The Limits of Abstraction

214 Pages·2002·9.35 MB·English

by  Kit Fine

THE LIMITS OF ABSTRACTION
What is abstraction? To what extent can it account for the existence
and identity of abstract objects? And to what extent can it be used as a
foundation for mathematics? Kit Fine provides rigorous and sys-
tematic answers to these questions along the lines proposed by
Frege, in a book concerned both with the technical development of
the subject and with its philosophical underpinnings.
Fine proposes an account of what it is for a principle of abstraction
to be acceptable, and these acceptable principles are exactly charac-
terized. A formal theory of abstraction is developed and shown to be
capable of providing a foundation for both arithmetic and analysis.
Fine argues that the usual attempts to see principles of abstraction as
forms of stipulative definition have been largely unsuccessful but
there may be other, more promising, ways of vindicating the various
forms of contextual definition.
The Limits of Abstraction breaks new ground both technically and
philosophically, and is essential reading for all those working on the
philosophy of mathematics.
'The text is essential reading for anyone interested not only in
abstractionist philosophies of mathematics, but the philosophy of
mathematics and language in general. The philosophical chapters
display a consistently high level of rigour and insight ... the new
philosophical problems raised are valuable and thought provoking,
and promise to be the basis for much philosophical discussion to
come.'
Roy Cook and Philip Ebert, British Journal of Philosophy of Science
Kit Fine is Professor of Philosophy at New York University.
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The Limits of Abstraction
KIT FINE
CLARENDON PRESS • OXFORD
OXFORD
UNIVERSITY PRESS
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Published in the United States
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©Kit Fine 1998, 2002
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Database right Oxford University Press (maker)
First published 2002
First published in paperback 2008
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without the prior permission in writing of Oxford University Press,
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British Library Cataloguing in Publication Data
Data available
Library of Congress Cataloging in Publication Data
Fine, Kit.
The limits of abstraction / Kit Fine.
p. cm.
Includes bibliographical references and index.
1. Mathematics-Philosophy. 2. Abstraction. I. Title
QA8.4 .F56 2002 510'.l-dc21 2002070134
ISBN 978-0-19-924618-2 (Hbk.)
978-0-19-953363-3 (Pbk.)
13579 10 8642
Typeset in 10.5 on 12 pt Minion
by SPI Publisher Services Ltd, Pondicherry, India
Printed in Great Britain by
Biddies Ltd., Guildford & King's Lynn
Preface
TH Is monograph has its genesis in a paper of the same name, written
in 1994 as a contribution to the proceedings of a conference on
the philosophy of mathematics that was held in Munich during the
preceding year. Because of various delays, these proceedings (The
Philosophy of Mathematics Today, edited by M. Schirn) were not
themselves published until 1998. Peter Momtchiloff from Clarendon
Press offered to publish an expanded version of the paper as a
separate monograph; and I was happy to agree. I have corrected
various errors in the original paper, improved the exposition here
and there, and incorporated some brief comments on the more
recent literature. The major change is the addition of a new part on
the context principle (which was omitted from the original paper for
lack of space).
The earlier discussion of the context principle contained both a
negative part, dealing with the difficulties in providing a proper
formulation of the principle, and a positive part, which attempted
to show how these difficulties might be met. I now appreciate that
the positive part calls for a new approach to the philosophy of
mathematics—what I call 'procedural postulationalism'—and that
discussion of it is best postponed to another occasion. I have
therefore presented only the criticisms from the negative part. But
it is important to bear in mind that these criticisms are intended as
the prolegomena to a more constructive account.
I am much indebted to the participants of the Munich confer-
ence—and especially to Boolos, Clark, Hale, Heck, and Wright—for
reawakening my interest in the topic of logicism. Preliminary ver-
sions of the paper were given at the third Austrian philosophy con-
ference in Salzburg, at a talk at the City University of New York, at a
philosophy of mathematics workshop at the University of California
at Los Angeles, and at a workshop on abstraction in St Andrews; and I
am grateful for the comments that I received at those meetings. I have
been greatly influenced by the writings of Michael Dummett and
Crispin Wright and have greatly benefited from the comments of
vi Preface
Tony Martin. Joshua Schechter read through the original published
paper and suggested many helpful improvements, both typographic
and substantive; and Sylvia Jaffrey, for OUP, provided careful copy-
editing of a disorderly text.
I am very grateful to John Burgess, Roy Cook, Philip Ebert, Stewart
Shapiro and Alan Weir for pointing out some infelicities and errors
in the original hardback edition of the book and I have attempted to
correct these (along with some other minor infelicites) in the present
paperback edition.
Contents
Introduction ix
I. Philosophical Introduction 1
1. Truth 3
2. Definition 15
3. Reconceptualization 35
4. Foundations 41
5. The Identity of Abstracts 46
II. The Context Principle 55
1. What is the Context Principle? 56
2. Completeness 60
3. The Caesar Problem 68
4. Referential Determinacy 77
5. Predicativity 81
6. The Possible Predicative Content of Hume's Law 90
III. The Analysis of Acceptability 101
1. Language and Logic 101
2. Models 105
3. Preliminary Results 107
4. Tenability 114
5. Generation 118
6. Categoricity 122
7. Invariance 138
8. Hyperinflation 156
9. Internalized Proofs 161
viii Contents
IV. The General Theory of Abstraction 165
1. The Systems 165
2. Semantics 175
3. Derivations 189
4. Further Work 191
References 193
Main Index 197
Index of First Occurrence of Formal Symbols and Definitions 200
Introduction
THE present monograph has been written more from a sense of
curiosity than commitment. I was fortunate enough to attend the
Munich Conference on the Philosophy of Mathematics in the Sum-
mer of 94 and to overhear a discussion of recent work on Frege's
approach to the foundations of mathematics. This led me to inves-
tigate certain technical problems connected with the approach; and
these led me, in their turn, to reflect on certain philosophical aspects
of the subject. I was concerned to see to what extent a Fregean theory
of abstraction could be developed and used as a foundation for
mathematics and to place the development of such a theory within
a general framework for dealing with questions of abstraction. To my
surprise, I discovered that there was a very natural way to develop a
Fregean theory of abstraction and that such a theory could be used to
provide a basis for both arithmetic and analysis. Given the context
principle, the logicist might then argue that the theory was capable of
yielding a philosophical foundation for mathematics, one that could
account both for our reference to various mathematical objects
and for our knowledge of various mathematical truths. I myself am
doubtful whether the theory can legitimately be put to this use. But,
all the same, there is surely considerable intrinsic interest in seeing
how the theory of abstraction might be developed and whether it
might be capable of embedding a significant portion of mathematics,
even if the theory itself is in need of further foundation.
The monograph is in four parts. The first is devoted to philoso-
phical matters and serves to explain the motivation for the technical
work and its significance. It is centred on three main questions: What
are the correct principles of abstraction? In what sense do they serve
to define the abstracts with which they deal? To what extent can they
provide a foundation for mathematics? The second part (omitted
from the original paper) discusses the context principle, both as a
general basis for setting up contextual definitions and in its particular
application to numbers. The third part proposes and investigates a set
of necessary and sufficient conditions for an abstraction principle to