> There is this thing in math called the "axiom of choice" which you can, quite literally, choose to accept or not.
No, there are these different mathematical structures: set theory with the axiom of choice, set theory with the negation of the axiom of choice, and set theory with neither. (More precisely, "Zermelo-Frankel set theory", since there are other set theories.) All of those structures were discovered. And, as I said, different humans invented different notations to describe these different structures.
Humans choose which of these structures to use for particular applications, yes. I don't think that process is either invention or discovery. Not everything humans do has to be an invention or a discovery.
No, there are these different mathematical structures: set theory with the axiom of choice, set theory with the negation of the axiom of choice, and set theory with neither. (More precisely, "Zermelo-Frankel set theory", since there are other set theories.) All of those structures were discovered. And, as I said, different humans invented different notations to describe these different structures.
Humans choose which of these structures to use for particular applications, yes. I don't think that process is either invention or discovery. Not everything humans do has to be an invention or a discovery.
Similar remarks apply to geometry.