Kant's Philosophy of Mathematics and the Greek Mathematical Tradition

Abstract

Kant made two intimately related claims that greatly influenced the philosophy of mathematics: first, mathematical cognition is synthetic a priori; second, mathematical cognition requires intuition for the content and the justification of mathematical concepts and propositions. Kant held that intuitions, like concepts, are a fundamental kind of representation. Intuitions belong (at least for humans) to the faculty of sensibility and represent spatial and temporal properties; concepts belong to the faculty of understanding. Kant contrasts intuitions and concepts by claiming that intuitions are singular representations that relate to objects immediately, while concepts are general representations that relate to objects mediately, that is, mediated by intuitions (A320/B376-77, A68/B93).1 It is therefore quite natural that some recent accounts of Kant's philosophy of mathematics have focused on the singularity and immediacy of intuition, and have argued that one or both play a central role in Kant's philosophy of mathematics.2 While not disagreeing with this approach or its fruitfulness, I would like to propose a very different one: I would like to consider the role of intuition in representing magnitudes, and in particular, the spatially extended magnitudes of geometrical constructions. Kant's theory of magnitudes has been largely overlooked; uncovering it complements recent work and gives us a more complete understanding of Kant's philosophy of mathematics. I shall argue that magnitudes are at the heart of Kant's theory of mathematical cognition. In particular, I shall argue that one of the aims of the theory is to explain our cognition of the mathematical properties of magnitudes, for which intuition is indispensable. Kant's treatment of magnitudes is, I maintain, strongly influenced by the Greek mathematical tradition. That tradition still had currency in Kant's time, allowing Kant to make allusions and tacit references to it. The best evidence for the influence of the Greek mathematical tradi-