Maths has the reputation of a a ‘pure’ discipline which sets both the practice and essence of mathematics apart from all other fields of inquiry. Mathematicians often use this ‘axiom’ to remind other academics of their superiority. Many people mention Gödel’s incompleteness theorems as a proof to the contrary but given that his is a purely mathematical argument, it probably isn’t the case. However, there are at least some aspects of mathematics that are subject to what in other disciplines would be called an empirical inquiry (if not an experiment). Moreover, these were used to establish the very foundations of mathematics. I can think of three. (I’m sure mathematicians would deny all of this - and it is possible, if not likely, that I am monumentally wrong.)

1/ zero - it’s always puzzled me greatly why it’s not possible to divide by zero (which is a surprisingly recent invention); from all the explanations I’ve hears is that the reason is because if we allow division by zero eventually other parts of mathematics would not work - but it took somebody ‘going to’ those regions of mathematics and discovering what happens if we divide by zero - well, that sounds pretty empirical to me 2/ set theory - why cannot a set be a member of itself? on the surface of it, there is no reason it should be; only if we take this premise and ‘experiment’ with it, does it turn out that a lot of things break (Russel’s paradox) - again, sounds pretty empirical to me 3/ Zeno’s paradoxes - these always intrigued me as the ultimate example of a mismatch between basic math and the world; to resolve this problem, we need to start making up lots of axioms (if perhaps marginal) to make math work

This thought is of no consequence to mathematics but it might be of more import to philosophy of science. Unless, this is complete non-sense (which statement is indicative of the intimidating power of pure and mysterious mathematics).