For example, ZFC and ZF.
I have come across the notion of pure and applied mathematics, and how the development of the former can (and is usually intended to) lead to the furtherance of the latter. In this case, how do we know that the axioms we do pure maths with are "sound"? For me, there is no point in doing something as rigorous as maths unless we are certain it could at least possibly provide a practical use to us, however subtle that use may be: otherwise, it is just an elaborate (and hugely unproductive) mind game.
I suppose my question boils down to why we do maths and what it is. If its a way of abstractly describing physical natures and phenomena (like Plato thought), I'd have thought you could test axioms with the scientific method, in which case there would be no confusion surrounding, for example, the millennium problems.
There is so much nuance about axioms and proofs, truth, falsity and their natures that really interests me, and I would like to know more about, but I don't know where to start.
I have heard of "metamathematics" as related a field of study, which is why it's included in the title, but I honestly don't know much about it. I also appreciate that ZFC and ZF may be bad examples, in which case please point out why.
Thank you!