Axioms are not discovered but decided. A nice set of axioms gives birth to interesting anduseful interactions which you are the one that you discover, although

Axioms are not discovered but decided. A nice set of axioms gives birth to interesting anduseful interactions which you are the one that you discover, although in practice there's is a bit of back-and-forth, as you decides the axiom based on what you are trying to achieve and they get changed or tuned as time pass. Bonus point if the new axioms play well with existing ones, and are useful to work on our reality.