I dunno if I'd call it dogma. For the most part, most working mathematicians I've run into basically don't care either way and adaptively adopt whichever philos

I dunno if I'd call it dogma. For the most part, most working mathematicians I've run into basically don't care either way and adaptively adopt whichever philosophy happens to match best with their intuition for a particular problem. This can change on a problem by problem basis. I wouldn't be surprised if a working mathematician hadn't heard of intuitionism or formalism. Interestingly, your latter point is exactly formalism (the "formal" part is referring to the formalized language), i.e. Hilbert's position. Formalism basically asserts that mathematics is a fancy string manipulation game.