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.
What I meant by “there is a very formal sense in which the calculus cannot be formalized.” is that you can’t have a complete and consistent axiomatization of the reals. I apologize for the ambiguity. But unlike finitism, frst order logic, boolean arithmetic, category theory, and I’m sure many more, calculus and arithmetic cannot be completely and consistently axiomatized.
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.)
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.
What I meant by “there is a very formal sense in which the calculus cannot be formalized.” is that you can’t have a complete and consistent axiomatization of the reals. I apologize for the ambiguity. But unlike finitism, frst order logic, boolean arithmetic, category theory, and I’m sure many more, calculus and arithmetic cannot be completely and consistently axiomatized.
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.)