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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An axiom is something that is assumed to be true and then you prove things based on those being true.
So it’s always true that a mathematical group has an identity element because that axiom is assumed to be true for groups and if it isn’t true, then it isn’t a group.
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.
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.
Mathematics is a construct of logic and has nothing to do with location.
I assume that what you meant is how do physicists know that their theories hold true in every region of the universe.
The answer is that they don’t know. Most assume this because it is the simplest assumption (Occam’s razor) and because it makes formulating theories easier. Furthermore, they haven’t yet found something that requires this to be true.
If you want a good example of all this in action; the early work of Stephen Hawking, Roger Penrose and others that showed there must be a singularity in a black hole and the Big Bang is all based on the axiom that general relativity is true. Hawking later argued that there probably aren’t actual singularise out there in the universe, on the basis that general relativity probably isn’t valid in those circumstances. That doesn’t make the original maths incorrect though.
Math is an abstract creation.
For example, there is euclidean geometry, elliptical geometry, and hyperbolic geometry. Each has a different set of axioms, and each has different applications in the real world.
So the real question is “how do you decide which mathematical model to use in the real world?”
And the simplest answer is “we use the model that most closely agrees with what we observe in reality”…
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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