To add to the discussion: the very dichotomy Philosophy x Mathematics didn't make sense to most thinkers of the past. That still holds true for many of today's philosophers - Philosophy is not a discipline, like Mathematics or Chemistry.
That being said, I could list tens of philosophers that satisfy your expectations: Frege, Russell, Tarski, Smullian; the list goes on and on. Now, for those that don't satisfy your expectations, but may be even more interesting to you for that exact reason, I'll mention Husserl, Wittgenstein and classics like Pythagoras and Plato (yes, they had a lot to say about "Mathematics" and ontology).
Based on this sentence alone: "Philosophy might elucidate the 'true meaning' of axioms and of definitions by examining their ontology in a wider context", I would definitely give Wittgenstein a try. You'd be surprised to know that the sun may not rise tomorrow...
