Math is only sometimes done that way. Often an interesting field gets axiomatized later. Calculus, arithmetic, geometry, algebra, all existed productively for c

Math is only sometimes done that way. Often an interesting field gets axiomatized later. Calculus, arithmetic, geometry, algebra, all existed productively for centuries before axiomatization. Same way you can build a programming language without a formal spec. Yes, you might find an ambiguous piece of code later, or paint yourself into a corner. But you can do quite a lot without axioms.