PROOF THEORY SYNTHESE LIBRARY

STUDIES IN EPISTEMOLOGY,

LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE

Managing Editor:

JAAKKO HINTIKKA, Boston University

Editors:

DIRK VAN DALEN, University of Utrecht, The Netherlands DONALD DAVIDSON, University of California, Berkeley THEO A.F. KUIPERS, University of Groningen, The Netherlands PATRICK SUPPES, Stanford University, California JAN WOLENSKI, iagiellonian University, Krakow, Poland

VOLUME 292 PROOF THEORY

History and Philosophical Significance

Edited by VINCENT F. HENDRICKS University of Copenhagen. Denmark STIG ANDUR PEDERSEN and KLAUS FROVIN J0RGENSEN University of Roskilde, Denmark

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. A c.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-5553-8 ISBN 978-94-017-2796-9 (eBook) DOI 10.1007/978-94-017-2796-9

Printed on acid-free paper

All Rights Reserved © 2000 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2000 Softcover reprint of the hardcover 1st edition 2000 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. CONTENTS

Preface ...... ix

Contributing Authors ...... , xi

HENDRICKS, PEDERSEN AND J0RGENSEN / Introduction ...... 1

PART 1. REVIEW OF PROOF THEORY

SOLOMON FEFERMAN / Highlights in Proof Theory ...... , 11 1. Review of Hilbert's Program and Finitary Proof Theory ...... , 11 2. Results of Finitary Proof Theory via Gentzen's L-Calculi ...... 14 3. Shifting Paradigms ...... 21 4. Countably Infinitary Methods (Getting the Most Out of Logic) ... 25 Notes ...... 29 References...... 29

PART 2. THE BACKGROUND OF HILBERT'S PROOF THEORY

LEO CORRY / The Empiricist Roots of Hilbert's Axiomatic Approach 35 1. Introduction...... 35 2. Heinrich Hertz...... 35 3. Carl Neumann...... 39 4. Paul Volkman...... 42 5. Physics and Geometry in Hilbert's Early Courses ...... 44 6. Grundlagen der Geometrie and its Aftermath...... 45 7. Concluding Remarks...... 50 Notes ...... 51 References...... 52

DAVID ROWE / The Calm Before the Storm: Hilbert's Early Views on Foundations '" ...... , 55 1. Hilbert's Early Career ...... , 56 2. Hilbert's Grundlagen der Geometrie ...... 63 3. Hilbert's Axiomatic Method and Frege's Critique ...... 71 4. Hilbert's Return to Foundations ...... 83 Notes ...... 87 References ...... , 88 vi

WILFRIED SIEG / Toward Finitist Proof Theory...... 95 1. Introduction...... 95 2. Background...... 96 3. Mathematical Logic ...... " 98 4. Constructive Number Theory ...... 101 5. Finitist Proof Theory...... 103 6. Remarks and Issues ...... 106 Notes ...... 108 References ...... 110

PART 3. BROUWER AND WEYL ON PROOF THEORY AND PHILOSOPHY OF MATHEMATICS

DIRK VAN DALEN / The Development of Brouwer's Intuitionism. . .. 117 1. Mysticism and the Dissertation...... 117 2. Issues and Topics in the Dissertation ...... 124 3. Place and Function of Logic ...... 128 4. The Introduction of Choice Sequences...... 135 5. The Impact of the Grundlagenstreit ...... 144 Notes ...... 148 References ...... 149

MORITZ EPPLE / Did Brouwer's Intuitionistic Analysis Satisfy its own Epistemological Standards? ...... 153 1. Introduction ...... 153 2. Some Comments on the Epistemological Standards of Intuitionism ...... " 160 3. The Problem of Solving the Fan Theorem ...... 161 4. Epistemological Discussion ...... " 166 5. Historical Discussion...... 169 6. Conclusion...... 172 Notes ...... 173 References...... 176

SOLOMON FEFERMAN / The Significance ofWeyl's Das Kontinuum 179

1. Weyl's System Reconstructed...... 187 2. Examples of Theorems ...... 192 3. Limitations of Weyl's System...... 192 Notes ...... 193 References...... 193 VB

ERHARD SCHOLZ / Herman Weyl on the Concept of Continuum. . .. 195 1. Introduction ...... 195 2. Two Different Continuum Concepts, 1918 and 1920/21 ...... 195 3. Space made up from "Infinitesimally Small Parts" ...... 200 4. Digression: Purely Infinitesimal Geometry and Field Theoretic Matter Explanation...... 202 5. "Free Emergence" in a Combinatorially Specifed Framework .... , 204 6. Problems for Weyl's Semi-Intuitionistic Approach to Manifolds .. 206 7. Symbol Systems and "Transcendent/ Transient" Reality ...... 209 Notes ...... 212 References ...... , 214

PART 4. MODERN VIEWS AND RESULTS FROM PROOF THEORY

SOLOMON FEFERMAN / Relationships between Constructive, Predicative and Classical Systems of Analysis...... 221 1. Informal Mathematical Part ...... 221 2. Metamathematical Part ...... , 229 Notes ...... 235 References...... 235

