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Mechanism, Mentalism, and Metamathematics Synthese Library MECHANISM, MENTALISM, AND METAMATHEMATICS SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE Managing Editor: JAAKKO HINTIKKA, Florida State University Editors: ROBER T S. COHEN, Boston University DONALD DAVIDSON, University o/Chicago GABRIEL NUCHELMANS, University 0/ Leyden WESLEY C. SALMON, University 0/ Arizona VOLUME 137 JUDSON CHAMBERS WEBB Boston University. Dept. 0/ Philosophy. Boston. Mass .• U.S.A. MECHANISM, MENT ALISM, AND MET AMA THEMA TICS An Essay on Finitism i Springer-Science+Business Media, B.V. Library of Congress Cataloging in Publication Data Webb, Judson Chambers, 1936- CII:J Mechanism, mentalism, and metamathematics. (Synthese library; v. 137) Bibliography: p. Includes indexes. 1. Metamathematics. I. Title. QA9.8.w4 510: 1 79-27819 ISBN 978-90-481-8357-9 ISBN 978-94-015-7653-6 (eBook) DOl 10.1007/978-94-015-7653-6 All Rights Reserved Copyright © 1980 by Springer Science+Business Media Dordrecht Originally published by D. Reidel Publishing Company, Dordrecht, Holland in 1980. Softcover reprint of the hardcover 1st edition 1980 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 informational storage and retrieval system, without written permission from the copyright owner TABLE OF CONTENTS PREFACE vii INTRODUCTION ix CHAPTER I / MECHANISM: SOME HISTORICAL NOTES I. Machines and Demons 2. Machines and Men 17 3. Machines, Arithmetic, and Logic 22 CHAPTER II / MIND, NUMBER, AND THE INFINITE 33 I. The Obligations of Infinity 33 2. Mind and Philosophy of Number 40 3. Dedekind's Theory of Arithmetic 46 4. Propositioning the Infinite 57 CHAPTER III / THE MENTAL, THE FINITE, AND THE FORMAL 72 1. Kronecker Versus Hilbert Versus Frege on Geometry 72 2. Logic, Intuition, and Mechanism in Hilbert's Geometry 88 3. The Problem of Solvability III 4. Consistency, Denumerability, and the Paradox of Richard 120 5. Frege, Later Hilbert, and the Problem of Formalization 132 6. Hilbert's Rule and the Phenomenology of Infinity 152 CHAPTER IV / EFFECTIVENESS MECHANIZED 175 1. Effectiveness, Diagonalization, and the Problem of Recursion 175 2. The Formalization of Diagonalization 189 3. The Generalization of Recursion 203 4. Church's Thesis and Artificial Intelligence 219 CONCLUDING SUMMARY 244 BIBLIOGRAPHY 248 INDEX OF NAMES 264 INDEX OF SUBJECTS 271 PREFACE This book grew out of a graduate student paper [261] in which I set down some criticisms of J. R. Lucas' attempt to refute mechanism by means of G6del's theorem. I had made several such abortive attempts myself and had become familiar with their pitfalls, and especially with the double­ edged nature of incompleteness arguments. My original idea was to model the refutation of mechanism on the almost universally accepted G6delian refutation of Hilbert's formalism, but I kept getting stuck on questions of mathematical philosophy which I found myself having to beg. A thorough study of the foundational works of Hilbert and Bernays finally convinced me that I had all too naively and uncritically bought this refutation of formalism. I did indeed discover points of surprisingly close contact between formalism and mechanism, but also that it was possible to under­ mine certain strong arguments against these positions precisely by invok­ ing G6del's and related work. I also began to realize that the Church­ Turing thesis itself is the principal bastion protecting mechanism, and that G6del's work was perhaps the best thing that ever happened to both mechanism and formalism. I pushed these lines of argument in my dis­ sertation with the patient help of my readers, Raymond Nelson and Howard Stein. I would especially like to thank the latter for many valuable criticisms of my dissertation as well as some helpful suggestions for reor­ ganizing it in the direction of the present book. Innumerable discussions with Jean van Heijenoort on the history and philosophy of mathematical logic have also been invaluable to me in writing this book. It would not exist at all, however, much less in its present form, but for Ilona Lappo: she has been its typist, its editor, its sounding board, and its muse. vii INTRODUCTION This is an essay on the significance of various metamathematical theorems, notably those of Godel and Church on incompleteness and decidability, for both psychology and the philosophy of mathematics. It is widely believed, for example, that Godel's incompleteness theorems refute the formalist philosophy of Hilbert. A less prevalent but not uncommon view - one which Godel himself inclined towards - sees in such theorems a refutation of mechanism in psychology, i.e. as showing that man is not, or could not be adequately modelled by, a machine. I shall criticize both of these views at length, and argue to the contrary that both the formalism of Hilbert and Bernays and the