Even for Mathematics though, which I think is where your point is strongest, this is not actually true. If you're a Mathematician in the nineteenth century (fo

Even for Mathematics though, which I think is where your point is strongest, this is not actually true. If you're a Mathematician in the nineteenth century (for example Peano) you know what a set is in some sense, and if pressed you'll admit that you don't really have a formal way to explain your intuition. Nevertheless you feel content to write about sets as if they're formally defined, even for the infinite sets. Turns out you've been relying on an unstated Assumption, the Axiom of Choice. When Ernst Zermelo and Abraham Fraenkel write down axioms for a coherent set theory they discover that oops, the system works fine either way with this regard to this Axiom, and many years later it was proved to be entirely independent, and yet mathematicians had been gaily assuming it's true, without saying so. So in a sense Peano and say Turing are working with different, slightly incompatible versions of Mathematics. Euclidean geometry works fine... but today you'll be told that our universe's geometry isn't actually Euclidean, so, if something big enough just doesn't work that's to be expected, Euclid's model is neat but it's not actually a scale model of our universe, our universe is much stranger.