The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview

8
Beyond Definitionism--But Not
Too Far Beyond

GEOFFREY HELLMAN


1. OVERVIEW

Our aim here is twofold: first, to identify the principal advantages and disadvantages of the definitionist foundational programme of predicativist mathematics (originating with Poincaré and Weyl, and developed by Fefermanet al.), and second, to propose a recipe for liberalizing that programme which overcomes the disadvantages identified while preserving most of the advantages, and which synthesizes (arguably the best of) nominalism, predicative flexible type theory, and modal-structuralism. The resulting system is of proof-theoretic strength exceeding that of full classical analysis (indeed, equal to that of a second-order theory of continua); a great deal of modern analysis--quite possibly all that finds scientific application--can be developed within it quite naturally, i.e. with minimal reliance on codings; and yet it presupposes little more than the notions 'nominalistic sum (or fusion) of atoms' and 'finite fusion of atoms', together with the mere logical possibility of a model of a very elementary theory of finite sets and classes (roughly equivalent to postulating the possibility of an ω-sequence of atoms). The system can be claimed to transcend certain critical limitations of predicativist analysis and certain awkwardness of known nominalistic approaches to mathematics at once, and would seem to provide a well- motivated alternative to Zermelo set theory as a framework for scientifically applicable mathematics.


2. THE VIRTUES AND VICES OF DEFINITIONISM

By 'definitionism' we refer to the view that 'all mathematical objects should be explicitly present[able] by finite symbolic expressions' ( Feferman 1985,

____________________
Work reported on here has been supported by the National Science Foundation, USA, Award No. SBER93-10667. I am grateful to John Burgess and Allen Hazen for helpful correspondence.

-215-

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