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 systematic 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 characterized. 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 will be essential reading for all who work on the philosophy of mathematics.

ISBN 0199533636   Pages 214 ISBN13 9780199533633 (What's this?)   Volumes 1 Publisher Oxford University Press   Weight (grammes) 285 Imprint Oxford University Press   Published in Oxford Format Paperback   Height (mm) 218 Publication date 22 May 2008   Width (mm) 140 Library of Congress QA8.4   Spine width (mm) 13 DEWEY 510.1   Academic level Undergraduate, Postgraduate DEWEY edition DC22

		Introduction
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
IV		The General Theory of Abstraction		165
1		The Systems		165
2		Semantics		175
3		Derivations		189
4		Further Work		191
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"It is a great book. In demanding but always succinct prose, it presents a wealth of distinctions, insights, and arguments."--Oystein Linnebo, Australasian Journal of Philosophy

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