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  • Print publication year: 2006
  • Online publication date: December 2009

4 - Implicit Definition

Summary

INTRODUCTION

The birth of conventionalism was inextricably linked to the emergence of the notion of implicit definition. As we saw in chapter 2, Poincaré justifies his construal of the axioms of geometry as conventions in terms of his proposal that they be viewed as disguised definitions rather than necessary truths. Although use of implicit definitions is not confined to conventionalists – Hilbert, for one, made extensive use of implicit definition in his Foundations of Geometry and later works without committing himself to conventionalism – the link between axioms and definitions, and thus between axioms and conventions, recurs in the literature. The logical positivists in particular were enthralled by the far-reaching implications of the construal of axioms as definitions: if we are as free to lay down axioms as we are to stipulate the meanings of terms in garden-variety definitions, conventionalism would appear to be vindicated. And if it works for geometry, why not seek to ground mathematical truth in general in definition, and thus in convention? Why not let definition serve as the basis for the entire sphere of a priori knowledge? In this chapter, I examine the notion of implicit definition and its putative connection to convention. In line with the approach taken in the other chapters, I argue for an account of implicit definition that does not rest on the idea that truth can be postulated ‘by convention’.