Truth Through Proof A Formalist Foundation

Truth Through Proof defends an anti-platonist philosophy of mathematics derived from game formalism. Classic formalists claimed implausibly that mathematical utterances are truth-valueless moves in a game. Alan Weir aims to develop a more satisfactory successor to game formalism utilising a widely accepted, broadly neo-Fregean framework, in which the proposition expressed by an utterance is a function of both sense and background circumstance. This framework allows for sentences whose truth-conditions are not representational, which are made true or false by conditions residing in the circumstances of utterances but not transparently in the sense.
Applications to projectivism and fiction pave the way for the claim that mathematical utterances are made true or false by the existence of concrete proofs or refutations, though these truth-making conditions form no part of their sense or informational content.
The position is compared with rivals, an account of the applicability of mathematics developed, and a new account of the nature of idealisation proffered in which it is argued that the finitistic limitations Godel placed on proofs are without rational justification. Finally a non-classical logical system is provided in which excluded middle fails, yet enough logical power remains to recapture the results of standard mathematics.

Alan Weir is Head of the Department of Philosophy at the University of Glasgow. His main research interests have been in philosophy of logic and mathematics, but he has also written on philosophy of language, epistemology and the theory of perception.

Introduction	p. 1
Metaphysics	p. 12
Traditional Realism	p. 12
Contemporary Realism	p. 14
Sense, Circumstance, World	p. 25
Ontological Reductionism	p. 39
Projectivism in the SCW Framework	p. 39
Snapshot Dispositions, Correction, Fiction	p. 47
Reduction	p. 56
A Map of the Terrain	p. 64
Neo-formalism	p. 68
An Initial Specification	p. 68
Contentful Language	p. 77
Quatification and Existence	p. 90
Objections and Comparisons	p. 99
Closure	p. 99
Relativism or Pluralism?	p. 101
Incompleteness	p. 118
Applying Mathematics	p. 127
Conservativeness	p. 127
*Comparison with Fictionalism	p. 136
Theoretical Language	p. 141
Promises, Promises	p. 151
Proof Set in Concrete	p. 152
Tokenism	p. 152
The Spectre of Strict Finitism	p. 176
Idealization Naturalized	p. 192
Idealization	p. 192
The Spectre of Finitism	p. 209
Logic	p. 222
Calculus versus Logic	p. 222
The Basic Semantics Again	p. 223
Logical Consequence	p. 226
The Conditional	p. 230
*Quantification and Infinitary Logic	p. 233
Classical Recapture	p. 235
Really Existing Mathematics	p. 239
Conclusion	p. 249
Appendix	p. 252
Minimax Rules	p. 252
Soundness of Strict Neo-classical Logic	p. 253
Infinitary Rules	p. 256
IS	p. 257
Prime Extensions	p. 258
Limitative Results in Infinitary Logics	p. 260
Bibliography	p. 262
Name Index	p. 273
Subject Index	p. 277
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Truth Through Proof A Formalist Foundation for Mathematics

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