Hilbert’s Program: An Essay on Mathematical Instrumentalism

198 Pages·1986·7 MB·English

by  Michael Detlefsen (auth.)

HILBERT'S PROGRAM
SYNTHESE LIBRARY
STUDIES IN EPISTEMOLOGY,
LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE
Managing Editor:
JAAKKO HINTIKKA, Florida State University, Tallahassee
Editors:
DONALD DAVIDSON, University of California, Berkeley
GABRIEL NUCHELMANS, University of Leyden
WES"LEY C. SALMON, University ofP ittsburgh
VOLUME 182
MICHAEL DETLEFSEN
Department of Philosophy, University of Notre Dame
HILBERT'S
PROGRAM
An Essay on Mathematical Instrumentalism
'"'
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging.in·Publication Data
Detlefsen, Michael, 1948-
Hilbert's program.
Bibliography: p.
lncludes index.
1. Mathematics-Philosophy. 2. Hilbert, David,
1862-1943. 1. Title.
QA9.2.D48 1986 510'.1 86-6460
ISBN 978-90-481-8420-0 ISBN 978-94-015-7731-1 (eBook)
DOI 10.1007/978-94-015-7731-1
AII Rights Reserved
© 1986 by Springer Science+Business Media Dordrecht
Originaily published by D. Reidel Publishing Company, Dordrecht, Holland in 1986
Softcover reprint of the hardcover 1s t edition 1986
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 permis sion from the copyright owner.
To Martha, Hans, Anna and Sara
T ABLE OF CONTENTS
PREFACE IX
ACKNOWLEDGEMENTS xiii
CHAPTER I: THE PHILOSOPHICAL FUNDAMENTALS
OF HILBERT'S PROGRAM 1
1. Introduction 1
2. Hilbertian Instrumentalism 3
3. Hilbert and Quine: Frege's Problem Revisited 24
4. Concluding Summary 34
5. Notes 36
CHAPTER II: A CLOSER LOOK AT THE PROBLEMS 45
1. Introduction 45
2. The Status of Induction 46
3. Poincare's Problem 59
4. The Dilution Problem 62
5. Notes 73
CHAPTER III: THE GODELIAN CHALLENGE 77
1. Introduction 77
2. The Standard Argument 78
3. The Stability Problem 80
4. Strict Instrumentalism 83
5. The Convergence Problem and the Problem of Strict
Instrumentalism 85
6. Conclusion 90
7. Notes 91
vii
viii TABLE OF CONTENTS
CHAPTER IV: TH E ST ABILITY PROBLEM 93
1. Introduction 93
2. The Standard Choice of '(/f 94
3. Arithmetization 97
4. Mostowski's Proposal 101
5. The Kreisel-Takeuti Proposal 113
6. The Classical Proposal? 124
7. Conclusion 129
8. Notes 129
CHAPTER V: THE CONVERGENCE PROBLEM AND
THE PROBLEM OF STRICT INSTRUMENTALISM 142
1. Introduction 142
2. Localization 145
3. The Problem of Strict Instrumentalism 148
4. The Convergence Problem 152
5. Conclusion 155
6. Notes 157
APPENDIX: HILBERT'S PROGRAM AND THE FIRST
THEOREM 161
REFERENCES 179
INDEX 183
PREFACE
Hilbert's Program was founded on a concern for the phenomenon
of paradox in mathematics. To Hilbert, the paradoxes, which are
at once both absurd and irresistible, revealed a deep philosophical
truth: namely, that there is a discrepancy between the laws accord
ing to which the mind of homo mathematicus works, and the laws
governing objective mathematical fact. Mathematical epistemology
is, therefore, to be seen as a struggle between a mind that naturally
works in one way and a reality that works in another. Knowledge
occurs when the two cooperate.
Conceived in this way, there are two basic alternatives for
mathematical epistemology: a skeptical position which maintains
either that mind and reality seldom or never come to agreement,
or that we have no very reliable way of telling when they do; and a
non-skeptical position which holds that there is significant agree
ment between mind and reality, and that their potential discrepan
cies can be detected, avoided, and thus kept in check. Of these
two, Hilbert clearly embraced the latter, and proposed a program
designed to vindicate the epistemological riches represented by
our natural, if non-literal, ways of thinking. Brouwer, on the other
hand, opted for a position closer (in Hilbert's opinion) to that of
the skeptic. Having decided that epistemological purity could
come only through sacrifice, he turned his back on his classical
heritage to accept a higher calling. And if that meant living a life of
epistemological poverty in the eyes of the world, then so be it;
poverty would find its consolation in purity.
To Hilbert, this was hollow piety. Homo mathematicus some
times goes astray, but he is not born in sin. Therefore, the answer
is not for him to deny his nature, but rather to take precautions to
