The Attack on Logicism

Abstract

Poincaré argued that logicism fails because the logicist cannot show that the epistemological source of the concepts required for the logicist derivation of arithmetic is really logic. Indeed, Poincaré believed that the concept of indefinite iterability — of the ability to iterate or repeat (for example, the application of a rule, like ‘+’) without stopping — is foundational, not only for arithmetic, but for all systematic thinking; and its epistemological source is synthetic a priori intuition. I call this intuition ‘arithmetic’ intuition; and the claim made by Poincaré is that it underlies all systematic thinking because it underlies our ability to generalise, as in our understanding of ‘and so on’, of potential infinity, ‘etc.’, etc. On this view, knowledge that the principle of induction is satisfied by the numbers is not strictly knowledge of logical or analytic truths, for induction is not merely part of the definition of ‘number’. Rather, any definition of ‘number’ requires the employment of induction (or a principle at least as strong as induction) outside of it, so to speak; and this means that to define ‘number’ we require prior epistemological access to (something strong enough to yield) induction. Poincaré therefore concludes that induction is a synthetic (a priori) principle, and it holds in any nonempirical domain which can be regarded as a model of an indefinitely iterable rule.