I see axiom of choice, and really LEM, as logic's equivalent to limit points in calculus. No, you can't calculate 0/0, but here's what the answer would be if you could. No, you can't prove the truthiness of this statement, but here's what it would be if you could.
I guess one could work in a brand of math whose axioms make defining and using limits impossible, which, maybe if formalization came before the invention of calculus, would make some 17th-century mathematicians feel more comfortable. Though I imagine it would make progress in physics challenging. I think the same about LEM/AoC. Given that almost every element in the power-set of reals is non-measurable, maybe stuff like Banach-Tarski is actually fundamental in real physics: it can't be predicted or computed, but it can be observed.
I guess one could work in a brand of math whose axioms make defining and using limits impossible, which, maybe if formalization came before the invention of calculus, would make some 17th-century mathematicians feel more comfortable. Though I imagine it would make progress in physics challenging. I think the same about LEM/AoC. Given that almost every element in the power-set of reals is non-measurable, maybe stuff like Banach-Tarski is actually fundamental in real physics: it can't be predicted or computed, but it can be observed.