Experiment and metaphysics : towards a resolution of the cosmological antinomies 9781900755290, 1900755297

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GRAD BD 113 . W763 2001

Experiment and Metaphysics Towards a Resolution o f the Cosmological Antinomies

Edgar Wind

LEGENDA European Humanities Research Centre I in i\/i»rcif\r

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E x p e r im e n t a n d M e t a p h y sic s T o w a r d s a R e so l u t io n of t h e C o s m o l o g ic a l A n t in o m ie s

THE EUROPEAN HUMANITIES RESEARCH CENTRE U N I V E R S I T Y OF O X FO R D T h e European Humanities Research Centre o f the University o f O xford organizes a range o f academ ic activities, including conferences and workshops, and publishes scholarly works under its own imprint, LEGENDA. W ithin O xford, the E H R C bridges, at the research level, the main humanities faculties: M odern Languages, English, M odern History, Literae Humaniores, M usic and Theology. T h e Centre stimulates interdisciplinary research collaboration throughout these subject areas and provides an O xford base fo r advanced researchers in the humanities. T h e C en tre’s publications programme focuses on m aking available the results o f advanced research in medieval and m odem languages and related interdisciplinary areas. A n Editorial Board, w hose members are drawn from across the British university system, covers the principal European languages. Tides include works on French, Germ an, Italian, Portuguese, Russian and Spanish literature. In addition, the E H R C co-publishes with the Society for French Studies, the British Comparative Literature Association and the M odern Humanities Research Association. T h e Centre also publishes Oxford German Studies and Film Studies, and has launched a Special Lecture Series under the LEGENDA imprint. Enquiries about the Centre’s publishing activities should be addressed to: Professor M alcolm B o w ie, Director Further information: Kareni Bannister, Senior Publications O fficer European Humanities Research Centre University o f O xford 47 Wellington Square, O xford o x i 2JF enquiries@ ehrc.ox.ac.uk w ww .ehrc.ox.ac.uk

LEGENDA EDITORIAL BOARD Chairman Professor M alcolm B ow ie, A ll Souls C ollege Professor Ian M aclean, A ll Souls C ollege (French) Professor M arian H obson Jeanneret, Q ueen M ary University o f London (French) Professor R itch ie R obertson, St Jo h n s C ollege (German) Professor Lesley Sharpe, University o f Bristol (German) D r D iego Zancani, Balliol C ollege (Italian) Professor D avid R obey, University o f R ead in g (Italian) D r Stephen Parkinson, Linacre C ollege (Portuguese) Professor H elder M acedo, K in g ’s C ollege London (Portuguese) Professor Gerald Smith, N e w C ollege (Russian) Professor D avid Shepherd, University o f Sheffield (Russian) D r D avid Pattison, M agdalen C ollege (Spanish) D r Alison Sinclair, Clare College, Cam bridge (Spanish) D r Elinor Shaffer, School o f Advanced Study, London (Comparative Literature) Volume Editor Professor N igel Palmer, St Edm und Hall Senior Publications Officer Kareni Bannister Publications Officer D r Graham Nelson

LEGENDA E u r o p e a n H u m a n itie s R e s e a r c h C e n t r e

University o f Oxford

Publication o f this volum e marks the opening in Sum mer 2001 o f the W ind R o o m o f the new Sackler Library in the University o f O xford. T h e room is named in honour o f Edgar W ind, the U niversity’s first Professor o f the H istory o f A rt (195 5-67). This handsome room, reminiscent o f a Renaissance studiolo, will house books from Professor W ind’s personal library and a selection he bought for the library o f the Departm ent o f the H istory o f A rt during his tenure. T h e arrangement o f his collection was influenced by the classification o f the Warburg Institute, where W ind had spent many years, and follows it in allow ing prim ary texts in original editions to share shelf space w ith m odern editions and critical apparatus. T h e collection thus constitutes a living bibliography and an outstanding resource in the fields o f iconology and iconography.

I share Lothario’s opinion: that the energy o f all the arts and sciences meets at one central point, and hope by the gods that I can provide nourishment for your enthusiasm even from the field o f mathematics [...]. A nother reason w hy I have given physics the preference is because here the connection is at its most visible. In physics you cannot conduct an experiment w ithout a hypothesis; every hypothesis, even the most limited, i f it is thought through systematically, leads to hypotheses concerning the whole, and is indeed based upon these, even if this is not realized by the person w ho employs it. F r i e d r i c h S c h l e g e l , Dialogue on Poetry

Edgar Wind

Experiment and Metaphysics Towards a Resolution of the Cosmological Antinomies ❖

E d g a r W in d 4

T ran slated b y C y r il E dw ards In tro d u ced b y M a tth ew R am pley

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LEGENDA European Humanities R esearch Centre University o f O xford

2001

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Published by the European Humanities Research Centre o f the University o f Oxford 47 Wellington Square Oxford OXl 2JF L E G E N D A is the publications imprint o f the European Humanities Research Centre IS B N 1 900155 29 7 First published 2001

A ll rights reserved. N o part o f this publication may be reproduced or disseminated or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, or stored in any retrieval system, or otherwise used in any manner whatsoever unthout the express permission o f the copyright owner British Library Cataloguing in Publication Data A C IP catalogue recordfo r this book is available from the British Library © European Humanities Research Centre o f the University o f Oxford 2001 L E G E N D A series designed by C ox Design Partnership, Witney, Oxon Printed in Great Britain by Information Press Eynsham Oxford 0X8 1JJ C h ief Copy-Editor: Genevieve Hawkins Editorial Consultant: Michael Wood

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CO N TEN TS ❖

Editorial Note Introduction by M atthew R am pley Author's Preface

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PART ONE: THE TH EO R Y OF THE EX PER IM EN T § i . The Circle in Physical Inquiry § 2. T he Elements o f Measurement and the M eaning o f the Claim to Accuracy § 3. Einsteins C oncept o f ‘Practical G eom etry’ § 4. Poincaré s Principle o f ‘Arbitrary Convention’ § 5. T h e Task o f the Experim ent § 6. Transformation and Em bodim ent § 7. T he ‘J udgem ents o f Appropriateness’ § 8. R e a l and Neutral Hypotheses § 9. T h e Cyclical Progression and its M ethodological Foundations § 10 . Metaphysics and Em pirical Experience § 1 1 . Transcendental Philosophy and Experim ental M ethod

7 io 13 16 18 22 29 31 33 38 46

P A R T T W O : T H E ‘E X P E R I M E N T A L R E D U C T I O N ’ O F T H E C O S M O L O G IC A L A N T IN O M IE S § 12 . T h e Em pirical C riteria o f Metaphysics

53

Chapter 1 : The Antinomy o f the Concept o f the World § 13 . Clarification o f the Q uestion o f the First Antinom y (Refutation o f R ussell’s Objection) § 14 . T he Mathematical Antinom y o f Euclidian Space § 15 . T he Physical Antinom y o f the N ew tonian System § 16 . T he Inevitability o f the N ew tonian Antinom y according to the D octrines o f Kant § 17 . Kant’s Interpretation o f Absolute Space

61 67 69 72 74

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§ 18 . §19 . §20. § 2 1.

M ethodological Conclusion T he Mathematical R esolution o f the Euclidian Antinom y T h e Physical R esolution o f the N ew tonian Antinom y T h e Principle o f Internal Delim itation

o ntents

77 78 81 85

Chapter 2: The Antinomy o f the Concept o f the Atom §22. T h e ‘Inner Lim it’ in the Process o f Division §23. Two Forms o f ‘U ncertainty’

87 88

Chapter 3 : Causality and Freedom §24. Heterogeneity and Necessity in the C onnection between Cause and Effect §25. §26. §27. §28. §29. §30. § 3 1.

T h e Temporal R elation o f Dynam ical C onnection T h e ‘Free Play’ o f the Present Linear and Configurai Progression o f Tim e Internal Delim itation and Indétermination ‘Constancy’ and ‘Em ergence’ T h e Beginning and the End o f Tim e Freedom under the C ondition o f Natural Law

93 97 103 104 107 111 116 120

Chapter 4: The Modality o f Events §32. T h e Measurability o f Chance as the O bject o f Interpretation

126

A ppendix: Letter o f 6 March 1933 from Edgar W ind to Oskar Siebeck

133

Bibliography

135

Index

144

ED ITO RIAL NOTE

The translation is based on the original edition o f Edgar Winds Das Experiment und die Metaphysik: Zur Auflösung der kosmologischen Antinomien, completed in typescript in 1929 as a Habilitationsschrift for the University o f Hamburg, accepted by the faculty on 29 November 1930, and published in 1934 by J. C. B. Mohr (Paul Siebeck) in Tübingen, with Winds preface dated 15 September 1933, as volume 3 o f the series ‘Beiträge zur Philosophie und ihrer Geschichte’. At the time o f writing a second edition o f the German text has just appeared, published by Suhrkamp in Stuttgart, edited by Bernhard Buschendorf and with an introduction by Brigitte Falkenburg. We are indebted to Dr Buschendorf for showing us the proofs o f the new edition, thus permitting us to take account o f the introduction and his editorial decisions while finalizing the text o f the English version. The type­ script o f the original, which contained some passages excised from the published version, is lost. The passage on p. 81 (with note 63) relating to Bolyai and Gauss is based on a handwritten addition in the author s copy made available to us by Margaret Wind; in the new German edition it is included as a footnote. Winds notes, which are sparse, have been augmented using information from two sources, additional information provided by the translator and the numerous bibliographical additions in the notes o f the Suhrkamp edition. The references cited in the footnotes have been reformulated according to the model ‘Einstein, Relativity (1920)’ , full details being relegated to the bibliography. Wherever possible, details o f English translations o f the works cited by Wind have been supplied. Additional bibliographical details contained in the Suhrkamp edition have been taken over selectively, in so far as they might be o f use to an English reader. Where material in the footnotes goes beyond what was cited in the original edition o f 1934 the new information is given in square brackets. In an appendix, a letter written by Wind to his publisher on 6 March 1933 is given in English translation. It provides important

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information about the sections which were omitted from the published edition as well as a statement about his objectives in writing the book, stressing that his analysis o f scientific method is intended to have implications for the humanities. For a more explicit development o f some o f these ideas see Winds article in the Festschrift for his teacher, Ernst Cassirer, ‘Some Points o f Contact’ (1936). The translation by Cyril Edwards has been edited to take account o f the ‘authorized’ English formulation o f some sections, published in advance o f the completion o f the German text, in a lecture given at Harvard on 14 September 1926 (Wind, ‘Experiment and Meta­ physics’ , 1927). For a further statement about the argument o f the book in English see Wind, ‘Can the Antinomies be Restated?’ (1934), in which he takes issue with a review o f his work by E. Nagel (1934). For the appendix, the footnotes, the bibliography, the index and the revision o f §§1, 2, 5 and 9, the responsibility is mine. The translator would particularly like to record his thanks to Helmut Grupe, Fred Marcus and Theo Livingstone for helping on individual points. I also wish to thank John Michael Krois, Graham Nelson, Matthew Rampley and Bill Williams for advising on various aspects o f the translation, and Kareni Bannister and the E H R C for taking on the planning and execution o f such a complex publishing project. My warmest thanks go to Margaret Wind, who gave her support and encouragement to this project from the outset. St Edmund Hall, Oxford

Nigel Palmer

IN TRO D U CTIO N

Matthew Rampley

Edgar Wind is best known as a scholar o f Renaissance culture and art, who fled Germany in 1933 and, having spent over twenty years as, first, Deputy Director o f the Warburg Institute in London, then Professor at various universities in the United States, was appointed in 1956 as the first Professor o f the History o f Art at the University o f Oxford, where he remained until his death in 19 71. Two o f his books, Pagan Mysteries in the Renaissance and Art and Anarchy, remain widely read, and the more recent publication o f collections o f his other articles has ensured that his name retains a prominent place in the history o f art historical scholarship.1 Thus it is an unfamiliar Edgar Wind we encounter in the pages o f Experiment and Metaphysics. His concern with the problems o f Kant’s cosmological antinomies, the metaphysical status o f the experiments o f physics and the impact o f non-Euclidean geometry or Einsteinian notions o f space-time are distant from those o f the Renaissance. Those who may have assumed that Winds interest in philosophy was restricted to philosophical aesthetics will be struck by his level o f engagement not only with fundamental epistemological questions in Kant but also with technical issues in contemporary physics and mathematics. As Wind himself admitted, citing David Hume, Experiment and Metaphysics fell ‘stillborn from the press’ . The reasons for this were varied. On the one hand it was pardy connected with the fact that Wind was already a refugee when the work appeared, and had thereby become shut off from the German philosophical community. In addition, as he states in the Preface, his book was at odds with what he perceived to be the tenor o f contemporary German 1 Wind, Pagan Mysteries in the Renaissance (1958); Art and Anarchy (1963); The Eloquence of Symbols (1983); Hume and the Heroic Portrait (1986). The Eloquence of Symbols also contains a detailed biographical memoir o f Wind by Hugh Lloyd-Jones.

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philosophy which, dominated by figures such as Martin Heidegger, Karl Jaspers, or Edmund Husserl, was moving away from traditional epistemological questions and towards concerns such as ‘being’, ‘existence’ , ‘understanding’ and ‘life’ .2 In fact, the work was favourably reviewed by Ernest Nagel, and it may be that the principal reason for its subsequent neglect was the fact that it did not fit easily into any recognizable philosophical school.3 Drawing on NeoKandan thinking, but distancing itself from some o f its fundamental tenets, and attempting to apply motifs from American pragmatism— specifically, the notion o f embodiment— to the distant concerns o f Kant s antinomies o f reason, Wind’s text was difficult to situate. Essentially, Experiment and Metaphysics undertakes two tasks. The first is to take issue with Kant’s notion o f the transcendental dialectic, according to which reason is restricted to judgements regarding the logic o f phenomenal appearances— as constructed by the laws o f the understanding. The second is a critical engagement with Kant’s presentation o f the antinomies o f reason— for Kant the most dramatic examples o f the limitations o f the transcendental dialectic— through employing models o f space and time that had developed in physics since Kant’s time. The lynchpin in these two enterprises is Wind’s theory o f the experiment, which argues for the reciprocity o f the empirical and the metaphysical, a reciprocity that is strikingly demonstrated by the procedure o f scientific experimentation. I shall explore each in turn. One o f the most important innovations o f Kant’s critical philosophy was the fundamental distinction between the world o f phenomenal appearances and that o f metaphysical ‘noumenaT truth. The latter would remain perpetually beyond cognition, and all questions regarding the nature o f ‘reality’ were actually motivated by debates around the nature o f the phenomenal existence as accessible to human knowledge and intuition. And because Kant believed he had determined the laws governing the mode o f appearing o f pheno­ mena in general— their conformity to the laws o f understanding and the spatio-temporal forms o f intuition— it was possible to outline in principle the character o f knowledge and its objects. Attempts to address the ultimate ‘reality’ o f the world were classed as dogmatic 2 Herbert Schnadelbach offers a clear overview o f the main philosophical currents o f this time in Philosophy in Germany (1984). 3 Nagel reviewed it in The Journal of Philosophy 31 (1934), 164-5.

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metaphysics, a misrecognition o f the limitations o f epistemology, which, strictly speaking, was confined to working out the logic o f appearances. Thus, the debate between Leibniz and the followers o f Newton Samuel Clarke, for example, over whether space simply consisted o f the set o f spatial relations between objects or whether it had a substantive reality o f its own, was resolved by Kant in the Critique of Pure Reason with the notion that while space had no substantial metaphysical reality, as a form o f intuition, it constituted a frame governing the system o f spatial relations, an la priori presentation that necessarily underlies outer appearances’ .4 Such a position was obviously o f revolutionary importance for an understanding o f the nature o f scientific inquiry, for this was now no longer concerned with probing the nature o f reality, but rather with the nature o f the real as constituted by the forms o f intuition and the categories o f understanding. In the introduction to the Critique of Pure Reason Kant states this quite clearly when he notes that ‘Thus even physics owes that very advantageous revolution in its way o f thinking to this idea: the idea that we must, in accordance with what reason itself puts into nature, seek in nature (not attribute to it fictitiously) whatever reason must learn from nature and would know nothing o f on its own’.5 Metaphysics is, in contrast, dismissed as an entirely speculative science. It is this idea that forms the starting point o f Wind’s discussion o f experimentation. Scientists are, for Wind, ‘enclosed in a finite world which can only be explored from within’ (p. 5), a position which the recent renewal o f Kantian thought by figures such as Heinrich Rickert, Hermann Cohen or Ernst Cassirer had only served to reaffirm. But against this Neo-Kantian orthodoxy, Wind argues that this circle o f immanence can be broken, indeed frequently is, through the procedure o f scientific experimentation. Drawing on ideas o f embodiment he encountered in the work o f the American pragmatist Sidney Hook, Wind argues that the process o f physical measurement, testing scientific theories against the empirical results o f experiments, implicidy provides answers to the larger metaphysical questions that Kant thought were properly beyond the bounds o f reason.6 His discussion focuses on the relation between the system o f mathematical * Kritik der reirteti Vernunfi, B 39/A 24. 5 Kritik der reinen Vernunfi, B xiii-xiv. 6 Winds debt to Pragmatism is explored in Krois, ‘Kunst und Wissenschaft’ (1998).

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and geometrical axioms and principles governing scientific inquiry, their translation into actual physical hypotheses, and the empirical properties o f the physical instruments used to test that system. While on the one hand experimentation is supposed to test the predictions o f scientific theory, for the instruments to function they are also expected to behave in accordance with the theoretical principles they are supposed to be testing. It is assumed, for example, that physical instruments o f measurement will display a spatio-temporal constancy commensurate with the rigidity o f mathematical units o f measurement. Wind writes o f physical instruments as being the empirical ‘symbols’ or embodiment o f mathematical concepts; the mathematical concept o f identity is symbolized by the coincidence o f two empirical physical bodies, and as Wind notes, this depends on the assumption that ‘if two measuring bodies are found to be “ coincident” , they may be regarded as “ equal” ’ (p. 14). This emphasis on the symbolic function o f physical instruments and measurement might be read as merely affirming the impossibility o f going beyond the logic o f phenomenal illusion, but Wind argues that it is through the process o f translating the system o f theoretical axioms and principles into propositions about the physical world that its validity is affirmed or refuted: ‘It is by means o f the experiment, therefore, that the ultimate decision is made as to whether a geometrical system is physically applicable or not’ (p. 16). At the heart o f Wind’s discussion is the distinction between mathematics and physics, or between mathe­ matical and applied geometry, one that it is now widely recognized Kant failed to acknowledge. As Carnap has argued, through this distinction it is possible to go beyond the limitations o f Kant’s transcendental dialectic, for while mathematical geometry, as a theory o f geometrical structures, has the a priori character Kant saw as the basis o f all knowledge, applied geometry is dependent on the results o f experimentation and thus although synthetic (in Kant’s use o f the term) cannot be a priori. Wind makes a similar argument drawing on Einstein’s concept o f ‘practical’ geometry, and he employs this notion in rebutting Henri Poincaré’s famous thesis regarding the role o f convention in geometry. Kant assumed that the a priori structure o f phenomenal space was Euclidean, for the simple reason that this was the only geometrical system known at the time he wrote the Critique of Pure Reason. However, the subsequent formulation o f non-Euclidean geometries

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by first Gauss, and then Riemann or Lobachevsky, posed the funda­ mental question o f which constituted a more accurate description o f space. Poincare’s solution to this dilemma was to claim that it is possible to construct multiple geometrical systems capable o f being translated into a body o f physical hypotheses that will accord with a set o f empirical observations. For Poincare the choice o f the particular geometry thus cannot rest on the experimental observations, but rather will be guided by convention. Again the key for Wind is the translation o f a geometrical system into the physical hypotheses accounting for a given set o f observational facts. Specifically, two different geometrical systems will lead to two different sets of axioms about the structure o f physical space, and these in turn will lead to two sets o f non-congruent physical hypotheses accounting for a given set o f empirical observations. Now although it may be possible to account for specific given data by using either set o f physical axioms (and, by extension, either geometrical system) this is no guarantee that each set o f axioms will be equally applicable to all subsequent empirical data. Indeed, Wind argues, following each set o f physical axioms it will be possible to posit an entire series o f further physical hypotheses, and it will be the function o f the experiment to test these other hypotheses and their empirical findings. Eventually the empirical findings that follow from two different sets o f physical axioms will diverge, at which point it will be possible to choose between them, and to judge which geometrical system is physically correct. Poincare’s disregard o f the importance o f the consequences when a mathematical geometrical system is embodied in a set o f physical axioms and their attendant hypotheses about the behaviour o f the physical world renders his conventionalism problematic for Wind. Poincare assumes that it is possible to construct a series o f logically coherent physical hypotheses matching a given set o f empirical findings with the same arbitrariness and ease with which one can construct a logically consistent theoretical geometrical system, and it is precisely this that Wind contests. He makes a similar criticism o f Carnap’s analysis o f space. For Carnap, given a specific empirical topological array (set o f empirical observations) it will be possible either to choose arbitrarily a system o f measurement such that only one possible geometrical system will present itself, or, alternatively, to choose arbitrarily a geometrical system such that the appropriate system o f measurement will unambiguously follow.

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While Wind admits that within the thought experiment o f the mathematician such reciprocity is logically possible, on translation into physical space this is no longer the case. One simply cannot arbitrarily devise a measuring instrument (i.e. the physical embodiment o f a particular measuring system) that will behave in accordance with the consequences o f applying a specific geometry to a given set o f topological data. One example from Carnap discussed by Wind is the consequence o f determining the earths curvature to be zero, following which the surface o f the earth would necessarily be conceived as infinite in extent. Carnap argues that this would be possible as long as a specific system o f measurement was adopted in accordance with this hypothesis. Choosing an arbitrary place as an Archimedean point from which to begin measurement o f the earth, the measuring instrument would, logically, have to have the property o f diminishing the further it moved away from its original Archimedean position, until, on reaching the earth’s circumference, it became infinitely small. As Wind notes, this thought experiment immediately invites one to wonder both whether such an instrument actually exists and whether, even if it could be constructed— which is highly doubtful— it would actually be o f any use as a measuring instrument, for it would lead to the conclusion that the earth was o f the same magnitude as the sun. Implicit in his criticisms is again Wind’s conviction o f the reciprocity o f the physical and the mathematical/geometrical, o f the empirical and the metaphysical. The findings o f the experiment determine the applicability o f the geometrical system to the physical world. Thus, the fact that in Carnap’s example the possibility o f even finding an instrument o f experimental measurement is low suggests that the adoption o f an arbitrary geometrical system in which the surface o f the earth is declared to be flat is deeply problematic. More generally, therefore, Wind is deeply sceptical o f the idea that one can, through arbitrary fiat, construct a logically consistent theoretical geometry and then, through manipulation o f the system o f measurement, ensure that it accounts for observable data. The experiment constitutes what Wind terms an experimentum crucis in which the physical validity of, for example, Euclidian or one o f any number o f non-Euclidean geometries can be tested. What had, in Kant, been regarded as the a priori conditions o f all further scientific inquiry are now susceptible to experimental testing and confirmation.

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Having expounded his general theory o f the significance o f the experiment Wind devotes the remainder o f his text to the consequences o f his notion o f the experiment for Kant s antinomies o f pure reason. Kant regarded the antinomies as a potent demonstration o f the limitations o f cognition inasmuch as any attempt to resolve fundamental metaphysical questions about the reality o f space and time, or o f freedom and necessity, would always end in aporia. Specifically, Kant demonstrated that it is possible to argue, for example, both that space and time are infinite and that they are finite, or that matter is both infinitely divisible and indivisible. Such contradictions were clear evidence for Kant that ‘when we apply our reason not merely— for the sake o f using the principles o f understanding— to objects o f experience, but venture to extend our reason beyond the bounds o f experience, then there arise subtly reasoning doctrines. These doctrines neither may hope to be confirmed in experience, nor need they fear being refuted in it.’7 In fact, Kant does anything but demonstrate that space and time are both infinite and finite, for he actually shows that one can deny both the infinity and finitude o f space and time, and denying one thing is not the same as asserting its opposite. However, Wind avoids this fine o f criticism, and instead explores the ways in which the results o f contemporary experimental physics— in particular, the development o f relativistic notions o f space-time— dissolve what Kant had deemed to be irresolvable contradictions. It had already been recognized before Wind that much o f the force o f Kant s argument rested on his assumption o f Newtonian physics, with its reliance on Euclidean geometry. The apparent contradictions addressed in the first antinomy, whereby if space is regarded as a totality it cannot be thought o f as infinite, and if conceived as infinite, cannot be regarded as a totality, are intimately related to the assumption o f the Euclidean structure o f space. But as Wind points out, if one uses a non-Euclidean geometrical model, such as Riemann s construction o f space as a curved surface, it is possible to maintain that space is both infinite and a totality. This, o f course, is only a theoretical solution to the antinomy, and the newly posited Riemannian space would be just as much beyond the reach o f phenomenal experience as the Euclidean model it was supposed to supplant. Following the lead o f Einstein, however, Wind is concerned with finding the experimental procedure 7 Kritik der reinen Vernunfi, B 448—9/A 421.

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that would test the Riemannian model o f space for its physical validity. The possibility o f such an undertaking, which Wind’s theory o f the experiment affirms, is indicated by the difficulties encountered in Newton, whose reliance on Euclid contradicts his own theory o f gravitation. What had hitherto been an entirely theoretical difficulty has now become a problem concerning physical hypotheses about the world, and Einstein (and Wind) were o f the opinion that experimental observations would be able to resolve the question as to whether the structure o f space was Euclidean and infinite, or non-Euclidean and finite. Specifically, the Newtonian assumption o f an infinite space leads to further assumptions about the distribution o f mass in the universe and, consequently, the gravitational forces operating in space. One cannot measure the average distribution o f mass in the universe for the purposes o f verification, but it is in principle possible, argues Einstein, to take a fragment such as the Milky Way and, calculating the velocity necessary to prevent the gravitational forces causing the Milky Way to implode, measure their velocity to test the Newtonian picture. The remaining three antinomies in Kant are also motivated by his dependence on Newton and classical mechanics, and Wind proceeds to argue that in each case what Kant regards as an insoluble philo­ sophical problem is in fact resolved by the experimental results o f contemporary physics. Thus the second antinomy, in which Kant argues that the notion o f infinite divisibility is both necessary and absurd, and the concept o f the atom a self-contradiction, is resolved by adopting the model developed in quantum theory o f the wave-particle duality, where the metaphysical question o f the nature o f quantum entities is supplanted by inquiry into how they behave in certain circumstances. Likewise with the third antinomy, concerned with the respective roles o f freedom and causality, in which, according to Wind, the apparent opposition o f the two is resolved by the abandonment o f linear notions o f time (which implicidy underpin Kant’s discussion) and the adoption o f a relativistic model o f space-time which, coupled with the quantum theory o f uncertainty, renders classical causality redundant. Wind’s argument that scientific experiments, though apparently restricted to merely empirical observations, in fact hold the potential for resolving metaphysical problems that Kant and the neo-Kantian philosophical tradition had deemed beyond cognition, is certainly a bold one. The important question is whether his position is a con­ vincing one, and to answer this it is necessary to separate out the various claims contained within his text. Specifically, two distinct

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issues are being addressed in Experiment and Metaphysics: the first concerns the role o f experimentation in highlighting the interdependence o f metaphysical and empirical scientific inquiry, while the second concerns the ability o f experiment to resolve disputes between competing cosmological theories, in particular, and physical theories in general. I shall deal with each in turn. Wind’s claims about the metaphysical significance o f empirical experiment rely on a conflation o f the cosmological and the metaphysical. Since the time o f publication o f Experiment and Metaphysics the positing o f a non-Euclidean spatial structure for the universe has become a physical commonplace, and hence what Kant had regarded as an insoluble paradox has been shown to be a question capable o f resolution. At the same time, Wind’s assertion that the resolution o f the first antinomy casts into doubt the entire epistemological basis o f Kant’s philosophy is rather less convincing. For Kant the purpose o f the first antinomy was to demonstrate that space and time are forms o f intuition and not realia. N ow although the apparent paradoxes o f this antinomy are presented with a solution by subsequent developments in geometry, mathematics and physics, they do not substantially damage Kant’s claim. Even if the structure o f space can now be determined for sure, from a Kantian perspective this still only concerns the structure o f phenomenal space and the phenomenal universe, not the noumenal thing-in-itself which, strictly speaking, may have no spatio-temporal character at all. Though the universe in its entirety cannot possibly be the object o f experience, the cosmological research Wind refers to remains limited to the phenomenal realm, scientific inquiry being guided by spatio-temporal forms o f intuition and the categories o f the understanding. As an answer to Kant, therefore, Wind’s argument runs into difficulties, though it demonstrates admirably the shortcomings flowing out o f the older philosopher’s reliance on Euclidean geometry and Newtonian physics. If, therefore, we lay to one side Wind’s claims as to the metaphysical significance o f experiment, we are still left with the substantial question o f the kinds o f inference that can be made from a given set o f empirical data, and the relation between such data and the formation o f general physical hypotheses. Traditionally, the question has been posed in the following way: given a set o f data, and two alternative (incommensurate) theories that are equally capable o f accommodating the data, how does one decide which theory to

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employ? The terrain o f philosophical debate has tended to be divided into two positions. The first would argue that physical theories being reducible to their observational consequences, the apparent dispute between two theories with the same observational data is in fact a dispute between two different ways o f formulating the same theory. The incommensurability o f two theories is thus merely apparent. The second, antireductionist position maintains that it is indeed possible to have two incommensurate theories with the same observational data, and that the choice is made according to one o f any number o f extrinsic criteria, such as convention, considerations o f elegance, simplicity and so forth. W inds solution does not belong to either o f these two broad positions. For while he maintains the antireductionist position that there may well be two incommensurable physical theories with identical observational data, he argues that this will not always be so, and that when the consequences of each physical theory are followed through, they will eventually predict sets o f data which are not identical. And at this point it will be possible, through experimentation, to establish which theory is commensurate with the set o f data. Implicit in Wind’s approach is the charge that this seemingly intractable problem in the epistemology o f science has been preoccupied with a pseudo-problem. The question to be answered, therefore, is whether Wind’s elegant solution resolves the issue once and for all, and whether experimentation has the redemptive powers attributed to it in his argument. A crucial element in Wind’s argument is his equation o f experimentation with quantitative measurement. The choice between two physical (and, for Wind, metaphysical) theories will be guided by the quantitative results o f experiments, which will test the predictions that flow from each theory. Moreover, while initially experimental observation may not be able to provide data that point conclusively to one theory over another, Wind’s argument is that eventually there will be sufficient divergence between the predicted data for it to be possible to make a choice. Wind also emphasizes the importance o f mathematical exactitude, arguing that although the predicted divergences between two sets o f physical hypotheses may be so small as to lie below the threshold o f observability, sufficiently sensitive measuring instruments will come to be constructed in order to determine which theory is ‘correct’ . The immediate issue, therefore, is whether quantitative data alone can ever conclusively prove one

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theory over another. This is highly doubtful, for an axiom o f modern physics, Heisenbergs uncertainty principle o f 1927, demonstrated that at the quantum level there will always be a certain degree o f inexacti­ tude. Moreover, mathematical exactitude, while central in the determination o f theory choice, is not the only criterion. A theory which is more exact with regard to one set o f observations may be less exact in another, or not have as wide an explanatory power as a competing theory. As Thomas Kuhn has argued, although the substitution o f the notion o f oxygen for that o f phlogiston accounted for one set o f observations, the older concept accounted for others that remained unresolved by oxygen theory.8 This is not to endorse an anti-reductionist view (such as that o f Poincare) that choice o f theory is only ever determined by extrinsic criteria; there will always be cases where quantitative data do decide conclusively which theory to use. At the same time, however, they can never be guaranteed to do so in all cases. In this regard it is also necessary to consider the role o f exactitude. Specifically, this kind o f exact congruence o f prediction and experimental result rarely occurs. Instead, experimental method and analysis proceed on the basis o f the notion o f ‘reasonable agreement,’ which can vary from field to field. There is a necessary tolerance o f inexactitude. This comment could be thought o f as belonging more in the field o f the sociology o f science than in that o f the philosophy o f experiment, except that inexactitude is a necessary condition o f scientific experiment— indeed, in the case o f quantum physics, is an intrinsic element in the theoretical hypothesis. At some finite stage, therefore, a limit always has to be set such that the level o f congruence between measurement and predicted result is deemed sufficient to confirm the theoretical hypothesis, for measurement is only ever an approximate process. Let us grant that there is a specific case in which the results o f experimental measurement unequivocally show that one theory is preferable to a competing alternative. Even then the question is not as clearly resolved as Wind assumes. Here we need only cite Duhem’s critique o f what Wind refers to as the experimentum cruris in physics.9 Duhem contends that the very idea o f the experimentum cruris relies on a false analogy between mathematics and science. For whereas in mathematics a single counter-example may falsify a mathematical Kuhn, ‘Objectivity, Value Judgement and Theory Choice’ (1977). 9 Duhem, The Aim and Stmcture of Physical Theory (1954).

