>>11357053Axioms dont necessarily have to be assumptions that you use to prove a certain statement. For instance we know that the statement "Every ring with identity has a maximal ideal" is equivalent to AC. But we know that AC is independent of ZF, this means that even if you removed AC from ZFC you would not actually be able to construct a ring with identity which does not have this property. That is, even if we disbelieve AC, we know through metamathematical methods that we cannot actually conceive of such pathological object. If you take intuitionistic point of view, this is the same as saying the object does not exist.
So even if you take a radical position, certain formalisms that might seem meaningless to you can still assert meaningful statements about a reality you consider meaningful. So a mathematical theory, in absence of any meaning, can still be useful on the basis of its formal consistency (which we can only know heuristically).
