>>16750741 (OP)
in mathematics and the philosophy of mathematics, i think that the very foundational question of what it means to be doing mathematics is largely ignored. this has been widely debated in the beginning of the last century during the foundational crisis of mathematics, where people have held very strong opinions on the ontological status of the objects of mathematical investigation. this debate has totally stifled by the early incompleteness, indenpendence and inconsistency results, and ever since mathematicians have stopped caring and settled for an indeterminate and inconclusive half-platonic/half-formalist attitude with an unquestioned acceptance of an incoherent set theoretic foundation: mathematics is seen as the study of mathematical objects somewhere out there, existing independently of the human mind, whose their existence however is seen as justified by some axiomatic, formal theory except when their existence is proven to be independent of the formal theory, then it's just unclear what to make of it and the internal response to this affair generally is "well, that's weird, i guess we will never know".
the question is widely considered very irrevelant by the vast majority of working mathematicians -- recently we have seen attempts at superseding the foundational framework of set theory by more modern and meaningful theories based on type theory and/or category theory such as sear / etcs, univalent foundations and the calculus of constructions. they all have in one way or another arisen from the desire to have a cleaner foundation, but they do very little in clarifying the epistemological nature of mathematics.
there is next to no willingness to question the most foundational assumptions, such as the law of excluded middle or the power set axiom, at least not in a serious, philosophical way because any dismissal of such assumptions is seen to render most of the current mathematics pointless.
> cont ..