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theory, the physical experiment does not function as a counter­ example in the mathematical sense. The process o f testing a specific theory through experimental measurement never involves a simple, direct relation between a physical hypothesis and a set o f observations, but rather is embedded in a complex network o f auxiliary theories that frame both the hypothesis to be tested and the conduct and analysis o f the experiment itself. Wind himself partially recognized this in his emphasis on the symbolic function o f measurement, but he did not follow it through to its consequences. Instead, Duhem claims, a physical hypothesis can be preserved, even in the face o f experimental measurements that seem to contradict it, by modifying the auxiliary theories, or by adducing new ones to account for the apparent anomaly. Wind himself refers to one such example, namely Seeliger’s theory o f the cosmic ‘absorption’ o f gravity, which he dismisses as an ad hoc attempt to resolve difficulties besetting Newton’s. Now although he may be correct regarding this particular example, this does not discredit Duhem’s general position, in which the relation between a specific hypothesis or theory and a set o f observational data is mediated by the multiple auxiliary theories framing the process o f experimental verification. For various reasons, therefore, Wind’s argument encounters a number o f difficulties. Yet although his general theory o f experimentation is problematic, his book offers an invaluable analysis o f a central part o f Kant’s Critique of Pure Reason. In particular, it highlights the extent to which Kant’s dependence on a Newtonian view o f the physical universe and its structure coloured his approach to fundamental philosophical questions. Wind does not manage to demonstrate that physical experimentation can provide the answers to putatively undecidable metaphysical questions, but he does show that the antinomies o f pure reason fail to achieve what Kant thought they did, namely, demonstrate the limitations o f the dialectic of reason and the limits o f cognition. For if the apparent paradoxes o f spatial and temporal finitude, or o f the infinite divisibility o f matter, are shown no longer to be paradoxes, the way is opened up to a reconsideration o f Kant’s argument that space and time are forms o f intuition and not realia. O f course, Wind does not prove that space and time are real, but he does demonstrate the weakness o f Kant’s argument that they cannot possibly be so. Ironically, despite his claims on behalf o f experimental measurement, Wind’s argument largely relies on theoretical physical models, which provide possible solutions to the

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Kantian antinomies on theoretical grounds alone. If his attempt to demonstrate the metaphysical significance o f empirical observation remains unconvincing, he does at least show the profound inter­ dependence o f science and philosophy. After the publication o f Experiment and Metaphysics Wind virtually gave up any further involvement with questions in the philosophy o f science, or indeed in philosophy generally. Apart from an essay on the relation between history and the sciences in a Festschrift for Ernst Cassirer which followed in 1936, and a polemic against Sartre in the Smith College Associated News, Wind immersed himself entirely in the study o f art history.10 Inevitably, we are faced with the challenge o f explaining this sudden break in his trajectory, and we can begin to explain it by setting his work within the broader intellectual milieu o f the time o f its publication. In the Preface he noted that ‘Although the subject o f this study is only indirecdy connected to the concerns o f the Kulturwissenschaftliche Bibliothek Warburg, it did evolve out o f the spirit o f this community’ (p. 6). Wind’s acquaintance with the circle o f scholars around Aby M. Warburg and his private research institute dates back to 1920 when, as a student at the University o f Hamburg, he came into contact with Erwin Panofsky, perhaps the most famous o f Warburg’s students, and Ernst Cassirer, often regarded as providing the philosophical foundations o f the Warburg Institute. This was built on in the summer o f 1927 when Wind met Warburg and became a research assistant at the library. The basis o f Aby Warburg’s researches was the question o f the social life o f symbols, their cognitive function and their role as bearers o f cultural memory. Cassirers own intellectual project had developed independently, but when he came into contact with Warburg it was clear that his philosophy o f symbolic forms, a historicizing appro­ priation o f Kantian epistemology, bore close affinities with Warburg’s own thinking.11 While the research output o f the Kulturwissenschaft­ liche Bibliothek Warburg and its successor, the Warburg Institute in London, focused predominandy on the role o f visual symbols in postclassical European culture, other forms o f symbolic representation were also studied. Panofsky’s famous study o f Perspective as Symbolic 10 The complete bibliography o f Wind’s work is included in Wind, The Eloquence of Symbols (rev. edn. 1993), 115-30 . 11 On Cassirer’s relation to the Warburg library see Habermas, ‘Befreiende Kraft’ (1997)-

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Form studied not only the history o f systems o f spatial representation from Antiquity to the Renaissance, but also the mathematical and geometrical concepts necessary for the development o f single-point perspective.12 Likewise, though more usually known for his studies o f myth, language or the history o f philosophy, Cassirer engaged both with historical issues in the philosophy o f science and with contemporary physics— publishing a study o f Einsteins theory o f relativity.13 Evidence o f this broader interest in the contemporary sciences within the Warburg library is also evident from its collecting policy o f the 1920s and early 1930s, which included both philosophical analyses o f contemporary sciences and works by key physicists such as Einstein, Max Bom and Niels Bohr. It is thus possible to construct an intellectual milieu centred on the scholarly activity o f the Warburg circle in which W ind’s interest in contemporary physics and the philosophy o f science could be linked in a variety o f ways with his subsequent art historical concerns. One should also take note o f the fact that the subject o f his doctoral dissertation o f 1922 was the relation between philosophy and history o f art.14 Wind therefore moved easily between the spheres o f philosophical aesthetics, philosophy o f science and the history o f art. More generally, too, there was a striking parallelism between scientific and art historical scholarship in the first few decades o f the twentieth century, when both were motivated by the philosophical issues raised by the application o f their respective methods. Within the history o f art, Wind was continuing a concern already followed through by Aby Warburg, and this itself was the continuation o f a debate amongst nineteenth-century scholars such as Willhelm Dilthey (18 3 3 -19 11) and Leopold von Ranke (1795-1886) over the philosophical status o f historical knowledge. The Warburg Institute, with its intertwining o f art history, aesthetics and anthropology, was thus a microcosm o f a wider current in German intellectual life, a pattern that repeated itself in the field o f the natural sciences. Some o f the leading figures in the development o f modern physics and mathematics were equally pre­ occupied with the philosophical problems o f their own discipline— 12 Panofsky, Perspective as Symbolic Form (1992). 13 Cassirer, Zur Einstein’schen Relativitätstheorie (1921). 14 An overview o f this was published under the tide Ästhetischer und kunst­ wissenschaftlicher Gegenstand. Ein Beitrag zur Methodologie der Kunstgeschichte (1924).

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indeed profound philosophical questions about the status o f scientific cognition were an intrinsic part o f both relativity and quantum theory.15 It is partly owing to this common conception o f the intertwining o f philosophical and practical knowledge that Wind himself saw parallels between scientific experimentation and art historical inquiry. In contrast to the predominandy neo-Kandan oudook o f Cassirer or the young Panofsky, Wind tended to the view that the symbolic forms o f the scientific (and metaphysical) representation o f the world could be tested against ‘reality’ . As noted above, this was guided by his nodon o f embodiment borrowed from Hook and, as Bernhard Buschendorf has indicated, also owed a considerable debt to Charles Peirce.16 Furthermore, Wind believed the idea o f testing a specific scientific explanation or representation to be equally applicable within the sphere o f art criticism. As he noted in Art and Anarchy: ‘There is one— and only one— test for the artistic relevance o f an interpretation: it must heighten our perception o f the object and thereby increase our aesthetic delight.’ 17 Indeed, his continuing interest in the lessons to be drawn from non-Euclidean geometry is evident in the concluding ‘ Observation on Method’ in Pagan Mysteries in the Renaissance, in which, defending his attention to exceptional rather than to commonplace historical phenomena, he notes that ‘In geometry, i f I may use a remote comparison, it is possible to arrive at Euclidean parallels by reducing the curvature o f a non-Euclidean space to zero, but it is impossible to arrive at a non-Euclidean space by starting out with Euclidean parallels. In the same way, it seems to be a lesson o f history that the commonplace may be understood as a reduction o f the exceptional, but that the exceptional cannot be understood by amplifying the commonplace.’ 18 B y this time, however, Wind was already writing for a different kind o f audience. As in the case o f his compatriot and contemporary Panofsky, the expressly philosophical concerns o f his earlier works had given way to a more orthodox historical form o f scholarship. Indeed, an entire generation o f émigré scholars o f necessity adapted to 15 See, e.g., Schrödinger, Über Indeterminismus in der Physik (1932); Heisenberg, Philosophie Problems 0/ Nuclear Science (1952); id., Physics and Philosophy (1959); Weyl, Philosophy of Mathematics and Natural Science (1949). 16 Buschendorf, ‘Wir ein sehr tüchtiges gegenseitiges Fördern’ (1985). 17 Wind, Art and Ananhy (3rd edn. 1985), 62. 18 Wind, Art and Ananhy (3rd edn. 1985), 238.

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the pragmatic culture o f the anglophone territories that offered them refuge, inducing, in the case o f art history, a philosophical slumber that lasted until the late 1970s. Philosophical questions pertaining to the interpretation o f art were left to the aestheticians, while the historians concentrated on the gathering o f historical knowledge, unconcerned with its status. The development o f the Warburg Institute in London out o f the Kulturwissenschaftliche Bibliothek Warburg in Hamburg is an exemplary case o f this cultural transformation, and it is interesting to note that Wind terminated his association with the Institute partly out o f dismay at the waning o f interest in fundamental philosophical questions o f methodology. Again, a parallel with the sciences can be detected. For as the centre o f scientific research shifted to the United States from the 1930s onwards, a similar dichotomization took place in art history, in which issues regarding the epistemology o f scientific inquiry and knowledge became the concerns o f the philosophers, while the scientists became increasingly absorbed in the perfection o f technical mathematical experimental methods, with litde regard for the wider metaphysical questions that occupied Wind’s contemporaries. As an essay in the philosophy o f science Experiment and Metaphysics may not provide the solutions Wind had hoped it would. Neverthe­ less it constitutes a significant addition to the body o f Kant criticism. More importandy, it is a remarkable document o f an intellectual moment when what Aby Warburg referred to as the disciplinary ‘border guards’ were briefly thwarted. It is a moment from which w e may still have much to learn.

A U TH O R’S PREFACE

The present work was completed in 1929 and submitted as a habitation thesis to the Faculty o f Philosophy at the University o f Hamburg. At that time external circumstances prevented it from being published. I am now publishing it without any major revisions. The developments in physics that have taken place in the inter­ vening time ought perhaps to have made it incumbent upon me to furnish a wider range o f examples and evidence. In the first chapter o f the second part I still take as a base Einstein’s understanding o f the universe as in a state o f equilibrium, whereas today we have at our disposal the whole range o f non-static models proposed by de Sitter, Friedmann and Lemaitre. There is also much richer material now available as evidence for the central idea o f §30 (‘The Beginning and the End o f Tim e’), which at the time seemed very daring to me, in that I derived it purely and simply from the logic o f the system.1 I have, however, resisted the temptation to conceal retrospectively how simple were the means I used, nor have I applied any superficial gloss to create the appearance o f a modernity which the basic structure o f the work does not possess. Whereas the zeal o f the practical inquirer is increased by the fact that every new discovery only spurs him on to replace it with an even newer one, philosophy should not permit itself to be infected by such inappropriate haste. When measured against the range o f cosmological questions raised by the problem o f the antinomies, the state o f research in 1929 is scarcely inferior to that in 1933» particularly as the problem has not previously been thought through from this philosophical perspective. N or have I paid any attention to the fact that the current tenor o f 1 Cf. the summary by Eddington, Expanding Universe (1933), especially pp. 45-60. [The developments to which Wind refers, which were pioneered by Einstein from 1905 onwards and came to fruition in the general theory o f relativity, are now standard in modern physics. For an up-to-date account see Wald, General Relativity (1984).]

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philosophy— for example, Jaspers and Heidegger— demands a mode o f presentation quite different from that which is offered here. The very untimeliness o f this study seemed to me its best justification. In the conviction that the dignity o f philosophy, which has for so long been lost, can only be restored if there is an undaunted struggle against the spirit o f the present time, which prefers declamatory broodings to elucidatory analysis, I have sought to posit unambiguous hypotheses and to support them with stringent proofs. The art o f conscientious argu­ mentation has not merely fallen into disrepute; worse still, it is in a state o f decline. I f the errors which I have no doubt incurred were to contribute towards reanimating the will to contradict, then even these would be o f greater value than all those philosophemes which are so deep or lofty that they are too nebulous even to be wrong. [Is it not here— in philosophy itself—that the guilt o f those most prominent representatives o f that ‘Idealism’ which is today con­ demned as ‘liberal’ lies? Possessing and enjoying a philosophy which they thought to be adequately anchored in a glorious tradition, at the critical moment they proved neither willing nor able to fulfil their logical obligations. Challenged to batde, they had the choice of weapons, but they preferred to behold from on high the development in which they ought to have intervened, to woo the enemy through tokens o f their favour, and, despite all their wisdom, to pin their hopes on the illusion that they might be able to make their peace even with this enemy. Thus they were conquered without entering battle, and even the fruits o f their defeat were lost; as no intellectual debate whatever took place, the victor absorbed absolutely nothing o f that critical mentality which was the best that the vanquished might have bequeathed to him. From the point o f view o f this development, the present work is reactionary in the most exact sense o f the word. It seeks to return to that point at which battle ought to have commenced, ought indeed to have done so if only for the sake o f clarity, long before the change in mentality set in. Whoever today finds the idealistic concept o f freedom too vacuous and shallow in its impotent universality is bury­ ing his head in the sand if he withdraws into gloomy Innerlichkeit, into an inner world in which he merely shrouds the problem that he purports to solve in the vapours emanating from that state. He would do better to turn to the cosmological foundations upon which that concept o f freedom was once erected, foundations which cannot be removed or changed without that concept itself becoming logically

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indefensible. Is it not the case that these foundations have long since collapsed? And is it not high time to pursue the logical reasons for this, that is, the philosophical errors o f construction— i f indeed there is any intention o f erecting a new edifice?]2 It is one o f the basic assumptions o f Idealism that there are, irrefutably, pressing questions whose solution lies in the infinite. Whether the world has always existed or whether it had a beginning, whether freedom exists in it or only causal determination, whether it is in spatial terms finite or infinite— ‘how might such matters be decided empirically?’ Precisely because experience can never give an answer to such questions, the individual is free to draw his own conclusions: he is ‘autonomous’. It is clear that the dialectic o f freedom is here based upon that o f the infinite. If just one of those questions to which experience can yield no answer were ultimately to be susceptible o f proof, then that would amount to a restriction o f the freedom o f the individual. He would no longer be able to decide o f his own volition whether he wished to regard himself as free or not, whether there is necessity in the world or merely contingency. His belief in autonomy in the field o f cosmology would be shaken. The basis for such tremors arose as early as 19 17, when Einstein deduced from his theory that the question o f whether the universe is spatially finite or infinite can be decided by physical means. Strangely enough, this assertion, direcdy contradicting as it does ‘transcendental dialectic’ , gave the Kantian philosophers no cause at all for concern. Convinced that the virtue o f critical philosophy lies in only ever 2 [These two paragraphs are deleted in Wind’s copy, indicating that they might be omitted from a future edition. It is not certain just how widely the group o f ‘Idealist’ (Neo-Kantian) philosophers attacked here is to be defined. Part o f the challenge is philosophical, namely that they failed to muster an appropriate response to the challenge o f Existentialist philosophy (in particular that ofjaspers and Heidegger), but it is also political. The criticism is certainly directed at Heinrich Rickert in Heidelberg, whose work Wind had reviewed in his article ‘Contemporary German Philosophy’ (1925), but it may also extend to others such as Bruno Bauch and the Hamburg professors who declined to stand by Bruno Snell in his protest against the National Socialist regime. After the ‘Reichsgesetz zur Wiederherstellung des Berufsbeamtentums’ o f 7 April 1933 Jews were no longer allowed to hold university posts. Wind’s own academic teacher, Ernst Cassirer, emigrated in March 1933, initially to England, where he held the post o f Chichele Lecturer at All Souls College, Oxford. For a discussion o f the background, and the question o f Wind’s relationship to Cassirer, see Bredekamp, ‘Falsehe Skischwiinge’ (1998), 207-10. I am grateful to J. M. Krois (Berlin) for discussion o f this passage.]

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reflecting on the axioms on which a science is founded, but never on its findings, they turned their attention exclusively to the question o f how Einstein’s unification o f space and time, achieved through the measuring process, might be compatible with the understanding o f space and time in ‘transcendental aesthetics’ . And after they had convinced themselves to their own satisfaction that ‘transcendental aesthetics’ could be salvaged through a clear distinction between ‘pure’ and ‘empirical’ intuition, they held, contrary to appearances, that transcendental dialectic was also implicitly secure. I have gone the opposite way. Plunged into uncertainty by the conflict between Einstein’s cosmology and transcendental dialectic, I asked how it might be possible for a physicist to adduce experimental criteria for the solution o f a problem which Kant, and even the fiercest opponents o f Kantianism, had held to be insoluble on an empirical basis. Is this extension o f the experimental process to areas in which it could not previously be applied an unjustifiable expansion?3 Or does the procedure which proves effective here show that we have cognitive means which have proved themselves in practice, but have not yet been sufficiently explored by critical epistemology? In exploring this second question I arrived at an observation which seems so obvious when once put into words that it scarcely requires proof: in Kant’s structuring of experience there is absolutely no logical place for the experiment. At precisely that point at which the verification o f an idea by experiments suggests itself to the inquirer, Kant refers to the ‘reality o f sensation’, which can never test out the formation o f categories, but only be subject to it. This view had to be challenged by a theory o f experimentation which renders intelligible the retrospective effect o f experience on the formation o f categories— a theory which explains that it is not by any means a paradox i f the conditions which make experience ‘possible in the first place’ can 3 Thus A. E. Taylor asserts that Einsteins cosmological deductions are artificially grafted on to his other theories. [See Taylors collective review o f Einstein, Relativity (1920), Eddington, Spaa, Time and Gravitation (1920), and Whitehead, Concept of Nature (1920), in Mind 30 (1921), here p. 78.] Whitehead argues along similar lines (An Enquiry concerning the Principles of Natural Knowledge, 1919; Concept of Nature, 1920; Principle of Relativity, 1922), although he traces the methodological error back to the ‘signal-theory’ o f the special theory o f relativity and desires to replace it by an alternative theory. For a criticism o f this view cf. Wind, ‘Mathemadk und Sinnesempfindung’ (1932), passim.

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themselves be reconfirmed by experience or refuted by it. This theory is based upon the principle that the instruments which serve to investigate the world are themselves integral parts o f the world that is the subject o f investigation, and are therefore affected in turn by the cognition which they themselves have transmitted. ‘Metaphysics’ (in the sense o f the doctrine o f the unknown whole) and ‘empirical science’ (the knowledge o f those parts accessible to us) are inextri­ cably linked in this cyclical pattern o f thought. For every experiment contains and tests a hypothesis concerning the whole, yet every decision concerning the whole leads, provided it is an actual decision and does not merely play with ideas, to experimental conclusions. The thesis o f this book is that, in spite o f transcendental dialectics, the questions posed in the ‘cosmological antinomies’ relate in this sense to ‘real’ decisions. Those philosophers who regard a yearning towards the infinite as their great heritage and see in dialectic the best method for satisfying this need will perforce oppose the result o f this investigation. Enclosed in a finite world which can only be explored ‘from within’, through the use o f those means supplied by that world itself, they will be on the lookout for a gap in the fence which will open up for them a view into the infinite. They will ask whether this whole theory, which invites us to avert our eyes from the infinite, is not after all conditioned by the result o f a historically limited inquiry which may prove out-of-date tomorrow. Should we allow ourselves to be intimidated by a finiteness proclaimed by the inquiries o f today, given the fact that these inquiries themselves only form one stage in an infinite process o f acquiring knowledge? To this I can only reply: I simply do not know whether cosmo­ logical inquiry will ever come to the point o f reinstating the prerogative o f infinity. I do, however, consider it methodologically unsound to construct a theory on the basis o f this ignorance. An appeal to an uncertain future does not invalidate the present, and anyone who acts as if he can arbitrarily free himself from the conditions o f his historical situation is taking an unjustifiable liberty. Even in philosophy it is the case that a given situation can only be overcome by a conscious decision to think it through to the end. Where an appeal is made to the infinity o f the process o f acquiring knowledge, and this position is taken as axiomatic, I fear that even this last refuge in the espousal o f an a priori position is now ruled out. M y question is: how can one know that the process o f acquiring

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knowledge is infinite? A reason for believing this would only exist if the world that is the subject o f investigation were itself infinite— infinite by comparison with the instruments which are available to us for investigating it. On that count we can know nothing a priori. Empirical inquiry, precisely when it reaches the point o f cosmic or atomic dimensions, has attached such importance to the hypothesis of infinity that philosophers would be well advised to adjust themselves to this as a possibility; it would seem to me preferable to adopt an epistemology which is not based on the presumption that the opposite view is certain. In the context o f cosmological questions, the maxim o f Charles Peirce, more than fifty years old now, has retained its validity: ‘Though in no possible state o f knowledge can any number be great enough to express the relation between the amount o f what rests unknown to the amount o f the known, yet it is unphilosophical to suppose that, with regard to any given question (which has any clear meaning), investigation would not bring forth a solution o f it, if it were carried far enough.’4 Although the subject o f this study is only indirecdy connected to the concerns o f the Kulturwissenschaftliche Bibliothek Warburg, it did evolve out o f the spirit o f this community, and I am indebted to the constant encouragement o f Fritz Saxl for the opportunity to publish it. I would also like to express my gratitude to the University o f Hamburg, and in particular to Erich M. Warburg for the subsidy with which they ensured its publication. Hamburg, 15 September 1933

4 [Peirce, ‘How to Make our Ideas Clear’ (1878), 301; repr. in The Essential Peirce (1992), I39- 40.J

PART O NE ❖

The Theory of the Experiment § 1. The Circle in Physical Inquiry5 All knowledge o f physical laws is based upon the results o f physical measurement. Physical measurements are, however, as physical processes, themselves subject to those laws which they are intended to elucidate. Therefore the physicist can only test and substantiate the accuracy o f an experiment by presupposing the knowledge o f those laws which the experiment intends to test. Thus classical mechanics, for example, employs scales and clocks for the determination o f its laws, concerning which it assumes that their metrical properties do not vary in motion. Yet this assumption itself relates to a mechanical law whose validity has been shown by modem physics to be highly dubious. Similarly, for the measurement o f heat certain liquid or gaseous bodies are employed which are assumed to expand in a uni­ form way within certain limits as heat increases. Yet this assumption itself is a proposition concerning the theory o f heat. In all these instances the physicist, if the validity o f his procedure is challenged, can only argue in circular fashion. He can only ‘guarantee’ the accuracy o f an experiment by basing it on the knowledge o f ‘universal’ laws. Yet he can only claim these laws to be ‘universal’ on the basis o f experiments whose accuracy is ‘guaranteed’ . The problem o f the circle has long been observed by philosophers. Comte in his Cours de philosophie positive derived from it the conclusion that scientific knowledge could only mature as a late fruit o f the human spirit, because the latter would never have made any 5 [The question o f the drculus vitiosus was first addressed, in much the same words as in the opening paragraph o f §i, in a lecture delivered by Wind at the Sixth International Congress o f Philosophy at Harvard in September 1926: ‘Experiment and Metaphysics’ (1927), 217. For an account o f the lecture, and the planned book, see the entry for the Department o f Philosophy in The University of North Carolina Record: Research in Progress, no. 247 (1927), 64.]

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progress if it had been caught up in an inescapable circle at the very beginning o f its development: ‘Ainsi, pressé entre la nécessité d’observer pour se former des théories réelles, et la nécessité non moins impérieuse de se créer des théories quelconques pour se livrer à des observations suivies, l’esprit humain, à sa naissance, se trouverait enfermé dans un cercle vicieux dont il n’aurait jamais eu aucun moyen de sortir’.6 Comte only adduces this observation, however, in order to prove that the positive stage o f science, which consciously restricts itself to the range o f this circle, was in fact preceded by a theological and metaphysical stage which in its naïve self-confidence ignored the circularity— indeed that, logically, it had to be anticipated by such a stage. The circle which is peculiar to all ‘positive’ cognition characterizes in his view a final stage (‘état définitif), into which all progressive development leads, but from which no further develop­ ment can arise. The progress o f science, after it has entered its positive stage, can therefore for Comte only consist in extending its scope, partly by discovering new matter which is not yet the province o f any branch o f scientific inquiry, partly by the gradual absorption o f the subject matter o f the ‘less advanced’ sciences. A methodological pro­ gression within the positive stage (for example, in mathematical physics, which was one o f the first branches o f science to reach this stage) would, however, be a contradiction in terms, as this stage o f evolution is only termed positive because in methodological terms it signifies an end-stage. Thus the original paradox o f scientific procedure is dissolved, as if by a spell— a spell that destroys its significance along with its problem­ atical nature. For now the aim and the process o f cognition are separated from each other by their very essence. The circle ceases to be false— for the simple reason that it no longer serves the process o f arriving at conclusions, but solely that o f the presentation o f conclusions which have already been reached. The actual process o f arriving at conclusions, however, is led by theological or metaphysical speculation, which cannot be accused o f circularity, for the very reason that it has as yet had no recourse whatever to purely logical procedure. 6 ‘Thus between the necessity o f observation for the formation o f genuine theories, and the no less pressing necessity o f constructing theories for the pursuit o f observation, the human mind must have found itself trapped in a vicious circle, from which it could never have escaped.’ Comte, Cours de philosophie positive, i (5th edn. 1907)» 5 [English transi. Clarke, The Essential Comte (1974.), 22].

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I f this solution is to remain valid, then it must never be possible to identify one position in the circle in which it is at once significant in logical terms and yet also effective as a method. In all those areas where it proves effective (in the speculations o f theology and meta­ physics), it cannot yet have begun to be logically significant. In all those areas where it forms the basis o f the method employed (in the system o f positive science), it must already have ceased to be part o f the procedure employed. The process o f scientific inquiry is thereby reduced to nothing. It is replaced by two illusions: on the one hand, the ghost o f ‘perfected’ science, the phantom o f a logical cloudcuckoo-land in which, without having to tread the laborious path o f elucidation, the laws are derived directly from the facts, and the facts from the laws; on the other hand, there is the image o f the human spirit which, without knowing its goal, wanders with all the confidence o f the somnambulist through the sequence o f stages which leads to that very cloud-cuckoo-land. It is only in such Utopian dreams, which contemplate the spirit either ‘at its birth’ (Tesprit humain, à sa naissance’) or in its ‘final state’ (‘état définitif’), that it is possible to believe in an escape from the circularity o f the scientific process, because the actual execution o f this process has been artificially brought to a standstill. Yet this only proves that once one has committed oneself to the process one inevitably falls victim to the circle. A particularly acute, not to say acrimonious, awareness o f this fact has evolved in contemporary physics. It was his awareness o f the circular basis o f physical observations that led Eddington to declare that all comprehensive physical laws express nothing but tautologies, since the knowledge expressed in them is rather like the insight o f an accountant who suddenly discovers that his books reveal a remarkable conformity to that law which states that for every item on the credit side an equal item appears somewhere else on the debit side.7 — Even if this pointed formulation will scarcely bear closer investigation (see §9 below), Eddington nevertheless grasps the basic idea, which leads him in a different context to a formulation which is so straightforward and forceful that it ceases to appear paradoxical and seems almost self-evident: T h e w orld o f physics is a w orld contemplated from within, surveyed by appliances w hich are part o f it and subject to its laws. W hat the world might

7 Eddington, Physical World (1928), 237-8.

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be deemed like i f probed in some supernatural manner by appliances not furnished by itself w e do not profess to know.8

The restriction for which Eddington argues, to a view o f the world based on instruments which have their place within this world and are therefore subject to the laws o f this world, is readily recognizable as the real root o f our whole problem. For it follows from this limitation that in order to make precise use o f these instruments we must know the cosmic laws to which they are subject. On the other hand, the reason for using them is precisely to discover these laws in the first place. §2. The Elements o f Measurement and the Meaning o f the Claim to Accuracy9 If it has been shown that the experiment’s claim to accuracy and the claim o f the law to be universal are based upon one another and mutually dependent, then in what follows it is our task to establish the meaning o f this peculiar and entirely circular mutual relationship. First, however, we must determine more precisely what exacdy the claim to accuracy involves. What elements can be distinguished in a measurement to which the problem o f accuracy might apply? And to which o f these components does the claim to accuracy, as we have just considered it, actually relate? In the attempt to answer these questions I shall restrict myself to the case o f mechanical measurement. As things are at their simplest in this area, it may be assumed that the complications that arise are minimal, that they can be varied and increased if other methods o f measure­ ment are applied, but can scarcely be reduced.— Three elements may be distinguished in each mechanical measurement: 1 . The presupposition o f a system o f axioms and theorems, defining the terms of measurement and exhibiting the ideal rules which govern their composition (e.g., the Euclidean system o f geometry). 2. The choice o f individual physical objects which represent the terms o f measurement available to empirical observation and thus serve as measuring instruments (e.g., the choice o f measuring-rods made o f a particular material, which are taken to be ‘rigid’ and to represent ‘straight lines’ in the sense o f Euclid). 8 Eddington, Physical World (1928), 225. 9 [The following pages overlap with the Harvard lecture: Wind, ‘Experiment and Metaphysics’ (1927), 2 17 -18 .J

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3. The application o f these measuring instruments to the objects which are to be measured (e.g., the act by which a certain measuring-rod is superimposed upon another physical body). N ow it is obvious that the third o f these three elements, the act o f measuring, is itself a physical event, the outcome o f which is purely a matter o f empirical observation. The two bodies involved either coincide or do not coincide. The coincidence or non-coincidence between their respective points can be stated— in the language o f Leibniz— as a ‘vérité de fait \ On the other hand, it is equally obvious that the first mentioned element, the geometrical system, can be exhibited only as a matter o f pure axiomatic reasoning. The theorems presented are either logically consistent or inconsistent. They must be intelligible as ‘vérités de raison . It follows, then, that, as long as we consider these two elements independendy o f each other (on the one hand the demonstration o f geometrical theorems, on the other the observation o f empirical coincidences), neither o f them contains any methodological puzzles. The claim o f accuracy refers in the one case to the consistency o f demonstration, in the other to the correctness o f observation, and both can be controlled within their respective fields. What makes the situation problematic is merely the relation which we must try to establish between the two if we wish (in accordance with the second point) to qualify physical objects as measuring instruments. To the bodies that coincide, and which we know only by empirical observation, we must attribute the geometrical properties which we have developed by axiomatic demonstration. It is at this point that the problem o f accuracy reveals its paradoxical meaning. For the choice o f instru­ ments can neither be justified logically by pure reasoning, nor be determined empirically by mere observation. It can be defined only as an act o f symbolic representation; which is to say that it is both arbitrary and purposive. It is arbitrary inasmuch as it imposes upon a physical body the function o f representing a system o f measurement. It is purposive because, owing to this procedure, the physical co­ incidences between this body and other bodies become indications o f metrical correspondences. The goal is to express these corres­ pondences in the form o f equations, so that the collection o f ‘vérités de fait’ becomes intelligible in terms o f a system o f ‘ vérités de raison . But here the question arises: how can we be certain that the equations expressing correspondences based on physical coincidences will be in harmony with equations deduced from geometrical axioms?

