Euclides ab omni naevo vindicatus

The issue is obscured by the fact that the word 'space' can be used in four different ways. It can be used, first, as a term of pure mathematics, as when mathematicians talk of an 'n-dimensional phase-space', an 'mdimensional vector-space', a 'three-dimensional projective space' or a 'two-dimensional Riemannian space'. In this sense the word 'space' means the totality of the abstract entities-the 'points'-implicitly defined by the axioms. There is no doubt that there exist, in this sense, non-Euclidean spaces, because all that is claimed by such an assertion is that sets of non-Euclidean axioms constitute possible implicit definitions of abstract entities, that is to say that some sets of non-Euclidean axioms are consistent. If Kant or any other philosopher had denied this, he would have been wrong; but Kant himself took care not to deny it,2 and there is little reason to suppose that any philosopher concerned about space has been using the word in this, the pure mathematician's, sense.