From the beginning of the nineteenth century, Kant's pure intuition had a rough time in analysis. The rigorization of the calculus banished intuition from the notions of function, continuity, limit, infinitesimal, and all else that had elicited Berkeley's justified complaint. The arithmetization of analysis cornered the pure intuition of time into arithmetic, where Frege would soon deal it a death blow (Chapter 4). Mathematics was not just the theory of abstract magnitudes, numbers, functions, and infinitesimals, however. It was also the science of space, geometry, and here Kantians could rest assured that intuition would never be dethroned. Or so it seemed for a while.

During the nineteenth century, geometry was the battleground of two major epistemological wars. The first, the subject of this chapter, concerned the role of pure intuition in knowledge; the second, surveyed in Chapter 7, took it for granted that that role was nil and questioned the nature of geometric concepts.