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Let us assume that we describe a circle by means o f a piece o f string whose length corresponds to the radius o f the circle, and that we then use the same piece o f string as a measure to determine the circum­ ference o f this circle. How can we predict in advance that the piece o f string, in order to run through the full length o f the circumference, must be applied 271 times in succession? We could only be certain o f that i f we knew that the length o f the string remains ‘the same’ throughout the whole process. We know, however, that the piece o f string changes its position in space and time, that it passes through certain physical stages o f motion. Would it not, then, be entirely conceivable that the laws o f motion to which it is thus subject should affect the invariance o f its length, and thus the identity o f the unit o f measurement embodied in it? Einstein did indeed show that if the act o f measurement takes place within a system which, regarded from a Galilean system o f relations, rotates in a plane and around the midpoint o f the circle which is to be measured, the piece o f string, as soon as it is placed around the circumference, undergoes a contraction lengthways according to the Lorentz-Fitzgerald formula, whereas as long as it forms the radius it retains its original length, as the contraction then only reduces its width. If, therefore, we first place the piece o f string along the radius, then along the circumference o f the circle, and on this basis— by correlating the results— determine the ratio o f the two measurements, then the length o f the circumference will yield a value which is greater than 2itr. The resulting equation, although founded upon precise calculation o f the coincidences, will be incomprehensible in terms o f Euclidian geometry. There is— theoretically— a double way out o f this dilemma. Either we remain with the preconceived Euclidian system, yet declare the measuring instrument to be an inappropriate embodiment o f this system; or we replace this system by that non-Euclidian system for whose embodiment precisely this instrument would be appropriate. In practical terms, however, in opting for the first way out one is con­ fronted with the frequendy insoluble task o f finding an appropriate measuring instrument that can be used in the given conditions. If the law which rendered the first instrument inappropriate is universally applicable, then one will not be able to find one that is more appropriate. This example may suffice to prove that we can only use a physical body as a measuring instrument if we anticipate the laws by which its

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physical behaviour is controlled. Only on the basis o f such anticipation can we regard the instrument chosen as an appropriate embodiment o f a preconceived geometrical concept; only thus can we expect the coincidences which arise when it is brought together with other bodies to correspond to the equations derived from geometrical axioms. Yet how can we trust such predictions? Is there a method by which it may be decided whether a given physical object which we have decided to use as a measuring instrument will adequately exhibit the geometrical concepts which we consider significant? This question has to be answered in two parts: 1 . How can we determine in general terms which physical properties correspond to a preconceived system o f geometrical concepts? 2. How we can decide in each individuai case whether the instruments we have chosen do actually possess these properties? §3. Einstein’s Concept o f ‘Practical Geometry* The first o f these two questions was in essence answered by Einstein in a lecture entided ‘Geometry and Experience’.10 In order to understand the way he looks at the subject, we may proceed from the simple and obvious fact that if we divide a metre rod into equal parts, we only ascribe to these parts the value o f measuring units because we make certain assumptions concerning the physical composition and manner o f behaviour o f the rod. We know, for example, that we can only regard the distances marked on the rod as equal to one another by regarding the physical behaviour o f the marked rod as invariant (in Einstein’s terms ‘practically rigid’). In so doing we presuppose, however, that the concept o f the mathematical unit o f measurement has its counterpart in the idea o f physical invariance (i.e., in the idea o f bodies which may be substituted for the ‘rigid’ bodies). In similar fashion we can prove (and this brings us closer to Einstein’s argument) that the principle o f identity, as presupposed by every mathematician for the employment o f his concepts, embraces the assumption o f constant space-time correspondences when it applies to the physical world. Whereas in the field o f mathematics every concept o f measurement is identical with itself, in physics it has 10 Einstein, Geometrie und Erfahrung (1921) [repr. in Akademie- Vorträge (1978), no. 19, 1- 8 ;'English fransi. Jeffery and Perrett in Einstein, Sidelights on Relativity (1922),

25-56].

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to be defined by the majority result o f a set o f appropriate measuring instruments which by virtue o f their physical existence will necessarily deviate in some way from one another, if only in their position in space and time. A body used for measuring ‘here and now’ is not identical with a body used for measuring ‘there and then’. If they are nevertheless regarded as equivalent to one another (i.e., as represen­ tatives o f one and the same concept o f measurement), the reason for this lies in the fact that they can be so superimposed upon one another that their corresponding points coincide exactly. This criterion could never have been accepted as valid if it had not been presupposed by definition that if two measuring bodies are found to be ‘coincident’ they may be regarded as ‘equal’. This definition presupposes for its own part, however, that if two measuring bodies coincide with one another once and anywhere, they can be brought to coincide with one another always and everywhere; for otherwise the concept ‘equal’ would forfeit its logical significance. This last proposition is, however, full o f expressly physical content. It expresses a law concerning the behaviour o f bodies which move and are brought into contact with one another— a hypothesis which Einstein actually seeks to verify by reference to empirical observations.11— At the same time this proposition amounts— clearly enough— to an almost literal translation o f the principle ofidentity into the language ofphysics. How then has this purely logical proposition suddenly acquired a physical significance? The answer is that it has been transposed into the realm o f space and time through the selection o f measuring instruments, for it is evidendy by this process that the logical concepts o f mathematics are made to designate physical properties. From the point o f view o f space and time, the mathematical point transforms itself into a point o f coincidence; coincidence becomes the criterion o f equivalence; and equations come to express physical correspondences, which ultimately go back to the principle o f physical constancy (i.e., 11 He transposes, according to the principles o f the theory o f relativity, the principle initially formulated for areas o f space to intervals o f time: ‘If this law were not valid for real clocks, the proper frequencies for the separate atoms of the same chemical element would not be in such exact agreement as experience demonstrates. The existence o f sharp spectral lines is a convincing experimental proof o f the abovementioned principle o f practical geometry.’ Einstein, Geometrie und Erfahrung (1921), 128 [Akademie-Vorträge (1978), no. 19, 5-6; English transl. Jeffery and Perrett (1922), 38].

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constancy which manifests itself in space-time transformations) in the same way as the correspondences expressed in mathematical equations go back to the principle o f logical identity. In the same moment in which this transposition is completed, axiomatic geometry is transposed into that form which Einstein designates ‘practical geometry’: the study o f the ‘possibilities o f relative disposition’ o f physical bodies which come into contact with one another in space and time. Whether or not these relations correspond to the equations o f Euclidian geometry depends on the laws which determine the contact between physical bodies. If we assume the physical validity o f Euclidian geometry, then we must (with Newton) presuppose certain fundamental laws o f physical behaviour, such as, for example, the ‘law o f inertia’ ,12 according to which we may only permit a select group o f relation systems. If, on the other hand, we assume certain principles o f physical behaviour to be secure, such as, for example, the Lorentz contraction, then we must deduce (with Einstein) that we must invoke non-Euclidian geometry in order to permit non-inertial frames as equivalent frames o f relation. The two cases complement each other, in that they show (1) how a system o f geometrical concepts, by being transposed to the orders o f space and time, indicates a corresponding group o f physical properties, yet (2) only the validity o f the corresponding physical laws can justify the existence o f these properties. Through the medium o f space and time a necessary connection is established between the selection o f a geometrical system and the assumption o f certain physical axioms. In order to decide, however, whether our geometrical selection is physically valid or not, we must test whether the physical axioms which it postulates lead to coincidences which can in fact be observed. The means whereby this test is to be carried out is: the experiment. In conclusion, the basic idea underlying the concept which has been sketched here can best be paraphrased by a quotation from Einstein: H o w can it be that mathematics, being after all a product o f human thought w hich is independent o f experience, is so admirably appropriate to the objects o f reality? Is human reason, then, w ithout experience, simply by the process o f thought, able to fathom the properties o f real things?

11 [Wind uses ‘Gesetz der Trägheit’ to refer to Newton’s First Law o f Motion, that a body continues in a straight line at a constant velocity unless some force acts upon it.]

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In m y opinion the answer to this question is, briefly, this:— As far as the laws o f mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.13

The two-stage process in the formation o f the physical judgement is clearly characterized here: through the schematism o f positing measurements in space and time, mathematical/apodictic propositions are transformed into physical/problematic propositions. By means o f the experiment, however, it can be established whether the physical/ problematic propositions may be transformed into physical/assertoric propositions. It is by means o f the experiment, therefore, that the ultimate decision is made as to whether a geometrical system is physically applicable or not. Before I approach the real proof o f the validity o f this hypothesis, I wish to consider a fundamental objection which has been raised against the very possibility o f its being advanced. §4. Poincaré’s Principle o f ‘Arbitrary Convention’ The discussion in the previous section, which is based essentially upon an analysis o f Einstein s method, is in evident contrast to the cele­ brated assertion o f Poincaré that any geometric system can be made to fit the coincidences observed in practice.14 As a geometrical system does not relate directly in any way to these coincidences, but only in combination with physical hypotheses, it follows that according to Poincares view these hypotheses can always be chosen in such a way that the conventionally agreed (most convenient) system in each case suits the observation. Given, therefore, a geometrical system (G) and a sum o f facts derived from experience (£), it ought always to be possible to find a system o f physical hypotheses (P) such that G + P = E. In this equation the meaning o f the symbols ‘ 4-’ and *=’ as functional relations (and expressions o f certain forms o f allocation) still requires exact interpretation. If we attempt such an interpretation, then it emerges that a triple limit is imposed upon the arbitrary application o f P :15 13 Einstein, Geometrie und Erfahrung (1921), 124 [Akademie-Vorträge (1978), no. 19, 1; English transi. Jeffery and Perrett (1922), 28]. 14 [Poincare, La Science et l’hypothèse (1904); English transi. Halsted (1913). Wind cites the German translation by Lindemann and Lindemann (3rd edn. 1914).] 15 Poincaré is quite aware o f the first two o f the three arguments that follow. The

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1. N ot every physical system can be combined with every geometrical system. On the contrary, the physical system chosen in each case must contain in its axioms propositions which expressly exhibit the relation to the geometrical system concerned. In classical mechanics, for example, it is the law o f inertia that secures for the ‘straight line’ that position o f prominence which it possesses in the system o f Euclidian geometry. N or is it a coincidence that at that moment when non-Euclidian geometry was introduced into the general theory o f relativity, the law o f inertia in its classical form was abandoned. In the combination G + P certain parts o f P are, therefore, preconditioned by the choice o f G. If nevertheless, after G has been determined, the choice o f P is to retain an arbitrary element, then within P the geometrically determined axioms (Pa), which predetermine the possible positional relations o f bodies, must be distinguished from the hypotheses linked with them (P/,), which describe the current distribution o f the bodies within this framework o f possibilities. And further, it is necessary to show that there is a degree o f arbitrariness in the positing o f P 2. In the choice o f Ph, however, the same phenomenon in relation to Pa repeats itself which we were able to observe for the choice o f Pa in relation to G. Not every physical hypothesis can be combined in the same way with every physical axiom. On the contrary, the range o f systematically admissible hypotheses is limited by the choice o f the axioms. From this it follows that if we proceed from two different geometrical systems G, we shall be led to two differ­ ent systems o f physical axioms Pa, which for their part bring two systems o f physical hypotheses Ph in their wake, which cannot be congruent with one another. If Poincare nevertheless believes that he can designate these two non-congruent systems o f hypotheses as equivalent, the reason is that the possibility still exists o f allowing the sum o f experiences E , which he presupposes, to be integrated into both systems. Both systems can still have in common that group o f propositions which expresses the given empirical data. In both systems, however, this group o f propositions, by dint o f the systematic relation in which they stand, must necessarily imply a group o f further propositions, which do not relate to already observed (i.e., ‘given’) facts, but rather to ‘ observ-able (i.e., really critical decision, therefore, lies with the third, but for an adequate explanation o f this the presentation o f the first two is necessary.

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‘postulated’) facts. And the question arises whether these new propositions, in which the two systems, being based on different axiomatics, must necessarily deviate from one another, are arbitrarily applicable or not. 3. The question is obviously to be answered in the negative. The experiment has to decide whether or not the facts postulated by P/,, which are not yet direcdy contained in E, are applicable. This shows, however, that Poincaré’s theory that every geometrical system can be combined with every empirical finding is only valid in the (hypothetical) event that this empirical finding E is thought o f as a fixed, mathematically ‘defined’ quantity, identical with itself, rather than as a systematically increasing quantity. If, by contrast, it is admitted that two different systems can only interpret the same empirical finding by implicidy referring to further empirical findings, in anticipation o f which they deviate from one another, then the difference between two such systems, provided that they are rigorously deduced, must be capable o f being embodied in an experimentum cruris.16 Yet— as indeed often happens— the difference can be so small that (at a certain stage in the inquiry) it lies beneath the limit o f what is observable. That, however, is not in itself a reason for declaring that both systems are ‘equally true’. The decision as to which o f the two is true must instead be deferred until the limit o f observation is extended by the instruments being made more precise, i.e., until it is really possible to carry out the decisive experiment. §5. The Task o f the Experiment17 What is it that an experiment is intended to test? Certainly, it is not meant to decide whether the mathematical demonstration which precedes the experiment has been logically consistent or not, for a test 16 In Poincaré s own terminology it might be said that by the rigid application o f E the propositions contained in Pf,, in so far as they go beyond £ itself, have been transformed illegitimately from ‘real’ hypotheses, i.e., hypotheses capable o f being contradicted by experience, into ‘neutral’ hypotheses. [For ‘real’ and ‘neutral hypotheses’ (‘hypothèses véritables et indifférentes’), see the definitions given by Poincaré, La Science et l’hypothèse (1904), 2-3, 180 -1; English transi. Halsted (1913), 28, 135-6.] 17 [This paragraph expands ideas set out, mosdy in the same words, in the Harvard lecture: Wind, ‘Experiment and Metaphysics’ (1927), 219-22. For Wind’s concept o f the experimentum crucis and the difference between his use o f the term ‘symbol’

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o f that kind could be carried out only on mathematical grounds. Neither can it serve to control whether the outcome o f the measure­ ment, the coincidence, has been correcdy observed or not, for this observation is supposed to answer, not to present, the problem. What is really tested is the physical presupposition on the basis o f which the outcome o f the measurement has been demonstrated in mathematical terms. This physical presupposition, however, is the very basis on which the measuring instrument has been constructed. If the outcome o f the experiment is, therefore, the one that was predicted, i f the geometrical demonstration proves to be physically effective, it follows not only that the corresponding physical law must be accepted as actually ‘valid’ , but also that the construction o f the measuring instrument must be considered ‘correct’ . We thus cannot escape the conclusion that the ultimate purpose o f the experiment is to test its own presupposition. But we may understand that this perfecdy ‘illogical’ task merely reflects the instrument’s meta-logical function— to register metrical values by reacting to (unknown) physical causes. In trying to conceive o f this function, we must face a startling methodological puzzle. The mere idea o f a measuring instrument seems to present a contradiction in terms, inasmuch as it suggests that the means o f inquiry can be identical with the objeds o f inquiry. The means are presumed to be unknown to the same extent as the objects o f inquiry; the objects are known to the same extent as the means employed in investigating them. However ‘unreasonable’ this assumption may seem, it is possessed in common by those two branches o f inquiry the methods o f which are usually considered as diametrically opposed: namely, physics and history. What has proved to be true o f the physical instrument can be shown to be true also o f the historical document. Just as the physical instrument is a physical object and therefore subject to laws which it intends to test, the historical document is itself an object o f historical inquiry; for it participates in the historical life which it helps to investigate. This fact has always been recognized by hermeneutical theory. Thus August Boeckh, for example, already saw ‘an unavoidable circular argument in the way that the various modes o f interpretation presuppose a real knowledge o f historical developments, whereas, on the other hand, such and Cassirers, which has sometimes been misunderstood, see Krois, ‘Kunst und Wissenschaft’ (1998); Buschendorf, ‘Zur Begründung der Kulturwissenschaft’ (1998), 237- 8.]

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knowledge can only be gained through the interpretation o f the source material as a whole’.18 Not only, however, must this fact be accepted as inevitable, but it must be required as an indispensable premiss. Unless the instrument or the document have a definite place within the physical or the historical world, they can tell us nothing authentic about them. But on the other hand it seems that just in being part o f physical nature and part o f history, they are disqualified from representing either o f them objectively. How can they reveal more than that part which is circumscribed by their individual status? How can we derive from them an objective understanding in which we conceive the parts in relation to the whole? The answer is that we establish this relation o f the part to the whole whenever we construct a measuring instrument, whenever we interpret a historical source. In order to construct the one and to interpret the other, we must preconceive the universal, physical or historical constellation o f which they form a part. We can preconceive it only by ordering the knowledge which has previously been gathered into a system. That is: we must make some primary (axiom­ atic) assumptions which enable us to correlate individual observations in a systematic fashion, and we must supplement the sum o f these observations by some secondary (hypothetical) assumptions which help to fill the missing links. If now, on the basis o f such a system, we construct an instrument, or interpret a source, the outcome o f the experiment (or the information derived from the source) can either confirm or contradict the system presupposed. In the one case the new physical or historical fact immediately enters the systematic relations previously established, and, by entering them, confirms not only the validity o f these relations as such, but also the accuracy o f the individual construction derived from them— i.e., the accuracy o f its own content. In the other case the ‘failure’ destroys the coherence o f the presupposed relations, and, by destroying them, disproves the validity o f the assumptions on which that particular individual construction had been founded. Such a crisis will always, however, amount to a breakthrough to new insights. ‘The physicist who has just renounced one o f his hypotheses’— says Poincare— ‘ought to be full o f joy, for he has found an unexpected opportunity for Tit

Elsenhans, Psychologie der Deutung (1904), 19. Cf. Boeckh, Encyklopadie und Methodologie (1877), 53-4, 83 ff.

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discovery.’ 19 The discovery presents itself as a discrepancy. The preconceived system must be altered in its secondary hypotheses and, if necessary, even in its primary axioms, so that it can assimilate the new fact. And after it has thus been rectified, it can be used as a basis for the more exact construction o f instruments or for the more accurate interpretation o f sources. The instruments and sources, thus renovated, will give a more accurate account o f the facts. And the facts, thus specified, will return to the system from which they emanated and will renew its obligation to assimilate them. Within this process we can observe how that which we usually call ‘fact’ is neither ultimate nor immediately given. The historical fact is revealed by a source, and sources need interpretation. The physical fact is registered by an instrument, and instruments are the result o f an act o f construction. The fact as such, therefore, reflects all the system­ atic problems o f construction and interpretation. And in appealing to facts, we unconsciously appeal, whether we admit it or not, to the systems which are responsible for their formulation. But, at the same time, we appeal to something more than that. We appeal to those occurrences which have tested the system; to those incidents in which a physical object that had been endowed with metrical significance lived up, as it were, to the logical standards implied in its metrical endowment. We appeal to those cases in which moving bodies have, through coincidence with other bodies, confirmed and specified those metrical relations which had been presupposed in the axiomatic system. To call the knowledge o f these occurrences empirical (in the sense o f something that can be experienced direcdy) would be absurd. For all that we know o f them we know only in terms o f the system which we have presupposed. To call it, on the other hand, a priori, would be equally absurd. For, although we know the meaning o f these occurrences only in terms o f the presupposed system, we cannot predict their occurrence. What they reveal to us is the answer to a question which we have presented in logical terms, but which we cannot answer by logical means. They inform us whether that which we have conceived as logical may be considered to be real. If we could ever resolve this problem by pure reasoning, we would not need to ask the question. But, since we are obliged to ask it, we must admit that, whatever the answer may be, we cannot dispute it by reasoning, but 19 I34-J

[Poincaré, La Science et l’hypothèse (1904), 178-9; English transi. Halsted (1913),

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can only try to understand it. Provided that an experiment has been executed with perfect methodological care— and such care includes the logical precision with which the question is asked— then its failure or its success must be regarded as nothing less than a metaphysical manifestation. To be able to cope with these manifestations is the art o f the physicist; to believe in the consistency o f their message, is his silent metaphysical creed. The method o f his art consists in testing a purely logical conception by provoking an entirely meta-logical act. The study o f the correspondence between the two, in the dialogue of question and answer, constitutes what we call the ‘physical world’ . The physicist himself may be shocked at the idea that he, the most exact o f all scientists, should share the company o f people who indulge in metaphysical speculations. But perhaps his distrust in metaphysics will be slighdy shaken when he is told that the very basis o f his scientific pride, the measuring instrument, is one o f the most striking examples o f a metaphysical symbol. As a physicist he will not hesitate to admit that the metrical significance o f his instruments is embodied in their physical construction. Yet if anything deserves the name o f metaphysical divination, then it is such an act of embodiment. §6. TYansformation and Embodiment The situation described above can best be clarified in detail if we take as our starting-point Carnap’s analysis o f physical space. Because it is formulated with extreme perspicacity and mathematical precision, it enables us to recognize clearly that point at which we must progress beyond the purely logical interpretation which he himself advocates.20 Carnap distinguishes between the following three elements in each physical measurement o f space: 1. the topological ‘array o f data’— Tatbestand (T), o f which the co­ incidences observed form a part; 2. the ‘positing o f a measure’— Majisetzung (M), by means o f which a measuring instrument is defined and by reference to which topo­ logical propositions acquire metrical significance; 3. the metrically determined ‘spatial structure’— Raumgefuge (R), constituted by the topological relationships (T) in regard to M and conceived as a geometrical system.

20 Carnap, Der Raum (1922).

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Between the three elements R, M and T there exists a relationship o f dependency such that, if two o f them are given, the third is un­ ambiguously defined thereby. We can therefore choose the measuring instrument freely as long as we are willing to recognize that spatial structure to be geometrically valid which derives from the relationship o f the topological data to this measuring instrument. Thus: R = / i ( M , T). Equally, however, we can choose the geometry freely, if we allow ourselves to determine the application o f the measuring instrument in such a way that the topological data can be structured metrically in accordance with the spatial structure. Thus: M = f 2 ( R ,T ). The third case, in which the geometry in combination with the establishment o f a system o f measurement determines the array o f data, i.e.: T = / 3 (R ,M ), is expressly declared by Carnap to be valid, for ‘it is this that forms the basis for the scientific description o f space: it is held that given a certain value for M, the physical spatial structures form a metrical structure R, and that on this basis the array o f data T, as experienced empirically, can be fully described in regard to its spatial relationships. Yet this third case differs most essentially from the others by the circumstance that while either R or M can be freely chosen, this is not the case with T: the array o f data is quite plainly given.’21 Following Carnap’s example, we shall for the time being ignore the third case and apply ourselves to the first two. The formula R = f \ (M , T) concurs entirely with our earlier deliberations, for we maintained that the form o f spatial structure cannot be chosen freely, but can be contradicted by the experimentally verifiable findings T, and in so doing we took as a basis a certain system o f measurement M .22 However, the statement M = f 2 (R , T) seems to contradict our arguments direcdy, for it asserts that the nature o f the system o f measurement can be determined by the choice o f the spatial structure just as readily as the nature o f the spatial structure can be Jl Carnap, Dtr Raum (1922), 54. 11 Cf. the use o f the ‘string’ for the measurement o f the circle, p. 12 above.

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determined by the choice o f a system o f measurement, and thus that the choice o f a spatial structure can never be contradicted by the facts o f experience, as there will always be a system o f measurement M which will accommodate the array o f data T to the form o f the chosen structure R. The mathematical validity o f this second proposition, badly though it falls out for us, cannot be doubted, any more than can that o f the first, for in mathematical terms every metrical spatial structure can be transformed into another by changing the system o f measurement, without the topology being thereby affected. Let me illustrate this, again following Carnaps example, with the case o f the measuring o f the Earth: Measured with an iron rod kept constantly at a certain temperature, the surface o f the Earth presents itself, disregarding the flattening at the poles, as the surface o f a sphere o f a certain finite circumference in Euclidian space. ‘It is now required that the surface o f the Earth E be regarded as a plane. As E, measured on the basis o f the usual measuring system M ,, proves to be a sphere, a different measuring system will be needed. Yet the freedom o f choice in respect o f the spatial structure still holds after the system o f measurement has been setded, as we are still free to determine what degree o f curvature the plane E shall have at various points and what is the nature o f the remaining space. We can thus, for example, determine that E is at all points to have a curvature o f zero. In that case the surface o f the Earth could be conceived as o f infinite magnitude.’ It can now be shown that ‘this seemingly strange view [...] does not in fact contradict any physical experience, not even the results o f geodetic measurements achieved by optical or mechanical means, provided only that these are interpreted in a different way to that which is customary, i.e., not on the basis o f the measuring system M j, but on that o f another, M e. M e would then have to be stated in something like the following form: “ These two points A and B on this iron rod mark a distance which is not to be regarded as invariant but (disregarding temperature, magnetization, etc., to varying degrees) as dependent above all, in varying ways, upon the place on E in which the rod has been placed ” Moreover, this dependency would have to be stated in such a way that one particular point on E , for example the place where the iron body is presendy situated, or the North Pole, is accorded a certain precedence.’23 23 Carnap, Der Raum (1922), 47 and 48.

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In order to visualize clearly the meaning o f these statements, we need only imagine how the new measuring instrument relates during the process o f measurement to the old one, the iron rod. Both the iron rod and the new instrument are protected in the same way from changes in heat and magnetic influences. In the preferred place— let us say the North Pole— they are placed one on top o f the other and prove to be o f equal length. Yet the further they move away from the North Pole in the course o f the measuring process, the more the new measuring instrument shrinks in relation to the iron rod. To put it another way, the iron rod, considered from the point o f view o f this new instrument, in the course o f its journey becomes bigger and bigger. And if it finally succeeds, as we all know, in measuring the circumference o f the Earth in a finite number o f steps, this is not surprising from the point o f view o f the second instrument, because for the latter the iron bar itself, after a finite number o f steps, has grown to an infinite magnitude. N or should it surprise us that the Earth presents itself as infinite to an instrument which does not parti­ cipate in this ‘growth’, for we know, after all, that this instrument has become smaller and smaller in relation to the iron rod and hence in relation to the Earth. Reference to such an instrument, however, prompts a twofold question: 1. Is it actually possible to find an instrument o f this nature? 2. Why, assuming that we could find it, should we wish to employ it for measuring? The first question need not trouble a mathematician. Strangely, however, it is nowhere posed by Carnap, who is, after all, concerned with physical space; probably because the question seemed too primitive to him. Yet it is only with this question that the thought experiment becomes a real experiment. (And is it not the true function o f thought experiments in physics to lead into real experi­ ments?) If a mathematician gives voice to the proposition: ‘ If, faced with a topological array o f data T, a spatial construct R is chosen on an arbitrary basis, then there exists a measuring system M, through which T can be brought into the form o f R \ then the word ‘exists’ expresses nothing other than the property o f logical definedness: the properties which an instrument must possess in order to function as M are unambiguously determined by the values o f T and R. But if a physicist gives voice to the same proposition, then the word ‘exists’ means not only that the properties which the instrument in question

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must have are known in terms o f their mathematical structure, but also that an instrument possessing these properties can be found in the real world. Through this pronouncement the mathematical propo­ sition is transformed into a physical hypothesis, a hypothesis which, depending upon the outcome o f the verifying experiment, can be proved true or false. Whether an instrument like that described above actually exists in the physical world— an instrument whose behaviour in relation to an iron rod has been determined in a mathematically unambiguous way— is an entirely reasonable experimental question. It follows from this, however, that the formulae established by Carnap— R = / j (Ai, T) and M = f 2 (R, T )— are not equally valid for physical space. The first formula expresses a necessary situation, the second a problematic one. If I take any physical measuring instrument M and structure a topological array o f data T metrically by reference to it, then the spatial construct R will necessarily follow. If, however, I freely choose the form o f the spatial construct R and seek to relate it to the topological data T, then it is not true that we can infer the existence o f a physical measuring instrument M, which will make the process o f measuring possible. Two different attempts have been made to circumvent this conclusion: Poincaré once jokingly declared ‘that any mechanic could construct instruments o f this kind if he cared to take the pains and make the outlay’.24 The question o f whether the postulated instru­ ment M exists or not is, according to this view, most easily solved by manufacturing one. Yet let us not underestimate the nature and scope o f the intervention which such a mechanic would have to carry out! As the behaviour o f the instrument would be subject to my will not merely in relation to the iron rod and the Earth, but in relation to absolutely all physical bodies in existence (for I am, after all, to be in a position to determine by the choice o f measurement the physical spatial construct as a whole), then this mechanic, in order to be able to manufacture the instrument, would have to be capable o f an inter­ vention o f cosmic dimensions. If he were capable o f controlling the behaviour o f the instrument not merely to a limited extent in relation to the iron rod, but to an arbitrary extent in relation to the Earth and the other planets, then he would have power over the stars themselves. N or can it be disputed that, given this presupposition, he would be 24 Poincaré, La Sàence et l’hypothèse, German transi. Lindemann and Lindemann (3rd edn. 1914), 85 [French edn. (1904), 104; English transi. Halsted (1913), 88].

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able to impose any geometrical system he liked upon the universe. Such power is logically consistent with the view o f the relationship between geometry and reality which I am here seeking to defend. There is, however, another escape route whereby our conclusion might be circumvented. The system o f measurement can be related not to a physical body but to ‘space’ , and it can be maintained that in physics absolutely no verifiable relation between the body and space exists. This alternative has again been most ingeniously pursued by Poincaré:25 I f w e construct a material circle, measure its radius and circumference, and see i f the ratio o f these two lengths is equal to 271, what shall w e have done? W e shall have made an experim ent on the properties o f the matter with w h ich we constructed this round thing, and that o f which the measure used was made. (La Science et l ’hypothèse V, 2) W ill you say that i f the experiments bear on the bodies, they bear at least upon the geom etric properties o f the bodies? B ut, first, what do you understand by geom etric properties o f the bodies? I assume that it is a question o f the relations o f the bodies with space; these properties are therefore inaccessible to experiments w hich bear only on the relations o f the bodies to one another. This alone would suffice to show that it cannot be properties such as these that are meant. (La Science et l ’hypothèse V, 7) W h en I say that such a part o f such a body is in contact w ith such a part o f such another body, I enunciate a proposition w hich concerns the mutual relations o f these two bodies and not their relations with space. (La Science et l ’hypothèse V, 7)

All these propositions are intended to prove the theory that the expression ‘geometrical properties o f physical bodies’ is meaningless. I f we accept this view, then it will indeed be possible to rethink any physical situation in a geometrically arbitrary fashion. If, for example, I compare a billiard ball with the terrestrial globe, then the relative magnitude o f the two manifests itself as a ‘relation between the two bodies’, and as this— according to the presupposed thesis— has absolutely nothing to do with ‘space’ , then I can maintain at any point that the two forms are ‘actually’ o f equal magnitude in space, but that in the zone which is accessible to observation physical powers are at work whereby one o f the balls experiences a contraction (or an 2J [Poincaré, La Science et l’hypothèse (1904), 92, 101, 101; English transi. Halsted (1913), 81, 86, 86.]

