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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Axioms are the rules of the game, and we deduce conclusions from the rules we decide. Just like the rules of chess don’t “exist” anywhere, it’s just that if you decide to not follow the rules then you’re playing a different game other than chess. If you decide that parallel lines meet somewhere, then you haven’t broken the universe or anything, you’re just not doing Euclidean geometry – you’re playing a different game.
Now, we construct and decide on what axioms to use based on how useful or fun we perceive them to be. We use Euclids axioms because they results that we deduce from them are useful in lots of ways. This doesn’t mean that they are true or false in some cosmic sense, just that in situations where the axioms seem to hold (like when you’re constructing a building) then the results are useful. The people who have to worry about whether or not their assumptions about the universe are true or not are physicists, but that’s a different issue.
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