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IN INTERPRETING THE IMPORTANT section of the Critique of Pure Reason entitled "Transcendental Exposition of the Concept of Space," it has long been standard to suppose that Kant offers a transcendental argument in support of his claim that we have a pure intuition of space. This argument has come to be known as the "argument from geometry" since its conclusion is meant to follow from an account of the possibility of our synthetic a priori cognition of the principles of Euclidean geometry. Thus, Kant has been characterized as having attempted to deduce a theory of space as pure intuition from an assumption about mathematical cognition.
Insofar as Kant's theory of space is typically taken to be a central tenet of his transcendental idealism, the "argument from geometry" has come to bear a heavy burden. Indeed, many if not most commentators agree that Kant's transcendental idealism is unsupported by the "argument from geometry," arguing that history has shown it to have buckled under the weight of developments in mathematics and mathematical rigor unforeseen to Kant. Even commentators who disagree about the ultimate status of Kant's idealist conclusions agree about the inability of the "argument from geometry" to sustain them. So, for example, Henry Allison holds that the arguments for transcendental idealism succeed independently of the failed "argument from geometry" while Paul Guyer indicts the former due to what he argues is their direct dependence on the latter.1
It is important to notice, however, that the standard interpretation of the "argument from geometry" takes Kant to be arguing in the "analytic" or "regressive" style that he assumes in his Prolegomena. Such an argument begins with some body of knowledge already known to have a certain character, such as mathematics, in order to "ascend to the sources, which are not yet known, and whose discovery not only will explain what is known already, but will also exhibit an area with many cognitions that all arise from these same sources."2 Thus, on the standard interpretation of the "argument from geometry," Kant analyzes our synthetic a priori knowledge of Euclidean geometry in order to discover what is not yet known: that we have a pure intuition of space. But this interpretation is in direct conflict with Kant's own stated claim...