Volume 21, Issue 1, Enero 2006
Signs, figures and time
Cavaillès on “intuition” in mathematics
This paper is concerned with Cavaillès’ account of “intuition” in mathematics. Cavaillès starts from Kant’s theory of constructions in intuition and then relies on various remarks by Hilbert to apply it to
modern mathematics. In this context, “intuition” includes the drawing of geometrical figures, the use of algebraic or logical signs and the generation of numbers as, for example, described by Brouwer. Cavaillès argues that mathematical practice can indeed be described as “constructions in intuition” but that these constructions are not imbedded in the space and in the time of our Sensibility, as Kant bclieved: They take place in other structures which are engendered in the history of mathematics. This leads Cavaillès to a critical discussion of both Hilbert’s and Brouwer’s foundational programs.