AoC is a hacky shortcut. It gives right answers when it is supposed to, and wrong answers when they don't care.
(Most, non-constructivist) Mathematicians are comfortable with it, because they don't care that it contradicts physical reality, as long as it doesn't contradict itself. They aren't physicists.
On the flip side, physics don't care about being mathematically correct, as long their errors approximate reality.
"comfort" isn't really useful here. You say you don't toss it out, so what does your discomfort mean? Is there some result you refuse to believe? Or do you simply prefer to study non-AoC math? I like algebra. That doesn't mean I'm not "comfortable" with geometry existing.
(Most, non-constructivist) Mathematicians are comfortable with it, because they don't care that it contradicts physical reality, as long as it doesn't contradict itself. They aren't physicists.
On the flip side, physics don't care about being mathematically correct, as long their errors approximate reality.
"comfort" isn't really useful here. You say you don't toss it out, so what does your discomfort mean? Is there some result you refuse to believe? Or do you simply prefer to study non-AoC math? I like algebra. That doesn't mean I'm not "comfortable" with geometry existing.