mathématiques, the name which would receive this book, and which remains unfinished today, is the fundamental work of Bourbaki and

 1

Weil 1992, 120.

Hilbert, Bourbaki and the Structural Image of Mathematics

227

its reason for being as a group. However, Bourbaki also published some notes in the Comptes Rendus of the Académie des Sciences, and a couple of papers more that contain Bourbaki’s main ideas about mathematics and the guidelines that the Elements would follow in the future.

Fig. 23-1 Photo taken at the Bourbaki founding congress in Besse-en-Chandesse, July 1935. Standing left to right: Henri Cartan, René de Possel, Jean Dieudonné, André Weil and Luc Olivier (a biologist). Seated left to right: a “guinea pig” (prospective Bourbaki member) named Mirles, Claude Chevalley and Szolem Mandelbrojt (1899-1983).

Bourbaki was formed initially by Henri Cartan, Claude Chevalley, Jean Delsarte, Jean Dieudonné and André Weil. These were “the true ‫ލ‬founding fathers,‫ ތ‬those who shaped Bourbaki and gave it much of their time and thoughts until they retired.”2 One of the main sources of inspiration for the group, was the approach followed by van der Waerden in his book Moderne Algebra. As Dieudonné explained in an address before the Romanian Society of Mathematics in 1968, “the Bourbaki treatise was modelled in the beginning on the excellent algebra treatise of van der Waerden […] Van der Waerden uses very precise language and has an extremely tight organization of the development of ideas and of the different parts of the book as a whole.”3 In fact, we could see the Bourbaki project as an attempt to extend to all branches of mathematics (set theory,

 2 3

Borel 1998, 374. Dieudonné 1970, 137.

Chapter Twenty-Three

228

algebra, topology, functional analysis, etc.) the conceptual and abstract approach, based on the underlying idea of mathematical structure, characteristic of van der Waerden’s book. Another major influence was Hilbert. In 1950, Dieudonné published an article in the journal American Mathematical Monthly, signed under the name Bourbaki, which has been considered as the group’s manifesto. In it, Dieudonné echoed Hilbert’s belief in the unity of mathematics, based on the universality of the axiomatic method, which would have enabled “the systematic study of the relations existing between different mathematical theories.”4 According to Bourbaki, it is the way in which mathematics is generated from a few axiomatic theories of various kinds, which constitutes its architecture and makes it intelligible. Later, Dieudonné would write the article “David Hilbert (1862-1943),” in which he affirmed that although it was true that there had been many partisans of the axiomatic method, “before Hilbert no one had pursued this program with so much decision and clarity, nor had anyone been able to emphasize so well the fundamental principle that in mathematics the very nature of the objects studied is not of interest; the relationships that exist between them are the only thing that matters.” 5 The axioms, insofar as they define these relationships, provide the architecture of the different branches of mathematics with which the mathematicians deal. In this sense, according to Dieudonné, the mastery of Hilbert would have had no paragon: More than by his ingenious discoveries, it is perhaps because of the character of his thought that Hilbert has exerted the deepest influence in the mathematical world: He taught the mathematicians to think axiomatically, that is to say, trying to reduce each theory to its stricter logical scheme, getting rid of the contingencies of the calculation [...] Due to his intense need to understand, due to his increasingly demanding intellectual integrity, for his untiring aspiration towards an increasingly unified, pure and free science, Hilbert has truly personified, for the generation between the two wars, the ideal of the mathematician.6

But Bourbaki went a step beyond Hilbert (and van der Waerden), when attributing a central role in his unified vision of mathematics to the notion, closely related to the axiomatic method, of mathematical structure:

 4

Bourbaki 1950, 222. Dieudonné 1983, 30. 6 Ibid., 32. 5

Hilbert, Bourbaki and the Structural Image of Mathematics

229

Each structure carries with it its own language, freighted with special intuitive references derived from the theories from which the axiomatic analysis described above has derived the structure […] What all this amounts to is that mathematics has less than ever been reduced to a purely mechanical game of isolated formulas; more than ever does intuition dominate in the genesis of discoveries. But henceforth, it possesses the powerful tools furnished by the theory of the great types of structures; in a single view, it sweeps over immense domains, now unified by the axiomatic method, but which were formerly in a completely chaotic state.7

Certainly, van der Waerden had set up a similar task in the field of algebra. Indeed, as we have already explained, van der Waerden’s book was an important source of inspiration in the early years of Bourbaki’s activity. In the thirties and forties, the new methods and concepts introduced by Emmy Noether and Emil Artin in Germany were not yet known in France, so the reading of van der Waerden’s book caused a profound impact. Following van der Waerden’s approach in algebra, “Bourbaki undertook the task of presenting the whole picture of mathematical knowledge in a systematic and unified fashion, within a standard system of notation, addressing similar questions, and using similar conceptual tools and methods in the different branches.” 8 However, there was a substantial difference between the approach and presentation of modern algebra by van der Waerden and the much more ambitious program of rewriting all of mathematics set up by Bourbaki. In fact, van der Waerden did not attempt to define at any time the notion of algebraic structure nor theorized about what we might call “structural research in algebra.” Bourbaki, on the other hand, not only defined the concept of mathematical structure in several places, but also developed an axiomatic and formal theory of structures through which “he” wanted to respond to the open questions posed by the central role of the concept of structure in mathematics.9 After the Second World War, Bourbaki resumed its activity and from 1947 onwards “he” continued publishing the volumes of Éléments de

 7

Bourbaki 1950, p. 227-28. Corry 1992, 320. 9 Bourbaki provides a formalization of the concept of structure in Chapter IV of the book on set theory. However, the following chapters do not make use of this formalization. In fact, Bourbaki’s “Theory of Sets, and particularly the concept of structure defined in it, are not essential to the contents of the Eléments. One can read and understand any book of Bourbaki’s treatise without first learning the theory of structures” (Corry 2004, 329-30). It is then in the “non formal” meaning of the word “structure”, which belongs to Bourbaki’s image of mathematics, where Bourbaki’s influence in the structural image of mathematics ultimately lies. 8

230

Chapter Twenty-Three

mathématiques, which had been stopped for five years when only a few chapters of the three first volumes had been published. When the project was resumed new young mathematicians joined the group. In particular, Roger Godement (1921-2016), Pierre Samuel (1921-2009), Jacques Dixmier (1924- ) and Jean-Pierre Serre (1926- ) joined Bourbaki in the late 40s. And, a little later, Samuel Eilenberg (1913-1998), Jean-Louis Koszul (1921- ) and Laurent Schwartz (1915-2002) also joined. In the mid-50s, another generation of young mathematicians joined, among them Serge Lang (1927-2005), Armand Borel (1923-2003), Alexandre Grothendieck (1928-2014), Francois Bruhat (1929-2007), Pierre Cartier (1932- ) and John Tate (1925- ). As remarked by Borel, “the fifties was a period of spreading influence of Bourbaki, both by the treatise and the research of members. Remember in particular the so-called French explosion in algebraic topology, the coherent sheaves in analytic geometry, then in algebraic geometry over, later in the abstract case, and homological algebra.”10 In 1958 the six projected volumes were finished. Grothendieck had proposed the year before, in the Bourbaki Congress of 1957, the elaboration of three volumes more about Homological Algebra, Elementary Topology and Varieties along the lines set in the six previous volumes. Grothendieck himself presented in the next congress a draft of two chapters of the third volume. Unfortunately, despite the successes achieved by Grothendieck and others, the project for the publication of new volumes of Éléments de mathématiques began to decline during the 70s, perhaps due to the difficulties of the new disciplines to be treated (homotopy, spectral theory of operators, etc.) and to the fact that these were disciplines still not solidified, to which it was difficult to apply the characteristic structuralist approach of Bourbaki. At the end of the twentieth and beginning of the twenty-first century, Bourbaki still exists but the group is not so active and influential as it was in the fifties and sixties. After the six volumes of Éléments de mathématiques were published in the fifties, four more volumes have been published since then: Commutative Algebra, Lie Groups and Algebras, Spectral Theory and Algebraic Topology. The most recent volume on Algebraic Topology was published in 2016, so although Bourbaki is far from being dead, it is also true that the volumes of Éléments de mathématique are no longer published at the same rate as before. Moreover, although the Bourbaki’s seminar are still held with the same

 10

Borel 1998, 376.

Hilbert, Bourbaki and the Structural Image of Mathematics

231

periodicity as before, today the mathematical community no longer speaks of Bourbaki and for many mathematicians Bourbaki is dead. The reasons for the decline of Bourbaki’s influence over time are multiple, some of them of an intrinsically mathematical nature: Today mathematics has expanded enormously in different and multiple directions, and Bourbakian emphasis on axioms and structures cannot easily accommodate various new branches of mathematics such as, for example, those arising from computer science. Also, unlike what happened in the middle of the last century, nowadays there is no consensus about the rooting of mathematics in a single theory such as set theory and there is more interest towards concrete geometry, combinatorics, algebraic topology and other branches of mathematics not touched upon by Bourbaki. Another important reason which can be related to the outdating of Bourbaki’s approach to mathematics is the emergence of Category theory: Today, no discussion of mathematical structures is complete without a discussion of category theory. Introduced around 1942 by Samuel Eilenberg (who would later become a member of Bourbaki) and Saunders MacLane, category theory provides an abstract and general framework for describing numerous mathematical situations and the connections between them [...] The language of categories and functors spread rapidly during the sixties. Some Bourbaki members put it to great use, including Eilenberg of course, but also Charles Ehresmann and especially Alexandre Grothendieck. Category theory, which is much more general than the structures described by Bourbaki in Éléments de mathématique, could have played an important part in the structural vision of mathematics, but Bourbaki did not update its Architecture des mathématiques. More importantly, the group did not manage to use categories in its treatise, despite the group’s numerous discussions and preliminary drafts on the subject. One of the reasons for this is that the task would have required a profound revision of the existing volumes.11

Although Bourbaki’s influence has decreased over time, there is no doubt that “his” name will survive in the Olympus of mathematicians. Not only has Bourbaki had a paramount influence in modernizing mathematics and clarifying its language and concepts during the second half of the twentieth century, but also some of “his” members have made, on an individual level but following the spirit of Bourbaki, impressive contributions to contemporary mathematics. This would be the case, for example, of the contributions of Laurent Schwartz, Jean-Pierre Serre,

 11

Mashaal 2006, 83-84.

232

Chapter Twenty-Three

Alexandre Grothendieck, Alain Connes (1947- ) or Jean-Christophe Yoccoz (1957-2016) just to cite some significant members of Bourbaki who were recipients of the Fields medal. Indeed, thanks to the works of Weil, Serre, Grothendieck, Pierre Deligne (1944- ) and other members and collaborators of the group, Bourbaki returned to France the hegemony in European mathematics, particularly in algebraic geometry, at least for a couple of decades.

