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 `n-dimensional vector-space', a `three-dimensional projective space' or a `twodimensional 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, iii 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
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    F. P. O'Gorman (1977). Poincaré's Conventionalism of Applied Geometry. Studies in History and Philosophy of Science Part A 8 (4):303-340.
    J. R. Lucas (1997). Comments: Reality and Time. International Studies in the Philosophy of Science 11 (1):97 – 108.
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