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SanguineEmpiricist comments on The Least Convenient Possible World - Less Wrong
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Riemannian geometry is not an axiomatic geometry in the same way that Euclidean geometry is, so it is not true that "the only difference is a single axiom." I think you are thinking of hyperbolic geometry. In any case, the geometry of spacetime according to the theory of general relativity is not any of these geometries, but it is instead a Lorentzian geometry. (I say "a" because the words "Riemannian" and "Lorentzian" both refer to classes of geometries rather than a single geometry -- for example, Euclidean geometry and hyperbolic geometry are both examples of Riemannian geometries.)
First i've heard of this, super interesting. Hmm. So what is the correct way to highlight the differences while still maintaining the historical angle? Continue w/ Riemannian geometry? Or just say what you have said, Lorentzian.
Special relativity is good enough for most purposes, which means that (a time slice of) the real universe is very nearly Euclidean. So if you are going to explain the geometry of the universe to someone, you might as well just say "very nearly Euclidean, except near objects with very high gravity such as stars and black holes".
I don't think it's helpful to compare with Euclid's postulates, they reflect a very different way of thinking about geometry than modern differential geometry.