EPILOGUE FROM GÖTTINGEN TO PRINCETON AND PARIS

In the late nineteenth and early twentieth centuries, the United States had spectacular economic growth. Some businessmen such as Andrew Carnegie, Henry Ford, John D. Rockefeller or Marshall Field gathered great fortunes that, at least to a certain extent, ended up returning to society through donations or philanthropic foundations. In 1925, Louis Bamberger and his sister Caroline Bamberger Fuld, owners of a department stores chain decided to sell it and to devote themselves to philanthropy. In 1929, just before the stock market crack, R. H. Macy, the owner of a famous department store in New York, purchased Bamberger’s department store chain. The Bambergers contacted Abraham Flexner, a well-known expert on higher education issues, to give them advice. Flexner, who had carefully studied institutions of recognized international prestige such as All Souls College in Oxford, the Collège de France in Paris, and the German universities of the late 19th century, particularly Göttingen, considered it necessary to create an advanced research institution in the United States. Flexner’s idea of a research institute where researchers had absolute freedom to carry out their research would immediately capture the interest of the Bambergers. After a series of discussions and correspondence, they abandoned their initial idea of a medical school and accepted Flexner’s plan to create an Institute for Advanced Studies, insisting that Flexner be its first director. Once Flexner’s plan was accepted, the Bambergers wanted to place the institution in Washington, the capital district, but Flexner convinced them that it would be better to place it in a quiet place like Princeton, next to the university of the same name, one of the oldest and most prestigious of the United States. Flexner also suggested that the Institute should begin with mathematics, which he considered the most difficult discipline and “antecedent” to all others. Thus, in the fall of 1932, Flexner publicly announced the creation of the first school of the Institute for Advanced Studies (IAS) at Princeton, the School of Mathematics. The first professors appointed by Flexner were Oswald Veblen and Albert Einstein. Flexner

234

Epilogue

also wanted to recruit Hermann Weyl, who at that time occupied Hilbert’s chair in Göttingen, but Weyl was still undecided about his future and would not join the IAS until the fall of 1933. Because the School of Mathematics still had no building, activities began in the spring of 1933 in the building of the Faculty of Mathematics of the University of Princeton. In 1933 Flexner recruited John von Neumann, who was then a visiting professor at Princeton University. In the IAS, von Neumann would make important contributions to game theory and computer science. He remained there until his death in 1957. Despite its name, the School of Mathematics included both mathematicians and physicists. Einstein was a physicist, and von Neumann had also made important contributions to physics. At the beginning of the academic year of 1933/34, the IAS already had five mathematicians and physicists of the first rank: James Alexander (1888-1971), Einstein, Veblen, Weyl and von Neumann, as well as about twenty visiting professors, some of them as prominent as the logician and mathematician Kurt Gödel. During 1934 and 1935 respectively, the theoretical physicists Paul Dirac and Wolfgang Pauli joined the staff of the School of Mathematics. We could say, then, that the IAS became a refuge for German and European researchers who fled Nazi Germany during the 1930s. Thanks to the arrival of these researchers at the IAS, Princeton replaced Göttingen as the world’s leading centre for mathematical and physical research. Obviously, this transfer of powers was even more noteworthy after the end of World War II. In 1947 the theoretical physicist Robert Oppenheimer was appointed director of the School of Mathematics. Oppenheimer was a charismatic leader and was able to attract the most outstanding theoretical physicists interested in particle physics research, and so Princeton became the new Mecca of research in theoretical physics around the world. The visiting professors at the IAS included internationally renowned figures such as Pauli, Dirac or Hideki Yukawa (1907-1981), as well as young researchers such as Murray Gell-Mann (1929- ), Geoffrey Chew (1924- , Francis Low (1921-2007), Yoichiro Nambu (1921-2015), and Cécile Morette (1922-2017). Oppenheimer also appointed as faculty members young physicists such as Abraham Pais (1918-2000), Freeman Dyson (1923- ), Tsung-Dao Lee (1926- ) and Chen Ning Yang (1922- ), the latter two winners of the Nobel Prize in Physics in 1957. The high number of physicists at the School of Mathematics made it advisable to create a School of Natural Sciences, that began work in 1966 and to which the physicists were assigned from then onwards. Also the School of Mathematics made some spectacular appointments under the direction of Oppenheimer such as Armand Borel,

From Göttingen to Princeton and Paris

Fig. 24-1 Kurt Gödel and Albert Einstein at Princeton, 1950

235

236

Epilogue

Deane Montgomery (1909-1992), Atle Selberg (1917-2007), André Weil and Hassler Whitney (1907-1989). Oppenheimer would retire as director of the IAS in 1966, having made Princeton an example for all American universities to follow. Despite the ravages caused by the Second World War in Europe, in the 1950s, some initiatives tried to continue with the best tradition of research and teaching of mathematics in Europe before the war. Perhaps the most fundamental initiative in this sense came from Bourbaki, which was favoured by the founding in 1958 of the Institut des Hautes Études Scientifiques (IHÉS), one of the most prestigious scientific institutions in Europe to date. IHÉS was founded by Leon Motchane, a business-man of Russian origin who loved mathematics, with the intention of creating an institution inspired and modelled upon the IAS of Princeton. With the help of Robert Oppenheimer, then director of the IAS, and with the financial support of various industrial groups and the French government, Motchane was able to inaugurate the IHÉS in 1958, a place where “everything would be organized so that the people had the highest degree of intellectual freedom possible.”1 Motchane’s original idea was to establish an institute dedicated to fundamental research in three areas: mathematics, theoretical physics and methodology of the human sciences, but unfortunately the last area never managed to have a place at the IHÉS. In spite of the economic difficulties of the first years, the IHÉS was remarkably successful in recruiting Jean Dieudonné and Alexandre Grothendieck for the first two vacancies of mathematics and in attracting as invited researchers some of the most brilliant mathematicians of the moment, such as Michael Atiyah (1929- ), Shiing-Shen Chern (19112004), Friedrich Hirzebruch (1927-2012) and André Weil. In 1960, the prestigious books series “Les Publications de l’IHÉS” was also launched, initially directed by Dieudonné. The creation of the IHÉS took place at the time that Bourbaki was exercising its maximum influence in France and worldwide. Dieudonné, one of the founding fathers of Bourbaki, and Grothendiek, one of the most prominent Bourbaki mathematicians active in the late 1950s, regularly gave seminars at the IHÉS since the beginning of its activity. Many other members of Bourbaki, such as Claude Chevalley, Jean-Pierre Serre and Armand Borel, attended Grothendiek’s seminars at the IHÉS. In 1971,

 1

In the words of his son Didier Motchane (http://www.ihes.fr/jsp/ site/ Portal. jsp? Page_id=20).

From Göttingen to Princeton and Paris

237

Pierre Cartier, another member of Bourbaki, began his long association with the IHÉS as a visiting professor. IHÉS was, at least during the first ten years of its existence, Bourbaki’s Göttingen. In the same way that, until the rise of Nazism, any mathematician or physicist who understood German minimally had as his greatest aspiration to work with Hilbert, Weyl and Noether at Göttingen, the IHÉS, located in Bures-sur-Yvette, next to Paris, would become the Mecca of French-speaking scientists (but also Anglophones, since Bourbaki has traditionally been bilingual) in the post-war period. From the first moment of its creation, the IHÉS has maintained strong ties with the international community of mathematicians and physicists around the world, especially the United States. In this way, the influence of Bourbaki and the IHÉS spread throughout the world, as had happened in the past with Hilbert and Göttingen. A clear example of this is found in the figure of Alexandre Grothendieck, winner in 1966 of the Fields medal for his work on algebraic geometry and the most representative mathematician of the third generation of Bourbaki. Assisted by Dieudonné, under whose direction he had received his doctoral degree, Grothendieck began working in 1958 on the book Éléments de géométrie algébrique (Elements of Algebraic Geometry), published in eight fascicles by the IHÉS between 1960 and 1967. He also ran the mythical Séminaire de géométrie algébrique (Seminar of Algebraic Geometry) at the IHÉS from 1960 to 1969, which would completely reshape the field of algebraic geometry. As one of the attendees at his seminar pointed out, his way of working was unique: He did not want to solve difficult or famous mathematical problems; his goal was to achieve a thorough and comprehensive understanding of the underlying structures in such a way that solutions to those problems fall like ripe fruit alone. In any case, the reformulation of algebraic geometry carried out by Grothendieck has made possible some spectacular results, such as the Riemann-Roch-Hirzebruch theorem (by Grothendieck himself), the famous Weil’s conjectures (Grothendieck again and Pierre Deligne), Mordell’s conjecture (for which Gerd Faltings (1954- ) received the Fields Medal in 1984) and, ultimately, a special case of the Taniyama-Shimura conjecture, from which follows the Last Fermat Theorem, demonstrated by Andrew Wiles (1953- ) with the help of Richard Taylor (1962- ) in 1995. In this sense, there is no doubt that the language and methods introduced by Grothendieck are part of the landscape of modern mathematics and one of the most important sources of research today. The influence of the IHÉS and Bourbaki through Grothendieck quickly spread throughout the world during the last quarter of the twentieth century.

238

Epilogue

Fig. 24-2 Alexandre Grothendieck lecturing at the Seminar of Algebraic Geometry, probably between 1962 and 1964, with Jean Dieudonné to his left and Claude Chevalley to his right

Oscar Zariski (1899-1986), a professor of mathematics at Harvard and one of the key figures in the extension and generalization of algebraic geometry, sent his students to the Grothendieck seminar to know about what was cooking there so that later, back at Harvard, they could explain the latest news they had learned. Among these students was David Mumford (1937- ), who extended Hilbert’s ideas on the theory of invariants, reformulating them in the language of Grothendieck’s algebraic geometry. This won him the Fields Medal in 1974 and helped to create a powerful research group in algebraic geometry in the United States. This is perhaps a good example of Hilbert’s lasting inheritance in contemporary mathematics and a good way to close this essay.

APPENDIX THE HILBERT PROBLEMS REVISITED

PROBLEM 1. Cantor’s problem on the cardinal number of the continuum. This problem asks for a proof of the continuum hypothesis (CH), that is, the conjecture that ʹՅబ ൌ Յଵ (see Chapter 11). In the next International Congress of Mathematicians at Heidelberg (1904), Julius König (18491913) delivered a lecture in which he argued that ʹՅబ is not an aleph, i.e., that the continuum cannot be well-ordered and so that CH was false. König’s astonishing argument stimulated some of those of the attendees to explore the matter further. By the next day Zermelo had found an error in König’s argument. Later the same year, Felix Hausdorff published an article where he claimed that Թ could be well-ordered and that every infinite cardinal was an aleph. By the same time, Zermelo published an epoch-making article in which he proved that every set can be wellordered. Three years later he published his famous axiomatization of set theory (see Chapter 13). In the meanwhile, Hausdorff developed the higher transfinite in his study of order types and cofinality. In this way, Hilbert’s first problem passed to the next generation of mathematicians. Zermelo’s proof that every set can be well-ordered rested on the controversial axiom of choice (it is actually equivalent to it), so it remained to prove the consistency of this axiom with the remaining axioms of set theory. This was done by Kurt Gödel in another epochmaking article published in 1938. In the same paper, Gödel established that if ZFC (Zermelo-Fraenkel set theory with the axiom of Choice) is consistent, then so is ZFC+CH. The consistency of ZFC+ ™ CH was established by Paul Cohen (1934-2007) in 1963 by a method called forcing. So, by Gödel’s and Cohen’s results, CH was finally shown to be independent of the axioms of ZFC. This was for Cohen (who was a formalist) the end of the story. For Gödel (who was a Platonist) it was not. He proposed to introduce the so-called axioms of infinity (or largecardinal axioms) in order to prove this and other hitherto undecidable sentences. Gödel’s program is still pursued nowadays, but without any

Appendix

240

definitive result regarding the proof of CH. For some mathematicians and philosophers CH is an undecidable sentence because it is not a welldefined problem, the view of Solomon Feferman (1928-2016), or because our experience with constructing models of ZFC+CH and ZFC+ ™ CH௅as argued by Joel Hamkins (1966- ).

