Kant's Philosophy of Mathematics Synthese Library

KANT'S PHILOSOPHY OF MATHEMATICS SYNTHESE LIBRARY

LOGIC, METHODOLOGY, AND PHll.OSOPHY OF SCIENCE

Managing Editor:

JAAKKO lßNTIKKA, Boston University

Editors:

DONALD DAVIDSON, University 0/ California. Berkeley GABRIEL NUCHELMANS, University 0/ Leyden WESLEY C. SALMON, University 0/ Pittsburgh

VOLUME219 KANT'S PHILOSOPHY OF MATHEMATICS

MODERN ESSAYS

Edited by CARLJ. POSY Duke University, North Carolina, U.s.A.

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. Library of Congress Cataloging-in-Publication Data

Kant's phllosophy of mathematlcs : modern essays I edlted by Carl ~. Posy. p. cm. --

ISBN 978-90-481-4105-0

Printed on acidlree paper

All Rights Reserved © 1992 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1992 Softcover reprint of the hardcover 1st edition 1992 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. TABLE OF CONTENTS

PREFACE vii

ACKNOWLEDGEMENTS ix

CARL J. POSY - Introduction: Mathematics in Kant's Critique of Pure Reason

I. CLASSIC PAPERS OF THE 1960'S AND 1970'S

JAAKKO HINTIKKA - Kant on the Mathematical Method 21 CHARLES PARSONS - Kant's Philosophy of Arithmetic 43 - Postscript 69 MANLEY THOMPSON - Singular Tenns and Intuitions in Kant's Epistemology 81 - Postscript 102 PHILIP KITCHER - Kant and the Foundations of Mathematics 109

11. RECENT WORK

CHARLES PARSONS - Arithmetic and the Categories 135 J. MICHAEL YOUNG - Construction, Schematism and Imagination 159 MICHAEL FRIEDMAN - Kant 's Theory of Geometry 177 STEPHEN BARKER - Kant's View of Geometry: a Partial Defense 221 ARTHUR MELNICK - The Geometry of a Fonn of Intuition 245 WILLIAM HARPER - Kant on Space, Empirical Realism, and the Foundations of Geometry 257 CARL J. POSY - Kant's Mathematical Realism 293 GORDON G. BRITTAN, JR. - Algebra and Intuition 315 JAAKKO HINTIKKA - Kant's Transcendental Method and His Theory of Mathematics 341

v vi TABLE OF CONTENTS

CONTRIBUTORS 361

INDEX 363 PREFACE

For the bulk of the twentieth century, Kant's philosophy of mathematics fell into low disrepute. His views suffered partly from the nineteenth and twenti• eth century challenges to the hegemony of Euclidean geometry. And their reputation declined partly because the great thinkers - RusselI, Hilbert, Brouwer and others - who shaped the twentieth century's philosophical approach to mathematics each took aim in his own way at central Kantian doctrines. To be sure, the early years of the twentieth century saw lively debates about Kant's mathematical views. (The famous interchanges between Russell and Couterat are but one instance of this.) But the generally negative assessment of Kant's philosophy of mathematics led naturally to general neglect. So, with scant exception, the middle of the century saw little creative work on this topic. The one main exception is the late Gottfried Martin's work. Martin's Arithmetik und Kombinatoric bei Kant appeared in 1938. Though the book's positive thesis - that Kant viewed arithmetic as a formal axiomatic science - is still highly controversial, its scholarship and its treasure trove of insights about Kant's way of thinking remain unsurpassed. It is indeed still the best starting place for anyone who wishes to do scholarly work on reactions to Kant's philosophy of mathematics. But for a long time Martin's work had sadly small impact on the wider philosophical community, and exploration of Kant's views about mathematics remained a philosophical backwater. All of that changed in the mid-1960's with the publication of a Iively debate about issues in Kantian mathematics between Charles Parsons and Jaakko Hintikka. The debate, which was fascinating in its own right, - each of these authors is an accomplished logician and asolid Kantian scholar - had the added effect of reigniting philosophical interest in Kant's thoughts about mathematics. A trickle of fascinating studies - inspired by the issues that Parsons and Hintikka raised - followed in the 1970's. Then in the next decade the movement grew into a full-fledged renaissance of interest in Kant's philosophy of mathematics. This was further sparked by a 1983 con• ference on the topic at Duke University.

