Comment by woolion

From the point of view of proof-theory, you can show that PA (arithmetic) is equivalent to the consistency of the omega cardinal (the countable infinite). Basically, everything line up quite well between things you want to be true and things that are true in that system. This equivalence breaks down with higher-order system such as System F, but it gives a system that may feel more natural, especially to programmers. The problem that explains the endurance of "formalism" is that there are so many things that you "want to be true" that can't be shown to be true in intuitionistic systems is a real issue. For instance, simply proving that a fast-growing function is total. You are fine with recurrence, but not if the function grows too fast? This sounds really stupid. But I don't think many people care that much, they'll just use whatever give them results.