Newton and Gauss and Euler did just fine without such solid foundations. If you get a PhD, very likely even a undergraduate degree in mathematics you cover this stuff, then (unless you choose foundations as your field) you go about doing statistics, or algebra (the higher kind), or analysis knowing you're working on solid fundamentals. It would be crazy if every time you proved something in one of those fields you had to state which derivation of real number you were using. And I guarantee at least 90% of PhD mathematicians could do so if they really needed to.