PROBLEM 2. The compatibility of the arithmetical axioms. The proof of the consistency of analysis (see Chapter 14) was the crux of the so-called Hilbert’s program (see Chapter 20). In 1924 W. Ackermann believed to have achieved a consistency proof of analysis, but when reviewing of this proof he realized that he had just proved the consistency of a fragment of arithmetic. In 1927, J. von Neumann improved the techniques introduced by Ackermann, although he still failed to get the desired proof of the consistency of analysis. In 1928 at the International Congress of Mathematicians held in Bologna, Hilbert considered that thanks to the work of Ackermann and von Neumann “the only remaining task consists in the proof of an elementary finiteness theorem that is purely arithmetical.”1 However, Gödel soon demonstrated that Hilbert’s aim of proving the consistency of analysis was unfeasible. More concretely, Gödel demonstrated in 1931 the incompleteness of every consistent and sufficiently strong theory T, such as PA (Peano arithmetic) or ZFC, and soon after he also showed informally the unprovability in such a theory of the statement formalizing “T is consistent.” This yielded a negative solution to Hilbert’s second problem for it established the impossibility for even weak theories like PA of demonstrating his own consistency. Positive results (using techniques that Hilbert surely would not have allowed) are due to Gerhard Gentzen in 1936 and Petr Sergeevich Novikov (19011975) in 1941.

PROBLEM 3. The equality of the volumes of two tetrahedra of equal bases and equal altitudes. The Wallace-Bolyai-Gerwien theorem, proved in the nineteenth century, says that if polygons ‫ ܣ‬and ‫ ܤ‬are of equal area, polygon ‫ ܣ‬may be cut into

 1

Mancosu 1998, 229.

The Hilbert Problems Revisited

241

triangles which may then be rearranged to form polygon ‫ܤ‬. Is there an analogous theorem for polyhedra ‫ ܣ‬and ‫ ?ܤ‬In other words, if ‫ ܣ‬and ‫ ܤ‬are polyhedra of the same height and base (and hence volume), is it always possible to cut ‫ ܣ‬into finitely many polyhedral pieces which may then be reassembled to yield ‫ ?ܤ‬Hilbert conjectured that this was impossible and that “it should be our task to give a rigorous proof of its impossibility.”2 The motivation behind Hilbert’s problem was the fact that in plane Euclidean geometry, the proof of the formula for the area of a triangle in terms of its base and height can be given by showing that any two such polygons can be transformed into each other by cutting and pasting. On the contrary, all the known proofs of the formula for the volume of a tetrahedron in terms of its base and height involved the calculus–in particular, the method of exhaustion. Hilbert’s third problem then asked for the possibility of finding a cut-and-paste argument in order to determine the volume of a tetrahedron. This problem was solved in the negative sense by Hilbert’s student Max Dehn in 1900, in fact before Hilbert’s lecture was delivered. It had been partially solved four years before by Raoul Bricard (1870-1943). Dehn’s proof relied on the observation that besides the volume there is one more quantity that remains invariant under cutting and pasting, the Dehn invariant. In higher dimensions the same problem can be studied and there are the Hadwiger invariants. In 1965 Jean Pierre Sydler (1921-1988) proved that in solid geometry the Dehn invariant is the only extra invariant besides volume.

PROBLEM 4. Problem of the straight line as the shortest distance between two points. As remarked by Hilbert this is a “problem relating to the foundations of geometry”3 and indeed one that he formulated precisely in axiomatic terms. More concretely, Hilbert asks for geometries that sit close to Euclidean geometry in which all axioms remain valid except the strong congruence axiom and this axiom is replaced by the requirement that straight lines are the shortest distance between two points. In the conclusion to his comments on Problem IV, Hilbert summarizes the problem by saying that it would be desirable to make a complete and

 2 3

Hilbert 1965, vol. 3, 302. Ibid.

242

Appendix

systematic study of all geometries in which shortest distances are realized by straight lines. This is a quite broad statement and hence the fourth problem it has been given different interpretations and generalizations. In particular, although Hilbert’s formulated the fourth problem as a foundational problem in an axiomatic setting, most modern approaches have interpreted it in terms of a metric setting. The first work on this direction was by Hilbert’s student Georg Hamel (1877-1954) in 1903. Hamel pointed out that the problem needed to be made more precise, and that one should ask for all Desarguesian spaces in which straight lines are the shortest distances between points.4 Nowadays, the problem is considered (basically) solved in the form of the following (generalized) theorem of Aleksei Pogorelov (1919-2002): Any ݊-dimensional Desarguesian space of class ‫ ܥ‬௡ାଶ ǡ ݊ ൒ ʹ, can be obtained by the B-B construction, a technique based upon integral geometry for obtaining Desarguesian spaces due to Wilhelm Blaschke (1885-1962) and Herbert Busemann.

PROBLEM 5. Sophus Lie’s concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group. This problem has played a key role in the development of the theory of topological groups. Hilbert’s fifth problem, like other Hilbert’s problems, does not have a unique interpretation. The most common formulation of Hilbert’s fifth problem is whether a locally Euclidean topological group is a Lie group. A Lie group is a group ‫ ܩ‬ൌ ሺ‫ܩ‬ǡήሻ which is also a smooth manifold, such that the group operations ή‫ ܩ ׷‬ൈ ‫ ܩ ื ܩ‬and ሺሻିଵ  ‫ื ܩ ׷‬ ‫ ܩ‬are smooth maps.5



4 A Desarguesian space is a space of geodesics ሺܺǡ ݀ሻሻ(a ‫ܩ‬-space) in which the role of geodesics is played by ordinary straight lines. The name comes from the fact that, for ݊ ൌ ʹ, it is required that Desargues theorem and its converse are valid; for dimension ݊ ൐ ʹ, it is required that any point in ܺ lies in one plane. These conditions are equivalent to Hilbert’s demand that we keep the strong congruence axiom and replace it by the requirement that ordinary lines are the shortest distance between points. 5 For a function to be smooth, it has to have continuous derivatives up to a certain order, say ݇. We say then that the function is ‫ ܥ‬௞ . So smoothness implies



The Hilbert Problems Revisited

243

Hilbert’s question is thus whether the requirement of smoothness in the definition of Lie group is redundant. To see the relation of Hilbert’s question so formulated to the above formulation, let us relax the notion of a Lie group to that of a topological group. A topological group is a group ‫ ܩ‬ൌ ሺ‫ܩ‬ǡήሻ, that is also a topological space, in such a way that the group operations ή‫ ܩ ׷‬ൈ ‫ ܩ ื ܩ‬and ሺሻିଵ  ‫ ܩ ื ܩ ׷‬are continuous. Clearly, every Lie group is a topological group (just erase the smooth structure). Furthermore, such topological groups are still locally Euclidean.6 The converse of this statement, i.e., that locally Euclidean topological groups are Lie groups, was established by Andrew Gleason (1921-2008), and by Deane Montgomery and Leo Zippin (1905-1995) in a couple of articles published in 1952.7 This solved Hilbert’s fifth problem, at least in its most common formulation given today.

PROBLEM 6. Mathematical treatment of the axioms of physics According to Hilbert “the investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theory of probabilities and mechanics.”8 To understand Hilbert’s statement of the problem we have to recall that Geometry itself was for Hilbert a physical science (see Chapter 3 and Chapter 7) and that he had already axiomatized it in 1900 (see Chapter 8), so that he could think of it as a model for the mathematical treatment of the axioms of other physical sciences. Hilbert’s approach to the foundations of physics was always via the axiomatic method. From 1912 to 1914 Hilbert applied the axiomatic method to the kinetic theory of gases and the elementary theory of radiation. By 1915 he applied it to what he then called the Foundations of Physics, that is, the formulation of a unified theory of gravitational fields

 continuity, but not the other way around. There are functions that are continuous everywhere, but nowhere differentiable. 6 A topological group is called locally Euclidean if it has a neighbourhood of the identity that is homeomorphic to a Euclidean space Թ௡ , i.e., if it is a topological manifold. 7 Gleason, A. M. “Groups without small subgroups,” Ann. of math., 56 (1952): 193–212. Montgomery, D. and Zippin, L. “Small subgroups of finite dimensional groups,” Ann. of math., 56 (1952): 213–241. 8 Hilbert 1965, vol. 3, 306.

Appendix

244

based on Mie’s theory of matter (see Chapter 16). In the opening paragraphs of his First Communication on the Foundations of Physics Hilbert announced that “in the following௅in the sense of the axiomatic method௅I would like to develop, essentially from two simple axioms, a new system of basic equations of physics, of ideal beauty and containing, I believe, simultaneously the solution to the problems of Einstein and of Mie.”9 These axioms are Mie’s Axiom of the World Function (Axiom I) and the Axiom of General Invariance (Axiom II), although in the proofs Hilbert uses a third axiom, the Axiom of Space and Time (Axiom III). Hilbert’s sixth problem has inspired several lines of research and its mathematical content has changed in time. It is thus more a “programmatic call” than a mathematical problem. As a response to this programmatic call many parts of the physical sciences have been axiomatized with great success. Regarding the two fields specifically mentioned by Hilbert in the statement of his programmatic call, mechanics and probability, the first was axiomatized quite early by Hamel in 1903, whereas the second was definitively axiomatized by Kolmogorov in 1933. Also, as a direct response to Hilbert’s call, Carathéodory axiomatizations of thermodynamics in 1909 and of special relativity in 1924 should be mentioned. Another independent axiomatization of special relativity was given by Alfred Robb (1873-1936) in 1914. As we already know, the first axiomatizations of quantum mechanics in terms of operators on Hilbert spaces were presented by Paul Dirac in 1930 and by John von Neumann in 1932. Modern axiomatizations of relativistic quantum field theory, such as the axiomatizations by Arthur Wightman (1922-2013) and by Rudolf Haag (1922-2016) and Daniel Kastler (1926-2015), are often viewed as realizations of Hilbert’s program of axiomatization of physical theories.

PROBLEM 7. Irrationality and transcendence of certain numbers. This problem asks whether “the expression ߙ ఉ , for an algebraic base ߙ and an irrational algebraic exponent ߚ , e. g., the number ʹξଶ or ݁ గ ൌ ݅ ିଶ௜ ǡ always represents a transcendental or at least an irrational number.”10 It was solved by Alexander Gelfond (1906-1968) in 1934 and Theodor Schneider (1911-1988) in 1935. In a previous paper of 1929 Gelfond “proved that ݁ గ is actually transcendental and indicated how his

 9

Hilbert 2009, 28-29. Hilbert 1965, vol. 3, 308.