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expansion) in relation to the other.26 Thus I can always define the geometrical relations o f congruence in whatever way I wish, provided only that I undertake to accumulate the ‘neutral’ physical hypotheses in a correspondingly arbitrary way. This doctrine is most simply refuted by adopting the procedure which Poincare regards as doomed to failure: by attempting to give the expression ‘geometrical properties o f physical bodies’ a meaning by means o f a definition. We declare that the mathematical space which embraces only imagined relations, has indeed nothing to do with the relations between physical bodies, and that physical space is nothing other than the very essence o f precisely these relations. Consequently physical space remains distinct from mathematical space in the clearest way imaginable, and Poincare’s endeavour to guard pure mathematics against ‘empiricism’ is in no way disturbed. On the other hand, it now makes good sense to ask what is the relationship between the observable relations that exist between those bodies which constitute physical space and the purely imagined relations o f a particular geometrical spatial structure. And this question leads in the last resort to that posed above: can the ‘there exists’ which the mathematician pronounces with regard to a particular system o f measurement be changed into the ‘there exists’ o f a physicist, who relates it to the existence o f the corresponding measuring instrument? Once the question has been put in this way, the answer is determined by the experiment and not by my arbitrary decision. If, on the other hand, I desire to preserve my freedom always to be able to determine what approach to adopt, then I can only choose between the two possibilities described above. Either I consider myself to be a demiurge who can alter the world by technological inter­ vention; then the mathematical transformation will in every case take on actual physical content; but nonsensically, in so doing, I transform the object which I desire to have cognizance o f through the act o f cognition. Or I consciously indulge in the transformation as a purely mathematical, theoretical game; then I will discover nothing new through it concerning the world, but am content to transpose a thought, once it has been conceived, from one language into another, remaining free to invent new linguistic modes o f expression as and when I see fit. 26 The example o f the balls is derived from Russell, as quoted by Whitehead, The Concept of Nature (1920), 122.

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It would be no easy matter to choose between these two possibilities. The first is for us impracticable, and moreover non­ sensical; the second is practicable, yet totally useless. In what, then, does the acknowledged cognitive value o f mathematical transform­ ations for progress in physics consist? §7. The ‘Judgements o f Appropriateness ' Let us admit that the ‘wild’ measuring instrument described above is not unthinkable. Let us even assume that it can be found! What can cause us to give it precedence over another instrument for the representation o f the world? The mathematicians have a very simple answer to this question. They say: ‘simplicity’ decides. No empirical criterion, but only a logical one may be used to decide whether it is ‘more correct’ to make use o f the ‘wild’ measuring instrument. Preference will always be given to whatever allows the laws o f nature to exhibit themselves in an especially simple way. Yet, from a purely logical point o f view, is ‘simplicity’ an unambiguous concept? Do we not have to ask: ‘simple for what purpose’? When Copernicus’ purpose was to describe the paths o f the planets, it was the Sun which as a point o f reference supplied simpler formulae than the Earth. But when Einstein sought to assess the structure o f space in the universe as a whole, he chose as a point o f reference the transmission o f light, because from that point o f view the relative velocities o f the stars became so extraordinarily small that one was justified in treating matter as existing in a state o f rest, and thereby created the possibility o f ‘arriving at a rough approxi­ mation as to the nature o f the universe as a whole’.27 Poincare only ever asked ‘simple for whomV, and thus he came (entirely logically^ to regard the choice o f measuring instrument as a mere convention.2 If, however, we ask the question ‘simple for what purpose?', then both question and answer have a real physical sense. The ‘wild’ instrument considered above would, even if its existence were as certain as that o f the Earth, be unsuitable for the measurement o f the circumference o f the Earth, because the Earth would grow to an infinite size in relation to it, and its circumference would therefore necessarily exhibit itself, 27 Einstein, Spezielle und allgemeine Relativitätstheorie (1917), §32, p. 75 [repr. in Colleded Papers, vi (1996), 500; English transl. Lawson (1920), 113]. 28 [Poincare, La Science et l’hypothise (1904), 173-8; English transl. Halsted (1913), 130- 3 ]

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when so measured, as ‘just as big’, for example, as the circumference o f the Sun. For other purposes, however, such an instrument, if it existed, would be o f greater use for precisely this reason. In thus declaring an instrument to be ‘appropriate’ or ‘inappropriate’ for a certain measurement, we are pronouncing a judgement concerning a proportion obtaining in the actual world. This judgement is, however, a hypothesis, for we can know nothing a priori concerning this ‘appropriateness’. Indeed this is that same question which was embodied in the construction o f the instrument, and the search for an answer to which is the prime purpose o f the experiment. The information about the world which we seek to find through the use o f an instrument is always at the same time information concerning the position o f that instrument in the world. I f the nature o f the world were self-evident we would have no need to look for this information in the first place. We could obtain all our information direcdy, and the concept o f the instrument would lose its meaning, the instrument itself lose its function. However, given that we form deductions about the world and do not merely observe it, and that we are dependent upon the use o f instruments, whose nature is no less unknown to us than the world to which they belong, the hypothesis present in the act o f embodiment remains indispensable for the process o f cognition; indeed, it necessarily precedes the recording o f the most simple data. It emerges from this, however, that the third o f Carnap’s formulae, which— following Carnap’s example— we have deliberately ignored so far, acquires a significance which Carnap himself does not appear to have recognized: T = / 3 (R ,M ). The right-hand side o f this equation— the bracketing together o f M and R — can be understood as a symbol o f the act o f embodiment: the combination o f a geometrical concept o f space with a physical measuring instrument through whose application the meaning o f the geometrical concept is to be proved empirically. The symbol T, which appears on the left-hand side o f the equation, however, would then denote the value which ought to arise from the combination o f M and R , but we do not know whether it will, as we have no power over the behaviour o f M (i.e., no complete knowledge o f it). To accept the quantity as ‘unambiguously given’ , as Carnap does, instead o f regarding it as postulated, but by no means certain, is to deny the

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peculiarly progressive character o f our cognition o f facts. In Carnap’s positing o f T the same constructivist concept o f experience is betrayed as we encountered in Poincare’s concept o f empirical data (E).2g Is it not here, perhaps, that the natural limit and the obvious fallacy o f all mathematical logic lies, no matter what direction it takes? §8. Real and Neutral Hypotheses Mathematical logicians have a strange habit when they find themselves talking about physical theories. They accept the ‘facts’ as established, spread them out before themselves and ask: how can I construct a logically coherent world out o f this material? They then discover, generally to their surprise, that the world can be constructed in many different ways, and report to their astonished fellow men that it is a matter o f ‘taste’ or ‘convenience’ whether preference is given to one system or another. Facts, they claim, have no say in this matter. Indeed, facts are conspicuous by their tolerance once they have been designated as certain. They are no longer even permitted to budge.30 In reality, however, they do shift ground. N o matter how rigid and obstinate they may appear (the English speak o f ‘hard and stubborn facts’), they can be compelled to answer questions— provided we put the questions correcdy and force them to answer in the correct way. We put the questions by thinking up transformations whose course and result is mathematically determinable. We compel them to answer by combining through an act o f embodiment a mathematical quantity with a physical one and observing whether the physical quantity undergoes those changes which the mathematical transformation prescribes for it. If this happens, then we can surrender ourselves to 19 Cf. pp. 18 above. 30 To exemplify this point it will suffice to cite the following sentences from Bertrand Russell, referring to the difference between Newtons and Einstein’s laws o f gravitation: ‘It is not quite clear why the man who uses forces with a conventional geometry should be regarded as making a “ mistake” , while the man who says that free particles travel in geodesics, and to justify himself has a queer geometry, is thought to be saying something substantially more accurate. [...] It if a mere question o f convenience whether we speak o f forces or not. Let it be conceded that the method o f the general theory o f relativity is better from a logico-aesthetic point o f view; I do not see, however, why we should regard it as any more “ true” . I am not considering, at die moment, the fact that Einstein’s law o f gravitation gives a slightly more accurate picture o f the phenomena than Newton’s, since this is not really relevant to the particular point at issue! The Analysis of Matter (1927), 76-7.

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the belief that the world really is as we think it is. If it does not happen, then, as Poincare himself puts it, we are making a discovery. The stubborn inquirer will always try to ward off such discoveries. If he possesses logical skill, he can always do so by introducing a sufficient number o f neutral hypotheses which are not in any way refutable by experience. Yet in order to link them in some way with experience he must interject somewhere a real hypothesis (no matter how inconspicuous it may be). And this hypothesis, precisely because it is real— i.e., because it is capable o f being transposed into factual statements— can explode at any moment. Yet through appropriate manipulation it is always possible to prevent the structure as a whole from collapsing as a result. A new, seemingly firmer hypothesis can always be inserted by a bit o f careful mending, in such a way that the neutral hypotheses which determine the picture as a whole survive intact through all these crises. These neutral hypotheses will never explode. But they do something else: they die out.31 Treated purely mathematically, the thinking out o f transformations thus remains a pretty game, but one which can only prove fruitful if man believes his destiny does not lie in his mirroring himself like a god in his own thoughts. Only in combination with an act o f embodiment does transformation become an instrument o f cognition, and the instrument becomes all the more powerful (and thus all the more dangerous) in proportion to the comprehensiveness with which the embodiment is attempted. O f course, an infinite intellect would have no need o f such an instrument. For such an intellect the judgements o f ‘appropriateness’ , which we express hypothetically in the choice o f the instrument, would have no meaning whatever. For that intellect the proposition that where infinite quantities are concerned the part is just as big as the whole would lose its applicability, for the simple reason that for 31 The question o f the viability o f scientific theories is only touched on here. It cannot be settled simply by reference to logical consistency and experimental testing, and it suggests rather that there are limits o f usefulness to the distinction between neutral and real hypotheses, i.e. between the efforts expended in logical analysis and what is actually learned. The result is that we are confronted with different degrees o f ‘closeness to reality’, and thus o f practical reliability. The point at which the accumulation o f neutral hypotheses becomes insupportable and provokes a crisis o f logistical ‘inflation’ can never be determined on a strictly logical basis, but only in terms o f the pragmatic-historical situation. The question ‘for whom?’, which was rejected above (cf. p. 29), is now joined by the question ‘for what purpose?’ . On this whole complex o f questions see Lewis, The Pragmatic Element in Knowledge (1926).

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that intellect there would be no such thing as ‘infinite quantities’. Even by applying the ‘wild’ instrument it could recognize that the Sun is bigger than the Earth. The transformation would never have anything new to say to such an intellect, only the same thing again and again. It would not be a means o f progressive cognition, but only ever a means o f mirroring. For us, however, who are nothing but a part o f this world, seeking to measure the totality o f the world from the conditional nature o f the part and with the conscious employment o f means conditioned by that part— for us the transformation acquires its meaning in that it opens up to us aspects o f the universe which are always new. It only does so, however, provided we venture to undertake the act o f embodiment, by which we underpin the formal mathematical game with the meaning o f reality, with the meaning that derives from a metaphysical question which can be answered on an empirical basis. §9. The Cyclical Progression and its Methodological Foundations32 We can now proceed to resolve the circle described in §1. In order to test again the results obtained thus far, we shall at first attempt to exclude the metaphysical element and remain within the purely logical sphere. Given this precondition, how would the process o f physical cognition present itself? There would be no means o f any kind to avoid the circle described in §1. Viewed from a logical standpoint, the claim o f universality in the conception o f laws is related to the claim o f accuracy in the conduct o f experiments as the validity o f a general premiss relates to the validity o f a particular conclusion. As a negative instance this correlation o f the two is, to be sure, o f the greatest importance for physical inquiry. If an experiment fails to test an individual hypothesis, this failure reflects not only upon the hypothesis as such, but on the entire system o f assumptions upon which it was founded. It criticizes not only the construction o f the individual instrument, but the whole system o f methodological rules from which this construction has been derived. In this sense it is no exaggeration to say that with every individual experiment a whole system o f experimentation comes to a 32 [§9 is a reworking o f the final section o f the Harvard lecture: Wind, ‘Experiment and Metaphysics’ (1927), 222-4.]

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test. But to make this correlation o f the individual and the universal the basis o f a positive method o f inference would be a fallacious procedure. We would deduce the individual claim from the universal one, and then test the validity o f the latter by showing that it can include the former, a subsumption which would only prove what the deduction had defined. As it is, however, we not only derive the individual claim from the universal system, but embody it by construction in a physical instrument. Then we let the instrument do its work, and see if the outcome can be subsumed under the universal premiss. What first appeared as a logical circle, and therefore as selfcontradictory, thus turns out to be a methodological cycle, and therefore self-regulating. This cycle can be illustrated in detail in the historical development which leads from the Newtonian system o f physics through the special to the general theory o f relativity: Newton’s concept o f absolute space and absolute time incorporates the following double requirement: 1. that all the Galilean frames o f reference (i.e., all those frames which are at rest relative to the fixed stars or are in uniform translation movement in relation to them) may be treated as equivalent for the formulation o f natural laws, and 2. that the various results o f measurements obtained within these frames can be translated into one another according to the Galilean equations (i.e., according to the addition theory o f velocities). These propositions formulate methodological rules intended to direct the process o f measurement and calculation. At the same time, however, they relate to physical circumstances which they presuppose actually to exist. The first proposition can appeal to the observation that earthly space is isotropic.33 The second states that the extension o f measuring bodies is not affected by motion. It was towards the second o f these assumptions that Einstein’s criticism was directed, on the basis o f purely physical evidence. The motive underlying the criticism, however, lay, as is well known, in a general methodological dilemma: Newton’s principle o f relativity, according to which all Galilean systems o f reference are held to be equivalent for the formulation o f 33 Cf. Einstein, Spezielle und allgemeine Relativitätstheorie (1917), §5, p. 10 [repr. in Colleded Papers, vi (1996), 434; English transl. Lawson (1920), 15].

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natural laws— a principle which had proved indispensable for the derivation o f mechanical laws— had entered into conflict with the law o f the constancy o f the velocity o f light, a law which for its part amounted to one o f the most unshakeable results o f electrodynamic research. From the point o f view o f the Newtonian principle o f relativity the speed o f light could not possibly be viewed as constant. If, on the other hand, one were to be persuaded by the evidence from electrodynamics to posit it as constant, then it could no longer be maintained that all Galilean frames o f reference are equivalent for the formulation o f natural laws. The unity o f the natural system was at stake. N o principle could be discovered which might claim to be generally valid.34 The way in which Einstein overcame this discrepancy typifies the physical method o f research. He satisfied the logical demand for universality by exposing a physical inexactitude in the formulation o f the Galilean equations, which affected the construction o f the respective measuring instruments. If we presuppose that all our observations o f physical processes o f motion are conveyed through the medium o f light, then we must take the speed o f light into account in the systematization o f these observations. The system o f Galilean equations, which, without taking the speed o f light into account, specifies the mathematical formulae whereby the results o f measurements in a given case can be transferred from one Galilean system to another, would only be accurate if light were propagated with infinite velocity, for then no differences in time would result during the transference by means o f measuring signals. As light, however, travels with finite speed, the Galilean equation, which records absolute spatial and temporal values, must be replaced by the Lorentz transformation, which by introducing the speed o f light as a limiting value into the formula relativizes the spatial and temporal values by reference to this limit. Einstein permits no doubt that this change has a factual physical significance. The Lorentz transformation is intended to give a more precise account o f what actually happens to measuring-rods and clocks according to the laws o f optics and mechanics, but it is interesting to observe how the result o f these physical expositions is 54 Cf. the account by Cassirer, Zmx Einsteiti'schen Relativitdtsthcorie (1921), 26-8 [repr. in Zur modernett Physik (1957), 2 1-2 ; English transl. Swabey and Swabey in Cassirer, Substance and Function (1923), 367-86].

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immediately turned into a methodological claim. The Lorentz transformation formulates, as Einstein himself puts it, ‘a definite mathematical condition that the theory o f relativity demands of a natural law’ .35 The proof o f accuracy has as its consequence the claim to universality. In the transition to the general theory o f relativity Einstein brought both these aspects o f physical inquiry to bear, but from the opposite point o f view. The motive for this transition lay in a purely methodological claim, namely that the laws o f nature should be conceived in such a way that all frames o f reference prove to be equally valid for their formulation. If a law lays claim to general validity, then it ought to express itself in equations which are universally covariant and no longer confined to the Galilean frames o f reference. From the outset, however, this logical requisite had a physical counterpart. It amounted to nothing less than the rejection o f the law o f inertia, for if the law o f inertia were valid, it would mean that nature singled out certain systems for the formulation o f its laws in preference to others. Einstein’s claim, by contrast, assumes that the natural laws can be apprehended in the same way from all the points o f the universe. If, for this claim to be fulfilled, Gaussian coordinates have to be introduced, then here again a specific physical assumption is present, for in their application to physics— no less than any other preconceived method o f construction— the Gaussian coordinates pre­ suppose the physical laws which control the measuring instruments. The sole difference lies in the fact that the preconception has become much more subde than previously. Einstein no longer presupposes the absolutely regular behaviour o f clocks and measuring-rods, which in Newton’s concept o f mechanics was accepted as a physical counter­ part o f the absolute values o f time and space. Nor does he presuppose that relative irregularity which was compensated for in the opticalmechanical world o f the special theory o f relativity by the Lorentz transformation. In the general theory o f relativity the forms o f the measuring bodies and the rhythms o f the clocks may be absolutely irregular. In order to fulfil the mathematical conditions o f the Gaussian system, however, the deviations resulting from this irregularity have to maintain a continuous course. The discrepancy in the numerical values resulting from adjacent measurements must be 35 Einstein, Spezielle und allgemeine Relativitätstheorie (19 17), §14, p. 29 [repr. in Collected Papers, vi (1996), 453; English transl. Lawson (1920), 43].

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infinitely small, so that, even after the reference frames are no longer conceived as fixed, they retain, as ‘molluscs’ , the property o f physical coherence.36 The results o f measurement may deviate within the system, but abrupt leaps are not envisaged. That this amounts to an absolutely radical physical assumption whose empirical validity can be limited or even disproved by experiments was voiced in all clarity by Einstein himself: ‘In contemplating the immediate future o f theo­ retical physics we ought not unconditionally to reject the possibility that the facts comprised in the quantum theory may set bounds to the field theory beyond which it cannot pass.’37 Thus the postulate o f continuity represents, in the extended form o f the Einsteinian theory, the last remnant o f those physical pre­ suppositions which had formed the basis o f the claim to accuracy in the Newtonian system and in the special theory o f relativity. This assumption in turn is only the physical counterpart o f the methodo­ logical claim to universality. After considering the mutual relationship between these two claims in one specific case, we can now finally resolve the antinomy described in §1 by setting forth the two principles which govern the cycle o f physical inquiry and which at first sight seem to contradict, but really supplement each other. 1. The claims o f universality and accuracy represent two different sides o f one and the same claim, and can therefore not be separated. 2. The claims o f universality and accuracy represent the same claim from two different sides, and must therefore be distinguished. The first o f these statements points out that the physicist cannot demonstrate the logical consistency o f the rules which are meant to conduct his inquiries without anticipating the corresponding physical laws which govern the objects o f his inquiry. This statement alone would be sufficient to destroy the suspicion o f a vicious circle. For if every methodological claim necessarily implies a physical hypothesis, the one cannot be inferred from the other, but both must be set forth simultaneously. The assumption o f such a simultaneous act, however, 36 [The expression ‘mollusc’ is used in Einstein’s discussion o f the general principles o f relativity: Spezielle und allgemeine Relativitätstheorie (1917), 68; repr. in Colleded Papers, vi (1996), 492; English transl. Lawson (1920), 100-1.] 37 Einstein, Äther und Relativitätstheorie (1920), 15 [repr. in Albert Einsteins Relativitätstheorie, ed. Meyenn (1990), 122-3; English transl. Jeffery and Perrett in Einstein, Sidelights in Relativity (1922), 23].

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seems to deliver us from one methodological dilemma only by throwing us into a new one. For if the claims o f universality and o f accuracy are at bottom only one and the same claim, their relation can be o f no logical significance. Being a purely symmetrical relation, it cannot serve as a basis for justifying a hypothesis or controlling an inference. It could explain a valid system only as a lucky guess, not as the outcome o f methodological progression. At this point the second statement reveals its positive meaning. It indicates that the two formally identical claims do diverge in content: the claim o f universality has a logical content and must therefore be judged with regard to its systematic consistency, while the claim of accuracy has a physical content which must be determined by the experiment. The claim o f universality thus becomes the motive o f systematization, which is a logical procedure based on the assumption o f axioms whose value is to be tested on the basis o f immanent criteria. Those axioms which are assumed within the field o f physics, however, are subject to physical control. Hence, the claim o f accuracy has to fulfil the function o f testing the validity o f axioms through the ‘embodiment’ o f the consequences in instruments— a test which will either succeed or fail. §10. Metaphysics and Empirical Experience It might at first glance appear surprising that I have designated that higher authority which provides the answer ‘yes’ or ‘no’ to the question embodied in an experiment a ‘metaphysical’ authority. Three reasons may, however, be given (even at this stage o f our deliberations) which justify this designation: i. Despite all the independence to which we may lay claim for our thinking, we have absolutely no power over this factor. When we speak o f autonomy o f thought, this can only refer to the formulation o f the question and the conceptual considerations which are associated with the answer. The answer itself, however, the ‘yes’ or ‘no’, is subject to different laws from the thought; the thought has to accept the answer, and must try to reconcile itself with it. Thus reason does not prescribe laws to nature, but derives them from it by questioning— a distinction which, no matter how much it is anticipated in the Kantian system,38 38 Cf. especially the preface to the second edition o f the Kritik der reirten Vemunjt (1787). [Wind cites the Critique from vol. iv o f Ernst Cassirer’s edition, published in

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is not carried through in the course o f that system. For this ‘questioning’ it is not sufficient to know what is meant by the question, but rather the technical means o f embodiment must be devised through which the answer may be forcibly extracted from nature. It is not sufficient simply to calculate the formulae which result from a certain way o f contemplating nature and give it its peculiar character as a form o f thought, but it is essential that, in addition, ‘the testing o f the question’ (as Planck expresses it) should take place, in order to establish ‘whether the entire process [...] is in reality no more than a feat o f calculation, or whether it does indeed have an actual physical basis’ , a meaning which is subjected to the ‘art o f experimentation’ and which emerges from the results o f the experiment as something real.39 It is precisely this decision, namely as to whether the meaning we seek can be said to be real or not, which can never be determined by a theoretical pronouncement. 2. The authority to which we appeal here is by its nature unknoum to us, for it manifests itself only in the act o f saying ‘yes’ or ‘no’, or in the formulation o f discrepancies which give rise to new questions. This ‘yes’ and this ‘no’, however, never provide our cognition with new content, particularly when they appear indisputable. If an experi­ ment ends in a negative result— as for example in the MichelsonMorley experiment— then the physicist sees in this a sign o f the ‘obstinacy o f the facts’: ‘Nature remained silent and refused to vouchsafe a reply.’ If, on the other hand, the experiment has a positive outcome— as for example in the testing o f the curvature o f light beams in the gravitational field o f the Sun predicted by Einstein— then nature’s answer follows in precisely that language in which the question was put. In such unambiguous cases in particular, the enrich­ ment o f our cognition does not amount to an extension in terms o f content, to a new, totally unexpected, sensory observation, but rather to the occurrence or non-occurrence o f a predicted event. It is not the content o f the actual which is o f new import for us, but the quality

1913. See also Kant (Akademie-Ausgabe), iii-iv (19 11); English transl. Pluhar (1996). Citations refer to the page numbers o f the second edition o f 1787 (B), and where appropriate to the first edition o f 1781 (A); the pagination o f A and B is preserved in the standard editions and translations.] 39 Planck, Entstehung der Quantentheorie (1920), 14 -15 ; repr. in Physikalische Rundblicke (1922), 148 ff. [cited by Wind]; [Physikalische Abhandlungen und Vorträge (1958), iii. 126; English transl. Jones and Wilhams in Planck, Survey (1960), 107].

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o f actual-ify. The criterion for the presence o f this quality is, however, no immanent possession o f our intellect. In order to determine it, therefore, we must appeal anew in each individual case to an unknown instance, arriving at a conclusion methodologically through an act o f embodiment. And it is precisely in this act that we must presuppose that its inner reality is unknown.40 3. I f a factor which plays a role in our process o f cognition is by its nature removed from our power and our knowledge, that should suffice to justify its being termed ‘metaphysical’. Furthermore this factor turns out to be indispensable for our process o f cognition. We are concerned not with an element which may occasionally occur, but with a principle which must occur regularly. Without this metaphysical sanction, which has often been termed ‘the sanction through empiri­ cism’ , in the belief that the illusion o f harmlessness may thereby be preserved, all mathematical-physical arguments are, in Planck’s words, ‘just a meaningless, formal game’. They share this fate with all other autonomous creations o f our intellect, far beyond the purely theo­ retical realm. Even an artistic conception only becomes a work o f art by dint o f being embodied and demonstrating its aesthetic validity through this act o f embodiment. Otherwise it remains a mere figment o f the imagination, which does not expose itself to the danger o f error and thus forfeits the opportunity to be given objective form. An ethical demand or a social law remain utopian thoughts up to the point where they are themselves embodied in actions (whether individual or collective in nature) and prove that they can retain their meaning in the complications to which those actions lead. No matter how well thought through a law may be, if it simply cannot be put into practice then it is just as null and void as a law which in the course o f its execution destroys its own meaning. It is not its effectiveness, in the shallowly pragmatic sense o f its success, which determines how realistic it is, but rather its capacity to be realized. Whether the case in point is an artistic creation or a scientific experiment, legal decrees or ethical decisions, the route to verification leads through crises and discrepancies, for the process o f realization 40 If people wish to assign to the ‘unknown’ the term ‘transcendent’, there is no objection to that. The transcendent would then, however, be not only, as it is frequently conceived, the unknown in an absolute sense— that concerning which nothing can be experienced— but also that ‘unknown’ which manifests itself and can therefore be defined as unknown only from the point o f view o f a historical stage in knowledge.

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always involves intervention in a world which is subject to different laws from the abstract idea; it means grappling with forces over which the idea has no power. Yet it is only through entering this world that the idea fulfils its meaning; only thus can it test its reality. The manner o f entering the world, the form o f embodiment, differs, o f course, depending upon the field o f intellectual endeavour. The intervention is one thing if it leads to an experiment which puts into practice the testing o f a natural law, another if it leads to a legal idea being implemented by the power o f the state; another again, if it leads to the representation o f an aesthetic vision in marble or bronze. Common to all these, however, is the fact that in the act o f embodiment a liCTdfJaais e is dXXo takes place, a transcendence o f the self which in the course o f so-called empirical experience finds its metaphysical confirmation or rejection. From these considerations a number o f deductions relating to the concept o f metaphysics and its relationship to the concept o f empirical experience emerge, which I propose to list briefly, with explanations, and to follow with a hypothesis which will be given more detailed justification in the second part o f this work. 1 Empirical experience and metaphysics are opposing concepts which are inseparably linked with one another in the mode of cognition available to a finite being\ An ‘infinite intellect’ is ignorant o f metaphysics, as its omniscience rejects any spheres that transcend direct insight. Nor will it acknowledge any form o f empirical experience, for its omnipotence will not entertain the use o f instruments as a means o f cognition for fear o f a contrary result.41 A finite intellect, on the other hand, is committed to the employment o f instruments, for it knows itself and the tools o f its cognition to be parts o f the world which it endeavours to comprehend, and must, therefore, employ some ‘part o f the world’ as an organ in order to grasp the whole o f the world. It has to direct its inquiry empirically to the parts accessible to it and to imagine metaphysically the whole which is hidden from it. The finite intellect can only research the parts by Unking them hypothetically to a whole that it senses must be there, and it must prove the hypothesis o f the whole by testing it 41 Cf. pp. 28 above.

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experimentally against its cognition o f the parts. Yet what continually renews the possibility o f this link between the whole and the part, between the hypothesis and the experiment, is the procedure o f embodiment in an instrument and the ‘judgements o f appropriateness’ which relate to that. The testing o f such judgements is the means whereby a finite intellect can attain to empirical answers through metaphysical questions. II The concept of the ‘infinite intellect* (which I have employed for the purposes o f demonstration) is not something that can be realized in metaphysical terms, but a mathematical metaphor. The impossibility of its objectivization is capable of analytical proof. We must be clear that, however paradoxical it may sound, the concept o f the infinite being is much weaker than that of the finite. Let us try, in so far as it is possible without self-contradiction, to imagine a ‘god’ o f infinite gifts: we shall find that he— like Locke’s ape— recites identical propositions. There can be no real cognition for him, as in his omniscience every insight is a reflection. N or can there be real action for him, as, given his omnipotence, every act is a metamorphosis. Such a god can only ever be imagined as playing a game, but not even, like an artist, playing the game seriously. Even beauty would never take on reality for him, as he would always be able to ascertain the perfection o f a figure, such as an artist projects into a piece o f marble, without first having to hew it out o f the marble as the artist does. This divine being, intended to be superhuman, proves on closer inspection to be half human— a fiction which originates when man is stripped o f that aspect which constitutes the seriousness o f his existence. For man decisions have consequences, as through these he intervenes in reality and is subject to the fate o f all reality, the fate o f becoming historical. For this god, however, decisions have no consequences, as he himself is the one who determines reality. Thereby, however, he suspends his own reality. This argument might jocularly be designated an ‘ontological rejection o f God’, for a being is demonstrated here in whose ‘essence’ his ‘non-existence’ lies contained— given the pre­ supposition that ‘heteronomy’ is the condition o f existence. The assertion that empiricism and metaphysics are reciprocal concepts may be viewed as a peculiar inference from this presupposition.

