PROOF, LOGIC AND FORMALIZATION This book addresses various problems associated with finding a philosophically satisfying account of mathematical proof. It brings together many of the most notable figures currently writing in the philosophy of mathematics in an attempt to explain why it is that mathematical proof is given prominence over other forms of mathematical justification. The difficulties that arise in accounts of proof range from the rightful role of logical inference and formalization to questions concerning the place of experience in proof and the possibility of eliminating impredicative reasoning from proof. The editor, Michael Detlefsen, has brought together an outstanding collection of essays, only two of which have previously appeared. Essential for philosophers and historians of mathematics, it is also recommended for philosophically inclined logicians and philosophers interested in the nature of reasoning and justification. A companion volume entitled Proof and Knowledge in Mathematics, edited by Michael Detlefsen, is also available from Routledge. PROOF, LOGIC AND FORMALIZATION PROOF, LOGIC AND FORMALIZATION Edited by Michael Detlefsen London and New York First published 1992 by Routledge 11 New Fetter Lane, London EC4P 4EE This edition published in the Taylor & Francis e-Library, 2005. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.” Simultaneously published in the USA and Canada by Routledge a division of Routledge, Chapman and Hall, Inc. 29 West 35th Street, New York, NY 10001 Selection and introductory material © 1992 Michael Detlefsen Individual chapters © 1992 the respective authors. All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing in Publication Data Proof, logic and formalization 1. Mathematics I. Detlefsen, Michael 511.3 Library of Congress Cataloging in Publication Data Proof, logic and formalization/edited by Michael Detlefsen. p. cm. Includes bibliographical references. 1. Proof theory. 2. Logic, Symbolic and mathematical. I. Detlefsen, Michael. QA9.54.P77 1992 511.3–dc20 91–17469 ISBN 0-203-98025-5 Master e-book ISBN ISBN 0-415-02335-1 (Print Edition) In memory of Kenneth Lee Steel, Born 18 March 1947, Lincoln, Nebraska, Died 28 February 1967, Bong Son, South Viet Nam. “Therefore I have uttered what I did not understand, things too wonderful for me, which I did not know. ‘Hear, and I will speak; I will question you, and you declare to me.’” Job 42 CONTENTS Notes on contributors vii Preface viii 1 PROOFS ABOUT PROOFS: A DEFENSE OF CLASSICAL LOGIC. PART I: THE AIMS OF CLASSICAL 1 LOGIC John P.Burgess 2 PROOFS AND EPISTEMIC STRUCTURE 8 Glen Helman 3 WHAT IS A PROOF? 22 Richard Tieszen 4 HOW TO SAY THINGS WITH FORMALISMS 31 David Auerbach 5 SOME CONSIDERATIONS ON ARITHMETICAL TRUTH AND THE ω -RULE 39 Daniel Isaacson 6 THE IMPREDICATIVITY OF INDUCTION 60 Charles Parsons 7 THREE INSUFFICIENTLY ATTENDED TO ASPECTS OF MOST MATHEMATICAL PROOFS: 71 PHENOMENOLOGICAL STUDIES Robert S.Tragesser 8 ON AN ALLEGED REFUTATION OF HILBERT’S PROGRAM USING GÖDEL’S FIRST 88 INCOMPLETENESS THEOREM Michael Detlefsen Index 104 NOTES ON CONTRIBUTORS David Auerbach teaches philosophy and logic at North Carolina State University in Raleigh. John P.Burgess received his doctorate from the Group in Logic and the Methodology of the Deductive Sciences at the University of California at Berkeley, and is now Professor of Philosophy at Princeton University. He is an editor of the Journal of Symbolic Logic and the Notre Dame Journal of Formal Logic, and a frequent contributor to journals and anthologies in mathematical and philosophical logic and the philosophy of mathematics. Michael Detlefsen is Professor of Philosophy at the University of Notre Dame and Editor-in-Chief of the Notre Dame Journal of Formal Logic. His areas of special interest are the philosophy of logic and mathematics. He is the author of Hubert’s Program: An Essay on Mathematical Instrumentalism (Reidel, 1986). Currently, he is working on a book on intuitionism, and coauthoring another with Michael Byrd on Gödel’s theorems. Glen Helman, formerly a member of the Philosophy Department of the Pennsylvania State University, is now Assistant Professor of Philosophy at Wabash College. His publications concern the equivalence of proofs and their relation to the extended lambda calculus. Daniel Isaacson is currently University Lecturer in the Philosophy of Mathematics at Oxford University and Fellow of Wolfson College, Oxford. He has been lecturer in philosophy at the University of Washington in Seattle and visiting professor of philosophy at the University of California, Berkeley, and has held research posts at the Rockefeller University in New York and St John’s College, Oxford. He was an undergraduate at Harvard, and took his doctorate at Oxford. Charles Parsons is Professor of Philosophy at Harvard University. From 1965 to 1989 he was on the faculty of