I attend Shimer College, and a few of my school-related friends are St. John's grads, so I've grown accustomed to friends opining on how one section of the great Books canon bears upon another section. One popular such connexion is between Nikolai Ivanovich Lobachevsky and Immanuel Kant, and it goes thusly:
When Kant wrote the Transcendental Aesthetic, he was unfortunately constrained by an insufficient understanding of geometry: he suffered under the delusion that Euclidean geometry was the only possible variety, whereas Lobachevsky has demonstrated that we can just as easily have geometries that do not satisfy the Euclidean postulates. Inasmuch, consequently, as the thesis that the One True Geometry must be Euclidean is fairly crucial to the Aesthetic, and the Aesthetic crucial to the rest of Kant's transcendental philosophy, the falsity of the first item renders the final pretty suspicious.
I wonder, however, how much Lobachevsky's discovery really speaks against Kant's doctrine in the Aesthetic. My chief difficulties with the argument are two.
Firstly, Kant regards the chief consequence of the Aesthetic for the theorems of Euclidean geometry to be that the former establishes a synthetic basis for the validity of the latter. Lobachevsky's insight, conversely, is more or less (unless I am profoundly misunderstanding him, which is completely possible) that certain Euclidean theorems (and the parallel postulate) can be denied without landing us in contradiction. Requiring a contradiction, however, is the hallmark only of the denial of analytic truths. Why, then, can't Kant simply say to Lobachevsky, "Yes, you're right: Euclidean geometry is not analytic. But I never said that it was! The reason I wrote the Aesthetic in the first place was to account for their being true necessarily without being true analytically. All you have done is to rigourously prove my contention that geometry does not present us with an analytic doctrine."?
Secondly, I sometimes hear it objected that Kant overlooks the possibility of non-Euclidean forms of intuition (a possibility supported, it is claimed, by the non-contradictoriness of Lobachevskian geometry), which is supposed to somehow threaten the Transcendental Aesthetic. But I don't see that the purported consequence follows. Kant never claims that he is presenting the forms of intuition for anything other than humans, and Kant's treatment of space will be true of humans if of anything. I seem to remember, in fact, him somewhere saying that a radically different being could be possessed of forms of intuition other than space and time, and he is certainly emphatic that there might well exist an intelligence possessed of intellectual intuition, which would thus not be affected with appearances and a fortiori not possess a means of organising them (i.e. a form of intuition like space or time). So, I don't see how on either account Kant's doctrine can be impugned by appeal to the authority of Lobachevsky.