 But first a word of warning. The logical fact that non-Euclidean geometry turned out to be consistent (relative to Euclidean geometry, or to real analysis) does not per se establish the invalidity of the Kantian viewpoint. Kant's views are compatible with the existence of logical alternatives to Euclidean geometry; after all, he was arguing about space as an intuition, not about mere conceptual possibilities; he was not a Leibnizian, and denied that geometry is analytic. If anything, a Kantian would expect that the alternative geometries are only an intellectual possibility, and that we cannot visualize them intuitively. A sophisticated elaboration of such a standpoint was Frege's in his famous correspondence with Hilbert (see the entry on the  Frege-Hilbert controversy).  But it's true that most of the relevant authors chose a different line of thought and abandoned the Kantian position.