Everyone once in a while I run across the opinion that non-Euclidean presents a serious problem for Kantian epistemology. While I've rebutted this notion before, it's common enough that I thought I'd have another go at explaining why it's a misconception.
For Kant we can't know the universe to be spatial "in itself" (as in "things-in-themselves"), Euclidean or Non-Euclidean or otherwise. Time and space are something supplied by our cognitive faculties. To say what really is the case here means we're making objective judgments about the world of object-appearances (which is to say we're correctly analyzing what we've already synthesized).
If we discover that the universe is actually, objectively (in the Kantian sense) non-Euclidean when our spatial intuition suggests it is Euclidean, then there is a conflict here between the faculties of understanding and intuition. If you've studied non-Euclidean geometry you'll readily see what this means: the denial of the parallel postulate violates our intuition (unless we model the new geometry within Euclidean geometry but as occurring on a hyperbolic surface); but it does not produce any logical inconsistency. And in fact this is the whole point of Kant calling our perception of Euclidean space "intuition": I have no other basis for the parallel postulate -- I cannot argue for it as following from a principle of logic or arithmetic; nor can I argue about it from some a posteriori discovery in physics about the nature of the world.
Nothing says that our spatial intuition has to be "right," either in the metaphysical thing-in-itself sense, or in the Kantian sense of always being confirmed by higher level theoretical judgments of the understanding, as in physics. And the following remains a fact: from the standpoint of cognitive science, we perceive the world in Euclidean terms. No discovery outside of cognitive science could change the fact that this is how we intuit the world. All of this is a way of saying that the space of physics is not the same thing as the space of ordinary, every-day experience (the former is conceptual and a posteriori, the latter is intuitive and a priori). If non-Euclidean geometry is useful for physics and is better at modeling "space," this means not that Kant must revise his concept of intuition as being of Euclidean space; rather, he must revise his conception of the relationship between understanding and intuition (accounting for the possibility of conflict and a posteriori, non-intuitive conceptions of space).
In the end, this conflict between understanding and intuition is not something Kant foresaw; and I doubt he would have liked it. But ultimately, his system doesn't depend on the absence of such a conflict any more than our discovery of the non-Euclidean nature of space requires that we intuit our living room in non-Euclidean terms.
-- Wes Alwan