Well maybe not quite that extreme. It's more analogous to the fact that most professional programmers do not fully understand the programming language they use

Well maybe not quite that extreme. It's more analogous to the fact that most professional programmers do not fully understand the programming language they use (nor honestly do they need to for the most part). Nonetheless, it's perhaps surprising for a beginning student of pure mathematics when all they can see is the sky-high demand for rigor that "professional" pure mathematics is seemingly content to hand-wave its foundations. (For the most part this hand-waving is the postrigorous stage described by Terence Tao https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/theres-more-to-... rather than the pre-rigorous hand-waving of the student).