Columbia University. He is the author of Mathematics in Philosophy (Cornell University Press, 1983) and of papers on proof theory, intensional logic, philosophy of logic and mathematics and Kant. Richard Tieszen received his Ph.D. in philosophy at Columbia University and is presently Assistant Professor at San Jose State University, located in the Silicon Valley. His research interests include logic and logic programming, philosophy of mathematics, and cognitive science. Robert S.Tragesser is currently Assistant Professor of Philosophy at Barnard College in Columbia University, and Research Associate at the CUNY Graduate Center. He is the author of two books, Phenomenology and Logic (Cornell University Press, 1977) and Husserl and Realism in Logic and Mathematics (Cambridge University Press, 1984). He is at work on a book, Logical Imagination, which will contain (among other things) an elaboration of the ideas presented in his paper in this volume. PREFACE Among the features of mathematical thought that most invite the philosopher’s attention is its regimentation of justificatory procedures; its canonization of proof as the preferred form of justification. This regimentation is made the more striking by the fact that it is clearly possible to have other kinds of justification (even some of very great strength) for mathematical propositions; a fact which at least suggests that concern for truth and/or certainty alone cannot account for the prominence given to proof within mathematics. This being so, arriving at some understanding of the nature and role of proof becomes one of the primary challenges facing the philosophy of mathematics. It is this challenge which forms the motivating concern of this volume, and to which it is intended to constitute a partial response (on a variety of different fronts). One possibility, of course, is that it signifies nothing distinctive in the epistemic aims of mathematics at all, but is rather the outgrowth of certain purely social or historical forces. However, since regimentation of justificatory practice is the kind of thing that is at all likely to arise only out of consideration for epistemic ideals, to attribute it to nonepistemic, socio-historical factors would require special argumentation, and there is currently little if any reason to suppose such argumentation might be given. Thus, accounts of the ascendancy of proof in mathematics that make appeal to some epistemic ideal(s) would appear to be more promising. One consideration which seems to have weighed heavily in favor of proof as the preferred form of justification is the traditional concern of mathematics for rigor. This drive, in turn, seems to have derived from one of two (in some ways related) impeti. The first of these—an old and persistent one, with a bloodline reaching from Aristotle through Leibniz and on down to Frege —s based on the idea that the aim of a justification is not merely, nor even primarily, to establish the certainty of its conclusion, but rather to identify the metaphysical grounds or origins. This induces a hierarchy of truths at the base of which are located the ur-truths or axioms. The epistemic ideal is to be able to trace a given truth all the way back to the ur- truths which found it, thus obtaining a complete account of why it is true. Rigor is essential to this enterprise because, without it, one cannot get a clear distillation of those truths that are being used (and perhaps even need to be used) to ground a given truth. A historically more influential idea—also associated with Aristotle (in particular, with his conception of a deductive or demonstrative science), as well as with Euclid and, in the modern period, with Descartes—has a less metaphysical, more purely epistemic foundational thrust. It too searches for ultimate truths, but not in the sense of final metaphysical grounds of truth. Rather, it seeks to find premises that are epistemically ultimate—truths, that is, that are (at least in certain respects) epistemically unsurpassed. In this type of scheme, the axioms are not taken to be metaphysically responsible for the truth of the theorems, though there is still a sense in which they may be said to show why they (i.e. the theorems) are true. More recently, these traditional conceptions have given way to less foundationally oriented conceptions of mathematical justification. Axioms are no longer taken as giving the metaphysical grounds for the theorems. Nor are they taken to be insurpassably, or even insurpassedly, evident. Rather, they are seen only as being evident enough (i) to make a search for further evidence seem unnecessary, or perhaps (ii) to remove sufficiently much of whatever rational doubt or indecision it seems possible and/or desirable to remove from the theorems, or maybe just (iii) to be attractively simple and economical, and evident enough to serve the purposes at hand, etc.