Who is Afraid of a Priori Synthetic Judgements? Mathematics, Intuition and Concepts

Abstract

Kant, starting from his analytic-synthetic distinction, holds that mathematical statements are synthetic a priori and that an extra-conceptual element takes part in their justification: the pure intuition. Many authors after Kant, specially those close to logical positivism, attempted to refine Kant's distinction and to challenge his position relative to the foundations of mathematics, defending that its axioms must be understood as implicit definitions, conventionally adopted and, in this sense, analytically justified without the intervention of the intuition. The decline of logical positivism is partly related to the devastating criticisms, Quine's particularly, to the analytic-synthetic distinction. This paper attempts to show that there are elements in the Kantian approach to mathematics, as synthetic a priori, and in his reference to intuition that are relevant for recent discussions in the philosophy of space-time. In order to do that, I defend a particular reading of the original distinction, argue that Quine's criticism is ineffective against the understanding of some statements as constitutive a priori and suggest that this involves interpreting certain structures as analogous to the Kantian intuition.