Philosophers of mathematics spent the 20th century what mathematics is all about and they did not settled on any potential definition.
Ok, I have to disagree with you here. What Godel established is that there's no complete axiom system. Every mathematical formalism will have statements P for which neither P nor ~P can be proven. This was not an unexpected result. What makes Godel's Incompleteness Theorem awesome is that it proved incompleteness, which few people thought possible.
Same with Turing and the Halting Problem. Almost no one actually believed that such a program (that could determine, algorithmically, if a program halted) existed. If one did, it would blow open all of mathematics. Turing proved, in a very elegant way, that it didn't.
What 20th-century mathematicians agree upon is that mathematical statements aren't "true" or "false" in an absolute, platonic sense, but that they are products of the axiom systems that generate them.
Whether the Axiom of Choice or Continuum Hypothesis are "true" is meaningless. These aren't mathematical "controversies" that have people yelling at each other saying that the other is wrong. They're axioms about infinity (specifically, uncountable infinity) that, although they have no physical correlates (you can't actually Banach-Tarski an orange) are logically independent of the "obvious" axioms. What logically independent means is that neither L nor ~L will generate a contradiction, and therefore neither has any absolute high ground.
Most mathematicians use AoC and CH for typical mathematics, but there are alternative mathematical worlds in which they don't hold, and those are interesting in their own right.
AoC is used indirectly in almost every place of pure mathematics, but I researched numerical methods for PDEs and Stochastic PDEs which is not pure mathematics, maybe someone who is much more intelligent than me will say where you will use AoC directly in this area outside of some theorems of Analysis that are used but well, I've never ever touched this same axiom again in my life and if I chose to spend my life as a researcher I doubt I will ever need it again to work and publish, and yet I was able to get a Ph.d in Applied Mathematics. The theorem that I know uses the AoC that I cited once but not exactly used is Banach–Alaoglu theorem.
As sbi put if you go to a department of mathematics that includes mathematicians, statisticians and applied mathematicians chances are that almost no one will know too much of set theory excluding some pure mathematicians, this was true on almost every department that I saw in my entire life. So there are mathematicians whose life are not spent trying to use abstractions everywhere. That was the point.