[commonlisp94]

Let's recall your original comment. You said we can't talk about reality correspondence of math because multiple logical systems can describe physical space equally well.

A triangle in a euclidean system is not the same shape as a triangle in spherical, etc. They are only equivalent in that they have an analogous logical definition in their separate systems.

They don't look the same. They don't describe the same physical phenonoma either. In other words, the physical correspondence is not the same, so there is no contradiction.

[mcguire]

What? I said nothing of the sort.

In what we laughingly call "normal" geometry, given a line L and a point P not on L, there is exactly one line L' through P parallel to L. In this case, the interior angles of a triangle sum to 180 degrees.

In hyperbolic geometry, there is more than one line L', L'', etc., through P that are parallel to L. In this case, the interior angles of a triangle sum to <180 degrees.

In elliptic geometry, there are no lines L' through P parallel to L. (You also have to alter Euclid's 2nd axiom, defining a line as extending indefinitely.) And in this case, the interior angles of a triangle sum to >180 degrees.

Triangles are triangles; they have three points and three line segments. Their properties, however, are inconsistent in the three axiomatic systems.

If you are operating in a relatively small area, say a bronze-age farmer's field, Euclid's axioms satisfy all your geometrizing needs. If, however, you are on the somewhat-spherical Earth and are dealing with a large enough area, you can make a triangle with three 90 degree interior angles. You can tell which geometry your "physical space" is dealing with, and if you use the wrong one then you will get wrong answers.

For more than 2000 years, geometers assumed their space was essentially Euclidian. It turns out that is not really true, and it also turns out that there are (now, common) situations where the difference matters. So when you do any applied math, you assert that your axioms "faithfully reflect reality". And you may be wrong. And you may not have any way of knowing you are wrong. And you may not even have the mathematical tools to know what "wrong" means.