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in Wherever the phenomenon of embodiment is encountered, it documents the achievement of a finite consciousness and bears theform of historicity. The act o f embodiment would have no meaning for an ‘infinite intellect’ capable o f surveying the whole o f the world. N or does it, however, have any meaning for a finite being which remains completely committed to its conditioned existence as a part o f the world. What this act presupposes is a finite being in whom reflective contemplation has commenced, i.e., whose consciousness is such that the concept o f the possibility o f a thing can be separated from the experience o f its tangibility and so can address itself to structures o f ideal significance which are in need o f realization. It is, after all, ideas, concepts, visions (in scholastic terms: universalia) which are to be embodied, and in confrontation with which the question o f whether they can be embodied or not acquires the status o f a problem in the first place. However, a being which fives in a state o f tension between the idea and its realization fives subject to the conditions o f historicity. N ot only does its conception o f ideas grow out o f a historical situation, but the means which it can employ for their embodiment are also historically conditioned. The act o f embodiment, however, in which the ideas and the means are united, goes beyond the given situation and becomes a force which furthers history, for in the assertion that these (historically given) means are appropriate for the embodiment o f these (historically conditioned) ideas there lies a hypothesis concerning the nature o f the world, a judgement, which, no matter to what extent it may arise from a historically conditioned perspective, transcends this historical situation in its claim to validity. It is one o f those judgements which, proceeding from the conditionality o f the part, concern themselves with the whole o f the world, and through their success or their failure alter the historical situation out o f which they grew. The progression o f experience which takes place in this continuously renewed contemplation o f ideas, linked with the continuously renewed testing o f them through embodiment, is thus itself a historical process, and, in order to be correctly understood, requires a modification o f the concept o f experience. ‘Experience’ is for a god a meaningless concept; for an animal which is ignorant o f the ambitions and embarrassments o f

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intelligence it is a biological process o f accustomization. For the human being, however, experience is testing through realization, i.e., history. It is therefore— in the exact sense o f the word— only for him that ‘memory’ exists. IV

In the history of the sciences ‘embodiment’ takes the form of a continuous process of selection between ‘real’ and ‘neutral’ hypotheses. In this context, those hypotheses are to be regarded as 'real’ which prove capable of confirmation or refutation by experience, whereas all those which are not exposed to this danger are to be regarded as ‘neutral’. The decision as to whether a certain hypothesis amounts to a real or a neutral hypothesis is itself a real hypothesis. To suggest that the decision as to whether a proposition can be proved by experience or not can itself be tested by experience is not by any means a circular argument, but only an indication that we are here concerned with one o f the ‘fundamental principles’, for the property o f self-reference is common to all ‘fundamental principles’. The law o f identity itself obeys the law o f identity, and the law o f contradiction is subject to the law o f contradiction. Thus the principle o f observational verification is itself subject to this same principle; the assertion that something is capable o f being tested by experience is itself capable o f being tested by experience. Planck described this state o f affairs in the following words: ‘The question whether a physical entity can in principle be observed, or whether a certain question is meaningful in a physical context, can never be answered a priori, but only from the standpoint o f a given theory. The distinction between different theories consists precisely in the fact that according to one theory a certain entity can in principle be observed, and a certain question have meaning in a physical context; while according to another theory this is not the case. For example, according to the theories o f Fresnel and Lorentz, with their assumption o f a stationary ether, the absolute velocity o f the Earth can in principle be observed; but according to the theory o f relativity it cannot; again, the absolute acceleration o f a body can in principle be observed according to Newtonian mechanics, but in relativity mechanics it cannot. Similarly the problem o f the construction o f a perpetuum mobile had a meaning before the principle o f the conservation o f energy was introduced, but ceased to have a meaning after its introduction. The choice between these

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two opposed theories depends not upon the nature o f the theories in themselves, but upon experience. Hence it is not sufficient to describe the superiority o f quantum mechanics, as opposed to classical mechanics, by saying that it confines itself to quantities and entities which can in principle be observed, for in its own way this is true also o f classical mechanics. We must specify the particular entities or quantities which, according to quantum mechanics, are or are not in principle observable; after this has been done it remains to demonstrate that what has been asserted agrees with experience.’42 v The distinction between real and neutral hypotheses is also applicable to the history ofphilosophy. Here it supplies a methodological tool whereby the truth content of philosophical systems can be critically extracted without their historical position being affected. The decision of a philosopher as to whether one of his pronouncements amounts to a ‘real’ or a ‘neutral’ hypothesis is itself a real hypothesis. On the other hand, the answer to the question whether the real or the neutral hypotheses are the ‘truly philosophical ones’ is itself a neutral hypothesis. The attempt has been made to draw a clear Une between philosophy and individual sciences through the claim that the progressive separation o f real from neutral scientific hypotheses amounts to a progressive emancipation o f science from philosophy. This suggestion has been taken up by philosophers working in epistemology. They left it to the scientists to torment themselves with real hypotheses, whilst regarding themselves as administrators o f ideas which are totally incapable o f contradiction by experience. Unfortunately, they overlooked the fact that the decision as to whether or not an idea can be refuted empirically is itself a real hypothesis. Thus something took place which really ought not to have done so: with the progression o f science one epistemological axiom after another has proved accessible to being tested by empirical experience, and the philosopher found himself reduced to smaller and smaller remnants o f ideas, to the provisional administration o f which he laid claim. This ludicrous rearguard 42 Planck, Weltbild (1929), 42-3 [repr. as ‘Zwanzig Jahre Arbeit’ (1929), 215; Physikalische Abhandlungen und Vorträge (1958), iii. 201; English transi. Johnston (1931), 45 - 7 ]-

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action has its basis in a concept o f ‘philosophy’ and ‘science’ which only allows the inquirer to regard a problem as ‘strictly scientific’ when it ceases to be ‘merely philosophical’ , whereas the privilege is conceded to the philosopher that he may only accept a question as being part o f his province when it ceases to be ‘merely scientific’ . But what is meant by this distinction? If it is intended to signify that a question which becomes accessible to proof by experience thereby ceases to be decidable a priori, then that is doubdess correct. If, however, the meaning is that philosophy by its nature only has to do with questions which are decidable a priori, and that philosophy’s interest in a question ceases once it can be confirmed or refuted empirically, then that in itself is a decision a priori which no one can be prevented from making, unless it is demonstrated to him that he himself would not accept the consequences o f this doctrine. There should be little difficulty in proving this in a specific case, for if someone asserts with regard to a philosophical idea that it can be neither confirmed nor refuted by experience, then only two possibilities exist: either he is right, and then this idea is incapable o f embodiment, and the idea remains a mere ‘thought-entity’. Or, on the other hand, he is wrong, and then something real has been accidentally expressed by means o f precisely this idea, and it thus succumbs to history. Those philo­ sophers, however, who would dearly love to express something real without thereby succumbing to the historical process are playing a futile game with the idea o f the infinite intellect. There are a number o f inferences which derive from these thoughts with regard to the assessment and criticism o f transcen­ dental philosophy, which I shall present in the form o f propositions with appended explanations. § 1 1 . Transcendental Philosophy and Experimental Method I Kant’s position on metaphysics (and in particular his verdict concerning the relationship between metaphysics and empirical experience) can be clearly defined by invoking the distinction between *\real’ and ‘neutral’ hypotheses. The meaning of this distinction is anticipated by Kant himself in the distinction between ‘analytics’ and ‘dialectic’, but it is consciously carried through in such a way that metaphysical hypotheses prove, as experience is

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constructed, to be either neutral, on the one hand, or, on the other hand, indispensable. This might virtually be regarded as a definition of the \transcendental m e t h o d i t sees empirical experience as founded upon metaphysical presuppositions, but it regards these presuppositions as neutral. They thus become ‘meaning-giving principles’, which make experience possible without themselves being refutable by experience. This theory of the neutrality of meaning-conferring principles— the theoretical basis of the Kantian concept of autonomy and freedom— is itself, however, a real hypothesis, i.e., it is capable of being confirmed or refuted by experience. It is immediately apparent that the concept o f the ‘neutral hypothesis’ relates to Kant’s concept o f ‘dialectic’ , for in Kant’s view those propositions are ‘dialectical’ in which reason enters into conflict with itself because it seeks to decide upon questions which go beyond the limits o f its use o f experience. If Poincaré, for example, were right in his hypothesis that the applicability o f a particular geometrical theory cannot be decided by experience, then the question whether the cosmos is Euclidian or nonEuclidian would be— in Kant’s terminology— a dialectical question. On the other hand, the transition from ‘dialectic’ to ‘analytics’ in Kant is based solely upon restricting the use o f reason to experience. It is unthinkable for Kant that the use o f reason might, above and beyond this, be tested by experience, as happens in the transition from neutral to real hypotheses, for the principles o f reason can, in his view, never be affected by experience. In the retreat from dialectic to analytics these principles transform themselves, without changing their logical structure, which is determined by the functions o f judgement, and are changed from being metaphysical (transcendental) principles— which demand that we follow through the sequence o f conditions to the point o f the unconditional and thus lead us astray into vacuous and contradictory propositions— into methodological (transcendental) principles, which teach us to restrict progress through the sequence o f conditions to the data o f experience, whereby our judgement is once and for all removed from any danger o f contradiction. This is because the only conceivable form o f contradiction— presupposed as self-evident in Kant— is reason s contradiction o f itself. In the transcendental use o f reason such a contradiction was inevitable, for there the attempt was made to think o f the opposing aims o f cognition (that directed towards the finite and that directed towards the infinite, that

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directed towards the simple and that directed towards the composite, etc.) as being united in one ‘thing in itselF, in which they perforce, as predicates, cancelled each other out. In empirical usage, however, this danger o f mutual exclusion cannot exist, for here the opposing aims o f cognition prove to be complementary ways o f cognizing objects, which are all the more incapable of contradicting one another in relation to these objects in that they themselves, after all, only amount to the guidelines according to which these objects are cognized. In this ‘restricted’ sense the principles o f reason are, therefore, refutable neither by themselves nor by experience. This theory, however— that the meaning-conferring principles o f experience cannot be refuted empirically— owes its per­ suasiveness, retained until the present day, to a purely analytical consideration: it relies upon the presumption that the assumption o f its opposite would include a self-contradiction. Indeed, the thought that the meaning-conferring principles which make experience ‘possible in the first place’ might themselves be con­ firmed or refuted by experience amounts to a circular argument. It is, however, precisely this circular argument which science passes through in full consciousness when it proceeds empirically, i.e., experimentally. As has been shown, this vicious circle is free o f all erroneous conclusions because by an act o f embodiment a metrical element has been inserted into it, through which a logical requirement has been transformed into a metaphysical question to which there are empirical answers. The embodiment in the instru­ ment and the ‘judgements of appropriateness’ which are linked with it characterize with irrefutable clarity a point at which transcendental philosophy enters into open conflict with the experimental method, as the latter here completes that circle o f thought which transcendental philosophy deems ‘unthinkable’ , and which it must continue to regard as unthinkable as long as it wishes to remain true to its basic concept o f autonomy. As the experimental method is put into practice, the belief that rational judgements cannot contradict themselves in the realm o f experience, because they there relate to the way in which objects are perceived and conceived, is also destroyed. This is because through the act o f embodiment that step is put into practice within the sphere o f experience which Kant believed he had prevented once and for all by limiting reason to experience: a judgement

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which expresses our way o f ‘cognizing’ objects is changed in meaning, by being transformed into a ‘judgement o f appro­ priateness’ , so that it relates to the properties o f these objects themselves. These objects, however, are no longer conceived as mere ‘thought-concepts’ (Kantian ‘things in themselves’), but as empirical instruments whose peculiar nature it is to serve as means for our cognition o f the world on the one hand, but on the other hand to be themselves parts o f the world which is to be cognized. This double relation affects the judgements which orient themselves by these instruments in the following way: for such time as they determine the use o f the instruments, their meaning is transcendental; as soon as they determine the appropriateness o f the instruments for this use, however, their meaning becomes transcendent. Before a judgement o f appropriateness can be made in the first place, the choice must be made (with Poincare) as to whether the world to which the instrument belongs is to be regarded as Euclidian or non-Euclidian, and the decision must be made (with Kant) as to whether the totality o f this world is to be viewed as finite or infinite, the elements o f the world as simple or composite, the events within this world as determinate or indeterminate, etc. These are all decisions which, precisely because they are transcendental (or, as Poincare would put it, conventional), cannot contradict themselves in the empirical sphere, either in Kant’s or in Poincare’s view. They do, however, contradict themselves when it comes to the point that, in determining the use o f the instruments, they also anticipate their behaviour—and this contradiction is inevitable. Once related to the behaviour o f the instruments, such alternatives as ‘Euclidian : non-Euclidian’ , ‘simple : composite’, ‘determinate : indeterminate’ exclude one another in such a way that one o f these assertions can only be true to the extent that the other, the opposing assertion, is false. In the system o f the judgements o f appropriateness’, which express the relationship between the instruments, these determinations are just as irreconcilable as in the metaphysical judgements which relate to Kant’s ‘things in themselves’ . In order to ‘rescue’ the meaning o f the transcendental doctrine, it would therefore be necessary to ‘prohibit’ all judgements o f appropriateness (and thus, any transcending o f reason by embodiment) in the same way that Kant prohibited metaphysical judgements, i.e., all transcendence o f reason by itself. But such a prohibition would certainly not do

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justice to the sense o f the Kantian doctrine, for the judgements of appropriateness are, after all, the means whereby— in accordance with Kant’s stipulation— the transcendental use o f reason is limited to experience. At the same time they denote that point at which— in total contrast to Kant’s concept o f the transcendental and his restriction to experience— empiricism and metaphysics meet. II

Kant’s attempt to secure the transcendental method against empirical and metaphysical objections is only meaningful as long as the sensualistic concept of empiricism and the ontological concept of metaphysics are presupposed. If, on the other hand, the concept of empiricism is reshaped according to the prescriptions of the transcendental method, this method loses its oum neutrality and submits itself to a transcendental authority. For Kant, precisely because, according to his definition, it related only to the way in which we have knowledge o f objects, the transcendental method was both empirically and metaphysically neutral. Therein lay for him its ethical pathos, and at the same time its theoretical investigative power. In Kant’s view the empiricist, when he waxed philosophical, was compelled to fall victim to the scepticism o f Hume and to despair o f the use o f concepts because o f excessive faith in the evidence o f his senses. The metaphysicist as viewed by Kant, however, untroubled by sensory perception, was clutching a manual o f Wolffian ontology in his hand and believed that merely by combining concepts he could make a dogmatic statement about reality. Faced with these two extreme positions of the philosophical spirit, which assume in the same way, although in opposite directions, one kind o f objectivity as certain, only then to devalue another, Kant asked under what conditions objects can be comprehended by us in the first place. And by demonstrating that the functions o f reason and sensory perception are equally necessary in order to construct an object for our consciousness, while neither o f the two is sufficient to effect this construction on its own, he opened the way, in presenting the mutual relationship between the different functions o f our cognition, to a whole set of tasks for philosophical thought, in whose vicinity it might feel safe from empirical and metaphysical attacks alike. From this position Kant could argue against the metaphysicist that without experience all ideas concerning reality are mere castles in the air— and against the empiricist he could argue that without ideas experience could not

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materialize at all. His own ‘transcendental’ doctrine, on the other hand, was intended to confer upon philosophy a new solidity in that it sought to establish its foundation in the form o f consciousness itself. In their exuberance, by denying the limits o f our cognition, the metaphysicists had transposed the foundation into the ‘things in themselves’ , which are totally unknown to us, whereas the despairing empiricists— in full consciousness o f these limits— had questioned whether it might be traced at all. Kant positioned it in the a priori conditions o f cognition, which, precisely because they made experience possible in the first place, were not refutable by any empirical content. The strange thing about this doctrine is that the philosophical certainty which it promises can only continue to exist for as long as that very belief in the concept o f metaphysics and empiricism is retained which it so emphatically demands should be dissolved. As long as the empiricist remains a sceptical sensualist and the metaphysicist concerns himself with dogmatic conceptual specu­ lations, i.e., for as long as both remain at the stage o f pre-critical philosophizing, for such time— but only for such time— the transcendental philosopher is secure against empirical and meta­ physical objections. At that point, however, at which the recognition o f Kant’s two theories: 1. that without the use o f reason an experience is absolutely impossible, 2. that only through experience can reason acquire true meaning, leads to a reformation o f the concepts o f the empirical and meta­ physical, two other theories o f Kant, which he links with these first two, must lose their validity, i.e.: 3. that the principles o f reason in their use o f experience may be regarded as certain a priori; 4. that in their metaphysical use they must be rejected as unfounded a priori. The empiricist who recognizes that every step in the field o f scientific inquiry is guided by anticipatory ideas can confront the philosopher, who not only supports him in this concept (cf. proposition 1) but moreover asserts the autonomy o f these ideas (cf. proposition 3), with the experimental process, in the course o f which ideas are tested by embodiment in such a way that their existence is no longer dependent upon themselves, but upon the assent o f a transcendental instance. The metaphysician, however,

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only needs to show that these are transcendental questions whose validity is tested in this fashion by experiments, and he will be able to maintain that metaphysical propositions do not necessarily need to be (as proposition 4 declares) meaningless formulae, in which reason falls victim to ‘transcendental semblance’. Rather, as proposition 2 requires, they can have a ‘real’ meaning— ‘real’ in the sense that the validity o f the metaphysical proposition can be reduced to the validity o f a proposition which is susceptible o f experimental verification. On the basis o f the theory o f the experiment developed above, the third and fourth o f Kant’s propositions must be confronted with the double hypothesis: 1. that the principles o f reason are subject in their empirical use to a metaphysical (transcendental) control; 2. that in their metaphysical (transcendental) use they are capable o f proof by empirical criteria. This may also serve as a basis for understanding the hypothesis which forms the subject o f the second part o f this work. In anticipation, it may be formulated in the following way: The problems posed in the ‘cosmological antinomies’ can be transformed from dialectical problems, i.e., neutral hypotheses, into analytical problems, i.e., real hypotheses. The metaphysical questions which they pose prove themselves capable o f being decided empirically through the application o f the theory o f the experiment.

PART TW O ❖

The ‘Experimental Reduction’ o f the Cosmological Antinomies ‘To man, propose this test— Thy body at its best, How far can that project thy soul on its lone way?’ B

ro w n in g,

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§12. The Empirical Criteria o f Metaphysics ‘We may blunder in various ways in metaphysics without any fear o f being detected in falsehood. For we can never be refuted by experience if we but avoid self-contradiction, which in synthetic, though purely fictitious propositions may be done whenever the concepts which we connect are mere ideas that cannot be given (as regards their whole content) in experience. For how can we make out by experience whether the world is from eternity or had a beginning, whether matter is infinitely divisible or consists o f simple parts? Such concepts cannot be given in any experience, however extensive, and consequently the falsehood o f either the affirmative or the negative proposition cannot be discovered by this touchstone.’43 Kant makes no bones here about his belief that metaphysical propositions are condemned to remain neutral hypotheses, as exper­ ience supplies no criterion to test their validity. For can experience— Kant asks— ever extend to the whole o f the world? Is not this whole itself a mere idea which can never ‘be realized in its entirety in experience’? 43 Prolegomena zu einer jeden kiinjtigen Metaphysik, die als Wissenschaft wird auftreten kônnen (1783), §52b [Kant (Akademie-Ausgabe), iv (19 11), 340; English transi. Carus and Ellington (1977), 81].

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H ow we stand on this question depends upon what meaning is associated with the expression ‘realized in experience*. If we understand by this that the idea in question is to be represented as an object of sensory perception, then it is indeed the case that experience cannot supply any criterion for metaphysical propositions; for the universe as a whole can never become the object o f sensory perception. If, however, it is thought that the idea in question is to be tested by experiment, then it is entirely within the realm o f possibility that experience may supply a criterion for metaphysical propositions; we would only have to be able to prove for this purpose that meta­ physical propositions can be embodied in instruments (i.e., reduced to experimental propositions). The question o f the empirical criteria o f metaphysics thus transforms itself into the more general question o f the extent to which judgements may be confirmed experimentally. Are propositions which relate to the universe as a whole capable o f being embodied at all? In the ‘Theory o f the Experiment’ (Part I) it was maintained that in every judgement o f appropriateness there is implicidy contained a judgement concerning the position o f the instrument in the universe, and thus concerning the character o f the universe as a whole, in so far as it affects the behaviour o f this instrument. In the light o f this theory every empirical judgement has a metaphysical meaning and is dependent for its validity upon the metaphysical sanction which lies in the ‘success’ o f the experiment. Conversely, if this theory is to prove correct, then an experimental meaning must be derivable from every metaphysical judgement, in so far as it is a real judgement. This means that it must make an empirically determinable difference if we choose the one or the other o f two opposing metaphysical hypotheses. If, therefore, Kant in each case advances two contra­ dictory propositions with regard to the concept o f the universe and the concept o f the atom, and the relation between causality and freedom and that between contingency and necessity, each o f which is held to be demonstrable on the grounds o f reason and neither refutable on the basis o f experience, then it is our task to investigate whether this dispute cannot after all be setded if, instead o f relying upon experience through sensory perception, which is indeed incapable o f deciding the question, we refer to experience through experimental investigation. That such reference to experiment can be fruitful is to be proved in what follows for all four o f the cosmic antinomies as advanced by

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Kant. In the interests o f clarity I shall first set out the propositions which are to be tested in Kant s own formulation:44 The world has a beginning in time and is also enclosed within bounds as regards space.

The world has no beginning and no bounds in space, but is infinite as regards both time and space.

Every composite substance in the world consists o f simple parts, and nothing at all exists but the simple or what is composed o f it.

N o composite thing in the world consists o f simple parts, and there exists in the world nothing simple at all.

The causality according to laws o f nature is not the only causality, from which the appearances o f the world can thus one and all be derived. In order to explain these appearances, it is necessary to assume also a causality through freedom.

There is no freedom, but everything in the world occurs solely according to laws o f nature.

There belongs to the world some­ thing that, either as its part or as its cause, is an absolutely necessary being.

There exists no absolutely necessary being at all, neither in the world nor outside the world, as its cause.

For each o f these antitheses a twofold investigation must be undertaken: 1. The subject o f the contradiction must be logically determined, i.e., the ‘tertium’ or middle term with reference to which the opposed propositions enter into conflict must be precisely defined. 2. This ‘tertium’ must be investigated to see whether it can be made the subject o f a real hypothesis, i.e., whether the propositions relating to it can be embodied in experimental form. The direction taken by such an exploration is not intrinsically alien to Kant’s own. Kant himself in the introduction to the second edition o f the Critique of Pure Reason said o f his own ‘method, which imitates that o f the investigator o f nature’, that it ‘consists in searching for the elements o f pure reason in what can be confirmed or refuted by an experiment\ 45 For him, o f course, since he regarded the limit o f 44 [Kritik der reineti Vemunjt, B 454 f./A 426 f.; B 462 f./A 434 f.; B 472 f./A 444 f.; B 480 f./A 452 f.J 45 Kritik der reinen Vemunjt, B xviii note.

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experience as the limit o f that which can be presented in sensory perception, the testing o f the propositions o f reason could not be a real experiment but only an experiment in the mind. For ‘the propositions o f pure reason, especially i f they venture beyond all bounds o f possible experience, cannot be tested by doing (as we do in natural science) an experiment with their objects. Hence testing such propositions will be feasible by doing an experiment with concepts and principles that we assume a priori!*6 Even with regard to these concepts and principles, however, the experiment could never pursue the aim o f testing the validity o f the ‘elements o f pure reason’ in the manner o f hypotheses; for ‘as regards certainty’ Kant declared himself bound by his own verdict: ‘that holding opinions is in no way permissible in this kind o f study; and that whatever in it so much as resembles a hypo­ thesis is contraband, which is not to be offered for sale at even the lowest price, but must be confiscated as soon as it is discovered. For any cognition that is to hold a priori proclaims on its own that it wants to be regarded as absolutely necessary.’47 Therefore it was only for appearance (to continue the parallel with natural science) that Kant introduced the distinction between appearances and things in themselves as a hypothesis in the preface, ‘even though in the treatise itself it will be proved, not hypothetically but apodictically, from the character o f our presentations o f space and time and from the elementary concepts o f the understanding’ .48 But this seeming corres­ pondence with the method o f the natural scientist was only rendered possible by the fact that the dialectical conflict o f reason yielded for Kant ‘a splendid touchstone’ by which to emphasize the obligatory nature o f that distinction. The analogy with the experiment lay for Kant in the fact ‘that the same objects can be contemplated from two different standpoints: on the one hand, for the sake o f experience, as objects o f the senses and o f the understanding; yet on the other hand [...] as objects that we merely think. N ow if it turns out that contemplating things from that twofold point o f view results in harmony with the principle o f reason, but that doing so from one and the same point o f view puts reason into an unavoidable conflict with itself, then the experiment decides in favour o f the correctness o f distinguishing the two points o f view.’49 46 Kritik der reinett 47 Kritik der reinett 48 Kritik der reinen 49 Kritik der reinen

Vernunfi, Vernunfi, Vernunfi, Vernunfi,

B A B B

xviii note. xv. xxii note. xix note.

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The proposition which, in this ‘experiment’ , is not to be tested so much as to be formulated with regard to its meaning (for in Kant’s view it is by no means necessary for it to be tested) is the distinction between ‘appearances’ and ‘things in themselves’ . The object, however, by reference to which this distinction is to retain its meaning and towards which in consequence the whole experiment is directed, is precisely that ‘tertium’ in which the conflicting propositions o f reason meet. The success o f the experiment thus consists in the fact that this middle term disintegrates as soon as it is contemplated from the point o f view o f Kant’s distinction between appearances and things in themselves. In the first two antinomies it proves to be a nonsensical concept, in the latter two an ambiguous one. For in the formation o f the expressions ‘world as a whole’ (the universe) and ‘world-element’ (the atom), it is implied that two determinations are linked in respect o f spacial extent which, once that Kantian distinction is introduced, cannot on any account be brought together in the concept o f one and the same property o f a thing: their capacity to be experienced and their determination in themselves. In the opposition o f ‘natural law’ and ‘freedom’, however (as in the corresponding antithesis o f ‘contingency’ and ‘necessity’), two kinds o f explanation are posited as incompatible, which with the aid o f that distinction are nevertheless applicable to one and the same thing, as the latter can be regarded in one respect as bound by nature and accidental, in another respect as free and necessary. Thus the key for the solution o f the conflict o f reason has been found. The conflicting propositions in the first two antinomies must, as they relate to a meaningless concept, both befalse. ‘ For the logical criterion o f the impossibility o f a concept consists in this, that if we presuppose it, two contradictory propositions both become false; consequendy, as no middle between them can be thought, nothing at all is thought by that concept.’ s° In the two latter antinomies, however, both thesis and antithesis can be true, as they relate to two different concepts and ‘are opposed to one another merely as a result o f misunderstanding’. Thus, by virtue o f the distinction between appearances and things in themselves, reason is brought into harmony with itself, and this success compels us to accept that distinction as true. ‘This experiment o f pure reason is very similar to that done in chemistry, which is called sometimes the experiment o f 50 Prolegomena, §52b [Kant (Akademie-Ausgabe), iv (1911), 341; English transi. Carus and Ellington (1977), 82].

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reduction, but generally the synthetic procedure. The analysis o f the meta­ physician has divided pure a priori cognition into two very hetero­ geneous elements, viz., such cognition o f things as appearances, and o f things in themselves. His dialectic recombines the two in order to yield agreement with reason’s necessary idea o f the unconditioned, and finds that this agreement can never be obtained except through that distinction, which is therefore a true one.’51 It is here, however, that for Kant, the man o f the Enlightenment, the interest in this whole conflict lies: impelled towards a solution on rational grounds, he compels us to discover and recognize a principle which proves the agreement o f reason with itself by a distinction which reveals to us the peculiarly limited nature o f our capacity for cognition. That the apodictically deducible limits o f cognition form an indispensable piece o f evidence in a process directed towards the self-justification o f reason— to prove this is the real aim o f that experiment in thought which is called the ‘resolution o f the anti­ nomies’ . In this experiment— however paradoxical it may sound— a rational reason for the limitation to experience is adduced. The purely theoretical resolution o f the antinomies, as carried out in this experiment o f thought, has recourse to twofold means: on the one hand it appeals to the ‘agreement with pure reason’ as the highest instance, in order to enforce the accord with a principle. On the other hand it supplies the apodictic deduction o f this very principle from the ‘nature o f our capacity for cognition’ . The first criterion supplies the requirement that the antinomy must be resolved by argument; the second criterion supplies the explanation as to how it can be resolved a priori. The purely theoretical treatment o f the antinomies only makes sense, therefore, if the following two presuppositions are accepted: if (i) a rational proof is considered adequate to solve a cosmological problem; if (2) we believe on the basis o f a fixed concept o f reason in the apodictic deducibility o f the conceptual distinction which renders the resolution possible. If this belief is abandoned, then we cannot continue with the experiment in thought, but must pass on to an actual experiment. The analysis o f concepts can then no longer serve to ‘resolve’ the ’tertium’ deductively on the basis o f axioms; instead it must try to render it precise in such a way that it is accessible to experimental proof. The method o f embodiment with its two distinct stages— (1) establishing the possible alternatives through conceptual 51 Kritik der reinen Vernunft, B xxi note.

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analysis; (2) deciding between these alternatives by relation to the experiment— thus reaches that exact point at which the distinction between appearances and things in themselves stands in Kant. We must therefore ask with regard to the first antinomy: what spatial structure is accorded to the universe if it is maintained that it can be thought neither as finite nor as infinite? And is this particular kind o f structure actually embodied in the world? Where the second antinomy is concerned, we must ask: exacdy what kind o f relation o f the whole to its parts is presupposed if it is asserted that the world may consist neither entirely o f simple parts nor entirely o f compound parts? And is it possible to realize precisely this kind o f relation between the whole and the part, this form o f composition, in an experiment? With regard to the third antinomy, too, this twofold questioning makes good sense. Even the causal principle can be seen to be inadequately justified when it is demonstrated that it fulfils a logical function as a principle o f explanation. Moreover, the question must be put as to whether it proves itself valid as a hypothesis concerning the character o f the world. This, however, depends upon whether and to what extent the meaning indicated by this principle may be objectivized through embodiment. From the point o f view o f Kant’s presuppositions this question is, indeed, incomprehensible; and recendy, too, there has been an attempt to reject it as meaningless. ‘The law o f causality is concerned’ , writes Hugo Bergmann, ‘not with an assertion— which would have to be either true or false— but with an instrument’ ; and he warns against ‘mistaking the symbol for the reality on the basis o f a misunderstanding o f the systematic signifi­ cance o f the categorical assumptions o f science’ .52 However, precisely with regard to this distinction between ‘assertion’ and ‘instrument’ , between hypothesis and category, it is impossible for us to call a halt, i f we wish to pursue the categorical assumptions o f science to the point o f their experimental consequences. For it is precisely in science that the use o f the instrument is based upon a hypothesis concerning its appropriateness; and every judgement concerning appropriateness is an assertion which must be either true or false, and which will prove to be either one or the other when the experimental method is put into practice. Only those who believe a priori in the ‘neutrality’ o f the categorical assumptions— whether because, in the Kantian sense, they 52 Bergmann, Kausalgesctz (1929), 74.

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regard their origins in reason as proof o f their autonomy, or because, like Poincare, they neutralize the facts o f experience in the face o f all categorical problems— can see a danger in ‘taking the symbol to be the reality’. On the other hand, those who approach every such assumption with the postulate o f embodiment know that there is contained in every specific type o f symbolization, in an epistemological context, a hypothesis concerning the reality which is to be com prehended, and they w ill attem pt to test its validity as a hypothesis

by reducing it to propositions susceptible o f experimental proof. In what follows such a reduction will be attempted for the antinomies.

CHAPTER

1

The Antinomy of the Concept o f the World §13. Clarification o f the Question o f the First Antinomy (Refutation o f RusselVs Objection) Kant’s first antinomy (in contrast to the other three) has a twofold subject; it relates both to time and to space. Hence the method o f proof for the thesis and antithesis divides itself into two parts, and the proof relating to time takes precedence over the proof relating to space. With regard to the substantiation o f the antithesis in which it is to be proved that the universe cannot be finite, the sequence o f proofs is irrelevant, for they are adduced independently o f one another. It will be shown that, in order to conceptualize a finite universe, it would be necessary to assume not merely a relation between appearances within space and within time, but also a relation to space and to time, both o f which must be considered empty; this is impossible, at least in an empirical procedure. However, the sequence o f the two proofs exercises a decisive influence on the proof o f the thesis (according to which the universe is spatially and temporally limited). The proof for space isfounded here upon the proof for time, in that it is shown that for the experience o f infinite space infinite time would be necessary. Time itself, however, cannot be infinite; for that would mean that at every given point in time an eternity would have ‘passed by’. ‘Therefore an infinite bygone world series is impossible, and hence a beginning o f the world is a necessary condition o f the world’s existence— which was the first point to be proved.’53 The mathematicians took umbrage at this proposition. Russell above all declared that, in logical terms, it is purely coincidental that we do not live long enough to be able to perfect an infinite series ‘by 53 [Kritik der reinen Vernunft, B 454.]

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successive synthesis’.54 The nature o f an infinite series is therefore, because o f this impossibility o f bringing it ‘extensionally’ [by enumer­ ating the items] to a conclusion, far from being adequately defined. An infinite series has indeed quite positive qualities, which may be derived ‘intensionally’ [by demonstrating that the items belong to a class] from their belonging to the class o f infinite quantities; for instance, the peculiarity that in this class the part is just as big as the whole. For the Kantian antinomy it may (in Russell’s view) be deduced that the proof for the thesis is erroneous; as it is possible to substitute an intensional definition o f the ‘completed infinity’ for the extensional definition, the concept o f a spatially and temporally infinite universe makes complete sense, and the antinomy is to be decided in favour o f the antithesis. That means, however, that there is in essence no antinomy whatever. Yet this solution itself contains a problem, for if Russell requires o f us that, in order to suspend the antinomy, we substitute the ‘intensional’ definition for the ‘extensional’ , there then arises the question o f the possibility and justification o f this very substitution. How do the two kinds o f definition relate to one another? In Russell’s view, the difference is purely psychological, and the examples which he tends to prefer in his writings are o f such a nature as to support this view. If we wish to define the character o f the inhabitants o f London, we have absolutely no need to go through them one by one, although we can o f course choose this path, if we so desire. It is much easier, however, to visualize the climatic and sociological conditions in which Londoners live, and derive from these that feature which ‘necessarily characterizes’ the ‘class’ o f those who live in London. Or— to take Russell’s favourite example— if we wish to study the social relationships in a monogamous state, there is no need to take individual count o f its male and female citizens in order to find that the number o f husbands and wives in such a state is equal. Instead, the numerical equality o f the spouses derives automatically from the very concept o f the monogamous state. Yet what— we have to ask— if this concept is not merely to be phrased as an idea, but verified by the facts? What i f I am not content to know that in every state which is 54 Russell, Principles of Mathematics (1903) [esp. 349, with a definition o f the terms ‘extensional’ and ‘intensional; and 458-61]. C f also id., Knowledge of the External World (1914), lecture vi [‘The problem o f infinity considered historically’, 153-82, esp. 156-60].

