It seems to me that as one goes higher up in mathematics, the proof of theorems get more involved and convoluted, that at some point one must postpone (or even give up?) understanding all prior theorems, and take them as “Black Boxes” instead? Is this true?
Personally, in an ideal situation, I would certainly like to understand and know (and remember) the proof of the theorems, but am slowly finding it starting to be an uphill (and near impossible) task.
In high school, it is still quite possible for the conscientious student to learn and understand the proofs of Pythagoras’ Theorem, limits proof of calculus, etc, even though it is not being taught in school. At the undergraduate level, things get harder, but I suppose it is still possible, to know all the necessary theorems proofs like Intermediate Value Theorem, Mean Value Theorem, etc.
As things go higher, the key theorems like Egorov’s Theorem, Lebesgue’s Differentiation Theorem, etc, have rather non-trivial proofs that span pages. I am sure there are more difficult theorems than these which have even more difficult proofs.
If I were to learn the full proof of the theorems first, before proceeding to learning other stuff, my progress would be painfully slow.
My question is: To do mathematics at a sufficiently high level, what are some tips to judge which theorems to learn in detail, and which to just take it like a “black box”? Any experiences to share?
I think it’s important to remember the key ideas in long proofs, as these may serve you later. Trying to remember every single equation in a long proof is a loss of time and energy. However, it is a good idea to memorize the statements of theorems in order to be able to recall them without having to reread them every single time. The thing is, the more you know by heart, the more you will be able to make links in your knowledge and recognize patterns.