>It is as if mathematicians would start with axioms. You do not start with axioms - you start with proofs. Only when you have found a bunch of related proofs, can you come up with axioms. You end with axioms.
This makes no sense. When you write a proof, you're already using axioms, you don't come up with them afterwards. Mathematicians usually start with conjectures, then either proves them or shows them to be undecidable and can be used as an axiom.