Note 182 — Chapter V · MAKROKOSMOS
Charles Francis Atkinson translation
His idea that the a priori-ness of space was proved by and through the unconditional validity of simple geometrical facts rests, as we have already remarked, on the all-too-popular notion that mathematics are either geometry or arithmetic. Now, even in Kant’s time the mathematic of the West had got far beyond this naïve scheme, which was a mere imitation of the Classical. Modern geometry bases itself not on space but on multiply-infinite number-manifolds—amongst which the three-dimensional is simply the undistinguished special case—and within these groups investigates functional formations with reference to their structure; that is, there is no longer any contact or even possibility of contact between any possible kind of sense-perception and mathematical facts in the domain of such extensions as these, and yet the demonstrability of the latter is in no wise impaired thereby. Mathematics, then, are independent of the perceived, and the question now is, how much of this famous demonstrability of the forms of perception is left when the artificiality of juxtaposing both in a supposedly single process of experience has been recognized.