Regarding space and time (and responding to Erik at http://partiallyexaminedlife.com/2010/05/14/episode-19-kant-what-can-we-know/):
Kant is explicitly worried about the same thing that troubled Leibniz, which is there is a discord between mathematics and the concrete — what we consciously see and touch in the world “out there.” Leibniz was concerned with the paradox of the continuum: that points can have no extension (otherwise an infinite number of points couldn’t lie on a line), and yet can’t be done away with without getting rid of everything important to mathematics (e.g., that two lines always intersect at a point — requiring aforesaid infinity). Hence this idea of an extension-less element being important to a branch of science (geometry broadly conceived) is troubling — it suggests that science in fact falsifies the world. And it is on par with Hume’s argument about causality. Kant explicitly notes in the Prolegomena that Leibniz’s paradox goes away with his view (actually Leibniz successfully handled it as well with his version of phenomenalism).
It goes away because the intuitions are constructive, not merely extractive. These are unconscious, a priori constructions, not merely a matter of what we see consciously. And so when we come to what has been constructed within space and time and analyze it, we get things that could only be mind-related out of it, e.g. points and other abstractions that don’t seem to belong to the world.
So in fact Kant’s argument is meant to deal precisely with the case of parallel lines and the fact that we don’t consciously “see” this sort of thing but get it only at the level of analysis of what’s grounded in intuition. So I think the criticism misses the point here: again, Kant’s spatial intuition is meant to deal with precisely these sorts of criticisms by dealing with space as a mental construction from which we get abstractions rather than as something that comes in through the senses — and so vitiating the problem of a conflict between what’s really in the world and what’s just in the mind. So we’re better off saying that the parallel postulate or the definition of points are grounded in intuition, not that they are themselves “intuitions” that require us to see invisible points or infinite lines.
When it comes to Dedekind, number theory, and the axiomatization of geometry, I think the fact that you can violate a postulate and still yield a consistent system actually lends support to the Kantian position in one way and vitiates it in another. Because it goes to show that the premises of every axiomatic system can either be arbitrarily chosen our must have some other source. It is just this source that Kant calls “intuition.” The axioms cannot themselves be analytically derived from anything, or they wouldn’t be axioms … unless we get interference from other systems. Which is to say that a posteriori observations can come into conflict with intuition.
I think the accurate criticism of Kant is to say that he didn’t conceive of the possibility of Geometric axioms being related inferentially to other areas of the physical sciences. Meaning that they could be modified by being grounded not in intuition but in back-calculation from the sciences to produce a geometry that satisfies another set of observations. To take a slightly related exmaple concerning the speed of light: You’ll see that in Einstein’s Electrodynamics of Moving Bodies, he preserves the seemingly inconsistent assertions of Galilean relativity and the constancy of the speed of light by warping space. If the speed of light is constant relative to all frames of reference, then it is the distances and times that go into speed that must differ between frames. So we have to alter something possibly grounded in intuition for the sake of meeting another empirical observation. (I’m not sure this is a violation of Geometrical intuition exactly — how do we categorize Galilean relativity?).
Kant didn’t conceive of the possibility of other empirical observations coming into conflict with spatial intuition. In other words, he didn’t conceive of an inner conflict in his system between scientific inquiry (grounded in the understanding) and spatial intuition. The system is also iterative: empirical observations framed (partly) by one set of intuitions can come back to demand modification of those intuitions — as a back-calculation that solves other problems. That’s the way I would frame the criticism of Kant’s views on space and time — not that he chose the wrong system of Geometry (since Euclidean geometry is applicable to everyday experience), but that he didn’t leave open the possibility of a conflict between scientific experience (at the level of the understanding) and spatiotemporal intuition; which is to say, he didn’t understand that intuitions of space and time have their limits of applicability when it comes to the sciences. The understanding rules with an iron fist.