10

The Hilbert Problems Revisited

245

method could be used to prove transcendentality whenever ߚ belongs to an imaginary quadratic field. The extension to real quadratic fields was given by C. L. Siegel (unpublished) and R. A. Kuzmin in 1930. Gelfond returned to the question later and in 1934 he could give a complete solution of Hilbert’s problem which was followed in 1935 by a more elementary solution obtained independently by Th. Schneider, one of Siegel’s pupils. The proof by Gelfond, though more advanced, has quite a simple basis and gives a beautiful example of teamwork between algebraic and analytical ideas.”11

PROBLEM 8. Problems of prime numbers. This problem deals with the “distribution of prime numbers” and more concretely with the Riemann Hypothesis, although Hilbert mentions some generalizations of it and also related problems such as the Goldbach conjecture. The Riemann hypothesis is the most famous and important of the unsolved conjectures in mathematics. It is a statement about the zeros of the so-called Riemann zeta function. As it is well known, the distribution of prime numbers ሺʹǡ ͵ǡ ͷǡ ͹ǡ ͳͳǡ ͳ͵ǡ ͳ͹ǡ ͳͻǡ ǥ ሻ among the natural numbers does not follow any regular pattern. However, Riemann observed that the frequency in which prime numbers appear is very closely related to the behavior of the function ߞሺ‫ݏ‬ሻ ൌ ͳ ൅ ͳΤʹ௦ ൅ ͳΤ͵௦ ൅ ͳΤͶ௦ ൅ ǥ , later called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting solutions of the equation ߞሺ‫ݏ‬ሻ ൌ Ͳ lie on a certain vertical straight line. Although this has been checked with computer aid for the first 1.5 billion zeroes, there is no proof that it is true for every interesting solution. According to many mathematicians, the Riemann Hypothesis is the most important open problem in pure mathematics, since its solution would shed light on the inextricable misteries of the distribution of prime numbers. It is one of the seven “Millennium Prize Problems” for which the the Clay Mathematics Institute has designated a $7 million prize fund, with $1 million allocated to the solution of each problem.12

 11

Hille 1942, 654. For a technical and complete exposition of the problem and the mathematical developments arising from it, see the Official Problem Description by E. Bombieri (http://www.claymath.org/sites/default/files/official_problem_description.pdf). 12

Appendix

246

As remarked by Hilbert, following Problem no. 8 come “three more special problems in number theory: one on the laws of reciprocity, one on Diophantine equations, and a third from the realm of quadratic forms.”13

PROBLEM 9. Proof of the most general law of reciprocity in any number field. We have extensively dealt with the origins of this problem in Chapter 4. As remarked there, Hilbert himself had contributed to the solution of this problem in 1895 and 1896 with a generalization of the reciprocity law by means of the norm residue symbol, later called Hilbert symbol. After Hilbert, Ph. Furtwängler, T. Takagi, H. Hasse and E. Artin made important contributions to the study of reciprocity laws. In 1927 Artin gave reciprocity laws for general number fields. Artin’s reciprocity law is the crux of class field theory. In 1950 Igor Shafarevich (1923-2017) settled the analogous question of reciprocity laws for function fields. Artin’s reciprocity law is usually considered a partial solution of Hilbert’s 9th problem, since it deals with Abelian extension of algebraic number fields. The analogue problem for non-Abelian extensions, which is intimately connected with Hilbert’s 12th problem, is still an open problem.

PROBLEM 10. Determination of the solvability of a Diophantine equation. A Diophantine equation is a polynomial equation ܲሺ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ ሻ with integers as coefficients. It is solvable if there are integral solutions. For example, the Fermat equation ‫ ݔ‬௡ ൅ ‫ ݕ‬௡ ൌ ‫ ݖ‬௡ for a given natural number n is a Diophantine equation which has infinitely many solutions for ݊ ൌ ͳǡ ʹ and no solutions for larger n. Hilbert tenth problem asks for a computing algorithm (a decision procedure) which tells of a given Diophantine equation whether it is solvable or not. The Russian mathematician Yuri Matiyasevich (1947- ), working on previous work by Martin Davis (1928), Julia Robinson (1919-1985) and Hilary Putnam (1926-2016), showed in 1970 that there is no such algorithm, so giving a negative solution to Hilbert’s tenth problem.

 13

Hilbert 1965, vol. 3, 310.

The Hilbert Problems Revisited

247

A couple of remarks are in order: First, as noted by Davis, “the way in which the problem has been resolved is very much in the spirit of Hilbert’s address in which he spoke of the conviction among mathematicians ‫ދ‬that every definite mathematical problem must necessarily be susceptible of a precise settlement, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution …‫]…[ ތ‬ Matiyasevich’s negative solution of Hilbert’s tenth problem is of just this character. It is […] a ‫ދ‬precise and completely satisfactory‫ ތ‬proof that no such solution is possible.”14 Second, it is also worth mentioning that the first steps towards Matiyasevich’s negative solution of Hilbert’s tenth problem were given by Gödel. For Gödel did not only introduce the general notion of recursiveness, but he also demonstrated that every recursive function can be defined by a finite number of existential and bounded universal quantifiers (which is essential in the MatiyasevichRobinson-Davis-Putnam proof of the unsolvability of Hilbert’s tenth problem).

PROBLEM 11. Quadratic forms with any algebraic numerical coefficients. Quadratic forms were first studied over Ժ, by all the great number theorists from Fermat to Dirichlet. A major portion of Gauss’s Disquisitiones Arithmeticae was devoted to the study of binary quadratic forms over the integers. By the late 19th century the concept was generalized since it was realized that it is easier to solve equations with coefficients in a field ‫ܭ‬ than in an integral domain like Ժ and that a firm understanding of the set of solutions in the fraction field of ‫ ܭ‬is prerequisite to understanding the set of solutions in ‫ ܭ‬itself. In this regard, a general theory of quadratic forms with Է-coefficients was developed by Minkowski in the 1880s and extended and completed by Hasse in his 1921 dissertation. This problem asks for the classification of quadratic forms over algebraic number fields. A quadratic form over a field ‫ ܭ‬is a polynomial ‫݌‬ሺ‫ݔ‬ሻ ൌ ‫݌‬ሺ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ ሻ ‫ܭ א‬ሾ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ ሿ, with coefficients in ‫ܭ‬, such that each monomial term has total degree ʹ, that is, ‫݌‬ሺ‫ݔ‬ሻ ൌ

෍ ܽ௜௝ ‫ݔ‬௜ ‫ݔ‬௝ ଵஸ௜ஸ௝ஸ௡

 14

Davis 1973, 233-34.

248

Appendix

with ܽ௜௝ ‫ܭ א‬. Let ‫݌‬ǡ ‫ ݍ‬be two ݊-ary quadratic forms. We say that they are equivalent if there is an invertible homogeneous linear substitution of the variables ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ which takes the form ‫ ݍ‬to the form ‫݌‬. For example, if ‫݌‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ is the form ‫ ݔ‬ଶ െ ‫ ݕ‬ଶ and ‫ݍ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ is ‫ݕݔ‬, the substitution ‫ ݔ‬฽ ‫ ݔ‬൅ ‫ݕ‬ǡ ‫ ݕ‬฽ ‫ ݔ‬െ ‫ ݕ‬changes ‫ ݍ‬to ‫ ݌‬since ‫ݍ‬ሺ‫ ݔ‬൅ ‫ݕ‬ǡ ‫ ݔ‬െ ‫ݕ‬ሻ ൌ ሺ‫ ݔ‬൅ ‫ݕ‬ሻሺ‫ ݔ‬െ ‫ݕ‬ሻ ൌ ‫ ݔ‬ଶ െ ‫ ݕ‬ଶ ൌ ‫݌‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ. As remarked by M. Hazewinkel, “the problem is to classify quadratic forms up to this equivalence. This was solved in [Hasse, 1924] by the Hasse–Minkowski theorem and the Hasse invariant. The theorem says that two quadratic forms over a number field ‫ ܭ‬are equivalent if and only if they are equivalent over all of the local field ‫ܭ‬௣ for all primes ‫ ݌‬of ‫ܭ‬. For instance for ‫ ܭ‬ൌ Է, the rational numbers, two forms over Է are equivalent if and only if they are equivalent over the extensions Թ, the real numbers, and the ‫݌‬-adic numbers Է௣ for all prime numbers ‫݌‬. This reduces the problem to classification over local fields, which is handled by the Hasse invariant (apart from rank and discriminant).”15

PROBLEM 12. Extension of Kronecker’s theorem on Abelian fields to any algebraic realm of rationality. This problem asks for the extension of the Kronecker-Weber theorem on Abelian extensions of the rational number to any base number field. As we already know, Kronecker-Weber theorem asserts that every finite Abelian extension of Է is a subfield of a cyclotomic field. Recall that a cyclotomic field is a number field of the form Էሺߦ௠ ሻ, where ߦ௠ ൌ ‡š’ሺʹߨ݅Τ݉ሻ (see Chapter 4). It was Kronecker’s Jugendtraum (dream of youth) to find similar functions whose values could generate analogues of cyclotomic fields over other number fields. The attempt to prove Kronecker Jugendtraum, i.e., Hilbert’s twelfth problem, has led to many important theories in mathematics, particularly class field theory and complex multiplication. The problem has been settled for certain number fields; e.g., imaginary quadratic fields and their generalization, the so-called CM fields.

 15

Hazewinkel 2005, 737. The paper mentioned is Hasse, H. “Äquivalenz quadratischer formeln in einem beliebigen algebraischer Zahlkörper,” J. reine und angew. Math., 153 (1924): 113-130.

The Hilbert Problems Revisited

249

Hilbert’s statement of Kronecker’s Jugendtraum in his twelfth problem first considers the special case whether all Abelian extensions of imaginary quadratic fields could be obtained by adjoining values of the elliptic modular function ݆ሺ‫ݖ‬ሻ. He then passes to the general case and asks for “the extension of Kronecker’s theorem to the case that, in place of the realm of rational numbers or of the imaginary quadratic field, any algebraic field whatever is laid down as realm of rationality.”16 The first problem was worked out intensively in the first decades of the twentieth century by Weber, Blumenthal, Hecke, Takagi and Hasse, among others. These works established the fact that a certain elliptic modular function ݆ሺ߬ሻ generates the Hilbert class field of ‫ܭ‬, where ‫ ܭ‬is an imaginary quadratic field. These results were generalized in the early fifties by Max Deuring (1907-1984), Shimura, Yutaka Taniyama (1927-1958), Weil and others to a much wider class of number fields, the CM fields௅so named by their connection with the theory of complex multiplication.

PROBLEM 13. Impossibility of the solution of the general equation of the seventh degree by means of functions of only two variables. This problem asks whether it is possible to write every continuous function of three variables as a superposition of continuous functions of two variables. A superposition is just a composition of functions. For example, ݂ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬ሻ ൌ ‫ ܨ‬ቀ݃ሺ‫ݔ‬ǡ ‫ݕ‬ሻǡ ݄൫݆ሺ‫ݔ‬ሻǡ ݇ሺ‫ݖ‬ሻ൯ቁ is a superposition of functions of one and two variables. The question is then whether all functions of three variables are representable by functions of two variables. Hilbert conjectured a negative answer: It is probable that the root of the equation of the seventh degree is a function of its coefficients which […] cannot be constructed by a finite number of insertions of functions of two arguments. To see this, the proof would be necessary that the equation of the seventh degree ݂ ଻ ൅ ‫ ݂ݔ‬ଷ ൅ ‫ ݂ݕ‬ଶ ൅ ‫ ݂ݖ‬൅ ͳ ൌ Ͳ is not solvable with the help of any continuous functions of only two arguments.17

 16 17

Hilbert 1965, vol. 3, 312. Ibid., 314.

Appendix

250

In 1954, Anatoli Vitushkin (1931-2004) proved that there are continuously differentiable functions which cannot be written as a superposition of continuously differentiable functions of three variables. This was a remarkable first result in line with Hilbert’s conjecture. However, Andrey Kolmogorov proved in 1956 that every function of ݊ variables can be written as a superposition of functions of three variables. The next year, his student Vladimir Arnol’d (1937-2010) reduced this number to two. These results culminated in 1957 with the so-called Kolmogorov Superposition theorem, which asserts that every continuous function of two or more variables can be written as a superposition of continuous functions of just one variable along with just one function of two variables, namely addition. This result gives a positive solution to the 13th problem, although a negative one to Hilbert’s conjecture about it. The problem is usually considered as partially resolved. This is because in his presentation of the 13th problem, Hilbert did not make clear what kind of functions he had in mind. Indeed, it seems from Hilbert’s 1927 paper “Über die Gleichung neunten Grades,” that he was originally thinking about algebraic functions when he posed this problem. This is also the interpretation given by Arnol’d and Shimura in the article Superposition of algebraic functions.18 For these kinds of functions the problem is still unresolved.