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monogamous the number of husbands and wives must be equal, but wish also to know whether in a given state, which is supposed to be monogamous, the number is really equal, in other words, whether monogamy actually prevails in that state? Then, presumably, my only solution is to take each o f the inhabitants individually (or to visualize in detail the means whereby the state compels its inhabitants to observe its laws). And the same applies to the inhabitants o f London, if I have attributed to them— on the basis o f some meteorological observations or other— some arbitrary properties, concerning which I now wish to establish whether they are correct or not. Extensional and intensional determinations thus fulfil entirely different functions. And i f one function is substituted for another, then it is not as a replacement but as a control. Determination by extension is collective and therefore specifically empirical (that is to say, based upon observation). Determination by intension is distributive and therefore specifically logical (that is to say, based upon definition). The concept o f the ‘world’ , however, which is questioned in the first antinomy, is compounded o f logical and empirical determinations. For it is not a question here o f the idea of a quantity which is to be comprehended as a totality independently o f the spatial and temporal conditions o f its capacity for being experienced, but o f the fact o f that quantity, which becomes the object o f experience in space and time. If, therefore, infinity can only be comprehended distributively as a positive property (by virtue o f logical definition), but never collectively (by virtue o f the empirical testing o f facts), then the proof is supplied that this ‘world’ cannot be regarded as truly infinite. Russell would have been absolutely correct in objecting to the method o f proof o f the thesis if Kant had wished to derive from the impossibility o f completing an infinite series in successive synthesis its incomprehensibility as a totality. Kant’s sole aim was, however, to prove that its completion could not be experienced. And this proof remains valid, despite Russell’s introduction o f Cantor’s concept o f the transfinite. For even if this concept makes the infinite comprehensible as a totality, in so far as it attributes to it logically definable properties distributively, yet it is in the nature o f these properties that they are collectively by no means comprehensible. In the relation o f the distributive to the collective (or, to use Russell’s own terminology, the relation o f the intensional to the extensional determination), the very opposition which Kant was concerned to formulate in his antinomy re-establishes itself. The control o f the one function by the other is

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the condition o f its empirical use. The possibility o f this control is, however, limited to a certain category o f objects, that is, to those which, in mathematical terms, represent ‘countable quantities’ . With regard to the inhabitants o f London and the monogamous beings it is therefore understandable if Russell says that the difference between extensional and intensional determination is purely psychological. For here the cognition gained in one way might just as readily have been gained in the other. For this reason the inhabitants o f London and marital relations in monogamous states are also— in the sense o f Kant’s epistemology— ‘appropriate for our empirical concept’ . The peculiar nature o f infinite quantities, however, as Russell himself teaches us, is thought to He precisely in the fact that by intension determinations are attributed to them, which cannot in principle be demonstrated by extension. It is absolutely impossible to verify by extension that the sum total o f all numbers is equal to the sum total o f the square o f all numbers; this is not because I may not live long enough to count the sum total o f these numbers, but because this sum total is by its nature not an extensionally countable quantity. Thus Russell’s introduction o f the concept o f the ‘transfinite’ into the discussion o f the first antinomy achieves something totally different from what was intended. Instead o f contradicting the proof of the thesis and thus securing the content o f the antithesis as thinkable, it supplies a new means for the formulation o f the thesis itself. It may be mentioned in passing that this means was actually employed for this purpose some considerable time ago. Wundt attempted to show through distinguishing between the ‘infinite’ (das Endlose) and the ‘transfinite’ {das Uberendliche) that the thesis and antithesis o f the first antinomy can be valid in the same way, as they are two distinct concepts o f infinity, whose applicability to the world is asserted in one case, but contested in the other. In the antithesis it is maintained that the world must be infinite; and this concept o f infinity is ‘compatible with the empirical view o f the world, because it assumes as given only one step beyond every given boundary, but not infinity itself’ . In the thesis it is maintained that the world cannot be transfinite; and this concept o f infinity is indeed incompatible with the empirical view o f the world, as ‘the transfinite is an absolutely transcendental concept’.55 55 Wundt, Kleine Schrijten (1 91 o), i. 116, cited from Nef, Die Philosophic Wilhelm Wundts (1923), 272-3. [Cf. Winds review o f N e f in The Journal of Philosophy 21 (1924), 498-502.]

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W rong as Russell was to assume that the proof o f the thesis is negated by the concept o f the transfinite, his logical perspicacity is nevertheless on firm ground when he suspects psychologizing in Kant’s use o f the concept o f ‘successive synthesis’ . Certainly, when applied to the passage o f time, this concept is absolutely indisputable, for time does actually present itself in the form o f a succession, and the impossibility o f conceiving o f time without a beginning, that is to say as constandy ‘passed and infinite’ , can be readily deduced from the impossibility o f completing an infinite series extensionally. With regard to space, however, the relation o f the parts to the whole has in the first place absolutely nothing to do with the problem o f ‘successive synthesis’ , ‘for the variety o f a world which is infinite in extent is simultaneously given’ . If Kant none the less believes that the impossibility o f comprehending space as an infinite whole must be deduced from the (already proven) impossibility o f thinking o f an infinite time series as completed, the reason for this is to be sought in the fact that he can visualize a determination by extension, that is, through the progression from the part to the whole, only as ‘successive addition’. Herein lies, however, a ii€Td0 a a is into the psychological. If it were valid to make a link between the concept o f space and the temporal feasibility o f the individual acts o f perception which make space ‘experience-able’ in terms o f its whole content, then we could never measure empirically the distance o f the fixed stars from the Earth. For we are incapable o f ever, in a man’s lifetime, successively living through the ‘light years’ in which we express this distance. Yet just as certainly as in this case we are perfecdy capable o f deducing and measuring by a process of inference a distance through which we could never pass, so the impossibility o f comprehending space as infinite must have a reason other than that we do not have at our disposal the time to construe it by ‘successive synthesis’ . With regard to space, there must be a possibility sui generis here, which derives not from the contingent relation o f space to time, but from the general laws o f extensional determination— from the extensional relation o f the part to the whole, which is applicable to space and time in equal but entirely independent manner. If indeed one contemplates space and time solely from the point o f view o f the extensional relation between the whole and the part, then for the first antinomy a formulation is obtained which is proof, I believe, against all objections o f the Russellian kind:

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If cosmic space is to embrace the whole o f space, then it cannot be a whole (i.e., extensionally completed) space. If it is to be a whole, extensionally completed space, then it cannot embrace the whole o f space, but only a part o f it. 2. If the series o f cosmic events is to embrace the whole o f time, then it cannot be formulated in the form o f a whole period o f time (which has a beginning and an end). If it is to form a whole period o f time (which has a beginning and an end), then it cannot embrace the whole o f time, but only a part o f it. From this kind o f formulation three conclusions are automatically derived: 1. The antinomy o f space is in terms o f its meaning, and therefore also in terms o f its proof, completely independent o f the antinomy of time, not despite the fact that it runs exacdy parallel with it, but because it does so. 2. It is necessary to complement the antinomy o f time, in content, so that it includes not merely the question o f the beginning o f the world, but also that o f the end o f the world, which is curiously lacking in Kant. 3. The root o f the antinomy lies in the irreconcilability o f two concepts o f wholeness, one o f which, viewed subjectively, designates the whole o f the world, while the other, understood in adjectival terms, judges the wholeness o f the world. The contradiction established in the third proposition cannot, however, as one might at first glance imagine, be resolved analytically by a separation o f the two concepts o f wholeness; for no offence against the proposition o f the contradiction can be discovered in the combination o f ideas o f a ‘whole Whole’, try as one will, even when a different meaning is accorded to the two concepts o f wholeness, as sense demands. Yet it is even less the case that dialectical games can be played with this contradiction, for example, by asserting jubilandy that it is in the nature o f the ‘Whole’ that it can never be ‘whole’; for in no way can it be deduced a priori from the concept o f the Whole that the predicate o f wholeness is not applicable to it. If this position is nevertheless maintained, then a middle term has already been interposed, in which both concepts contradict one another. But the responsibility for the contradiction then lies with the middle term, not with the concepts. The logical structure o f the antinomy is indeed to be judged in precisely this way. The root o f the contradiction is to be sought in the

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middle concept, in which the two concepts o f wholeness are united. Our next task is therefore to define the peculiar nature o f this middle concept, and, after this has been done, to ask whether the choice o f this medium cannot be understood as a hypothesis whose validity can be rendered dependent upon the result o f an experiment. §14. The Mathematical Antinomy o f Euclidian Space In so far as infinity implies the impossibility o f extensional completion, a universe which is to be comprehended as a closed whole must necessarily be finite. If, however, finiteness involves the positing o f given limits, then something which is finite is by nature a part o f a whole which is more comprehensive than itself. Therefore a universe which is to be comprehended as an all-encompassing whole must necessarily be infinite. Kant resolved this antinomy by explaining that we can only comprehend existence as conditioned and that we are therefore bound to become entangled in contradictions once we posit the unconditioned as existent. All our cognition is based upon the application o f categories to empirical data, and this application consists in the continuous process o f Unking one empirical piece o f data with another. Any idea which presents this continuous process o f cross-referring in the series o f conditions as completed— and such an idea is necessarily contained in the very concept o f a universe— transcends the borders o f human cognition, for it causes us to forget that this cognition can only be concerned with the correlation o f phenomena, and that it cannot direct itself towards ‘things in themselves’. The thought o f completing the process o f cognition in such a way that it encompasses the totality o f conditions, and hence the unconditioned, can only be admitted as a regulative principle— as an infinite task which is a constant source o f inspiration to inquiry precisely because it can never be fulfilled. Measured against the powers o f human cognition, this idea must, however, appear out o f proportion. Its content must prove either too small or too big for our comprehension, and this is indeed the case with regard to both the alternatives relating to the spatial extent o f the universe. ‘ For if the world is infinite and unbounded, then it is too large for any possible empirical concept. If the world is finite and bounded, then you righdy go on to ask, What determines this boundary? [...] Therefore a bounded world is far too small for your concept.’ 56 56 Kritik dtr reinett Vemunjt, B 515.

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The dilemma that Kant presents here is a necessary conclusion if the Euclidian system of geometry is accepted. Space as construed according to Euclidian ideas (that is, to employ a modern expression, space with a degree of curvature o f zero) can only be determined as finite if it is bordered by outer limits; on the other hand, it can only be comprehended as unlimited if it is defined as infinite. Boundaries, however, cannot be assigned to a universe which is supposed to be allencompassing; infinity cannot be assigned to a universe which is supposed to present a closed whole. The definition o f a universe as closed and all-encompassing thus combines two concepts which are mutually exclusive according to the assumptions o f Euclidian geometry. It follows that if we accept Euclidian geometry as physically valid, then we cannot retain the concept o f a universe, as the latter combines in itself both closedness and boundlessness. If, on the other hand, we accept the concept o f the universe as physically meaningful, then we cannot retain Euclidian geometry, as closedness and boundlessness are self-contradictory therein. For Kant there was no choice here. The second alternative did not exist for him. The only known geometry was Euclidian. Moreover, it served as the basis for the Newtonian system o f physics. But the thought o f abandoning the mathematical propositions which had proved an indispensable precondition for exact science in favour o f a metaphysical idea would have direcdy contradicted the spirit o f the critique which Kant sought to introduce to philosophy. The only feasible path for him lay in proving on the basis o f Euclidian geometry that the idea o f a closed and all-encompassing universe was a mere ‘noumenon’ and therefore void. The capacity to form such void concepts, which prove illusory and contradictory, was explained by Kant on the basis o f the conflict in the human spirit— a spirit which, endowed with intellectual concepts and destined to apply them to phenomena, feels itself constandy driven to go beyond the area o f its own experience by employing those concepts, and to think o f ‘things in themselves’, whereas it is only granted phenomena. The distinction between ‘appearances’ (Erscheinungen) and ‘things in themselves’ (Dinge an sich) is thus introduced as the metaphysical assumption on the basis o f which mathematics and physics can be accorded the greatest possible degree o f certainty within the area o f human cognition, yet only because insurmountable limits have been imposed upon this very cognition.

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Whatever one’s view o f this metaphysical conclusion, it has to be admitted that the analysis on which it is based has unconditional validity. A system o f physics which ascribes universal applicability to Euclidian geometry leads irrefutably to a cosmological antinomy, which can only be resolved on the basis o f metaphysical assumptions. If the concepts o f Euclidian geometry are generally applicable to physical phenomena, then the concept o f a physical universe becomes a contradiction in terms and demands a metaphysical treatment. § 1 5. The Physical Antinomy o f the Newtonian System Obviously it was not this kind o f consideration which caused the physicists to modify the Newtonian system and make use o f nonEuclidian geometry. From a physical standpoint the cosmological antimonies are only o f secondary interest. The physicist is not so much concerned with the question o f how a certain system o f geometry can be reconcilable with the idea o f the universe; rather he seeks to determine to what degree this geometrical system empowers him to explore and describe the laws and the distribution o f matter within that universe. What concerns him is therefore not so much the compatibility o f Euclidian space with a concept o f the cosmos com­ prehensible to the human intellect, as the compatibility o f Euclidian space with the actual behaviour and actual distribution o f matter. If matter from the point o f view o f Euclidian geometry is spread fairly uniformly over the observable parts o f the universe, then there is no reason to doubt that physical space is Euclidian; it is best left to the philosophers to worry about the proof that Euclidian space is incom­ patible with the idea o f a physical universe, and that hence the concept o f a physical universe must be rejected. It is a priori impossible that discussions o f this kind might call into question the validity o f the Newtonian system in physics, for these observations are themselves based upon the assumption that the Newtonian system is physically valid, that is, that physical space is regarded as Euclidian. The real motive for the critique o f the Newtonian doctrine lay in its own stricdy physical conclusions. The distribution o f matter as described by Newton seemed to find itself at odds with the assumption o f Euclidian physical space. For the Newtonian laws, while presupposing Euclidian physical space, render such a relatively

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uniform distribution o f matter within such space impossible.57 Since, according to Newton’s theory, the number o f lines o f force increases in proportion to the mass where they are centred, the assumption o f an average density o f matter within the universe would lead to the conclusion that the gravitational forces increase directly in proportion to spatial extension. In a Euclidian universe the extension o f space would gradually increase to the point o f infinity. As, however, given the assumption o f a uniform distribution o f matter, the intensity o f the corresponding fields o f force would increase in the same proportion, such an infinite universe could not possibly exist. It follows from this that for as long as we retain the Euclidian concept o f space, we must admit that Newtonian laws are incompatible with a uniform distribution o f matter, and we must choose between the two alternatives. The difficulty is that, whatever choice we make, we encounter insuperable contradictions. If we assume, intending to carry through the concept o f a Euclidian universe, that it is certain that the space described by Newton is uniformly filled with matter, then we must modify the Newtonian law and declare that for great distances the force o f attraction between bodies decreases by a factor greater than 1 /r2. This is Seeliger’s celebrated hypothesis o f the ‘absorption’ o f gravitation in the cosmos. Its disadvantage lies in the fact that because it is invented ad hoc in order to meet a theoretical difficulty, it leads to a complication o f the Newtonian formulae which can neither be justified on the basis o f methodological assumptions nor tested by experiments. Consequently it has not found general assent among physicists. The alternative consists in retaining the Newtonian laws in their original form and accordingly abandoning the idea o f a relatively uniform distribution o f matter. This view has, however, to accept the paradox that the distribution o f matter which it postulates is incongruent with the concept o f space which it presupposes. An infinite space, which has been expressly defined as homogeneous in its structure, is filled with matter in such a way that it obtains a kind o f 57 Einstein, Spezielle und allgemeine Relatiuitatstheorie (1917), §30, pp. 7 1- 2 [repr. in Collected Papers, vi (1996), 495-6; English transl. Lawson (1920), 105-7: ‘Cosmo­ logical difficulties o f Newtons theory’]. Cf. the references compiled in Henderson’s paper ‘Is the Universe Finite?’ (1925), in particular the seminal essay by Seeliger, ‘Uber das Newton’sche Gravitationsgesetz’ (1895). [See also the controversy with J. Wilsing: Seeliger, ‘Uber eine Kritik’ (1895); ‘Bemerkung zur vorstehenden Erwiderung’ (1895).]

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central point at which the density o f matter achieves its maximum. The further one moves away from this central point, the more the density declines, until it finally disappears into an infinite void. The physicist who, without questioning the assumptions o f this doctrine, seeks to visualize its physical consequences, will be somewhat discom­ fited by the idea o f a universe which floats like a finite island in an infinite ocean o f space. Light, or some other moveable substance emitted by the stars, would wander off into infinity in such a universe without ever returning in any form to the system from which it emanated; the system would perish as a consequence o f such a systematic process o f depletion. Yet quite apart from these disastrous conclusions, which, however terrifying they may appear, may nevertheless be entirely true in physical terms, this doctrine appears to expose itself to a purely logical critique in that it is derived from an extremely dubious conceptual system. The idea o f defining a given region o f space as a physical midpoint, and another region as physically empty, seems to indicate that Euclidian space does not accord with the Newtonian relations between matter. If physical space is accorded a midpoint, then this contradicts the assumption that its structure is homogeneously Euclidian. If Euclidian space is found to be empty over huge areas, then this contradicts the assumption that it is physically real. The mere thought o f empty physical space seems to signify a self-contradiction, for it seems to define space as physical, only to deny it all physical attributes. Thus we remain on the horns o f a dilemma: if we accept the uniform distribution o f matter, then we can solve the theoretical problem by accepting Seeligers suggestion. Then, however, we must modify Newton’s laws in a manner for which all empirical proofs are lacking. If, on the other hand, we reject Seeliger’s suggestion and retain the Newtonian idea of an irregular density o f matter, then we are led to a concept o f the universe which is logically untenable. Both decisions, however, have one common assumption: the E u clid ian concept o f space. B o th are therefore affected in the same

way by the cosmological antinomy as developed by Kant, and the methodologically important question arises o f whether the Kantian resolution o f this antinomy is applicable to both in the same way, or whether it speaks for one decision and against the other. The question is: what is the relation between the physical dilemma, in the resolution o f which the alternatives diverge, and the philosophical dilemma which remains common to them both?

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§ 16 . The Inevitability o f the Newtonian Antinomy according to the Doctrines o f Kant Let as assume that there might be a methodological way to verify Seeligers alternative. The Euclidian structure o f physical space could then be regarded as guaranteed. Wandering through the universe, we would keep on encountering new stars at fairly regular intervals, and this approximately uniform distribution o f matter would permit us to prolong our journey through Euclidian space for an indefinite distance without ever returning to the same point. In the course of this experience we would form the (valid) view that the physical world is infinite; yet as soon as we wanted to attempt to define it as a universe we would have to capitulate, faced with the impossibility o f comprehending it as a closed whole. In other words, the physical validity o f Euclidian space would expose the Kantian antinomy as inescapable. We could only resolve it by following Kant into the realm o f metaphysical speculation. Would the same apply if we were to choose the other alternative and presuppose an irregular distribution o f matter in Euclidian physical space? In that case the mathematical antinomy, which Kant had defined as a disproportion between the idea o f a Euclidian universe and the determinations o f empirical comprehensibility, would be complemented by a physical antinomy, which indicates a disproportion between Euclidian physical space and the distribution o f matter in this space. The fact that we cannot comprehend Euclidian space as simultaneously closed and all-encompassing would not be the only puzzle. In addition, we would not be able to view space as simultaneously physical and empty. The question is: how does this new antinomy relate to the earlier one? Do the two corroborate each other? Are they detrimental to each other? Or is one irrelevant to the other? The second antinomy is a purely physical one; until it is physically resolved, Euclidian space cannot be finally regarded as physically filled. But the physical validity o f Euclidian space was the assumption by virtue o f which the Kantian antinomy became inescapable. It follows that until this validity is firmly proved the inevitability o f Kant’s argument— that is, the necessity to follow him into the realm of metaphysical discussion— must remain an open question. From these deliberations it might be concluded that Seeligers

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alternative accords better with Kant than Newtons view. A superficial glance at Kants treatment o f the problem, however, suffices to prove that this would be in open contradiction to his doctrine. His resolution o f the antinomy is based upon the declaration that absolute space formulates a mere idea, not an object that really exists. Accordingly, any theory which assumes a uniform distribution o f matter in absolute space must appear inherently absurd, for it pre­ supposes that a mere idea can serve as a container o f matter. Looked at from the point o f view o f Kant’s assumptions, it is no accident that a hypothesis such as Seeliger’s could find no empirical proofs. The impossibility o f testing it experimentally is merely an expression o f the fact that it is methodologically incomprehensible. On the other hand, it is precisely that antinomy inherent in the Newtonian system which leads us to anticipate the Kantian deduction; Kant proves that it is methodologically inevitable. The disproportion between the Newtonian distribution o f matter and the Euclidian system o f space appears as the exact and necessary counterpart o f the disproportion existing in the mind between the restriction to empirical data, which are given in limited conditions, and the mind’s being endowed with transcendental functions which spur it to go beyond these conditions. If we acknowledge this correspondence, then we must draw the conclusion that Kants mathematical antinomy can only retain its original meaning in association with the Newtonian physical antinomy. And we must accept the significant fact that Kant expressly excludes the possibility o f that case which, from a modern point of view, would expose his mathematical antinomy as inescapable, that is, a complete union o f physical matter with Euclidian space. His philosophy requires that the union between these two remain incomplete. The objection that it might be possible to complete this union in such a way as to avoid the antinomy entirely could only have occurred to a thinker who acknowledged that Euclidian geometry might possibly be modified. Kant, however, convinced that Euclidian geometry was incapable o f any modification, felt secure in the assumption that the dilemma o f the Newtonian doctrine was by its very nature irresolvable. His philosophy was so structured that it could absorb the Newtonian system not despite, but because of the antinomy inherent in it. This becomes clearest in Kants treatment o f the Newtonian concept o f absolute space.

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§ 17 . Kant’s Interpretation o f Absolute Space In the ‘Metaphysical Foundations o f Natural Science’ Kant expounds his view that empty (absolute) physical space cannot possibly be comprehended as existent, but only defined as an idea: ‘For it is only the idea o f a space, in which I abstract from all particular matter (that makes it an object o f experience) in order to think in such a space the material, or every empirical, space still as movable, and thereby to think o f motion not merely unilaterally as an absolute predicate but always reciprocally as a merely relative one. Such space is, then, nothing at all belonging to the existence o f things; but it belongs merely to the determination o f concepts, and hence no empty space exists.’58 The proof that the concept o f empty space is itself an empty concept is here constructed upon exacdy the same assumption as the proof that the idea o f a physical universe is a mere ‘noumenon’ . It might even be said that the critique o f the concept o f empty space is just one more example o f the method employed in the treatment of the cosmological antinomies. For the implications o f the idea o f a physical universe for spatial extent are exacdy those which apply to the concept o f empty or absolute space in relation to physical motion: it formulates the idea o f the absolute and unconditional, which arises in the mind whenever it imagines the series o f conditions as being fulfilled. To desire to form a concept o f this ‘absolute’ is in Kant’s view just as unavoidable as to acknowledge the obligation that the concept is empty (that is, unverifiable). Kant is therefore fighting against two kinds o f theoreticians: 1. against those who, recognizing that absolute space can never be the object o f experience, feel themselves justified in rejecting outright the idea o f this space. 2. against those who, recognizing that the idea o f absolute space must be accepted, feel themselves justified in making o f it an actual object. The two theses, (1) that absolute space cannot become an object o f experience, and (2) that it is indispensable as an idea, only contradict each other if they are viewed dogmatically. Once they are interpreted critically, they are mutually complementary. Kant’s critical interpre­ tation reads as follows: 58 Metaphysische Anfangsgriinde der Naturwissmschaft: Kant, ed. Cassirer, iv (1913), 476 [Kant (Akademie-Ausgabe), iv (19 11), $63: English transl. Ellington (1970), 132].

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Whatever the physical status o f absolute space, reason would seem to require that it be presupposed as an idea. For matter requires space, in order to be able to move in it; and that entity, within which what fills it (i.e., matter) moves, may be regarded as static and empty. But this state o f absolute rest can never become the object o f experience, for we experience space merely in its material state o f being filled. A material space, however (that is to say, a space which is defined by a given constellation o f material bodies), can only ever be relatively at rest, for once it is thought as being contained in a greater space and the latter is at rest, then the former will present itself as moving. This greater space, however, ‘admits o f being arranged in just the same way as regards a still more enlarged space, and so on to infinity without ever by experience arriving at an immovable (immaterial) space with regard to which motion or rest could be attributed absolutely to any matter. Rather, the concept o f these relational determinations will have to be constantly changed according as the movable is considered to be related to one or the other o f these spaces. Now, the condition for regarding something as at rest or moved is always in turn conditioned to infinity in relative space; from this fact the following things become clear. Firstly, all motion or rest can be merely relative and neither can be absolute, i.e., matter can be thought o f as being in motion or at rest only in relation to matter and never as regards mere space without matter. Therefore, absolute motion, i.e., such as is thought o f without any reference o f one matter to another, is simply impossible. Secondly, for this very reason no concept o f motion or rest in relative space and valid for every appearance is possible. But a space must be thought o f in which this relative space can be thought o f as moved; the determination o f such a space does not further depend on any other empirical space and hence is not again conditioned— that is, an absolute space, to which all relative motions can be referred, must be thought of. [...] Absolute space is, then, necessary not as a concept o f an actual object but as an idea that is to serve as a rule for considering all motion therein only as relative.’59 The idea o f absolute or empty space appears here as a regulative principle in order to carry the relativization o f motion in relation to material elements as far as its final consequences, as an ideal concept, S9 [Metaphysische Anfangsgriinde der Natururissenschaft: Kant, ed. Cassirer, iv (1913), 4 7 1-3; Kant (Akademie-Ausgabe), iv (19 11), 559-60; English transl. Ellington (1970), I25-7-]

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which in truly Kantian manner becomes the source o f knowledge by its demarcation o f the bounds o f knowledge. It is clear that this is a philosophical interpretation o f the Newtonian system which goes beyond Newton s own position. Indeed it might almost seem as i f the declaration that Newton’s absolute space cannot possibly exist, but must be defined as a mere idea, anticipates that argument which gave the theory o f relativity its basis.60 Yet to think along those lines would be to forget that Kant, in displaying the conflict within the Newtonian system, had absolutely no intention o f overcoming it by dint o f physical means, but sought to define it as a philosophical necessity. For him the antinomy in the concept o f absolute space was a proof o f his own conviction that the human spirit, which, after all, is only capable o f understanding what it grasps as conditioned, feels itself constandy driven to direct itself towards the absolute. Empty space, which, deprived o f all matter, can never become an object o f experience, nevertheless remains in his view a ‘necessary rational concept’. As an infinite task, which is of service to the inquirer as a yardstick and as a stimulus precisely because it can never be completely fulfilled, it can cause conceptual difficulties only to those who do not understand its philosophical significance. The philosopher, however, who ponders upon the presuppositions o f this concept will only alert attention to the dilemma contained therein in order to prove that, rationally, nothing else could be expected. ‘And thus the metaphysical doctrine o f matter ends with what is empty and with what is, for precisely that reason, the inconceivable, and thus this doctrine suffers much the same fate as all other attempts o f reason when, in going back to principles, it aspires to the first causes o f things. Reason’s nature is such that it can never conceive anything except in so far as it is determined under given conditions. Consequently, inasmuch as it can neither rest with the conditioned nor make the unconditioned comprehensible, nothing remains for it, when thirst for knowledge invites it to grasp the absolute totality o f all conditions, but to turn back from objects to itself, in order to investigate and determine the ultimate boundary o f the capacity upon which it is dependent, rather than investigating and determining the ultimate boundary o f things.’61 60 Cf. e.g. Cassirer, Zur Einstein’schen Relativitätstheorie (1921), 82-3 [repr. in Zur modernen Physik (1957), 75-6; English transl. Swabey and Swabey in Cassirer, Substance and Fundion (1923), 416-17). 61 Metaphysische Anfangsgründe der Naturwissenschaft Kant, ed. Cassirer, iv (1913), 478 [Kant (Akademie-Ausgabe), iv (19x1), 564-5; English transl. Ellington (1970), 134].

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In the ‘incomprehensibility’ o f empty space— the dilemma o f the Newtonian system— the ‘fate o f reason’ is fulfilled, a fate unalterable, as Kant’s critical insight held. In the nature o f the human mind, in which ‘two very dissimilar elements’ further one another to the same extent as they restrict one another, he thought he had found a formula which was capable o f justifying the conflict in Newton’s basic con­ cepts to the same extent as it found itself confirmed in the empirical successes o f the Newtonian system. And indeed, for as long as Newton’s system sufficed to explain mechanical phenomena there was no empirical reason to doubt the validity o f Kant’s epistemology. As long as Kant’s concept o f human reason remained unquestioned, the paradox contained in Newton’s basic concepts was too well founded philosophically to be capable, or in need, o f modification. § 18 . Methodological Conclusion There can be no doubt that Kant, by linking his philosophy so closely with the Newtonian system o f physics, both strengthened and endangered his own position. His philosophy explained a problem which was a most disturbing riddle even for physicists. Yet it explained it as if it were absolutely insoluble on physical grounds. N o physicist could give unqualified assent to such a decision. For as long as the distribution o f matter in space remained a physical problem, the definition o f the relation between space and matter had to be amongst the tasks to be solved by physics. The concession that this definition can only be given in a paradoxical form, which in the last resort requires a philosophical interpretation for its own justification, would have amounted to an admission o f defeat. For it would have meant that physics would have to subject itself anew to the patronage o f philosophy, in order to be taught a lesson concerning the limits which it may not exceed. However much a philosopher might point to the fact that such ‘patronage’ is dictated by the nature o f our consciousness, no physicist could really accept such a situation. For the question still remained o f how the philosopher himself knew anything about the nature o f our consciousness— if not by virtue o f the study o f his achievements? The physicist could then with a clear conscience attempt to achieve what the philosopher had defined as impossible. A positive result would force the philosopher to modify his preconceived view o f the possibilities determined by the nature o f human consciousness.

