Since axioms can’t be proved, how do mathematicians know that they’re always true independently of the location? If they don’t, how do they cope with that?
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You seem to be a bit confused.
an axiom is something the mathematician states as “in my world/gedankenexperiment this holds true and from that I can deduce”.
for example Euclidean geometry, the stuff you probably have been told in school about triangles and squares and angles and that stuff…contains an axiom that basically says that two parallel lines never intersect (or only at infinity),
and it is 100% accurate when you use it on flat 2d surfaces and it allows you to calculate a lot of stuff. however, just pick up any ball or globe and you see that it doesn’t hold true there, there you can have parallel lines (for example going north to south) that will intersect (at the south pole)
and basically mathematicians love those kind of things, when they find out that you can take away one axiom and still get something meaningful that can allow you to describe stuff.
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