Poincaré, Kant, and the Scope of Mathematical Intuition

Review of Metaphysics 62 (4):779-801 (2009)
Abstract
Today it is no news to point out that Kant’s doctrine of space as a form of intuition is motivated by epistemological considerations independent of his commitment to Euclidean geometry. These considerations surface—apparently without his own recognition—in Poincaré’s, Science and Hypothesis, the very work that helped turn analytically-minded philosophers away from the Critique. I argue that we should view Poincaré as refining Kant’s doctrine of space as the form of intuition, even as we see both views as arbitrarily limited—in Kant’s case, to Euclidean transformations, and, in Poincaré's, to geometries of constant curvature. Both run together the question whether space is an a priori form of intuition with the question whether there are a priori constraints on its applied geometry. At his best, Kant sees—what Poincaré does not—that they are connected by a form of cognition consisting in the intuition of homogeneous pluralities, of totalities apprehended as unities
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