Comment by scoofy
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Axioms should capture the rules we can assume without justification.
But in most cases we want them to reflect the rules of the real world in the sense that statements derived from those axioms reflect our observations. That part (reflect observations) can be separated by many levels of abstractions, etc. I would not try visualizing general statements on Lie algebras or spectral theorems, but those abstractions serve the same goal -- help derive conclusions that apply in the real world. My 2c.
Some math is not about the real world. This is not my kettle of fish, but I have heard of some general topology research directions that discusses properties of topological spaces that probably do not exist at all. Those (according to a friend whose advisor worked in the area) are pretty sad affairs, with only 5-10 people in the world who understand or care about this particular sliver of the math.
But some math is about providing tools (again, likely via levels of abstractions) for understanding the world and, to me, this is the "real math".
This is a personal view, not an absolute position. I started on the other side of this fence and during my pure math PhD regularly picked fights with our buddies doing physics PhD arguing that mathematics is self-sufficient and does not need any validation from other sciences. But over the next 30 years gradually went to the other side and now think that my original view leads to splintering into gazillion tiny slivers that do not care about anything else; not even about adjacent slivers. Which leads to degeneration. My 2c.
Math is done backwards.
We know what kind of results we want to be true, and then we search for the minimum number of axioms which can deliver that.
There is another goal in addition to minimizing the number/complexity of axioms. Some "axioms" like induction actually introduce an infinite family of assumptions, a so-called schema. So in addition to working backwards from our (incomplete) knowledge, we find certain axioms let us make arguments that are obviously valid but would be formally very tedious without them.