More specifically, the correctness of the proof (at least in the triangles case) is common sense, coming up with the proof is not.
The integrals idea gets sketchy. Try it with e^(1/x). It’s just a composition of functions so reverse the chain rule then deal with any extra terms that come up. Of course, it’s not integrable. There’s not really any utility in overextending common sense to include things that might or might not work. And you’re very close to implying “it’s common sense” is a proof for things that sound obvious but aren’t.
Sure. And I’m of the opinion that it is only common sense after you’ve done quite a lot of the work of developing a level of intuition for mathematical objects that most people, including a significant proportion of high school math teachers, never got.
More specifically, the correctness of the proof (at least in the triangles case) is common sense, coming up with the proof is not.
The integrals idea gets sketchy. Try it with e^(1/x). It’s just a composition of functions so reverse the chain rule then deal with any extra terms that come up. Of course, it’s not integrable. There’s not really any utility in overextending common sense to include things that might or might not work. And you’re very close to implying “it’s common sense” is a proof for things that sound obvious but aren’t.
Sure. And I’m of the opinion that it is only common sense after you’ve done quite a lot of the work of developing a level of intuition for mathematical objects that most people, including a significant proportion of high school math teachers, never got.