mblase suggests that this as a possible response to my challenge in

math is not a social construct. I claim that it is not.

Euclid knew what he was doing. He didn't call the fifth postulate an "axiom", he called it a "postulate". He suspected it should be a logical consequence of the axioms, but was unable to prove it. This makes sense -- we know that Euclid was wrong, and that it is indeed independent (what Euclid would call "an axiom").

In any case, he realised that he needed it, and that he could not prove it, so he it stated outright. In particular, this means that everything in __The Elements__ implicitly depends on it; Euclid did not hide this fact. And if you're performing e.g. spherical geometry on the half sphere, you can immediately see that Euclid's results do not necessarily follow, since the fifth postulate does not hold on a sphere.

And, in fact, Eratosthenes (who came from roughly the same culture, although later) *did* do spherical geometry! And guess what -- he didn't make use of results in Euclidian geometry that aren't true on a sphere! This did not interfere with his work (he measured the diameter of the Earth based on empirical measurements conducted in Egypt). He was able to do all of this because of the fundamental nature of mathematical law: it is true in some domain, and that domain is precisely defined. (The same is not true of physics, where the nature of the approximation can give rise to problems).

So you see Euclid's results are absolute truths: **IF** you accept the axioms and postulates, **THEN** the sum of the angles in a triangle is 2 right angles, there are precisely 5 regular polyhedra, the laws of similarity (homotheticity) hold, etc. And if you do not accept the axioms and postulates (e.g. because you're trying to navigate a ship, and you know the fifth postulate does not hold), then none of these results follows (although they might still be true, and even provable by some other argument).

Let's go back to the result about the sum of angles in a triangle. It does not hold without the fifth postulate. Then again, it does not hold for quadrangles, pentagons, hexagons or any other polygon. This does not mean that its truth is relative or depends on a cultural context. It merely means that it is in the nature of theorems that the result does not necessarily follow if the assumptions do not hold. And that is not surprising.

In fact, we even see that culture, which supposedly "forced" Euclid to consider only plane geometry with straightedge and compass, was no match for mathematics: Euclid *knew* he had to state his assumptions precisely; his __Elements__ is based on taking this set of assumptions as far as was possible in his age.