mechanism of Hobbes, La Mettrie, and Turing are on balance supported and strengthened in various ways by these so-called 'limitative' theorems. Incompleteness, I shall argue, cannot coherently be interpreted as our inability to formalize our notion of num­ ber as such, but rather as our inability to completely describe the behavior of certain machines. I shall also explore the close relation, pointed out by N. Wiener and G. Kreisel and implied by their common metamathematical support, between formalism and mechanism. Indeed, J. Lucas conceded, in a re­ joinder to my earlier criticisms [261] of his attempt to refute mechanism using Godel's incompleteness, that It is inherent in my program that I cannot prove to the formalist that he cannot be represented by a formal system, but only enable a reasonable man to see that in fact he is not. But that, however achieved, is enough to refute mechanism ([177], p. 312). I hope to show, however, that formalism is a reasonable position, and that, insofar as Hilbert was a formalist, he was a very reasonable man, especially in the light of GOdel's work. Prior to this work Poincare had objected to Hilbert's formalism that his formal systems for number theory would be complete and hence sterile for research, while Brouwer had ob­ jected that, since Hilbert's mechanical formal systems represented only the hollow shell of mathematical language and not mathematical thought itself, their study could lead to nothing of mathematical value; in partic­ ular, he felt that a proof of their consistency would be so easy as to be trivial! This is not unrelated to classical objections to mechanism on the IX x INTRODUCTION grounds that machine behavior, being predictable in principle, was no match for the unpredictability of human behavior. A full understanding of Hilbert's formalism is impossible without a close study of his work on the foundations of geometry, which contains the germ of all his later ideas for the foundations of mathematics. In fact, Hilbert developed a program for the study of geometry which was in some ways more extensive than his later one for arithmetic and set-theory. At the heart of his mathematical philosophy lies a certain 'Parallelisierung' of geometry and arithmetic with which he opposed the prevalent belief of his day that true rigor resided only in arithmetic and analysis. And we shall see that the issue of mechanism arises in a more full-blooded and interest­ ing way in Hilbert's geometry than it does for the arithmetical theories of his later program. Another criticism leveled against Hilbert was that his later program required clarification of such hopelessly un mathematical ideas as 'decid­ ability in a finite number of steps'. Most of us would nowadays accept such analyses as Turing's as showing that such notions can be treated in a purely mathematical way. In fact, Church's thesis, especially in Turing's form, plays a central role in my whole present approach. In my previous criticism of Lucas, I suggested that since the generality of G6del's incom­ pleteness depends on Church's thesis, one could insulate mechanism against the incompleteness argument simply by giving up that thesis. After all, some rejected it in any case. But this suggestion was quite mis­ guided, not only for having underestimated the evidence for Church's thesis, but also for having not fully appreciated the force of Church's thesis as a restricted form of mechanism itself. Indeed, this thesis implies that however much creative insight may have gone into the discovery of an algorithm, and however abstract the concepts or objects be in terms of which it is defined or specified, as long as it is effective it will be mechaniz­ able after all. In the words of Post, this thesis is our "axiom of reducibility for finite operations". In fact, it was only when Post came to appreciate this aspect of Church's thesis that he gave up his own incompleteness argument against mechanism! The thesis does not say, of course, that all mental operations are effective, but by implying the mechanizability of those that are, it shows that mechanism cannot be effectively refuted. I will spell this out in various ways in subsequent chapters. By focussing my analysis on Turing's thesis I try to avoid having to consider such implausible constructions as 'the G6del sentence of a man', supposed to be a machine, which hang over this literature like a fog, INTRODUCTION xi lending it an unreal and forced air. Individual Godel-sentences of a formal­ ism are, after all, of no special significance in and of themselves: what is important is that the whole set of them is not recursively enumerable, the significance of which is formulated already by Church's thesis itself. In short, only the 'mass phenomena' of undecidability, if anything in logic, is of ultimate importance for mechanism.
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