PROBLEM 14. Proof of the finiteness of certain complete systems of functions. The fourteenth problem may be formulated as follows: Let ݇ be a field and ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ algebraically independent elements over ݇. Let ‫ ܭ‬be a subfield of ݇ሾ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ ሿ containing ‫ܭ‬. Is the ring ݇ሾ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ ሿ ‫ ܭ ת‬finitely generated over ݇? Hilbert’s motivation for this problem came from its positive answer in the case of algebraic invariant theory: If Šƒ”ሺ݇ሻ ൌ Ͳ, ‫ ܩ‬is a linear algebraic group acting on a polynomial ring ݇ሾ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ ሿ and ‫ ܭ‬is the field of ‫ܩ‬-invariant rational functions, then ݇ሾ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ ሿ ‫ ܭ ת‬is the ring of ‫ܩ‬invariant polynomials over ݇. As we already know, Hilbert had proved that this ring is finitely generated (see Chapter 2). However, the answer to the 14th problem is negative: Masayoshi Nagata (1927-2008) found in

 18

Browder 1976, 45-46.

The Hilbert Problems Revisited

251

1959 a counterexample, also in this setting of rings of invariants, showing that ݇ሾ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ ሿ ‫ ܭ ת‬may require an infinite number of generators. As David Mumford has written in his article on “Hilbert’s Fourteenth Problem. The Finite Generation of Subrings such as Rings of Invariants,” “it would appear that after Hilbert’s discovery of the extremely general finiteness principle on which his proof in the invariant case was based, namely “Hilbert’s basis theorem” on the finite generation of all ideals in ݇ሾ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ ሿ, Hilbert was overly optimistic about finiteness results in other algebraic contexts. However my belief is that it was not at all a blind alley: that on the one hand its failure reveals some very significant and farreaching subtleties in the category of varieties; and that the search for cases where it and related geometric questions are correct is a very important area of research in algebraic geometry.”19 This was indeed the case as the intriguing 1954 article “Interpretations algebrico-geometriques du quatorzieme probleme de Hilbert” by Oscar Zariski witnessed.

PROBLEM 15. Rigorous foundation of Schubert’s enumerative calculus. This problem asks for the justification of Hermann Schubert’s (18481911) enumerative calculus and the verification of the numbers he obtained. Hilbert’s statement of the problem is as follows: To establish rigorously and with an exact determination of the limits of their validity those geometrical numbers which Schubert especially has determined on the basis of the so-called principle of special position, or conservation of number, by means of the enumerative calculus developed by him.20

The Schubert Calculus is a formal calculus in which geometric conditions of figures are represented by algebraic symbols in order to solve problems in enumerative geometry. An example of the kind of problems the Schubert calculus deals with is the following: How many linear subspaces of projective space satisfy incidence conditions imposed by other linear

 19

Ibid., 431-32. Hilbert 1965, vol. 3, 312. Schubert’s treatment of enumerative geometry rested heavily on the principle of special position or conservation of number. This principle was introduced in the context of projective geometry by Poncelet, who called it the principle of continuity (see Chapter 3). Despite the criticism of Cauchy and Study, among others, the principle came into widespread use. 20

252

Appendix

subspaces? Or, to put another example: How many lines in projective 3space meet 4 given lines? The Schubert calculus originated in the work of Michel Chasles in the enumerative theory of conics and was systematized and extended by Schubert in his 1879 treatise Kalkül der abzählenden Geometrie (Calculus of enumerative geometry). The justification of Schubert’s enumerative calculus was a major theme of 20th century algebraic geometry and Intersection Theory provides a satisfactory framework for the treatment of the foundations of the Schubert calculus. However, as remarked by S. L. Kleiman in the article “Problem 15. Rigorous Foundation of Schubert’s Enumerative Calculus,” “to claim that the development of an intersection theory provides a complete solution to Hilbert’s fifteenth problem is to give the problem a narrow interpretation, for an intersection theory is only the first step on the road toward the validation of the geometrical numbers of classical enumerative geometry, and in both the statement and the explanation of the problem Hilbert makes clear his interest in the efficient production of accurate geometric numbers.”21 For example, in his 1879 book, Schubert finds the number 666, 841, 048 of quadric surfaces tangent to 9 given quadric surfaces in 3space, and the number 5, 819, 539, 783, 680 of twisted cubic curves tangent to 12 given quadric surfaces in 3-space. However, up to date “we cannot vouch for the accuracy of these two spectacular numbers, nor do we even know whether Schubert’s method is basically sound.”22

PROBLEM 16. Problem of the topology of algebraic curves and surfaces. The problem, as stated by Hilbert, splits in two different parts. The first part asks for an investigation of the relative positions of the branches of real algebraic curves and surfaces of degree n. In 1876 Axel Harnack (1851-1888) had found that algebraic curves in the real projective plane of degree n could have no more than ሺ݊ଶ െ ͵݊ ൅ ͶሻΤʹ separated connected components. He also showed how to construct curves with this maximum number of components, the so-called M-curves. In his investigation of the M-curves of degree 6, Hilbert found that the 11 components were always disposed according to certain constrains. The first part of problem 16 then asks for a thorough investigation of the possible configurations of the components of the M-curves and for similar

 21 22

Browder 1976, 455-56. Ibid., 445.

The Hilbert Problems Revisited

253

investigation of surfaces with the maximum number of components. For algebraic curves of degree 6 the problem was finally solved in 1970 by Dmitrii Gudkov (1918-1992). For curves of degree 8 the problem remains unsolved. The second part of the problem asks for the determination of the maximum number of limit cycles in a planar polynomial vector field of degree n and an investigation of their relative positions. In 1991/92 Yulij Ilyashenko (1943- ) and Jean Écalle (1950- ) proved independently that every planar polynomial vector field has only finitely many limit cycles. However, the question whether there exists a maximum number of limit cycles of planar polynomial vector fields of degree n remains unsolved even for quadratic polynomials. As remarked by Ilyashenko, “there were several attempts to solve it that failed. Yet the problem inspired significant progress in the geometric theory of planar differential equations, as well as bifurcation theory, normal forms, foliations and some topics in algebraic geometry.”23

PROBLEM 17. Expression of definite forms by squares. Hilbert’s 17th problem is concerned with the representation of positive definite rational functions as a sum of squares. A rational function ݂ሺ‫ݔ‬ଵ ǡ ǥ ǡ ‫ݔ‬௡ ሻ is positive definite if it takes non-negative values in all points of Թ௡ where it is defined. Hilbert conjectured that such a function can be written as a sum of squares of rational functions with real coefficients. The case ݊ ൌ ʹ had been settled by Hilbert in a paper of 1893. The general case was solved affirmatively by Emil Artin in 1927. More concretely, Artin proved Hilbert’s conjecture for rational functions defined over real closed fields (such as, for example, the real numbers) and over arbitrary fields under the condition that they have only Archimedean orderings (this is the case, for example, of algebraic number fields). Artin proof was not constructive and used the machinery of the Artin-Schreier theory of formally real fields. A constructive solution to Hilbert’s 17th problem was given by Charles N. Delzell (1953- ) in 1984.

 23

Ilyashenko 2002, 301.

254

Appendix

PROBLEM 18. Building up of space from congruent polyhedral. This problem raises three different questions. The first question asks whether there are only finitely many types of subgroups of the group ‫ܧ‬ሺ݊ሻ of isometries of Թ௡  with compact fundamental domain. It was answered affirmatively by Ludwig Bieberbach in 1910, who also gave estimates of the number of these subgroups in 2-space and 3-space. These subgroups are now called Bieberbach groups. The second question asks whether Euclidean space can be filled without overlap by congruent copies of a certain polyhedron which is not the fundamental domain for any Bieberbach group. It was answered in the negative by Karl Reinhardt (1895-1941) in 1928 by means of a 3dimensional counter-example. A simpler 2-dimensional counter-example was given by Heinrich Heesch (1906-1995) in 1935. Finally, Hilbert raises the following question: “How can one arrange most densely in space an infinite number of solids with the same given form, e.g., spheres with given radii […], that is, how can one so fit them together that the ratio of the filled to the unfilled space may be as great as possible?”24 As J. Milnor has written in the article “Hilbert’s Problem 18: On Crystalographic Groups, Fundamental Domains, and on Sphere Packing”: “For 2-dimensional disks this problem has been solved by Thue and Fejes Tóth, who showed that the expected hexagonal (or honeycomb) packing of circular disks in the plane is the densest possible. However, the corresponding problem in 3 dimensions remains unsolved. This is scandalous situation since the (presumably) correct answer has been known since the time of Gauss […] All that is missed is a proof.”25

PROBLEM 19. Are the solutions of the regular problems in the calculus of variations always necessarily analytic? Problems 19, 20 and 23 deal with the calculus of variations. As noted by E. Bombieri in the paper “Variational Problems and Elliptic Functions,” “this problem of regularity, together with the problem of existence of solutions, form two central questions in the theory of variational

 24

Hilbert 1965, vol. 3, 319. Browder 1976, 500. For a more updated view of current research trying to answer this question see Hazewinkel (2005, 740). 25

The Hilbert Problems Revisited

255

problems.”26 These are precisely the questions asked in problems 19 and 20 respectively. The calculus of variations is a field of mathematical analysis concerned with the problem of finding functions for which the value of certain functionals (usually definite integrals involving functions and their derivatives) is either the largest or the smallest possible. Functions that maximize or minimize functionals may be found using the Euler-Lagrange equation. By a regular variational problem Hilbert understands a variational problem whose Euler-Lagrange equation is an elliptic partial differential equation with analytic coefficients. Hilbert’s 19th problem asks whether the solutions of these equations always have analytic coefficients. It was answered affirmatively by Sergei Bernstein in his 1904 thesis, where he showed that ‫ ܥ‬ଷ solutions of non-linear elliptic analytic equations in 2 variables are analytic. According to Bombieri, “Bernstein result was later improved and generalized to several variables and elliptic systems by the work of several authors, among which Gevrey, Giraud, Lichtenstein, H. Lewy, E. Hopf, T. Rado, I. Petrowsky and Bernstein himself.”27

PROBLEM 20. The general problem of boundary values. In this problem, Hilbert turns his attention to the question of the existence of solutions of partial differential equations when the boundary values are prescribed. More specifically, he asks whether every regular variational problem does not have a solution, provided certain assumptions regarding given boundary conditions are satisfied […] and provided also if need be that the notion of a solution shall be suitably extended.”28 As examples of boundary conditions, Hilbert mentions that the functions concerned are continuous or have one or more derivatives. As remarked by J. Serrin in his paper “The Solvability of Boundary Value Problems”: Several main ideas have become dominant in studying the existence of solutions of elliptic partial differential equations satisfying given boundary conditions, namely: I. Continuation of solutions along a parameter, for which the problem varies from a known situation to a desired one.

 26

Browder 1976, 526. Ibid. 28 Hilbert 1965, vol. 3, 322. 27

256

Appendix II. The a priori estimation of the magnitude of the partial derivatives, depending only on the structure of the equation, the boundary data, and the domain. III. A functional-analytic approach, guaranteeing the existence of a weak, or generalized solution of the given problem.29

The main agent in the dissemination of the first two approaches was Sergei Bernstein in a series of papers written between 1910 and 1912. His work was clarified by Juliusz Schauder (1899-1943) in 1934 and complemented by the work of Georges Giraud (1889-1943). The third approach was initiated by Hilbert and brought to the so-called direct method of the calculus of variations, which was later developed by Richard Courant, Leonida Tonelli (1885-1946) and Charles Morrey (1907-1984), among others.30

PROBLEM 21. Proof of the existence of linear differential equations having a prescribed monodromic group. More specifically, Hilbert’s 21th problem is to show that “there always exists a linear differential equation of the Fuchsian class, with given singular points and monodromic group.”31 Linear differential equations defined in an open, connected set ܵ in the complex plane have a monodromy group, which is a linear representation of the fundamental group of ܵ, summarising all the analytic continuations round loops within ܵ. Literally, Hilbert’s problem is the inverse problem, namely that of constructing the equation, given a monodromic representation. However, it should be noted that although Hilbert talks of “equation” what he had in mind was undoubtedly “system of equations,” since it was already known at that time that for equations the problem had a negative answer. So Hilbert’s problem asks whether there always exists a Fuchsian system with given singularities and monodromy. In a paper of 1908, Josip Plemelj (1873-1967) answered affirmatively a problem analogue to Hilbert’s 21th problem concerning regular systems instead of Fuchsian

 29

Browder 1976, 509. Much more could be said about the mathematical research in the twentieth century inspired by Hilbert’s 19th and 20th problems. We refer the interested reader to the papers by Serrin and Bombieri already mentioned (In Browder 1976, 507-24 and 525-35 respectively). 31 Hilbert 1965, vol. 3, 322. 30

The Hilbert Problems Revisited

257

ones. For many years this was thought also a positive solution to Hilbert’s problem. However, in 1989 Andrei Bolibruch (1950-2003) found a counterexample. He constructed a monodromic system with Fuchsian singularities and a not Fuchsian one (thus a not Fuchsian system, although a regular one), so giving a definitive negative solution to Hilbert’s 21th problem.

