"Mathematics is not a deductive science—that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork. You want to find out what the facts are, and what you do is in that respect similar to what a laboratory technician does."-Paul Halmos.
Following the argument of Paul Halmos, which I agree with, I don't think mathematics as done by mathematicians proceeds in an axiomatic manner. Now, I wonder whether we can provide foundations for mathematical reasoning as done by humans.
I think this might have relevance to automated theorem discovery or 'automated conjecturing' if such a field exists.