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This proved indeed to be the fate o f the Kantian doctrine o f consciousness. Post-Kantian speculation has been inspired time and again by its inner conflicts; critics have constantly challenged its inner contradictions. Yet that speculation only lost ground, and such criticism only gained its firm basis, at that moment when the Kantian theory o f human consciousness found its mathematical-physical corollary usurped by the modification o f Newtonian mechanics. It would be a mistake, however, to believe (as so often is the case) that by merely establishing this fact the whole question has been dealt with, and that one can proceed to a new agenda. From a philosophical point o f view it is by no means sufficient merely to know that the Newtonian concept o f the relation between space and matter has been revised in modern physics, but it is necessary for a meaningful critique o f Kant to understand how this revision (in Kantian terms) became ‘possible’. It is not sufficient to say that here an assertion which Kant held to be indisputable has collapsed, and therefore the whole o f the Kantian system has collapsed. Still less may it be maintained by a loyal Kantian that it is precisely the form o f this collapse which proves that Kant’s doctrine is true in a deeper sense than he himself imagined. Things are not so simple. Instead the question must be put: how can Kant’s concepts be modified in order to do justice to this one modification? And what is the retrospective impact o f this one modification upon the Kantian system as a whole? It has been shown that it was precisely the impossibility o f recon­ ciling the actual distribution o f matter with the Euclidian concept o f space which served to support the Kantian system. It remains to be shown how the discovery o f non-Euclidian geometry has made possible the accommodation o f spatial structures to the actual distribution o f matter, and to what extent this achievement undermines the Kantian position. §19. The Mathematical Resolution o f the Euclidian Antinomy Modern axiomatics has discovered that by a modification o f Euclid's axiom o f parallel lines the possibilities o f geometrical deduction are not so much destroyed as extended. By this process Euclidian geometry is defined as a particular type o f geometrical argumentation whose logical significance and limitation is determined by its relation to the other possible types. It is one o f the peculiarities ofEuclidian geometry that it can construct

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curved surfaces only by utilizing the third dimension. In order, for example, to describe the two-dimensional relations on the surface o f a sphere, it must assume the existence o f the sphere as a three-dimensional body. As, however, the surface o f the sphere only extends in two dimensions, its geometry ought also to be capable o f being treated by means o f two-dimensional determinations— a treatment which is not supported by any knowledge o f the more complex qualities o f a third dimension! Thus Gauss invented a system o f co-ordinates, expressly devised for the elementary treatment o f surfaces with curvature, which diverges from the Cartesian system o f co-ordinates in that it makes no use o f the Euclidian concept o f parallel lines. This can be regarded as an expression o f the fact that the geometry o f curved surfaces, i f it is derived from concepts which are primitive rather than complex, is necessarily non-Euclidian. The sum o f the angles o f a triangle in such a geometry deviates from 2 R ,62 and the deviation decreases in proportion to the magnitude o f the triangles. In so doing, however, Gauss not only succeeded in the logical tour de force o f treating given two-dimensional manifolds without refer­ ence to the third dimension; he also simultaneously created a new group o f mathematical tools, which can in turn be employed for the construction o f the third dimension. These same instrumental means, which were first discovered by the elimination o f the third dimension, now provide new means for its conquest. For after a non-Euclidian continuum o f two dimensions has been constructed it can be extended to a continuum o f three dimensions, and thus leads to a construction o f corporeal space, which is established on the basis o f the form o f the curve and the curved surface in the same way as the form o f Euclidian space is based on the shape o f the straight lines and the plane. This space, which represents the three-dimensional analogue to the surface o f the sphere— spherical or Riemannian space— has the property of being simultaneously finite and un­ bounded. Just as the surface o f a sphere is finite, although if we wished to imagine that two-dimensional beings move in it they could wander around there without ever coming up against a boundary, so too spherical space has a finite volume, although three-dimensional beings could wander through it without ever reaching an outer limit. 62 [2R is a conventional notation for two right angles (180°), sometimes used in the formulation o f Euclid’s 32nd proposition in book i o f the Elements; cf., e.g., Bunt, Jones and Bedient, Historical Roots (1976), 173.]

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The only way in which they could experience the finiteness o f their space would be if, after they had traversed it in a straight line— ‘straight’ in the sense o f practical geometry— for a finite, even if very long duration, they were to observe that they were returning to their point o f departure. A universe conceived in accordance with this concept o f space is obviously free o f the antinomy described by Kant, for it would be at once closed and all-encompassing, and would thus fulfil the two conditions which are irreconcilable within Euclidian geometry. One could travel through this universe in all directions without ever meeting a boundary erected from outside; hence it would not be ‘too small for our understanding’. At the same time it would not be ‘too big’ for our understanding, for the extent o f the journey would be finite. The Kantian antinomy has, however, by no means been resolved once and for all by the discovery o f Riemannian space. All that has been proved is that space which is simultaneously finite and unbounded is as a concept non-contradictory. It remains to be decided whether that which we have comprehended as meaningful may be regarded as actual. Riemannian space could, after all, still be a mere ‘noumenon’ , to use Kant’s terminology, an empty concept to which no experience corresponds. This objection is the more worthy o f note because the method by which Riemannian space was discovered permits a great number o f other conceptions o f space as equally meaningful. If one o f these is given precedence on the basis o f physical experiments, then all the rest are in this context implicitly defined as empty, or at least as unreal to the extent that they can only be permitted as approximations to an exact description in specially defined conditions. In the field o f mathematical discussion nonEuclidian geometry has thus broadened rather than restricted the scope o f the Kantian antinomy. For whereas in Kant’s case the anti­ nomy pointed to a decision which had to be made within Euclidian geometry, it now embraces the task o f choosing between different possible geometries, one o f which is Euclidian. The prospect o f forming a meaningful (non-contradictory) concept o f space, which is also physically verifiable, is confronted by the increase in the number o f concepts o f space which as purely mathematical systems are meaningful and non-contradictory, but are condemned to remain physically empty. Moreover, the possibility still remains that Euclidian geometry, which was responsible for the Kantian antinomy, could prove itself to be physically valid, that is, that the physical evidence

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might favour that form o f space which, applied to the universe as a whole, reveals an inner contradiction. In these circumstances the development o f a method which can decide whether a meaningful concept is physically real or not seems just as important as the discovery o f those meaningful concepts themselves. For the extension o f the ‘possibilities’ which is achieved by the one form o f discovery requires, precisely because o f the abundance o f alternatives it supplies, a criterion o f ‘actuality’ which points to a quite different method o f discovery. [When in 1832 Janos Bolyai published his Appendix scientiam spatii absolute veram exhibens, in which he proved that the choice between a Euclidian spatial system (E) and a non-Euclidian spatial system (5 ) is a priori undecidable (a priori haud unquam decidenda), Gauss expressed his assent in the following words: ‘It is precisely in the impossibility o f deciding a priori between Z and S that the clearest proof lies that Kant was wrong in maintaining that space is only aform o f our intuition’ .]63 §20. The Physical Resolution o f the Newtonian Antinomy We shall proceed, as before, on the basis o f the two-dimensional analogy o f Riemannian space, that is to say, from the surface o f a sphere, and imagine that two-dimensional beings inhabit this surface. In order to discover the finitude o f their world they could employ only such criteria as belong to the province o f this world; they could not, for example, look down on this world ‘from the outside’ and so determine that they live inside a spherical surface, for the third dimension is not, after all, supposed to be accessible to them. They could, it is true, undertake a journey round the world and deduce from the fact that it leads them back to their point o f departure that their world is finite. But perhaps the lifespan o f such beings would not suffice for such a journey— not to mention other obstacles. What other means might they then employ for that purpose? There remains the Gaussian method, which states that no acquaintance at all with the third dimension is necessary for the study 63 [Letter to Farkas (Wolfgang) Bolyai, dated 6 March 1832, in Gauss, Werke, viii (1900), 220-4, here 224; cf. W. and J. Bolyai, Geometrische Untersuchungen (1913), i. 85-97, here 94. For the ‘Appendix scientiam spatii absolute veram exhibens’ see Bolyai, ‘Appendix’ (1832); English transl. Halsted in Bonola, Non-Euclidian Geometry (1955), supplement; Wind cites the author’s own German translation in W. and J. Bolyai, Geometrische Untersuchungen (1913), ii. 183-219.]

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o f a two-dimensional continuum, that one can indeed dispense with any previous knowledge o f the totality o f the continuum that is to be investigated. These two-dimensional beings would thus only need to undertake systematic measurements within their limited range. They could, for example, construct a circle and measure its circumference. And if the length o f the periphery were to deviate from 271 times the length o f the radius, whilst increasing in proportion to the size o f the circle, they could deduce from this that they lived within a nonEuclidian world, and they could even calculate, by contemplating the extent o f deviation, the approximate size o f this world. This consideration is in turn applicable to three-dimensional space. The decision as to whether this space is finite or not will then be dependent upon the results o f actual physical measurements. Given geometrical concepts will have to be embodied in given physical instruments, and it will prove possible to decide on the basis o f carrying out well-defined geometrical constructions (such as the construction o f spheres with a systematically increasing diameter), with the help o f these instruments, whether the equations which result from these constructions correspond to the Euclidian, the Riemannian or some other concept o f space. The procedure de­ scribed here is precisely that which Einstein has defined as ‘practical geometry’ , the principles o f which are expounded in the first part o f the present work (§3). The geometrical theorems take on a definable physical significance because the concept o f mathematical equivalence is assigned to the physical criteria o f coincidence through the method o f making measurements in space and time. Geometrical theorems, by dint o f this interpretation, become expressions o f physical facts. They describe the conditions under which physical bodies come into contact with one another— conditions which in the last resort are determined by the behaviour and distribution o f matter. The result is a union between matter and space, which overcomes the dilemma of the Newtonian system once and for all. The structure o f space can no longer be in conflict with the distribution o f matter, for by dint o f the ‘embodiment’ and by means o f the ‘judgements o f appropriateness’ the structure o f space is expressly defined as the geometrical expression o f the distribution o f matter. Yet just as certainly as the theoretical difficulties are resolved by this definition, so too the definition itself presents a practical problem of almost phantastic dimensions. For it requires that the distinction between the different possible structures o f space be defined as a

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distinction in the (hypothetical) distribution o f matter, and that an experiment can be specified which may determine which o f these different hypotheses is valid. Einstein himself, in the lecture in which he proved that ‘the question o f whether the universe is spatially finite or not is an entirely meaningful question in the sense o f practical geometry’ , discusses in detail both the theoretical and experimental aspects o f this problem:64 I do not even consider it impossible that this question will be answered before long by astronomy. Let us call to mind what the general theory o f relativity teaches in this respect. It offers two possibilities: 1. T h e universe is spatially infinite. This can be so only i f the average spatial density o f the matter in universal space, concentrated in the stars, vanishes, i.e., i f the ratio o f the total mass o f the stars to the magnitude o f the space through w hich they are scattered approximates indefinitely to the value zero w hen the spaces taken into consideration are constantly greater and greater. 2. T h e universe is spatially finite. This must be so, i f there is a mean density o f the ponderable matter in universal space differing from zero. T h e smaller that mean density, the greater is the volum e o f universal space. [...] Experience alone can finally decide w hich o f the two possibilities is realized in nature. H o w can experience furnish an answer? A t first it might seem possible to determine the mean density o f matter by observation o f that part o f the universe w hich is accessible to our perception. This hope is illusory. T h e distribution o f the visible stars is extrem ely irregular, so that we o n no account may venture to set down the mean density o f star-matter in the universe as equal, let us say, to the mean density in the M ilky Way. In any case, however great the space exam ined may be, we could not feel convinced that there were no more stars beyond that space. So it seems impossible to estimate the mean density. B u t there is another road, w hich seems to me more practicable, although it also presents great difficulties. For i f we inquire into the deviations shown b y the consequences o f the general theory o f relativity w hich are accessible to experience, w hen these are compared with the consequences o f the N ew ton ian theory, we first o f all find a deviation w hich shows itself in close proxim ity to gravitating mass, and has been confirm ed in the case o f the planet M ercury. B u t i f the universe is spatially finite there is a second deviation from the N ew tonian theory, w hich, in the language o f the N ew tonian theory, may be expressed thus:— T h e gravitational field is in its

64 Einstein, Geometrie und Erfahrung (1921), 129-30 [repr. in Akademie- Vorträge (1978), 6-8; English transl. Jeffery and Perrett in Einstein, Sidelights on Relativity (1922), 41- 5]-

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nature such as i f it were produced, not only by the ponderable masses, but also by a mass-density o f negative sign, distributed uniformly throughout space. Since this virtual mass-density would have to be enormously small, it could make its presence felt only in gravitating systems o f very great extent. Assuming that we know, let us say, the statistical distribution o f the stars in the M ilky Way, as well as their masses, then by N ew ton’s law we can calculate the gravitational field and the mean velocities which the stars must have, so that the M ilky Way should not collapse under the mutual attraction o f its stars, but should maintain its actual extent. N ow if the actual velocities o f the stars, which can, o f course, be measured, were smaller than the calculated velocities, we should have a proof that the actual attractions at great distances are smaller than by Newton’s law. From such a deviation it could be proved indirecdy that the universe is finite. It would even be possible to estimate its spatial magnitude.

This quotation is o f especial interest for us because two possibilities o f empirical verification are discussed here, one o f which is rejected as ‘illusory’ in the context o f cosmological decisions, whilst the other is admitted as meaningful. The rejected method is evidendy the same as that which Kant too rejects when he says that ideas which relate to the world as a whole can never ‘be given in experience in terms o f their whole content’.65 The average density o f ponderable matter in space cannot in principle be determined ‘by the observation o f that part o f the universe which is accessible to our perception’ , for it might always be suspected— ‘no matter how great the space that may have been examined’— that the proportions differ outside that space. Yet there is another method o f empirical verification, as emerges from Einstein’s words. Instead o f attempting ‘to produce the entire content o f the idea in experience’ (which is absolutely impossible), we regard it as a hypothesis concerning the structure o f the whole and examine how it reflects upon the behaviour o f the parts. A demonstrable ‘deviation’ in the behaviour o f the parts thus becomes the criterion by which the structure o f the whole can be measured. The question o f whether the result o f such a test will decide in favour o f or against the finiteness o f the universe66 is much less important for our problem than the fact that such tests can be devised 65 Cf. § 11. 66 The experimental decision seems today to advocate a process o f evolution which leads the world through systematically accreting stages o f finiteness to infinity, and thus to material self-destruction. (Cf. the latest development in these studies mentioned in the preface, and §30 below.)

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in some meaningful sense. For since in the demonstration o f the metaphysical aspects o f the experimental method we were not so much concerned with a specific subject o f science as with the logic o f the scientific procedure, we do not need— in this context— to worry about the result o f the experiments, but only about their methodological significance. It does not matter whether the answers are ‘yes’ or ‘no’— the point is that answers which amount to either ‘yes’ or ‘no’ can be devised and extracted. This alone suffices to justify the assertion that the physicist, through the embodiment o f geometrical concepts in physical objects, provokes an answer which is o f such a kind that it can decide even cosmological problems (that is to say, problems which relate to the physical world as a whole) on an experimental basis. §21. The Principle o f Internal Delimitation67 The method which has led to the resolution o f the first antinomy consists o f the following steps: 1. studying the variability o f geometrical axioms, which (a) restricts the antinomy to the Euclidian concept o f space, and (b) proves the possibility o f non-Euclidian concepts o f space, with regard to which the antinomy has no validity. 2. deciding between these rival concepts o f geometry through the method o f embodiment, which (a) accords the geometrical propositions a physical significance, and (b) tests by experiment and observation whether or not they are empirically valid as physical propositions. It can thus be shown, both in the mathematical and in the physical area, that the antinomy is linked to the method o f external delimit­ ation— a process which must lead to a dialectical contradiction as soon as it is applied to the world as a whole. This whole can, however, be grasped in a manner entirely free o f contradiction— disregarding all dialectic— if the principle of internal delimitation, the setting o f limits 67 [‘Internal delimitation’ renders the term ‘innere Grenzsetzung’ and refers to the setting o f limits or boundaries ‘from within’, from the perspective o f that part o f the ‘whole’ which is accessible to empirical observation. Wind, in his article ‘Can the Antinomies be Restated?’ (1934), l 77> translates it differently and writes o f the ‘method o f “ implicit determination” (“ innere Grenzsetzung”) which defines the relation o f the “ part” to the “ whole” in such a way that any proposition concerning the structure o f the “ whole” can be tested in terms o f the behaviour o f the “ parts” ’.]

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from within, is applied. In the mathematical area this means giving preference to those structures whose intrinsic principle o f con­ struction already contains a criterion for the determination o f their extent. In the physical area this means the application o f an experi­ mental process which investigates the structure o f the whole on the basis o f the behaviour o f the parts. In philosophical terms, however, this principle is only the formal expression o f the basic fact that we ourselves, together with our instruments, belong as parts to that world which is to be cognized and can therefore only investigate it ‘from the inside’— on the basis o f criteria supplied to us by the world itself. The ‘extraterrestrial point o f view’ is an absurdity. This principle o f internal delimitation will also prove valid with regard to the remaining antinomies.

CHAPTER

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The Antinomy of the Concept o f the Atom §22. The *Inner Lim it’ in the Process o f Division N ow that the first antinomy has been treated in detail, the second can be dealt with more cursorily, as it involves quite simply the direct inversion o f the first. It attempts to establish the same paradox for the divisibility o f physical space, and thus for the concept o f the physical atom, as seemed to be valid for the possibility o f extension o f physical space, and thus for the concept o f the physical universe. The atom and the universe (the minimum and maximum points o f extension) are paradoxical concepts when considered from the Kantian point o f view. For both spatial division and spatial accretion are processes o f conceptual organization which are related only to the forms o f ‘pure intuition’ ; precisely because such conceptual organization does not relate to actual things, but only to our way o f looking at the things, it can be extended indefinitely. There is no methodologically deter­ mined limit for either o f these. If, nevertheless, an atom is defined as the product o f a division which cannot be further subdivided, or the universe as the product o f an accretion to which nothing further can be added, then in both these concepts a process is treated as completed which by its very nature cannot be concluded; and an object is constructed which can only be characterized by contradictory attributes. In order to be able to define an atom as an object at all, it must be possible to arrive at it through a finite number o f stages o f division. However, an object which is arrived at through a finite number o f stages o f division cannot be comprehended as an ultimate, final unit, for there is no sound reason why the process o f division should not have been continued. Just as the concept o f the physical universe seemed either too big or not big enough for our empirical comprehension, so the idea o f the existing atom is either too small or not small enough for our empirical understanding. In the one case it is too limited not

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to be exceeded; in the other too unlimited for us to be able to grasp it at all. Kant therefore rejected the whole concept as a mere ‘noumenon’ , having absolutely no relationship to any existing objects, but able only to serve as a regulative principle for scientific advance. Here too, the root o f the antinomy has to be sought in the middle term— in the specific form o f the process o f division, which is assumed both in the thesis and in the antithesis. Here too it must be asked whether the impossibility of completion, demonstrated by Kant, is not perhaps restricted to just oneform o f division— and whether another form could not be developed as a plausible alternative and tested by physical criteria. Is it possible to carry out a division in such a way that in the process we never come up against an external limit, and yet, after a finite number o f steps, have covered the range o f divisibility? Is ‘internal delimitation’ conceivable in the process o f division? §23. Two Forms o f ‘ Uncertainty * Looked at from a physical point o f view, the principle o f internal delimitation means that the structure o f a physical whole takes effect in an observable way in the behaviour o f the structures o f the parts belonging to it. If therefore a partial structure o f the cosmos is sufficiendy known (for example, the distribution o f the stars in the Milky Way and their masses), then a criterion obtains for deciding between two different theories concerning the extent o f the universe. On the other hand, the case can also arise where the way a whole behaves is known, and one seeks to determine the structure o f the parts belong­ ing to that whole. The aim is to decide whether within this whole an infinite or only a finite process o f division can be carried out. If the principle o f internal delimitation is to be applied here too, then it must be required that those limiting conditions that determine the nature and the range o f its divisibility and thus the structure o f its ‘ultimate parts’ are contained in the material matter o f this whole. It is quantum mechanics which tells us how such a reduction o f the determinations o f the part to the qualities o f the whole can be thought and carried out, how a criterion is to be found in the character o f the whole which can decide the extent to which it is divisible. To quote Planck: / n

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Hitherto it had been believed that the only kind o f causality with which any system o f physics could operate was one in which all the events o f the physical world— by which, as usual, I mean not the real world but the world­ view o f physics— might be explained as being composed o f local events taking place in a number o f individual and infinitely small parts o f space. It was further believed that each o f these elementary events was completely determined by a set o f laws without respect to the other events; and was determined exclusively by the local events in its immediate temporal and spatial vicinity. Let us take a concrete instance o f sufficiently general application. We will assume that the physical system under consideration consists o f a system o f particles moving in a conservative field o f force o f constant total energy. Then according to classical physics each individual particle at any time is in a definite state; that is, it has a definite position and a definite velocity, and its movement can be calculated with perfect exactness from its initial state and from the local properties o f the field o f force in those parts o f space through which the particle passes in the course o f its movement. If these data are known, we need know nothing else about the remaining properties o f the system o f particles under consideration. In modern mechanics matters are wholly different. According to modern mechanics, merely local relations are no more sufficient for the formulation o f the law o f motion than would be the microscopic investigation o f the different parts o f a picture in order to make clear its meaning. On the contrary, it is impossible to obtain an adequate version o f the laws for which we are looking, unless the physical system is regarded as a whole. According to modern mechanics, each individual particle o f the system, in a certain sense, at any one time exists simultaneously in every part o f the space occupied by the system. This simultaneous existence applies not merely to the field o f force with which it is surrounded, but also to its mass and its charge. Thus we see that nothing less is at stake here than the concept o f the particle, the most elementary concept o f classical mechanics. We are compelled to give up the earlier essential meaning o f this idea; only in a number o f special borderline cases can we retain it. [...] Thus we reach the following result: in classical physics the physical system under consideration is divided spatially into a number o f smallest parts; by this means the motion o f material bodies is traced back to the motion o f their component particles, the latter being assumed to be unchangeable. In other words, the explanation is based upon a theory o f corpuscles. Quantum physics, on the other hand, analyses all motion into individual and periodic material waves, which are taken to correspond to the characteristic vibrations and characteristic functions o f the system in question; in this way it is based upon wave-mechanics. Accordingly in classical mechanics the simplest motion is that o f an individual particle,

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whereas in quantum mechanics the simplest motion is that o f a simple periodic wave; according to the first, the entire motion o f a body is taken as being the totality o f the motions o f its component particles; whereas according to the second it consists in the joint effect o f all kinds o f periodic material waves. To illustrate the difference between these two views, we may once more refer to the vibrations o f a stretched cord. O n the one hand these vibrations may be imagined as consisting o f the sum o f the motions o f the different particles o f the cord, where each particle is in motion independendy o f all the rest and in accordance with the force acting upon it, which in turn depends upon the local curvature o f the cord. On the other hand the process o f vibration may be analysed into the fundamental and upper partial vibrations o f the cord, where each vibration affects the cord in its totality and the sum total o f vibration is the most general kind o f motion taking place in the cord.

— At first sight it might be imagined that these two ways o f present­ ing the matter are equally valid. For from a theoretical point o f view it ought to be just as possible to describe the process o f vibration of the cord through the movement o f its individual particles, as to derive the position o f these particles from observing the process o f vibration as a whole; in reality, however, these two procedures are mutually exclusive, because the experimental method, which is the basis for carrying out the second procedure, annuls the empirical significance o f that concept o f the particle which was assumed to be empirically meaningful in the first procedure. Planck himself describes this state o f affairs as follows: ‘The following question must now be asked: if motion is to be analysed not into particles, but into material waves, what is the procedure o f wave-mechanics when it is called upon to describe the motion o f a single particle which occupies a given position at a given time?’ The answer lies in the rejection o f the question: ‘It appears immediately that such a description cannot be made in any exact sense. Wave-mechanics possesses only one means o f defining the position of a particle, or more generally the position o f a definite point in configuration space; this consists in superimposing a group o f individual waves o f the system, in such a manner that their wave-functions cancel each other by interference everywhere within configuration space, and intensify each other only at the point in question. In this case the probability o f all the other configuration points would be equal to zero, and would be equal to i only for the one point in question. In order to isolate this point completely we should, however, require infinitely small wave-lengths and

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consequently infinitely great momentum. Therefore, in order to obtain a result which would be even somewhat useful, we should have to begin by substituting for the definite configuration point a finite (though still small) region o f configuration space, or so-called wavegroup; this sufficiently expresses the fact that ascertaining the position o f a configuration point is always in the wave theory affected by some sort o f uncertainty.’69 The peculiar nature o f this ‘uncertainty’ , however, lies in the fact that it does not mark a limit which is imposed on the experimental process at a certain stage in its evolution, and which has to be pushed back as experience progresses further, but that the determination o f such a limit is to be understood as a result o f the experimental inquiry itself, as a physical proposition which, having evolved out o f the system o f wave-mechanics, claims the same degree o f validity for itself as is due to that system as a whole on the basis o f experimental proofs. ‘It had always been known, o f course, that every measurement is subject to a certain amount o f inaccuracy; but it had always been assumed that an improvement in method would lead to an unlimited improvement in accuracy. Now a limit is to be imposed in principle on the accuracy o f measurement, and what is most curious is that this limit does not affect magnitude, position and velocity separately, but only when these are combined together.’ From the way in which, in wave-mechanics, impulse and position are mutually dependent it necessarily follows that ‘the position o f a measuring point within a finite area is always uncertain’.70 Now, however, we are in a position to contemplate the very case which delivers the key to the physical resolution o f the second antinomy. For the two kinds o f ‘uncertainty’ distinguished by Planck, one o f which is quite common in classical mechanics, whereas the other, as Planck himself attests, presents ‘something totally unheard o f in classical mechanics’, when applied to the question o f the smallest possible magnitude o f extent, signify two kinds o f limitation to a finite area o f space. If uncertainty is defined in such a way ‘that by ap­ propriate refinement o f the methods o f measurement the exactitude 69 [Planck, Weltbild (1929), 30, 33-4; repr. as ‘Zwanzig Jahre Arbeit’ (1929), 208; Physikalische Abhandlungen und Vorträge (1958), iii. 194, 196; English transl. Johnston

(193 0 , 32 , 36 - 7 ]. 70 [Planck, Weltbild (1929), 35-6; repr. as ‘Zwanzig Jahre Arbeit’ (1929), 2 11; Physikalische Abhandlungen und Vorträge (1958), iii. 197; English transl. Johnston (1931), 38-9I.

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can be increased to an unlimited extent’ , then finiteness in the process o f spatial division only demarcates that degree o f precision which scientific inquiry has reached at a given stage o f its development. If, on the other hand, the uncertainty derives from the fundamental status o f measurement as a numerically determinable quantum, then in the assertion o f finiteness there lies a real hypothesis concerning the structure o f the actual materials which are to be measured. Moreover, whereas in the first case it is a question o f a regulative principle which denies fundamentally that there can be any empirical criterion for the distinction between finite and infinite divisibility, in the second case finiteness is asserted as a matter o f principle precisely on the basis of empirical criteria.— Heisenberg’s uncertainty principle, whereby ‘the product o f the uncertainty o f position and the uncertainty o f momentum is at least as great as Planck’s constant’,71 acquires in the context o f the second antinomy the same significance which was accorded in the treatment o f the first antinomy to Einstein’s proposition concerning the relation between the spatial extent o f the world and the mean density of matter. Just as in that case a physical law was demonstrated, according to which the maximum can be determined (whereby the assertion that there is indeed such a maximum simultaneously acquired an empirically comprehensible meaning), in the same way a physical law is now proposed which provides an empirical criterion for establishing the minimum and thus transforms the question o f whether there is indeed such a minimum into a question o f fact. In both cases this reduction o f an apparendy dialectical question to an experimental one is achieved by introducing a system o f measurement into the equations which express those laws, which, viewed as an expression o f a physical state, becomes the empirical criterion whereby the finiteness or infinitude o f extension or division can be measured. In the first case this is the mean density o f matter p, o f which it is required that, if the world is to be finite, it should have a value deviating from zero. In the second case it is Planck’s constant h, which, by virtue o f the fact that it occurs as an irreducible constant, yields sufficient reason to dispute the infinite divisibility o f matter by reference to a natural law. 71 [Cf. Heisenberg, ‘Kinematik und Mechanik’ (1927), esp. 175; repr. in Cesammelte Werke, Series A, Part I (1985), 481; cf. id., Physical Principles of the Quantum Theory (1930), esp. 13 -14 and 48 ff.; repr. in Cesammelte Werke, Series B (1984), 12 3 -4 , 131 ff.j

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Causality and Freedom §24. Heterogeneity and Necessity in the Connection between Cause and Effect In the first two antinomies Kant applies the distinction between ‘appearances’ and ‘things in themselves’ in order to annul the concept o f the physical universe and the physical atom, and to show that the world can be thought neither as finite nor as infinite, neither as consisting o f simple parts nor o f composite ones, on the grounds that the concept o f a closed and all-embracing universe, or o f a simple and yet extended atom, is self-contradictory. In the case o f the third antinomy he uses it to an entirely positive purpose. It is intended to prove that the two propositions: 1. Everything in the world occurs solely in accordance with the laws o f nature 2. There are in the world causes originating in free will are not mutually exclusive, but complementary, in that two different kinds o f causality are to be distinguished: 1. causality in the pheno­ menon, where every effect within its area o f occurrence is dependent upon a cause which precedes it temporally, and which itself, in turn, in order to be effective as a cause, needs a cause preceding it; and 2. causality of the phenomena, where a cause is assigned to every effect outside their area o f occurrence, which ‘as to its causality, must not stand under time-determinations o f its state’ , but rather be ‘a faculty o f starting events spontaneously (sponte)’ .72 The middle term, in which thesis and antithesis meet, is the ‘connection o f cause and effect’. And it is due, in Kant’s view, to the peculiar nature o f this ‘dynamical’ connection that it leads (if regarded from the point o f view o f the distinction between ‘appearances’ and ‘things in themselves’), in contrast to the mathematical forms o f 72 Prolegomena, §53 [Kant (Akademie-Ausgabe), iv (1911), 344; English transl. Carus and Ellington (1977), 84].

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connection from which the antinomies o f the magnitude o f extension derive, not to a nonsensical, but to an equivocal concept: ‘Any mathematical connection necessarily presupposes homogeneity o f what is connected (in the concept o f magnitude), while the dynamical connection by no means requires this. When we have to deal with extended magnitudes, all the parts must be homogeneous with one another and with the whole. But, in the connection o f cause and effect, homogeneity may indeed likewise be found, but is not necessary, for the concept o f causality (by means o f which something is posited through something else quite different from it) does not in the least require it.’73 This view is expressed even more clearly in the ‘Comment on the Thesis’, for there, to the objection that it is absolutely impossible to imagine causality through freedom, Kant replies abrupdy that ‘in causality according to natural laws we must likewise setde for cognizing a priori that such a causality must be presupposed, even though we do not comprehend in any way the possibility whereby through a certain [thing’s] existence the existence o f another is posited’ .74 The fundamental difference between cause and effect— precisely because, as Hume has shown, it makes a demon­ stration o f the relation o f cause and effect impossible— thus permits variation in the application o f the concept o f causality. This variation makes it seem justifiable to understand by the single concept ‘causal’ on the one hand the necessary connection o f two different objects as they are perceived by the senses in time, on the other the necessary connection o f two different aspects o f the event in a single object. Whereas such heterogeneity o f the elements that are to be connected makes possible the application o f the causal concept in a number o f ways, it is, however, in the necessity o f the connection as such that the identity o f the causal concept itself is guaranteed. This need to make links is therefore, as an idea, at the heart o f both the Kantian concept o f nature and the Kantian concept o f freedom. I f an action is carried out, in its empirical appearance, as an act o f free will, it is infallibly determined in its singularity by the fact that it derives from pure reason. As an ethical action which has its ‘beginning’ in the autonomy o f the will, its outcome can be such and no other; for what is to happen in certain conditions according to the maxims o f reason, 73 Prolegomena, §53 [Kant (Akademie-Ausgabe), iv (19 11), 343; English transl. Carus and Ellington (1977), 83-4]. 74 Kritik der reinen Vemunjt, B 476.

