The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview

12
Constructive Existence Claims
MICHAEL DETLEFSEN

1. INTRODUCTION

It is a commonplace of constructivist thought that a claim that an object of a certain kind exists is to be backed by an explicit display or exhibition of an object that is manifestly of that kind. Let us refer to this requirement as the exhibition condition. The main objective of this essay is to examine this requirement and thereby to arrive at a better understanding of its epistemic character and of the role that it plays in the two main constructivist philosophies of this century--the intuitionist programme of Brouwer and Weyl, and the finitist programme of Hilbert. Gödel gave what is the currently prevailing conception of the relationship between the intuitionist and finitist conceptions of the exhibition condition in his 1958 essay 'Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes':

Bernays . . . teaches us to distinguish two components of the finitary viewpoint; namely, first, the constructive element which consists in the demand that we be permitted to speak of mathematical objects (mathematischen Objekten) only to the extent that we can exhibit or through construction actually produce them; secondly, the specifically finitist element, which demands in addition that the objects about which one makes statements ( Aussagen), with which the constructions are executed, and which one obtains by their means, are 'intuitive'. In the final analysis, that means that they are spatio-temporal configurations ( Anordnungen) of elements whose constitution ( Beschaffenheit) is irrelevant aside from whether they are identical or distinct. (In contrast to this, the objects of intuitionist logic are meaningful statements and proofs (sinnvolle Aussagen und Beweise).) (p. 282, my translation)1

So, Gödel conceived of the exhibition condition itself--that is, the demand that claims of existence be backed either by an explicit exhibition of an object of the kind claimed to exist, or by a specification of a means of effecting such an exhibition--as the common feature which qualifies both finitism and intuitionism as forms of constructivism. He believed them to be distinguished not by different conceptions of the nature of exhibition or

____________________
1
See Kreisel ( 1958-9, 148) and Parsons ( 1967, 205) for statements of similar views (indeed, views which may well be simply affirmations of Gödel's).

-307-

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