vii

The present anthology is designed first of all to chronicle and publicize this renaissance. But hopefully it will also show the importance of Kantian think• ing for general issues in the philosophy of mathematics. Part I includes a pair of the seminal papers by Hintikka and Parsons. It also includes inftuential papers by Manley Thompson and Philip Kitcher which followed up on this initial work and which appeared in the ensuing decade. Parsons and Thompson have added more recent postscripts to their papers. Part 11 contains a selection of papers that represent the ftowering of interest in Kant's philosophy of mathematics during the 1980's and the beginning of the 1990's. It includes new work by both Hintikka and Parsons. I hope that the essays in this volume will demonstrate that Kant's views about mathematics are interesting, that they are defensible (or at least respectable in the light of our current knowledge) and that they are insepara• ble from his overall philosophical system. Indeed, a leisurely thumb through the volume will readily testify to the fascination that many modern Kant scholars and philosophers of mathematics have for Kant's mathematical views. As for defensibility, you will find in the pages that follow quite a few attempts to vindicate or reconstruct Kantian doctrines and arguments from a modern perspective. In fact, far from viewing Kant's work on mathematics as dead, the volume contains several attempts to connect his views with quite recent findings in physical science, cognitive science and mathematical logic. Finally, you will find the volume rich and diverse with suggestions connect• ing Kant's mathematical views with his overall philosophy. This collection contains a mixture of scholarship and speculation, recon• struction and close textual analysis. It represents a fair sampie of the current state of thinking on Kant's philosophy of mathematics. To be sure, you will find no unanimity on any single point; and it is hard to tell which, if any, of the perspectives and interpretations presented here will prevail. This very diversity and vigorous debate, however, are themselves the signs of a vibrant field of study.

Durham, 1991 ACKNOWLEDGMENTS

Part of the editorial and clerical costs for this anthology were underwritten by a generous grant from the Duke University Research Council. The editor is also indebted to Tod Davis and Marianne Grochowski for editorial assistance. The editor gratefully acknowledges the permission given by the authors, editors and copyright holders of the following publications to reproduce them in original or revised form in this anthology:

"Kant on the Mathematical Method" was first published in The Monist, volume 51, 1967, pages 352-375. It is copyright © 1967 The Monist, and is reproduced here by kind permission of Jaakko Hintikka and the publishers of The Monist, La Salle, Illinois, 61301.

"Kant's Philosophy of Arithmetic" was first published in Sidney Morgenbesser, Patrick Suppes, and Morton White, eds., Philosophy, Science and Method: Essays in Honor 0/ Ernest Nagel, 1969, St. Martin's Press. It is reproduced here by kind permission of Charles Parsons and the editors.

The Postscript to "Kant's Philosophy of Arithmetic" was first published in Charles Parsons, Mathematics in Philosophy: Selected Essays, copyright 1983, Cornell University Press. It is reproduced here by kind permission of Charles Parsons and Cornell University Press.

"Singular Terms and Intuitions in Kant's Epistemology" was originally pub• lished in the Review 0/ Metaphysics, Volume 26, copyright 1972, it .is repnnted here by kind permission of Manley Thompson and the Review 0/ Metaphysics.

"Kant and the Foundations of Mathematics" was first published in The Philosophical Review, Volume 84, copyright 1975. It is reprinted here by kind permission of Philip Kitcher and The Philosophical Review.

"Arithmetic and the Categories" was first published in Topoi volume 3, copy• right 1984. It is reprinted here by kind permission of Charles Parsons and of Topoi.

"Kant's Mathematical Realism," was first published in The Monist, volume 67, pages 115-134. It is copyright © 1984 The Monist, and is reproduced here by kind permission of the publishers of The Monist, La SaUe, Illinois, 61301.