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regardless o f whether it can happen in the conditions o f nature or not, results by apodictic necessity (in Kants view) from the categorical imperative. If, on the other hand, ‘this same’ action is regarded as an event in nature, then, regardless o f whether or not it is to happen according to the laws o f freedom, it is necessarily predetermined by the state o f nature which preceded it. For this is how Kant himself describes causality according to the laws o f nature: everything that happens presupposes a previous state, which it inevitably succeeds, according to a rule. These two determinations— the heterogeneity o f cause and effect and the necessary form o f their connection— are decisive for Kants formulation and treatment o f the third antinomy. The first is indispensable for him, if he is to resolve the antinomy, for it permits him to transpose into the realm o f the suprasensible the ‘causes through freedom’ , which are nowhere to be found within the world o f that which can be experienced, as there every cause must present itself as the effect o f another cause which precedes it in time. It is, however, only in its application to time that the second determination, the necessary form o f the connection, supplies the basis for that dynamical concept o f the natural law from which the antinomy derives. Only if one sees the essence o f the natural law as being that ‘everything that happens presupposes a previous state, which it inevitably succeeds, according to a rule’— only then does that dialectic described by Kant come into being: that in order to comprehend each state in the series o f causes and effects as sufficiently determined, it is necessary to postulate that the series itself be complete, and thus posit a first cause, but that this cause itself, in order to be thought as a natural cause, can in turn only be grasped as the effect o f a preceding cause, not, therefore, as the first cause. The significance o f the concept o f Euclidian geometry for the first antinomy thus corresponds to that o f the concept o f the dynamical law for the third antinomy. As in the former case, the antinomy here becomes inescapable, if the concept on which it is based proves to be the only one that can be justified. There is, then, no place in nature for freedom as a phenomenon or fact that can be experienced; it can only find its justification and its place as a practical idea in the suprasensible, which can never become a proper object o f our cognition. Furthermore, and even more significandy for the purposes o f cosmological investigation, nature itself, as long as it is ruled unrestrictedly by dynamical laws, becomes an object which can by no

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means be completely understood in terms o f itself. As every state must refer back to a preceding state as its cause, but no state, as long as the dynamical laws are upheld, can be assumed to be the first state, every attempt to grasp nature as a whole leads beyond nature into the suprasensible. ‘Hence the proposition’ , as Kant puts it, ‘in its unlimited universality, whereby any causality is possible only accord­ ing to natural laws, contradicts itself.’75 Physics, however, on the basis o f experimental evidence and theo­ retical deliberations, has recently raised fundamental questions as to the unconditional validity o f dynamical laws. It has formed the concept of the ‘statistical law’, which was originally intended to stand as a complement next to the dynamical law, but instead has developed more and more into a rival which threatens to suppress its opponent altogether. The idea o f the necessary connection, which for Kant determines in general terms the very essence o f conformity with laws and hence forms the real ‘tertium’ in the antinomy between the law of nature and freedom, here finds itself contrasted with a possible alternative, in the first instance an alternative to the concept o f natural law (i.e., to connections in time). For the concept o f the statistical law casts doubt precisely upon that inevitability of consequence, which is absolutely inseparable from the concept o f causal determination and hence from the Kantian concept o f the natural event. The opposition between the two ways of looking at things has been aptly character­ ized by Heisenberg: ‘In the precise formulation o f the causal law: if we know the present exactly, we can calculate the future— it is not the secondary clause, but the presupposition that is wrong. We cannot in principle acquaint ourselves with the present in all stages o f determ­ ination.’76 In this context we should not imagine, for example, a fortuitous boundary, which is imposed upon our cognizing o f the present from outside, and which we can push forward indefinitely in the process o f cognition— as if, as it were, ‘behind the [...] statistical world another “ actual” world were to conceal itself, in which the causal law is valid’. The formulation ‘in principle’ , on the contrary, implies clearly that the concept o f the ‘present’ by its very nature excludes unambiguous definition. In effect, the opposition between the two ways 75 Kritik der reinen Vernunft, B 474. 76 [Cf. Heisenberg, ‘Kinematik und Mechanik’ (1927), 197; repr. in Cesammelte Werke, Series A, Part I (1985), 503; cf. id., ‘Grundprinzipien’ (1927); repr. in Gesammelte Werke, Abt. C, Bd. I (1984), 21.]

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o f looking at things can be traced back to the opposition between two different kinds o f temporal determination, one o f which permits causal determination as possible, while the other radically rejects it. I shall first contrast these two kinds o f determination in the sequence o f time as possible alternatives, in order then to establish the criterion which decides their actuality. §25. The Temporal Relation o f Dynamical Connection That form o f temporal sequence which permits causal determination may be elucidated by the model shown in the following:

c

C' —>

Past A

Future O

O'

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D'

B

Fig. 1 The line A B, which is o f undefined length, designates the series o f points in time which an observer positioned at O has run through in succession, or will run through. The arrow indicates the direction from the past to the future as it presents itself to this observer. The vertical line C D , drawn through the point O, dividing the past o f the observer from his future, signifies for him the ‘present’; all events which for the observer are ‘now’ can be understood to he along this line. This ‘now’ is therefore a ‘slice’ , which cannot be extended temporally— a momentary cross-section through the world, which comprises a particular total constellation o f bodies in space. If the observer imagines a second such section made through O ' (a point in his ‘future’) and calls this slice C 'D ', then if he knows both the laws o f the interrelation

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between the bodies and the magnitude o f the interval o f time which lies between Oand O ', he can calculate in advance the total constellation C 'D ' on the basis o f the constellation CD. In this presentation, however, one important point has remained unexplained. How does the observer actually know that O ' lies in the future? B y what criteria can he indeed make the distinction between past and future which he so unhesitatingly assumes? Naturally, in his consciousness he possesses a criterion by means o f which he can distinguish ‘present’ experiences from those which are given to him in the modality o f recollection or expectation. That, however, contributes nothing to physics. If he really wishes to derive a future physical state from a present one, then he must be able to advance from the temporal sequence o f his personal experiences to the temporal sequence of physical events. In order to do that, however, he must first know on what basis in physical events the temporal sequence can be recognized. This is precisely the question that Kant posed himself in the ‘Second Analogy’. How can one progress from the ‘subjective succession o f apprehension’ to the ‘objective succession o f appear­ ances’? Kant’s response is that this is possible by recognizing how one appearance succeeds another in accordance with a rule. However, for this to happen, as Kant emphasizes, the mediation o f a concept o f under­ standing is necessary; ‘and here this concept is that o f the relation of cause and effect; o f these two, the cause is what determines the effect in time, and determines it as the consequence’ .77 Kant distinguishes accordingly between the *order o f time’ and the ‘progression o f time’, where the relation between cause and effect which finds expression in the order o f time is made so independent o f the progression o f the experience-based course o f time that ‘the relation remains even if no time has elapsed’ . ‘The time between the causality o f the cause and the cause’s direct effect may be extremely brief (i.e., simultaneous), but yet the relation o f the cause to the effect always remains determinable in terms o f time. If I consider as cause a [lead] ball that lies on a stuffed cushion and makes an indentation in it, then this cause is simultaneous with the effect. But I none the less distinguish the two by the time relation o f their dynamical connection. For if I lay the ball on the cushion, then the previous smooth shape o f the cushion is succeeded by the indentation; but if the cushion has an indentation (no matter from where), then this is not succeeded by a lead ball.’78 77 [Kritik der reinen Vemunjt, B 234.J 78 Kritik der reinen Vemunjt, B 248-9.

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There are, however, peculiar circumstances attached to the ‘time relation o f the dynamical connection’, which in Kant’s view enables the observer to distinguish objectively between the earlier and the later state. As has been expressly emphasized (by Kant himself), it has in essence absolutely nothing to do with the progression o f time. Nor, therefore, can such concepts as past, present and future play any part here. The ‘order o f time’ which is in question is based upon a general rule. It refers to any time, to time in general; and precisely in so far as it refers the concrete progression o f time to an abstract order it seems to annul the temporal progression, rather than give it a direction. Yet Kant s example seems to aim at a necessary connection between the definition o f a temporal direction and the dynamical succession; for the process which he describes is presented by him as non-reversible. Yet it is clear that this is deceptive, for Kant, in the hypothetical reversal o f the process, annulled the assumed completeness o f the dynamical conditions by disregarding the ball and the action related to it; hence the example proves fundamentally nothing other than that a dynamical process can only be lawfully derived if its conditions are given in full. The real meaning o f the example can then be effordessly annulled by a complete reversal o f the process that has been described. The proposition put forth by Kant: if I lay the ball on the cushion, then the previous smooth shape o f it is succeeded by the indentation,

can be juxtaposed with the other proposition: If I pick up the ball from the cushion, then the indentation is succeeded by the smooth shape.

And if it really is a question o f dynamical processes (which is open to doubt), then the second proposition is just as valid as the first. It follows, however, that if the process encounters an observer in the ‘sequence o f time’ , the dynamical law gives him no kind o f guiding point for deciding whether the ball is laid down or picked up, whether the smooth shape succeeds the indentation, or the indentation succeeds the smooth shape. The whole question o f ‘earlier’ or ‘later’ has, from a dynamical point o f view, no meaning.79 79 Kant asked: does the indentation succeed the ball, or the ball succeed the indentation? He obtained an unambiguous answer; not because the dynamical connection, as he believed, is by nature asymmetrical, but because he had selected asymmetrically from the system o f the dynamical relations two points o f contact

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That observer, however, whose intention it was to derive the later state dynamically from the earlier, now has to admit, for better or for worse, that when judged by the criterion o f the dynamical connection he is proceeding in an absolutely arbitrary manner if he regards that state whose conditions he posits as dynamically ‘given’ to be the earlier or present state, and that which he intends to derive as the later or future state. The distinction between present and future, or, in more general terms, between ‘earlier1 and ‘later’ (the arrow which in fig. i attributes to the distance O O ' a direction as well as a magnitude), introduces what amounts to a novel point o f view as far as the dynamical connection is concerned, and cannot be justified or understood on that basis. This new point o f view is, however, indispensable if the relation o f the dynamical connection is to yield an actual order o f time, not merely the schema for a possible order o f time. In order to obtain an empirical meaning, the ‘order’ has to return to the ‘progression’ from which it has freed itself. Yet how can the ‘progression’ be subsumed under the ‘order’? For this purpose a Majisetzung, the positing o f a system of measurement, is necessary. From within the realm o f empirical occur­ rence a process must first be arbitrarily selected, whose progression in time is regarded as ‘constant’, and which by its own direction simultaneously indicates the direction which leads from the past to the future. B y systematically relating all other processes to this one (although there remains the task o f determining more precisely what is meant in detail by such a ‘systematic relation’), the whole time progression is structured in terms o f a given temporal structure: some processes prove just as constant as this one, others deviate in a measurable way from this constancy; some prove to be ‘simultaneous’ which are contained in a process o f movement and isolated them from the rest. Admittedly, this procedure had a sound methodological basis. The ‘change o f state’ was for Kant, who takes as his starting point Newton’s concept o f dynamics, an ‘action* which assumes ‘activity and force’; and this ‘activity and force’ he placed in the ball. ‘Now, we do not a priori have the least concept as to how anything can be changed at all, i.e., how it is possible that one state occurring at one point o f time can be succeeded by an opposite state occurring at another point o f time. This concept requires knowledge o f actual forces— e.g., knowledge o f the motive forces, or, which is the same thing, o f certain successive appearances (as motions) indicating such forces— and such knowledge can be given only empirically’ [Kritik der reinen Vemunfi, B 252/A 206-7]. The whole argument o f the second analogy depends upon Newton’s assumption that ‘motions which indicate such forces’ exist.

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with the given process, others precede it in time or follow it. In order to be able to display the actual existence o f all these interrelations, however, it does not suffice to know the measure for determining time that was fixed when the process was selected, but it must also be possible to make use o f this metric standard as a worthwhile measuring instrument by applying it. to other processes. The measuring process must be capable o f being ‘repeated’, for it must also be possible to decide with regard to processes which precede or follow it whether they are just as ‘constant’ as the original process. The measuring process must be capable o f duplication, for it must be possible to establish even for processes which are spatially distant from it, whether these are just as ‘constant’. Moreover, what is valid for the transference o f the criterion o f ‘constancy’ is no less valid where the transference o f such determinations as ‘simultaneous’ , ‘successive’, etc. is concerned. The specification o f a means o f transference (or application to other situations) constitutes, therefore, an indispensable part o f the process whenever a complete metric standard is established, and, just like the choice o f the process that is to be transferred, it can in the first place be determined arbitrarily. This arbitrariness ceases, however, once it is maintained that on the basis o f a given standard (which implies a process and a method o f applying it), the events as they elapse structure themselves to form a given temporal structure. For there lies in this assertion a real hypothesis concerning the nature o f this very progression; and this hypothesis is either true or false. The case is exacdy the same as that which we described above with regard to the determination o f spatial structure.80 For the determination o f a physical temporal system also presupposes three different elements which enter into a functional relationship with one another: 1. a temporal structure— Zeitgefuge (Z); 2. the positing o f a system of measurement— Maflsetzung (M); 3. an array o f ‘chronographical’ data— chronographischer Tatbestand (7 ). Z is the schema o f a possible order o f time, for example that o f the ‘dynamical connection’ , as depicted in fig. 1. M is the positing o f a system o f measurement, such as that described above. T embraces the observed coincidences in a way that is neutral with regard to all metrical calibrations, in that it says nothing o f the events to be described except that one o f them lies ‘between’ two others, that two 80 Cf. §2, and in particular §6.

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o f them He ‘outside each another’ , that two o f them ‘coincide’ , and the like.81 The relation between Z, M and T is, however, as was the case with space, o f such a nature that if two o f these elements are given, the third is unambiguously determined thereby. Hence the temporal structure and the system o f measurement can be freely chosen, but in so doing we anticipate a specific array o f chronographical data and expose ourselves to the danger o f being contradicted by observation. We can choose the temporal structure freely, too, and attempt to relate it to a given set o f chronographical data, but for this purpose a system o f measurement must be postulated which may in fact be quite unrealizable in the real world. The third way is to adopt a heuristic procedure; on the basis o f an arbitrarily chosen, yet actual system of measurement, we seek to structure the chronographical data, in order to see what kind o f a temporal structure emerges. N ow we can explain precisely why our observer had such difficulties in arriving at a physical system o f time. He had taken as his starting-point a very specific temporal structure (the system o f the dynamical connection), had in a quite undefined way— how could it be otherwise?— interpreted the array o f chronographical data as a ‘succession o f apprehension’; what, however, he had not under­ taken— the truly decisive step— was to establish the standard of measurement. Does a system o f measurement4exist’ which is o f such a nature that it can recast the observable chronographical data in the form o f the temporal dynamical connection?82 This is precisely the question to which Heisenberg responded with such an emphatic negative when he declared that it is fundamentally impossible to know ‘the present in its entirety’. Every attempt to procure actuality for the dynamical temporal system by applying a system o f measurement does indeed incur a fundamental difficulty. In what follows an attempt will be made to expound the principle o f this difficulty in as general a form as possible.

Q.

For an example o f such a purely chronographical description cf. the table compiled by C. D. Broad in chapter ii (‘The general problem o f time and change’) of his Scientific Thought (1923), 55. 82 The word ‘exist’ is to be understood in the physical, not merely in the mathematical sense. Cf. §6.

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§26. The Tree Play’ of the Present Instead o f presupposing, as we have done up to now, an observer who believes that he knows what the ‘present’ means in physics, and on the basis o f this presumed knowledge makes a section through the world which cleanly separates the past from the future, let us imagine an observer who knows nothing o f the present other than that it lies ‘between’ the past and the future, yet who is capable o f forming an unambiguous physical concept o f the difference between past and future by positing a metric standard. From the point O, where he is ‘now’ situated, he distinguishes between two quite different parts o f the world: (a) that part o f the world from which effects can reach O; (b) that part o f the world which effects proceeding from O can reach. He calls the first part the past, the second the future. The world, to the extent to which the observer enters direcdy into contact with it, is described completely by this division. Every signal which he observes from O connects him with something past. The star which he ‘now’ sees is mediated to him by rays o f light which require time in order to reach him. Conversely, however, every signal which he himself gives off travels into the future. Sent out ‘now’, it links him with something that will be later. The ‘present’, however, regarded from this point o f view, is that which is absolutely unconnected, unmediated: on the one hand O itself, which requires no mediation in respect to itself, on the other hand that region which is beyond reach from O, which can be defined negatively inasmuch as effects cannot reach O from that region, nor can effects proceeding from O reach it. The peculiar nature o f this neutral region, whose existence nevertheless manifests itself inasmuch as effects from it can reach into O’s future, just as it can itself receive effects from O’s past, can be illustrated by the diagram in fig. 2.83 Instead o f dividing the past unambiguously from the future by a straight vertical line, here the present is indicated by a double wedge, which inserts itself (from above and below in the diagram) between past and future and meets the point O at an angle (Z .E O G and Z.FO H ) the magnitude o f which stands in an inverse relation to the velocity o f propagation o f the middle term by which O is connected with the world. 83 For the configuration o f the time cone cf. Weyl, ‘Zeitverhaltnisse im Kosmos’ (1927); Whitehead, Concept of Nature (1920), 3 1; Eddington, Physical World (1928), 48.

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H Fig. 2

If a middle term existed which expanded in space with infinite velocity, then the two straight fines E H and G F would combine to form a single straight fine, which, like C D in fig. I , would meet AB at right angles, at the point O; the ‘wedges’ would converge in a vertical ‘section’ . As, however, light which is employed as the middle term is propagated with a velocity which, even while at its maximum, is nevertheless finite, past and future form two distinct cones with a common tip O— two ‘cones o f fight’ between which lies the ‘dark­ ness’ o f the present. Every point in the world which, seen from O, belongs to this dark region, has its own ‘now’ , from which the cones o f light emanate. These can, however, only meet the world-fine on which O lies either in O’s future or in O’s past, and thus O itself, seen from these world-points, is also a ‘dark point’. The ‘present’ , according to this model, is never unambiguously determinable from O. Yet it is the present itself which contains those elements which condition O’s future. This future will, therefore, be undetermined in the same degree as the present is indeterminable. The system o f the dynamical connection appears inapplicable because the concept o f ‘now’ is fundamentally polyvalent. §27. Linear and Configurai Progression o f Time In this polyvalence o f the ‘now’, which emerges as an inevitable consequence from the use o f signals, we possess the exact counterpart

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of that unambiguity o f the now, which was required as the indispensable premiss o f the dynamical connection. On the one hand, a linear succession o f layers, each o f which is supposed to embrace the world in a moment in such a way that all points, being defined by this moment, enter into an unambiguous relation to one another. On the other hand, a configural succession o f zones, which accompany a single point through time and in each case embrace that part o f the world which does not ‘at the moment’ stand in an unambiguous relation to the point. In these two cases the word ‘moment’ is being employed in a different sense. In the linear succession it denotes an ideal requirement concerning whose capacity to be fulfilled nothing has as yet been pronounced. In the configurative succession it denotes a physical context which limits the area o f that which may be fulfilled. Yet this opposition does not seem insuperable. The linear succession, as already indicated, can be viewed as a borderline case o f the config­ urative— as that case in which the middle term which mediates between the individual points is propagated through space with infinite velocity. If this case were realizable, then the required ‘section’ could actually be implemented. The ‘now’-connection could be directly (that is, actually in the now) be established. As, however, this condition cannot actually be fulfilled, the ‘section’ expands again and again to form a ‘wedge’ , and the twofold question arises: 1. Are we to regard the ‘section’ as the ideal case, as the norm according to which we must ‘correct’ the actual observations? 2. O r are we to dismiss it as Utopian because it does not take account o f the actual conditions? The decision cannot be taken on an arbitrary basis, but each o f the two alternatives gives rise to a different hypothesis concerning the structure o f the matter under consideration, so that it depends upon the matter actually being dealt with whether the one alternative applies or the other. If the decision is in favour o f the first case, then we assume that the mediating middle term has in fact nothing in common with the object under investigation, that it is imposed upon it externally like a veil which conceals the real situation from us, and through which we must gradually penetrate in order to reach the object itself. Here the physical world is seen as a constellation o f bodies, and the medium is the [if] ou which is to be overcome, and which we can best overcome by resolving it too into a constellation o f bodies. The dynamical view here clearly takes on the character, in its experimental aspect, o f a corpuscular hypothesis, and indeed the

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concept o f the ‘initial state’ o f a system, as assumed by the dynamics, only makes sense if we are concerned with material points or bodies which we may regard as ‘rigid’ . If, on the other hand, we decide in favour o f the second alternative, then the middle term must itself be regarded as a part o f the object under investigation and every definition o f the physical object must be rejected which requires the exclusion o f the mediating processes. For this investigation the uncertainties caused by the process of mediation do not amount to an ‘accidental difficulty’ , which has to be corrected out, but (as Eddington expressed it) to an ‘elaborately arranged conspiracy which is intended to prevent us from seeing something which does not exist’ .84 If in the determination o f space­ time relations we hence meet, contrary to our expectations, a multi­ plicity o f possible interpretations, then this must cause us to question whether we are not perhaps deceiving ourselves concerning the nature o f the object, if we expect unambiguity. Unconsciously we are perhaps assuming a form o f temporal relation which may meaningfully exist between ‘rigid’ bodies, whereas it is fundamentally unsuitable in the case o f a wave process, which has to be linked to a period. It is evident that the first alternative has proved valid precisely in the theory o f celestial motions, where indeed the modifications determ­ ined by the finite velocity o f the propagation o f light necessarily seemed to be an element that had to be corrected out. Here the special theory o f relativity sought and found a middle way. An ‘absolute’ correction is, according to this theory, impracticable, because every measurement takes place within the physical world, in which there is no middle term with an infinite velocity o f propagation. As it is, however, possible to view light as a medium whose velocity o f propagation, although of finite magnitude, is nevertheless the same for all reference frames, then through this middle term a consensus becomes possible concerning the rules according to which the ‘now’-concepts diverge from one another in the ‘dark’ zone, and the ambiguity can be mastered. Even by this means, however, the ‘wedge’ can never be reduced to a ‘section’ . Yet it may, as it were, be split up into a ‘shaft’ o f sections, each representing a relatively valid connection to the ‘now’, and all o f which can be converted into one another through regular transformation formulae. Relativization here becomes a means o f adapting the concept o f the 84 [Eddington, Physical World (1928), 224.]

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‘rigid’ body to the configurative progression o f time. Even after the relatively rigid system has been replaced by the ‘molluscs’, it is feasible by retaining the concept o f the ‘material point’ to rescue the ‘dynamical law’ in terms o f relativity, despite all the uncertainties o f its configuration. Atomic theory has gone exacdy the opposite way. Instead o f rendering the concept o f the ‘rigid’ body ever more flexible through relativization, until ultimately all ambiguous configuration transforms itself into a system o f unambiguously allocated points o f coincidence, it has made precisely the concept o f the point ever more dependent on the concept o f configuration, until ultimately the material point as such has become the product o f the superposition o f waves. This reversal o f the process becomes immediately understandable if we reflect that in cosmic investigation we progress from the part to the whole, but in atomic research from the whole to the part, thus applying the ‘principle o f internal delimitation’ in the opposite direction. The second process, however, reflects back upon the first, for it is precisely that concept o f the point which field theory retained as the last element which is eroded and annulled by quantum theory. §28. Internal Delimitation and Indetermination A grave methodological objection may be raised against the whole approach adopted in the last few paragraphs. It seems extremely adventurous to place the macroscopic uncertainty o f the nowconnection, resulting from the finite velocity o f the processes o f mediation, on the same footing as that uncertainty which arises in the microscopic context, which is based upon the principle o f superposition o f waves, or even to attempt to trace it back in the same way to the configural concept o f time as their common root. The physicist will shake his head doubtfully if a model such as the time cone, which was introduced as a macroscopic phenomenon, is taken as a starting-point with which deliberations concerning microscopic structures are then linked. Moreover, these doubts cannot be immediately appeased by reference to the fact that the intention is only to demonstrate by means o f this model a configuration o f a quite general kind, whose fundamental characteristics are preserved in principle even if they are transformed by inverting the direction o f determination in order to consider microscopic conditions. There is, however, an entirely different consideration which serves to justify our

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procedure, and this indeed seems to me decisive (for the present purpose): the two types o f ‘uncertainty’ reveal not merely a structural correspondence, but they also have a common basis. In both cases the room for variation arises because the instruments with which we investigate the physical world themselves belong to this world. If we could fix the cosmic distances at a point in time ‘from the outside’ , then we would not be dependent upon light as the medium o f measurement; the polyvalence determined by the finite speed o f light would vanish without trace, and the need for compensation by relativization would become superfluous. If we could also determine the position o f a particle ‘from the outside’, then we could dispense with the superposition of waves as a means o f determination, and the uncertainty principle regarding position and momentum would not posit the quantum as the lower limit o f our attempt to obtain greater precision. We ‘could’ , but yet we know that this ‘could’ is nonsense; nonsense not only in factual terms, because the harsh fact is that we cannot get out o f the world in which we find ourselves, but also logically nonsense, because the concept o f a measurement from the outside is a self-contradiction. Every measurement expresses a propor­ tional relationship between the measuring instrument and what is measured by it, and both elements o f the proportion must be commensurable. Just as it is impossible to measure the intensity o f pain by using a metre rule, or the weight o f a body on a colour scale, so too it is meaningless to look for an extraterrestrial metric for the determination o f the proportions prevailing in the world. Only in the world itself are those instruments to be found by which the world can be measured. Hence every judgement concerning the position o f the instruments relative to the world is simultaneously a judgement concerning the metrical proportions existing in the world. If, therefore, the systematic use o f rays o f light in cosmic measurements leads to polyvalence in the determination o f space and time, and if, furthermore, the systematic superposition o f waves in atomic measurements leads to polyvalence in the determination o f a particle, then to establish these uncertainties does not by any means amount to a lament over the inadequacy o f our instruments, which by dint o f their physical nature prevent us achieving an adequate comprehension o f the unambiguously determined natural event; rather this brings with it the positive assertion that, precisely in so far as the structure o f the world which is to be cognized expresses itself in this characteristic nature o f the instruments, the adequate use o f the instruments

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instructs us that the unambiguous relations we presumed to exist in the world do not exist at all. The concept o f indetermination, which is prevalent in the cognition of these uncertainties, acquires from this point o f view an entirely positive meaning. It forms the exact opposite to the concept o f finiteness which (in the first and second antinomy) posited an inner limit to spatial extension and spatial division. Just as in that case infinity was excluded not merely relatively, but absolutely, here too it is not a question o f an external limit o f determination, which we encounter in the process o f cognition and which we push further and further forward, without ever being able to annul it, but o f a fundamental alternative to the concept o f necessary connection which is here presupposed. The probability calculus which gives the laws o f statistics their form is misunderstood if in it we see only a provisional limitation o f the certainty required by the dynamical law. It is a question o f an entirely different form o f certainty. The opposition between these two forms can best be made clear by comparing their relationship to ‘uncertainty’. In the dynamical law certainty requires the absolute exclusion o f the uncertain. In the statistical law certainty requires knowledge o f the degree o f uncertainty. Measured by the requirements o f the dynamical law, every statistical law must, therefore, appear insufficiendy determined. Measured by the metrics o f the statistical law, that certainty to which a dynamical law lays claim can prove deceptive. In their application, however, the two forms o f law stand in an asymmetrical relation to each other, for a dynamical law can be exposed as statistical if that relation which it sets out as necessary is proved to be only very probable; a statistical law, however, cannot be exposed as dynamical; instead the connection which is to be proved necessary must be sought at a level below the merely probable. In each individual case, therefore, the annulment o f a dynamical law by a statistical law means an actual change, whereas the possibility o f transforming this statistical law for its part into a dynamical law presupposes the existence o f a deeper layer, and— until the existence o f this layer is proved— must remain an empty postulate. It is possible to guard against even this empty postulate, however, if the reasons for the conversion o f the dynamical law into a statistical law are at the same time reasons for asserting that here the lowest layer has been reached. It is then a question o f a real hypothesis concerning the nature o f matter, which— like every real hypothesis— is either true or false; its truth or falsehood cannot,

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however, be asserted or disputed by argument, but must be proved on an experimental basis. In quantum mechanics, in particular, it is this tendency which explains the vigour with which Heisenberg rejects ‘the assumption that behind the [...] statistical world there hides another “ real” world in which the law o f causality is valid’ as ‘unfruitful’ and ‘meaningless’ speculation: ‘On the contrary, the true state o f affairs can be much better characterized as follows: since all experiments are subject to the laws o f quantum mechanics, it is therefore through quantum mechanics that the invalidity o f the law o f causality can be definitively established.’85 In this proposition the apparendy dialectical question o f the third antinomy is quite clearly reduced to an empirical question. Anyone wishing hereafter to assert the causal determinateness (or even merely the causal determinability) o f world occurrence will not be able to retreat into speculative arguments, but must find an experiment which, in express opposition to Heisenberg’s hypothesis, is not subject to the laws o f quantum mechanics. Only if such an experiment can be carried out, that is, if there is a procedure determinable as a ‘deviation’ from quantum mechanics, can the proposition put forth by Heisenberg be regarded as contradicted.86 85 (Heisenberg, ‘Kinematik und Mechanik’ (1927), 197; repr. in Gesammelte Schriften, Series A, Part I (1985), 503.] 8 It is most striking that it is precisely the great theoreticians o f the older generation, Planck and Einstein, who seek to avoid the ultimate consequence o f this approach, championing a return to the rigid concept o f causality by means o f arguments which are in essence entirely ‘Idealist’ and far removed from their earlier methodological doctrine. Thus Planck posits a purely ideal realm o f thought events in which the law o f causality retains its validity in opposition to the realm o f actual measurements; cf. Planck, Kausalbegriff (1932), esp. 21 ff. [repr. in Physikalische Abhandlungen und Vorträge (1958), iii. 237-9]. He does so despite his own challenge directed at the ‘axiomaticians’ to translate the ‘equation’ into ‘measurement’; Planck, Weltbild (1929), esp. 12 [repr. as ‘Zwanzig Jahre Arbeit’ (1929), 197; Physikalische Abhandlungen und Vorträge (1958), iii. 183; English transl. Johnston (1931), 11- 12 ] . Einstein too, although he was originally only willing to admit a concept as meaningful in physical terms if it included directions for measurement (cf. Spezielle und allgemeine Relativitätstheorie [1917]), has recently argued that, while causality as an empirical proposition has lost its meaning, nevertheless it must be preserved as a ‘form o f the theoretical system’, for no other principle possesses the same stringency and ‘simplicity’. (See for example his letter to Sir Herbert Samuel, printed in the latter’s Presidential Address to the British Institute of Philosophy [1932], 33.) But are these arguments, put forward as a mere programme, so very different and so very much better than those with which the battle against applying non-Euclidian geometry was once waged?

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§29. ‘Constancy’ and ‘Emergence’ Considered formally, a statistical regularity o f nature isjust as conceivable as a dynamical regularity. If it is asserted that the one form o f regularity can be reduced to the other, then a hypothesis concerning the structure o f matter is already present. In order to clarify the peculiar nature o f these hypotheses, I will assume in the first place that the ‘actual’ natural laws are dynamical, and examine how, given this assumption, the relation between dynamical and statistical laws presents itself. Dynamical laws, when they are subjected to experimental testing, require the fulfilment o f a well-defined set o f conditions and maintain that if these conditions are fulfilled, a given result will necessarily occur. Their claim to general validity is based upon their conditional form, which requires that the original constellation o f factors be unambiguously determinable. Statistical laws, on the other hand, deal with processes so complex that the original constellation o f factors cannot be described by means o f a well-defined set o f conditions, but only by means o f a range o f alternative sets. The result cannot therefore be predicted with absolute certainty, but can only be calculated by the means o f the theory o f probability. For each o f the alternative sets o f presuppositions the result can be predetermined with certainty, provided their conditions are so unambiguously defined that they can be comprehended under a dynamical law. The probability o f the occurrence o f precisely these conditions, however, can only be expressed in terms o f the numerical relation between the one case and the totality o f possible cases. The certainty o f the dynamical law thus serves as a basis for calculating the probability o f the statistical law. The following example, taken from a lecture given by Planck,87 may serve to illustrate the case in more detail: if we throw a hot piece o f iron into a vessel containing cold water, then we expect the iron to cool and the water to heat up until the temperature o f both becomes equal. This is a statistical, not a dynamical law, for heat must be understood as a molecular movement, and the different degrees o f temperature must be traceable back to the different speeds o f the molecules in motion. If the molecules o f the hot iron meet those o f 87 Planck, Dynamische und statistische Gesetzmäßigkeit (1914), 1 1 - 1 3 ; repr. in Physikalische Rundblicke (1922), 91 [cited by Wind]; [Physikalische Abhandlungen und Vorträge (1958), iii. 81-2; English transl. Jones and Williams in Planck, Survey (i960), 59-