see that he does not succumb to the false beauty of paradox. This
way, he can live a life not only of holiness, but also of abundance;
ix
x PREFACE
and such a life is surely preferable to that of the needless self
denial preached by the intuitionists. In this spirit, Hilbert boldly
set forth his program to vindicate the patterns of reasoning to
which homo mathematicus'cognitive nature predisposes him.
This program had two parts; one descriptive, the other justifica
tory. The aim of the former was to produce the sort of description
of our natural ways of mathematical reasoning that makes precise
evaluation of that reasoning possible. For Hilbert, this amounted
to formalizing mathematical thought in the manner so ably
illustrated by the work of Russell and Whitehead. Thus, he was led
to say that
The formula game that Brouwer so deprecates has, besides its mathematical
value, an important general philosophical significance. For this formula game is
carried out according to certain definite rules, in which the technique of our
thinking is expressed. These rules form a closed system that can be discovered
and definitively stated. The fundamental idea of my proof theory is none other
than to describe the activity of our understanding, to make a protocol of the
rules according to which our thinking actually proceeds. Thinking, it so
happens, parallels speaking and writing: we form statements and place them one
behind another. If any totality of observations and phenomena deserves to be
made the object of a serious and thorough investigation, it is this one ...
(Hilbert [1927), p. 475)
The aim of the justificatory part of Hilbert's Program was to
produce a finitary proof of the reliability of our natural modes of
mathematical reasoning. And, as is well known, it is this part of
the program that became the target both of a massive technical
assault by Godel's Theorems, and of a sustained philosophical
attack based on anti-Cartesian epistemological views. In our
opinion, most of this criticism is mistaken. We take Hilbert's
Program, as reconstructed here, to be a philosophically sophisti
cated and convincing defense of mathematical instrumentalism.
And we also believe that it can withstand the usual technical
criticisms based on Godel's Theorems. These are the central
themes of this essay, and the main reasons for writing it.
In Chapter I we set out the chief philosophical problems
confronting Hilbert's instrumentalist outlook. These include (1) a
PREFACE xi
problem of Frege's concerning the compatibility of treating a
mathematical proof both as a purcly formal cntity and as a means
of knowledge towards the proposition expressed by its conclusion,
(2) a problem of Poincare's concerning a possible vicious circu
larity in Hilbert's metamathematical use of induction, and (3) a
problem concerning how the instrumentalist can be sure that in
taking the epistemic shortcuts provided by his instrument, he does
not dilute the epistemic quality of the product that results from its
use.
In Chapter II we show how Hilbert's Program is equipped to
deal with the first two problems, and how it is out of concern for
the third problem that its finitism is best motivated. The rest of the
main body of the text is given to a discussion of the third problem,
which we call the Dilution Problem.
This problem is given urgency by GCidel's Second Theorem,
which threatens its solvability by allegedly ruling out a finitary
proof of the reliability of all but the weakest systems of mathe
matics. The argument underlying this allegation is set out in
considerable detail in Chapter III, and its more serious inade
quacies are exposed in Chapters IV and V. Finally, in the Appen
dix we contend with some of the supposedly anti-Hilbertian con
sequences of Godel's First Theorem, dismissing them as
unfounded.
The end result, we hope, is an essay which helps the philoso
pher of mathematics and the philosophically interested logician or
mathematician to come to a better understanding and appreciation
of Hilbert's Program and mathematical instrumentalism generally.
If it goes even a little way toward counteracting the common
misunderstandings of GCidel's Theorems and the crude charges of
formalism that have sullied the literature on Hilbert's Program, we
will consider it a success.