PROBLEM 22. Uniformization of analytic relations by means of automorphic functions. Classical uniformization theory, developed mainly during the last two decades of the 19th century and the first decade of the 20th, was concerned with proving that every algebraic or analytic curve can be uniformized, that is, represented parametrically by single-valued (or “uniform”) functions. Some of the most illustrious mathematicians of that time were involved on the topic: Friedrich Schottky (1851-1935), Weierstrass, Klein, Schwarz, Hilbert and Poincaré, among others. The complex 1-dimensional case was solved by Henri Poincaré and Paul Koebe in 1907 in the form of the general uniformization theorem, which asserts that if a Riemann surface (i.e., a connected 1-dimensional complex manifold) is homeomorphic to an open subset of the complex sphere, then it is conformally equivalent to an open subset of the complex sphere (a conformal mapping is a holomorphic bijection between Riemann surfaces). For higher complex dimensions, Hilbert’s 22nd problem, as well as various of its generalizations, is still open.

PROBLEM 23. Further development of the methods of the calculus of variations. As remarked by Hilbert, in contrast with the other 22 problems, the 23rd is not a definite and specific problem, but rather a general problem, namely “the indication of a branch of mathematics repeatedly mentioned in this lecture௅which, in spite of the considerable advancement lately given it by Weierstrass, does not receive the general appreciation which, in my opinion, is its due௅I mean the calculus of variations.”32 As J. J. Gray has

 32

Ibid., 323-24.

258

Appendix

noted, “after the brisk presentation of the previous four problems, the lengthy disquisition here suggests not only that Hilbert found the topic very important but that he was beginning to have quite precise ideas about what could be done to advance it.”33 An excellent and illuminating exposition on the development of the calculus of variations before and after 1900 is to be found in the article “Hilbert’s Twenty-Third Problem. Extensions of the Calculus of Variations” by G. Stampachia.34 Some modern extensions and links of the calculus of variations with theories such as optimal control theory and dynamic programming, the theory of minimal differential geometric objects such as geodesics, minimal surfaces and Plateau’s problem, the theory of variational inequalities and convex analysis are also noteworthy.35

 33

Gray 2000, 122-24. In Browder (1976, 611-628). 35 See Hazewinkel (2005, 742) for references. 34

REFERENCE LIST

Alexandroff, P. S. “In Memory of Emmy Noether.” In Dick, 1981, 153179 (also in Brewer and Smith,1982, 99-111). Bell, E. T. “Fifty Years of Algebra in America, 1888-1938.” In Semicentennial Addresses of the AMS, vol II. New York: American Mathematical Society, 1938, 1-34. Bliss, G. A., and Dickson, L. E. “Biographical Memoir of Eliakim Hastings Moore. 1862-1932.” National Academy of Sciences of the United States of America. Biographical Memoirs. Volume XVII. Fifth Memoir, 1935. Borel, A. “Twenty-Five Years with Nicolas Bourbaki, 1949-1973.” Notices of the AMS, 45, no. 3(1998): 373-380. Born, M. “How I became a Physicist.” Bulletin of the Atomic Scientists, XXI, no. 7(1965): 3-6. —. My Life: Recollections of a Nobel Laureate. New York: Scribner, 1978. Bourbaki, N. “The Architecture of Mathematics.” The American Mathematical Monthly, 57, no. 4(1950): 221-232. Brewer, J. W. and Smith, M. K. (eds.). Emmy Noether: A Tribute to Her Life and Work. New York: Marcel Dekker, 1982. Brouwer, L. E. J. Collected Works (vol. 1) (A. Heyting, ed.). Amsterdam: North-Holland, 1975. Cayley, A. “On linear transformations”. Cambridge and Dublin Mathematical Journal 1(1846): 104-122. Corry, L. “Nicolas Bourbaki and the Concept of Mathematical Structure,” Synthese, 92, no. 3(1992): 315-348. —. Modern Algebra and the Rise of Mathematical Structures. Basel and Boston: Birkhäuser, 1996. —. “David Hilbert: Geometry and Physics: 1900-1915.” In Gray J. J. (ed.), The Symbolic Universe: Geometry and Physics (1890-1930), 145-188. Oxford: University Press, 1999. —. David Hilbert and the Axiomatization of Physics (1898-1918). The Netherlands: Kluwer Academic Publishers, 2004. Corry, L., Renn, J. and Stachel, J. “Belated decision in the Hilbert-Einstein priority dispute.” Science 278(1997): 1270-1273.

260

Reference List

Cox, D. A. Primes of the form ‫ ݔ‬ଶ ൅ ݊‫ ݕ‬ଶ . Fermat, Class Field Theory, and Complex Multiplication (2nd ed.). New York: Wiley, 2013. Crilly, T. “Invariant theory”. In Grattan-Guinness, I. (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, vol. I, 787–793. London: Routledge, 1994. Curbera, G. P. Mathematicians of the world, unite! The International Congress of Mathematicians࣓a human endeavour. Wellesley, MA: A K Peters, 2009. Davis, Martin: “Hilbert’s Tenth Problem is Unsolvable.” The American Mathematical Monthly, vol. 80 (1973):233-269. Dick, A. Emmy Noether, 1882-1935. Boston: Birkhäuser, 1981. Dieudonné, J. “The work of Nicholas Bourbaki.” The American Mathematical Monthly, 77 (1970): 134-145. —. History of Functional Analysis. Amsterdam: North-Holland, 1981. —. “David Hilbert (1862-1943).” Rev. Integr. Temas Mat. 2, no. 1(1983): 27-33. Ehrlich, Ph. (ed.). Real Numbers, Generalizations of the Reals, and Theories of Continua. Dordrecht: Kluwer Academic Publishers (Synthese Lybrary, 242). General Introduction by the Editor: vii-xxxii, 1994. Ewald, W. B. (ed.). From Kant to Hilbert. A Source Book in the Foundations of Mathematics. 2 vols. Oxford: Clarendon Press, 1996. Fraenkel, A. A. Recollections of a Jewish Mathematician in Germany. Birkhäuser, 2016. Frege, G. Wissenschaftlicher Briefwechsel. Hamburg: F. Meiner, 1976. Frei, G. (ed.). Der Briefwechsel David Hilbert-Felix Klein (1886-1918). Göttingen: Vandenhoeck&Ruprecht,1985. Freudenthal, H. “The main Trends in the Foundations of Geometry in the 19th Century.” In Nagel, E., Suppes P., and Tarski, A. (eds.), Logic, Methodology and Philosophy of Science: Proceedings of the 1960 International Congress. Stanford: University Press, 613–621, 1962. Gauss, F. G. Disquisitiones Arithmeticae. Leipzig: Fleischer, 1801. PURL: http://resolver.sub.uni-goettingen.de/purl?PPN235993352. Gödel, K. Collected Works, Vol. I. Publications 1929–1936 (S. Feferman, et al., eds.). New York: Oxford University Press, 1986. Grattan-Guinness, I. “A sideways Look at Hilbert’s Twenty-three Problems of 1900.” Notices of the AMS, 47, no. 7(2000): 752-57. Gray, J. J. The Hilbert Challenge. Oxford: University Press, 2000. Hazewinkel, M. “David Hilbert, Paper on ‫ދ‬Mathematical Problems‫ތ‬ (1901)”. In Grattan-Guinness, I. (ed.), Landmark Writings in Western Mathematics 1640-1940. Amsterdam: Elsevier, 2005.

Hilbert, Göttingen and the Development of Modern Mathematics

261

Hecke, E. Vorlesungen über die Theorie der algebraischen Zahlen. Leipzig: Akademische Verlagsgesellschaft, 1923. Heijenoort, J. van (ed.). From Frege to Gödel. A source book in Mathematical Logic, 1879-1931. Cambridge, MA: Harvard University Press, 1967. Hesseling, D. E. Gnomes in the Fog. The Reception of Brouwer’s Intuitionism in the 1920s. Basel: Birkhäuser, 2003. Hilbert, D. “Über den Zahlbegriff.” Jahresbericht der Deutschen Mathematiker-Vereinigung, 8(1900): 180-183. —. Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. Leipzig: Teubner, 1912. —. “Logische Principien des mathematischen Denkens.” Vorlesungen. Summer-Semester 1905. Lecture notes by Ernst Hellinger. Unpublished manuscript. Bibliothek, Mathematisches Institut, Universität Göttingen (1905) —. “Elemente und Prinzipien Fragen der Mathematik.” Vorlesungen. Summer-Semester 1910. Lecture notes by Richard Courant. Unpublished manuscript. Bibliothek, Mathematisches Institut, Universität Göttingen (1910). —. “Prinzipien der Mathematik.” Vorlesungen. Winter-Semester 1917-18. Lecture notes by Paul Bernays. Bibliothek, Mathematisches Institut, Universität Göttingen (1917/18). —. “Grundlagen der Mathematik.” Vorlesungen. Winter-Semester 192122. Lecture notes by Paul Bernays. Bibliothek, Mathematisches Institut, Universität Göttingen (1921/22). —. Gesammelte Abhandlungen. Berlin: Springer. 3 vols. (vol 1: 1932, vol 2: 1933, vol. 3: 1935) (Reedition: 1965. New York: Chelsea). —. Hilbert’s Invariant Theory Papers. Brookline, Mass.: Math Sci Press, 1978 (translated by M. Ackerman; comments by R. Hermann). —. Theory of Algebraic Invariants. New York: Cambridge University Press, 1994. —. The Theory of Algebraic Number Fields. Berlin: Springer, 1998. —. Lectures on the Foundations of Geometry, 1891-1902. (M. Hallet and U. Majer, eds.). Berlin: Springer, 2004. —. Lectures on the Foundations of Physics, 1915-1927. (T. Sauer and U. Majer, eds.). Berlin: Springer, 2009. —. Lectures on the Foundations of Arithmetic and Logic, 1917-1933. (W. Ewald and W. Sieg, eds.). Berlin: Springer (2013). Hilbert, D. and Ackermann, W. Grundzüge der theoretischen Logik. Berlin: Springer, 1928. (Reedition: 1938).

