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MECHANISM, MENTALISM, AND SYNTHESE LIBRARY

STUDIES IN EPISTEMOLOGY,

LOGIC, , AND OF

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 . 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

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, , and 22

CHAPTER II / MIND, , AND THE INFINITE 33 I. The Obligations of 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 72 2. Logic, Intuition, and Mechanism in Hilbert's Geometry 88 3. The Problem of Solvability III 4. , Denumerability, and the 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 175 2. The Formalization of Diagonalization 189 3. The Generalization of Recursion 203 4. Church's Thesis and 219

CONCLUDING SUMMARY 244

BIBLIOGRAPHY 248 INDEX OF 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 . 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 , 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 on the history and philosophy of 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 , notably those of Godel and Church on incompleteness and , for both psychology and the philosophy of . 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 ­ 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 , 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 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 and not mathematical itself, their study could lead to nothing of mathematical value; in partic­ ular, he felt that a 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 of Hilbert's formalism is impossible without a close study of his work on the , 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 , 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 " 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 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. Thus any role played by Turing's theory in psychology will have to be somewhat analogous to that played by thermodynamics in physics. That is, just as this branch of physics deals very generally with both capabilities and limitations of machines, such as energy conversion and dissipation respec­ tively, which are established independently of the physical details of their realizations, so Turing's theory established both the existence of universal machines and the non-existence of any other machine predicting their halting behavior, all from very general finiteness assumptions. I shall' elaborate this in Chapter I by comparing L. Szilard's attempt to exorcise Maxwell's demon with Turing's exorcism of what I will call 'Hilbert's demon'. As I have already mentioned, not everyone accepts Church's thesis or Turing's argument for it. In fact, Kreisel has complained of the un­ deserved 'glamour' which surrounds Turing's analysis, arguing that he only analyzed mechanically effective and not humanly effective functions. Specifically, both Kreisel and Godel have claimed that Turing's finite­ state hypothesis is unjustified for the latter notion. The claim that the mind of a human has somehow an infinite number of states has also been used by R. Abelson to refute the mind-body identity thesis. Part of my criticism of these infinitistic assumptions and arguments, as well as my defense 'of the finiteness principles of Hilbert and Turing at which they are directed, turns on a close analysis of Dedekind's classic work on number, recursion, and infinity. The essential point is that, although it built profitably upon his admirable analyses of number and recursion, modern logic - including Brouwer - could not accept Dedekind's ingeni­ ous 'proof' for the existence of an infinite set based upon a consideration . of the of thought. We have long since replaced his proof by an axiom of infinity. Moreover, a careful examination of the difficulties of Dedekind's abortive proof leads one to see similar and related difficulties in the arguments of Kreisel, Godel, and Abelson for the infinity of mental states. Given the failure of all attempts so far to prove the infinity of either our , , or mental states, finitism may yet win out without ever firing a shot. In this connection we may do well to ponder a xii INTRODUCTION remark of Nelson Goodman. Goodman distinguishes his brand of nomi­ nalism from finitism which he admits is nevertheless "a friendly companion of nominalism". In answering a challenge of Church that he show his nominalistic adequate to salvage the main results of 'non-finitist syntax', including Godel's incompleteness theorem, Goodman remarks that

Admiration for Gtidel's incompleteness theorem does not make the result any more welcome. If incompleteness depends essentially upon those aspects of arithmetic or its supposed foundations that are not finitistic or nominalistic, this argues for rather than against finitism or nominalism. Gtidel's result stands as a reductio ad absurdum giving notice of an anomaly that calls for correction. Incompleteness is no more to be cherished for the sake of Gtidel's theorem than is crime for the sake of detection; banishment of crime and incompleteness to the realm of fiction would hardly be a matter for regret ([104], p. 154).

As for myself, I have come to cherish incompleteness for the support it lends to mechanism generally, and to Turing's thesis in particular. l}ut in trying to work out the details of this striking development I have more than once been struck by the make-believe air in which parts of the subject flourish. Perhaps indeed, it cannot be made to satisfy finitist scruples .completely. If not, we may yet, in all honesty, have to banish it to a realm of fiction where, though shorn of , its beauty may still shine. Finally, a word about my historical approach. Originally, I had planned to treat these matters entirely systematically, especially in view of the modem technicalities involved in this material. But the more I tried to sort out and understand the arguments, to sift claim and counterclaim, the more I found that most of the central figures, however original they might have seemed, had really gotten key ideas from their teachers and predecessors - sometimes very obscure ones at that. And, most important­ ly, time after time I found that, because of my ignorance of these antece­ dents, I had not, nor could have, really understood those ideas. All the logical analysis in the world will not reveal the intentions behind ideas, and without these intentions one all too easily misunderstands and mis­ judges the ideas and theories of a writer no longer living. Moreover, I found that many of the arguments I thought had necessarily arisen in con­ nection with modem developments had in fact already been made long ago, and in contexts in which their strengths and weaknesses stood more nakedly exposed. On the other hand, one also finds that current ideas and results can illuminate older and crustier ideas. The lesson seems to be this: we cannot fully understand our own conceptual scheme without plumbing INTRODUCTION XlIl its historical roots, but in order to appreciate those roots we may well have to filter them back through our own ideas. This is somewhat analgous to the relation of mutual dependence between quantum theory and clas­ sical physics urged by Bohr. In any case, it has become clear to me that many of the positions and ideas dealt with below, especially with respect to mechanism and formalism, have been badly misunderstood and mis­ represented by those who have ignored their history. The 'genetic fallacy' is only fallacious in straw-man cases.