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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The current answers are good, but I think they’re maybe giving you a slightly narrow perspecitve. It’s true that axioms don’t have to be, well, “true”, and that mathematicians happily switch between different and contradictory sets of axioms to study the consequences of picking different ones. However, some axioms definitely seem more interesting and useful than others, and some seem to capture real-world phenomena very well. So the axioms that people study aren’t completely arbitrary – they’re the result of a process that is presumably influenced to some extent by the local conditions in our part of the universe, plus the structure of our brains, our history, our culture, etc.
Lots of people have thought about this and have come up with different perspectives, but there is no universal agreement. Some people think that all those factors are pretty tiny and that maths is essentially just an arbitrary game. Some people think that it is closely tied to how our brains and minds work, and that another intelligent species might produce completely different mathematical ideas. Some people think that it is closely tied to the real world, either in the sense that the real world fundamentally follows mathematical rules, or in the sense that we have developed mathematical ideas to mirror real-world stuff.
Anyway, when it comes to *applying* maths to do real-world stuff, you’re more in the territory of science. You need to do experiments to demonstrate that a mathematical model approximately fits a real-world system, then you can study the model and derive conclusions about it, then ideally you also test those conclusions. Once you’ve tested a model to death, you can start using it with more confidence, and only need to test it again when you extrapolate it into some regime where nobody has done experiments on it before.
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