262

Reference List

Hille, E. “Gelfond’s Solution of Hilbert’s Seventh Problem.” The American Mathematical Monthly, 49, no. 10 (1942), 654-661. Huntington, E. V. “A complete set of postulates for the theory of absolute continuous magnitude.” Transactions of the American Mathematical Society, 3(1902): 264-279. —. “Simplified definition of a group.” Bulletin of the American Mathematical Society, 8 (1901-02): 296-300. Ilyashenko, Y. “Centennial History of Hilbert’s 16th Problem,” Bulletin of the American Mathematical Society, vol. 39, no. 3(2002): 301-354. Kimberling, C. H. “Emmy Noether.” The American Mathematical Monthly, 79, no. 2 (1972):136-149. Klein, F. Vergleichende Betrachtungen über neuere geometrische Forschungen. Erlangen: Verlag von Andreas Deichert, 1872. —. Lectures on Mathematics. AMS Chelsea publishing, 1894. https://archive.org/stream/134256628#page/n5/mode/2up. —. Vorlesungen Über die Entwicklung der Mathematik im 19. Jahrhundert. Berlin: Verlag von Julius Springer, 1926. Knoebel, A., Lodder, J., Laubenbacher, R., and Pengelley, D. Patterns in Prime Numbers: The Quadratic Reciprocity Law. In Mathematical Masterpieces. Undergraduate Texts in Mathematics. New York: Springer, 2007. Kummer, E. Collected works, vol. 1 (ed. A. Weil) Berlin: Springer, 1975. Lemmermeyer, F. Reciprocity Laws: From Euler to Eisenstein. Berlin: Springer, 2000. Lewis, D. W. “David Hilbert and the Theory of Algebraic Invariants.” Irish Math. Soc. Bull., 33(1994): 42-54. Mancosu, P. From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s. New York: Oxford University Press, 1998. —. “Between Russell and Hilbert: Behmann on the Foundations of Mathematics.” The Bulletin of Symbolic Logic 5, no. 3(1999): 303-330. Mashaal, M. Bourbaki: A Secret Society of Mathematicians. Providence, RI: The American Mathematical Society, 2006. Moore, E. H. “On the projective axioms of Geometry”. Transactions of the American Mathematical Society, vol 3, no. 1 (1902):142-158. Parshall, K. H. “Toward a history of nineteenth-century invariant theory.” In Rowe, D. E., McCleary, J. (eds.), The History of Modern Mathematics, vol. I, 157–206. Boston: Academic Press, 1989. —. “How We Got Where We Are. An International Overview of Mathematics in National Contexts (1875-1900).” Notices of the AMS, 43, no. 3(1996): 287-296.

Hilbert, Göttingen and the Development of Modern Mathematics

263

Parshall, K. H. and Rowe, D. E. The emergence of the American Mathematical Research Community 1876-1900. J. J. Sylvester, Felix Klein and E. H. Moore. American Mathematical Society and London Mathematical Society. History of Mathematics, volume 8, 1994 (reprinted 1997). Pasch, M. Vorlesungen über Neuere Geometrie. Leipzig: B. G. Teubner, 1882 (second edition, with additions: 1912). Peckhaus, V. Hilbertsprogramm und Kritische Philosophie. Das Göttinger Modell interdisziplinärer Zusammenarbeit zwischen Mathematik und Philosophie. Göttingen: Vandenhoeck & Ruprecht, 1990. —. “Logic in Transition: The Logical Calculi of Hilbert (1905) and Zermelo (1908).” In Prawitz, D. and Westertähl, D. (eds.). Logic and Philosophy of Science in Uppsala. Synthese Lybrary 236. Dordrecht: Kluwer, 1994. Poncelet, J. V. “Considérations philosophiques et techniques sur le principe de continuité dans les lois géométriques.” In J. V. Poncelet, Applications d’analyse et de géométrie qui ont servi de principal fondement au Traité des propriétés projectives des figures, t. 2, 296364. Paris: Gauthier-Villars, 1864. Reid, C. Hilbert. New York: Springer, 1970 (Reedition: 1996). Reid, L. W. The Elements of the Theory of Algebraic Numbers. New York: MacMillan, 1910. (With an introduction by David Hilbert). Roquette, P. David Hilbert in Königsberg. Vortrag am 30.9.2002 an der Mathematischen Fakultat in Kaliningrad, 2002. Roselló, J. From Foundations to Philosophy of Mathematics. An Historical Account of their Development in XX Century and Beyond. Newcastle: Cambridge Scholars Publishing, 2012. Rowe, D. E. “‫ލ‬Jewish Mathematics‫ ތ‬at Gottingen in the Era of Felix Klein.” Isis, 77, no. 3(1986): 422-449. —. “Klein, Hilbert, and the Gottingen Mathematical Tradition.” Osiris, 2nd Series, vol. 5, Science in Germany: The Intersection of Institutional and Intellectual Issues (1989): 186-213. —. “From Königsberg to Göttingen. A sketch of Hilbert’s early career.” The Mathematical Intelligencer, 25, no. 2(2003): 44-50. —. “Making Mathematics in an Oral Culture: Göttingen in the Era of Klein and Hilbert.” Science in Context 17, 1/2(2004): 85–129. —. A Richer Picture of Mathematics. Springer, 2018. DOI: 10.1007/978-3319-67819-1. Rüdenberg, L. and Zassenhaus, H. (eds.). Hermann Minkowski Briefe an David Hilbert. Berlin: Springer, 1973.

264

Reference List

Scanlan, M. “Who were the American Postulate Theorists? The Journal of Symbolic Logic, 56, no. 3(1991): 981-1002. Schappacher, N. “David Hilbert, Report on Algebraic Number Fields (‘Zahlbericht’) (1897). In Grattan-Guinness (ed.). Landmark Writings in Western Mathematics 1640-1940. Amsterdam: Elsevier, 2005. Schulmann, R. et al., (eds.). The Collected Papers of Albert Einstein (CPAE), vol. 8, The Berlin Years: Correspondence, 1914-1918. Princeton: University Press, 1998. Seelig, C. Albert Einstein: A documentary biography. London: Staples Press,1956. Segal, S. L. “Mathematics and German politics: The national socialist experience.” Historia Mathematica, 13, no. 2(1986): 118-135. Sieg, W. Hilbert’s Programs and Beyond. Oxford: University Press, 2013. Siegmund Schultze, R. Mathematicians Fleeing from Nazi Germany: Individual Fates and Global Impact. Princeton, N.J./Oxford: Princeton University Press, 2009. Staudt, K. G. Ch. von. Geometrie der Lage. Nürnberg: Bauer und Raspe, 1847. Toepell, M. “The origin and further development of Hilbert Grundlagen der Geometrie.” Le Matematiche, LV, Supplemento no. 1(2000): 207226. Veblen, O. “A system of axioms for geometry.” Transactions of the American Mathematical Society (July 1904): 342-384. Waerden, B. L. van der. “The school of Hilbert and Emmy Noether.” Bull. London Math. Soc, 15 (1983): 1-7. —. “Nachruf auf Emmy Noether.” Mathematische Annalen 111(1935): 17. English translation in Dick, 1981,100-111. Weil, A. The Apprenticeship of a Mathematician. Basel: Birkhäuser (1992). Weyl, H. “Emmy Noether.” Scripta mathematica 3, no. 3(1935): 201-220. English translation in Dick, 1981, 112-152. —. The classical groups. Their invariants and representations. Princeton (NJ): Princeton University Press (1939). —. “David Hilbert and his mathematical work.” Bull. Amer. Math. Soc., 50, no. 9(1944): 612-654. Wiener, N. Ex-prodigy: My Childhood and Youth. New York: Simon&Schuster, 1953. Wolfson, P. R. “George Boole and the origins of invariant theory.” Historia Mathematica 35(2008): 37-46.

Hilbert, Göttingen and the Development of Modern Mathematics

265

Zermelo, E. “Untersuchungen über die Grundlagen der Mengenlehre, I.” Mathematische Annalen 65 (1908): 261-81. English translation in van Heijenoort, 1967, 200-215.

AUTHOR INDEX

A Abel, Niels Henrik 68 Abraham, Max 139 Ackermann, Wilhelm 96, 163, 16668, 200, 202, 240 Alexander, James 234 Alexandroff, Pavel 205, 208-9, 212, 217 Appell, Paul 12 Archimedes 73, 75-7, 81-2, 85-6 Arnol’d, Vladimir 250 Aronhold, Siegfried 17 Artin, Emil 50-2, 69, 109, 203, 208, 210-12, 217, 229, 246, 253 Atiyah, Michael 236 B Baldus, Richard 175, 177 Bär, Richard 156-57, Becker, Oskar 175, 177 Behmann, Heinrich 138, 160 Beltrami, Eugenio 75 Bernays, Paul 96, 157, 159-63, 166, 175-77, 188, 193, 208, 216-17, 221 Bernstein, Sergei 112, 118, 219, 255 Bieberbach, Ludwig 177-78, 223, 254 Birkhoff, David 91, 225 Blackett, Patrick 187 Blaschke, Wilhelm 242 Blumenthal, Otto 1, 36, 66, 71, 80, 118, 129, 142, 178-79, 205, 249 Böhm, Friedrich 152 Bohr, Niels 151, 178, 180, 183, 18687 Bolibruch, Andrei 257 Boltzmann, Ludwig 140

Bolyai, János 73, 75, 86, 140, 144, 146, 240 Bolza, Oskar 89-91, 118 Bolzano, Bernard 124 Boole, George 17 Borel, Armand 230, 234, 236 Borel, Émile 225 Born, Max 4, 129-30, 136-37, 139, 141, 151-52, 183-84, 186-90, 217 Brianchon, Charles Jules 29 Bricard, Raoul 241 Brouwer, Luitzen Egbertus Jan 17180, 194, 199, 201 Bruhat, Francois 230 Burnside, William 212 Busemann, Herbert 217, 242 C Cantor, Georg 2, 99-100, 107-08, 111, 123-25, 127, 172, 174, 239 Carathéodory, Constantin 129, 146, 152, 165, 178-79, 244 Carnap, Rudolf 96, 179, 200-01 Carnot, Lazare 27 Cartan, Henri 226-27 Cartier, Pierre 230, 237 Cauchy, Agustin-Louis 30, 124, 251 Cayley, Arthur 17, 19, 21, 55, 86, 88, 95, 210, 212 Chasles, Michel 29, 252 Chern, Shiing-Shen 236 Chevalley, Claude 52, 225-27, 236, 238 Chew, Geoffrey 234 Clebsch, Alfred 10, 17, 21, 53-6, 58-9, 61, 89 Cohen, Paul 239

268

Author Index

Compton, Karl Taylor 187 Connes, Alain 231 Courant, Richard 5, 118, 129, 141, 146, 178, 183, 185-89, 207, 217, 219, 223, 256 Crelle, August Leopold 61 Curie, Marie 149 Curie, Paul 149 D Davis, Martin 246-47 Debye, Peter 144, 151-52, 157, 165, 182-83, 187, 207 Dedekind, Richard 2, 9, 40-1, 47, 49, 51, 98-9, 124-25, 131, 21012 Dehn, Max 222, 241 Deligne, Pierre 232, 237 Delsarte, Jean 226-27 Delzell, Charles 253 Desargues, Girard 27, 33, 36, 242 Descartes, René 29 Deuring, Max 212, 249 Dickson, Leonard E. 91, 94, 210, 212 Dieudonné, Jean 120, 226-28, 236, 238 Diophantus 41 Dirac, Paul 187-191, 234, 244 Dirichlet, Peter Gustav Lejeune 40, 47, 49, 53, 97-8, 102, 117, 141, 147, 247 Dixmier, Jacques 230 Dresden, Arnold 177 Dyck, Walther von 178 Dyson, Freeman 234 E Écalle, Jean 253 Ehrenfest, Paul 183 Ehrlich, Paul 152 Eilenberg, Samuel 230-31 Einstein, Albert 139-42, 149, 15257, 178-79, 189, 233-35 Eisenstein, Ferdinand Gotthold 40, 44, 46-7

