The key to understanding Poincaré's philosophy of mathematics is to realise that he defends Kant's epistemological view that mathematics is synthetic a priori, but that the details of Poincaré's theory of the 'synthetic a priori" are quite distinct from Kant's. Indeed, Poincaré's theory might be sufficiently distinct so as not to be subject to the pitfalls to which Kant's theory is so vulnerable.¹ Since Poincaré adopts the Kantian terminology, though he adapts the theory, my first task is to outline Kant's theory, to enable the necessary comparison to be made. Poincaré explicitly rejects Kant's thesis that Euclidean geometry is synthetic a priori. He even disagrees with Kant's more minimal thesis that the three-dimensionality of space is synthetic a priori. He holds that these are, rather, "conventional" matters. However, he follows Kant in asserting that the theorems and the acceptable axioms of pure number-theoretic mathematics have the synthetic a priori status.