I am not arguing about the possibility of formalizing the calculus, I am talking about the necessity. There have been proofs without formal systems and there will continue to be. All of these proofs might be formalize-able, but they were proofs before they were shown to be such.
People built buildings before mechanics and materials science, but at some point in the development of the technology, you need those to get any further. It’s the same with mathematics. The formal apparatus isn’t what people do mathematics in, but it’s a necessary foundation. Without it you get informal arguments that no-one is quite sure are really valid, as was the case for calculus before the epsilon-delta definition of a limit was worked out. (It was more than a century later before someone was able to make rigorous the original talk of infinitesimals.)
People built buildings before mechanics and materials science, but at some point in the development of the technology, you need those to get any further. It’s the same with mathematics. The formal apparatus isn’t what people do mathematics in, but it’s a necessary foundation. Without it you get informal arguments that no-one is quite sure are really valid, as was the case for calculus before the epsilon-delta definition of a limit was worked out. (It was more than a century later before someone was able to make rigorous the original talk of infinitesimals.)