I don't think I would agree with the general statement that science gives us axioms. In normal usage, "axioms" are not the same as scientific "givens" or "data" even though it makes sense to interpret the word that way. In math and logic, the axioms we use are typically very foundational. We might say, "the empty set exists," or "for all x, it holds that x = x." There's no need to invoke science for these sorts of axioms and we can make many useful conclusions without it.
You can create any set of axioms you want, but unless they carry a reference to something that happen "outside of Math", at the "Real World", they are only useful for nerd-sniping.
