It has been many years since I took geometry in college, so I had remembered incorrectly - I had thought that the axioms of Euclidean geometry need to be modified as you go through different numbers of dimensions (and the same thing with the other basic types of geometry). I see now that the basic axioms of Euclidean geometry (or non-Euclidian geometries) don't change as the number of dimensions increases.

It's the axioms which are the core of what I'm talking about, not the conclusions that are drawn about geometric shapes based on those axioms. Even though I was off about the effect of more dimensions, it doesn't change the underlying point I was making - that mathematicians don't consider one geometry to be more true or false than another geometry just because it is axiomatically different from another. And there are axiomatic differences between the three different kinds of geometry. The differences are:

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Login1.) Euclidean: Given a line L and a point P not on L, there is exactly one line passing through P, parallel to L.

2.) Hyperbolic: Given a line L and a point P not on L, there are at least two lines passing through P, parallel to L.

3.) Elliptic: Given a line L and a point P not on L, there are no lines passing through P, parallel to L.

It is the difference in those axioms in each of the three basic types of geometry that differentiate the three geometries. But mathematicians don't consider one type of geometry "true" and the other two "false". They are just different geometric systems. That's what I meant when I said the mathematicians are "agnostic" about one valid axiomatic system relative to another valid axiomatic system.