Enriques, Federigo 80 Euclid 30, 72-5, 80-1, 84, 98 Euler, Leonard 42-3, 142, 255 Ewald, Paul 151 F Faltings, Gerd 237 Feferman, Solomon 240 Fenchel, Werner 217 Fermat, Pierre de 29, 41-2, 46, 142, 144, 237, 246-47 Finsler, Paul 175, 177 Fischer, Emil 152 Fischer, Ernst 121, 206 Fourier, Joseph 53, 116, 117 Fraenkel, Abraham 4, 96, 129, 175, 177 Franck, James 183, 186-87, 217 Fréchet, Maurice 119, 121 Fredholm, Erik Ivar 113-16, 119 Frege, Gottlob 77, 100, 124-25, 163, 171 Freudenthal, Hans 222 Fricke, Robert 58, 129 Frobenius, Ferdinand Georg 27, 210, 212 Fuchs, Lazarus 9, 15, 65, 110, 117 Fueter, Rudolf 70, 118 Furtwängler, Philipp 52, 69, 200, 246 G Galois, Evarist 1, 13, 50 Gauss, Carl Friedrich 1, 16-7, 40-4, 47, 49-50, 53, 60-1, 72, 175, 247, 254 Gelfond, Alexander 244-45 Gell-mann, Murray 234 Gentzen, Gerhard 217, 240 Gergonne, Joseph 80 Giraud, Georges 255-56 Gleason, Andrew 243 Glivenko, Valerii 177 Gödel, Kurt 2, 96, 127, 169, 180, 200-03, 215-16, 234-35, 239-40, 247

Hilbert, Göttingen and the Development of Modern Mathematics Godement, Roger 230 Gomperz, Heinrich 200 Gordan, Paul 12, 15-17, 20-1, 56, 146, 206 Goursat, Édouard 226 Grelling, Kurt 118, 177 Grossmann, Marcel 140 Grothendieck, Alexandre 230-32, 236-38 Gudkov, Dimitrii 253 H Haag, Rudolf 244 Haar, Alfréd 118, 152 Hahn, Hans 200, 215 Hamel, Georg 118, 242, 244 Hamkins, Joel 240 Hardy, Godfrey Harold 143, 186, 217 Harnack, Axel 252 Hasse, Helmut 49-50, 52, 69, 212, 222-23, 246-49 Hausdorff, Felix 119, 239 Hecke, Erich 41, 49, 118, 129, 151, 179, 211, 222, 249 Heckmann, Gustav 184 Heesch, Heinrich 254 Heisenberg, Werner 4, 183, 186-89 Hellinger, Ernst 118, 120, 130, 137, 141 Helmholtz, Hermann von 15, 77 Hensel, Kurt 49, 96 Herbrand, Jacques 225 Herglotz, Gustav 129, 139, 188, 217, 223 Hermite, Charles 12, 20, 39, 109 Hertz, Heinrich 37, 79, 149 Hertz, Paul 151, 217, 221 Hesse, Otto 9 Heyting, Arendt 177, 179-80 Hirzebruch, Friedrich 236 Hölder, Otto 178 Holmgren, Erik 113 Houtermans, Fritz 187 Huntington, Edward 93-6

269

Hurwitz, Adolf 10-2, 27-8, 49, 58, 67, 103, 147, 159, 166 Husserl, Edmund 144 I Ilyashenko, Yulij 253 J Jacobi, Carl 8, 44, 47, 49 John, Fritz 217 Jordan, Camille 56 Jordan, Pascual 183, 187-90 K Kaluza, Theodor 223 Kármán, Theodor von 129, 151, 178 Kastler, Daniel 244 Kirchoff, Gustav 9 Klein, Felix 3, 10, 12, 15, 20, 27, 34, 40, 53-66, 72, 80, 86, 88-90, 95, 110, 117-18, 130, 140, 14445, 152-53, 176, 185, 189, 20507, 217, 223, 257 Koebe, Paul 129, 257 Kolmogorov, Andrey 177, 244, 250 König. Julius 239 Koszul, Jean-Louis 230 Kramers, Hans 183 Kronecker, Leopold 15-6, 21, 40-1, 47, 49-51, 53-4, 68-70, 89, 98, 109, 111, 174-75, 248-49 Kummer, Ernst 40, 44, 46-7, 50-1, 53, 67 L Lagrange, Joseph Louis 16, 43, 14243, 255 Landau, Edmund 65, 144-46, 152, 185-86, 207, 217-18, 222-23 Landé, Alfred 183 Lang, Serge 230 Lasker, Emanuel 211 Laue, Max von 139, 187 Lee, Tsung-Dao 234 Legendre, Adrien-Marie 43-5 Leibniz, Gottfried Wilhelm 125

270

Author Index

Lenard, Philipp 182 Leray, Jean 225 Lewy, Hans 217, 221, 255 Lie, Sophus 11, 55-6, 77, 89, 111, 207, 217, 230, 242 Lindemann, Ferdinand von 10-1, 39, 109 Littlewood, John Edensor 143 Lobachevski, Nikolai 73, 75-7, 86 Loewy, Alfred 96 Lorentz, Hendrik 139, 151 Low, Francis 234

Neumann, Franz 8 Neumann, John von 4, 120-21, 129, 177, 179, 188, 190-91, 215, 234, 240, 244 Nevanlinna, Rolf 223 Newton, Isaac 125 Noether, Emmy 4, 22, 51-2, 153, 182, 186, 203, 205-13, 217, 219-23, 229, 237 Noether, Max 205 Nordheim, Lothar 184, 187-90 Novikov, Petr Sergeevich 240

M Macaulay, Francis 211 Madelung, Erwin 151 Magnus, Wilhelm 217 Maier, Heinrich 146 Mandelbrojt, Szolem 227 Maschke, Heinrich 89-91 Matiyasevich, Yuri 246-47 Maxwell, James 149 Menger, Karl 200 Mie, Gustav 152-54, 244 Minkowski, Hermann 1, 4, 7, 10-2, 21, 24, 27, 36, 39-41, 65, 67, 103-04, 112, 117-18, 130, 13943, 149, 205, 217, 223, 247-48 Misch, Georg 144 Misses, Richard von 218, 222 Mittag-Leffler, Gösta 145 Monge, Gaspard 27 Montgomery, Deane 236, 243 Moore, Eliakim Hastings 88-94 Morette, Cécile 234 Morrey, Charles 256 Mumford, David 238, 251

O Ocagne, Maurice d’ 109-10 Oppenheimer, Robert 187, 234, 236

N Nagata, Masayoshi 250 Nambu, Yoichiro 234 Nelson, Leonard 144, 146, 160, 182, 184 Nernst, Walther 140, 151-52 Neugebauer, Otto 217, 221 Neumann, Carl 61, 98, 116

P Padoa, Alessandro 110 Pais, Abraham 234 Pasch, Moritz 30, 34, 36-7, 80, 95 Pauli, Wolfgang 183, 187, 234 Pauling, Linus 187 Peano, Giuseppe 95, 99, 110, 123, 132-33, 240 Picard, Émile 12 Pieri, Mario 80, 95 Planck, Max 4, 58, 123, 140-41, 149, 151-52 Playfair, John 75 Plemelj, Josip 256 Plücker, Julius 11, 54 Pogorelov, Aleksei 242 Pohl, Robert 183 Poincaré, Henri 12, 56, 58, 77, 103, 110, 116-17, 134-35, 138-39, 140, 144-46, 160, 171, 194, 199, 225, 257 Poisson, Siméon 53 Pólya, Georg 159-60 Poncelet, Jean-Victor 27, 29-30, 73, 251 Possel, Renné de 226-27 Prandtl, Ludwig 65, 130, 152, 184 Putnam, Hilary 246

Hilbert, Göttingen and the Development of Modern Mathematics R Reich, Max 183 Reid, Leigh Wilber 51 Reinhardt, Karl 254 Rellich, Franz 217 Reye, Theodor 35 Richelot, Friedrich 9 Riecke, Eduard 130 Riemann, Bernhard 1, 50, 53, 60, 73, 75, 77, 97, 109, 111, 125, 154, 178, 237, 245, 257 Riesz, Frigyes 121 Ritz, Walter 140 Robb, Alfred 244 Robinson, Julia 246-47 Röntgen, Wilhelm 149, 152 Rosenhain, Johann G. 8 Runge, Carl 58, 65, 118, 130, 14546, 152, 183, 188, 217, 223 Russell, Bertrand 123-25, 127, 131, 137-38, 160, 163-65, 171, 201 S Saalschütz, Louis 8 Samuel, Pierre 230 Schafarevich, Igor 246 Schauder, Juliusz 256 Scherk, Peter 217 Schlick, Moritz 200-01 Schmidt, Arnold 217 Schmidt, Erhard 116, 118-19, 121, 188, 206, 212 Schneider, Theodor 244-45 Schottky, Friedrich 257 Schreier, Otto 211-12, 253 Schrödinger, Erwin 188-90 Schubert, Hermann 111, 251-52 Schur, Friedrich 91-2 Schwartz, Laurent 230-31 Schwarz, Hermann 15, 89, 257 Schwarzschild, Karl 65, 130, 205 Selberg, Atle 236 Serre, Jean-Pierre 230-32, 236 Shimura, Goro 70, 237, 249-50 Siegel, Carl Ludwig 186, 223, 245 Simon, Hermann Theodor 130

271

Simpson, Stephen 203 Skolem, Thoralf 129, 177 Smoluchowski, Marian von 151 Sommer, Julius 51 Sommerfeld, Arnold 129, 139, 151, 178, 183-84 Speiser, Andreas 118, 212 Stark, Johannes 182 Staudt, Karl von 30, 35 Steinitz, Ernst 209-12 Study, Eduard 12, 20, 251 Sydler, Jean Pierre 241 Sylvester, James Joseph 12, 17, 21, 88 T Takagi, Teiji 52, 69-70, 118, 246, 249 Taniyama, Yutaka 237, 249 Tarski, Alfred 95-6 Tate, John 230 Taussky-Todd, Olga 217 Taylor, Richard 237 Teichmüller, Oswald 221, 223 Thomson, Joseph John 149 Toeplitz, Otto 129, 141 Tonelli, Leonida 256 Tornier, Erhard 222-23 V Vahlen, Theodor 223 Veblen, Oswald 77, 91, 93-6, 100, 221, 233-34 Veronese, Giuseppe 73, 80, 86 Vitushkin, Anatoli 250 Voigt, Woldemar 130 W Waerden, Baertel Leender van der 21-2, 64, 203, 205, 209, 211-12, 227-29 Waring, Edward 1, 142-43, 147 Wassermann, August von 152 Wavre, Rolin 177

272

Author Index

Weber, Heinrich 9-10, 40, 49-50, 60, 63, 68-70, 72, 89, 93-4, 21012, 248-49 Weber, Werner 221, 223 Wedderburn, Joseph 210, 212 Weierstrass, Karl 15-6, 53, 55, 89, 97-8, 125, 131, 176, 257 Weil, André 50, 52, 225-27, 232, 236-37, 249 Weyl, Hermann 4, 24, 49, 52, 11718, 120, 129, 146, 173-77, 185, 194, 205-08, 217-18, 221-24, 237 Whitehead, Alfred North 127, 13738, 160, 163-65 Whitney, Hassler 236 Wiechert, Emil 65, 72, 130, 139, 152 Wien, Wilhelm 152, 182 Wiener, Hermann 36, 72 Wiener, Norbert 65 Wightman, Arthur 244 Wiles, Andrew 237 Witt, Ernst 208, 212, 217 Wolff, Julius 175, 177

Y Yang, Chen Ning 234 Yoccoz, Jean-Cristophe 231 Yukawa, Hideki 234 Z Zariski, Oscar 238, 251 Zermelo, Ernst 296, 107, 123-24, 126-27, 129, 131, 137, 146, 151, 171, 239 Zippin, Leo 243 Zolotarev, Yegor 47