Reason and “rational intuition”

It will be painfully obvious to the reader that what has been said so far about the epistemology of mathematics, even arithmetic, is very incomplete. In §45, it was argued that very likely intuitive knowledge has rather narrow limits. Many would hold that that is due to the very restricted character of the conception of intuition developed in Chapter 5. Occasional reference has been made to conceptions of intuition of a quite different nature, which might promise to lead further than the one we have developed. I propose in this section and the next to approach the problem of mathematical knowledge from a different direction, not beginning with intuition at all, but leading very quickly to a place where some other conceptions of intuition, in particular that of Kurt Gödel, can be situated. I will begin with some very general considerations about Reason, not at all specific to mathematics.

The relevance of Reason to our inquiry is indicated by a long tradition of regarding mathematics as rational knowledge, echoed specifically by Kant in a remark at the beginning of one of the discussions of mathematics in the Critique of Pure Reason. He describes mathematics as “rational cognition from the construction of concepts” (A713/B741). Construction of concepts is construction in intuition, and it is here that intuition takes its place in Kant's account of mathematics.