Index ...... 237 PREFACE

hiS volume in the Synthese Library Series is the result of a conference T held at the University of Roskilde, Denmark, October 31st-November 1st, 1997. The aim was to provide a forum within which philosophers, math• ematicians, logicians and historians of mathematics could exchange ideas pertaining to the historical and philosophical development of proof theory. Hence the conference was called Proof Theory: History and Philosophical Significance. To quote from the conference abstract: Proof theory was developed as part of Hilberts Programme. According to Hilberts Programme one could provide mathematics with a firm and se• cure foundation by formalizing all of mathematics and subsequently prove consistency of these formal systems by finitistic means. Hence proof theory was developed as a formal tool through which this goal should be fulfilled. It is well known that Hilbert's Programme in its original form was unfeasible mainly due to Gtldel's incompleteness theorems. Additionally it proved impossible to formalize all of mathematics and impossible to even prove the consistency of relatively simple formalized fragments of mathematics by finitistic methods. In spite of these problems, Gentzen showed that by extending Hilbert's proof theory it would be possible to prove the consistency of interesting formal systems, perhaps not by finitis• tic methods but still by methods of minimal strength. This generalization of Hilbert's original programme has fueled modern proof theory which is a rich part of mathematical logic with many significant implications for the philosophy of mathematics. Although a completely secure justification of mathematics is impossi• ble it is, however, possible to achieve many fundamental partial results con• cerning relative consistency of theories, concerning strength of axiomatic systems and finally c~ncerning the relationship between constructive, pred• icative and classical systems of analysis. The purpose of this meeting is to track the history of proof theory and its role in the analysis of the philosophical foundations of mathemat• ics from its first primitive form in Hilbert's original Programme to its modern highly articulated form. Accordingly, the emphasis will be on his• torical and epistemological important episodes in the development of proof theory, not on technical aspects. All lectures will be of such a nature that they can be followed by mathematicians and philosophers without any professional training in proof theory but provided with general knowledge of fundamental issues. The editors would like to thank the invited speakers including Prof. Solomon Feferman (Stanford University), Prof. Wilfried Sieg (Carnegie Mel• lon University), Prof. Dirk van Dalen (University of Utrecht), Prof. David Rowe (University of Mainz), Prof. Leo Corry (Tel Aviv University), Prof. ix x PREFACE

Moritz Epple (University of Mainz) and Prof. Erhard Scholz (University of Wuppertal) for contributing, in the most lucid and encouraging way, to the fulfillment of the conference aim. The editors are also grateful to the invited speakers for making their contributions available for publication. The conference was organized by the Danish Network on the History and Philosophy of Mathematics. The editors would like to thank the net• work's organizing committee consisting of Prof. Kirsti Andersen (University of Aarhus), Prof. Jesper LUtzen (University of Copenhagen), Dr. Tinne Hoff Kjeldsen (University of Roskilde) and the committee's secretary Lise Mariane Jeppesen (University of Roskilde). In turn, the network was only made pos• sible by a research grant from the Danish Natural Science Research Council for which the editors and the network remain very grateful. Finally, the editors are indebted to Prof. J0rgen Larsen (University of Roskilde), who introduced us to the finer points of typesetting in J¥IEX 2c.

Vincent F. Hendricks Stig Andur Pedersen

Klaus Frovin J0rgensen

Roskilde CONTRIBUTING AUTHORS

In order of appearence:

Solomon Feferman is Professor of Mathematics and Philosophy and Patrick Suppes Family Professor of Humanities and Sciences at Stanford Univer• sity.He is the author of In the Light of Logic and editor-in-chief of the Col• lected Works of Kurt Codel. He is noted for his many contributions to logic and the foundations of mathematics, in particular in the areas of proof theory and predicative systems of mathematics.

Leo Corry teaches at the Cohn Institute for History and Philosophy of Science, Tel Aviv University. His main research field is the history of math• ematics in the early 20th century.

David E. Rowe teaches history of mathematics and exact sciences at Mainz University. His principal research interests center around mathematics in Germany in the period from 1800 to 1945. Alongside foundations issues he has been studying parallel developments in the foundations of physics con• nected with relativity theory. In connection with the latter ,he has worked as a contributing editor for the Einstein Editorial Project at Boston University since 1998.

Wilfried Sieg is Professor in the Department of Philosophy at Carnegie Mellon University, Pittsburgh. He has been pursuing proof theoretic issues ever since his graduate studies at Stanford University: consistency proofs for subsystems of analysis, characterization of provably total functions for fragments of arithmetic, and automated search for natural deduction proofs in logic. The mathematical work has been complemented by historical and philosophical analyses of the foundational work of Hilbert and Bernays, but also of the emergemce of the concept of effective calculability.

Dirk Van Dalen is Professor in the History of Logic and Philosophy of Mathematics at Utrecht Univer~ity. He is in charge of the Brouwer Project for preparing and editing the unpublished manuscripts and correspondence of Brouwer. He has published the first volume of the Brouwer Biography. In his contribution to the current volume L.E.J. Brouwer, "Intuitionistische Betrachtungen liber den Formalismus", K on Ned. Ak. Wet. Proceedings 31: 374-379 is reproduced here by kind permission of Koninklijke Neder• landse Akademie van Wetenschappen. "Uber die Bedeutung des Satzes vom ausgescholssenen Dritten in der Mathematik insbesondere in der Funktio• nentheorie", Journal fUr die reine und angewandte Mathematik 154: 1-8 is reproduced here by kind permission of Journal fUr die reine und angewandte Mathematik. xi xu CONTRIBUTING AUTHORS

Moritz Epple presently holds a Heisenberg fellowship in the History of Mathematics. After graduating in physics and philosophy, he received a PhD in mathematical physics from Tuebingen University. Since then, he has been teaching history of science and history of mathematics at the universities of Mainz and Bonn.

Erhard Scholz is Professor of History of Mathematics at the Department of Mathematics, University Wuppertal. His main research interests include the history of the 19th and 20th centuries. He contributes to the Hausdorff edition (Bonn). Additional research interests concentrate on the historical interrelation between mathematics and physics, centered around the work of Hermann Weyl.