Timeline for Why are mathematical proofs so hard?
Current License: CC BY-SA 4.0
9 events
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| Oct 13, 2020 at 15:58 | comment | added | j_random_hacker | I think it's more accurate to say that the way proofs are written down -- as neat sequences of definitions, lemmas and theorems -- is optimised for enabling a reader to verify correctness, not to learn how to do proofs, and has very little to do with the process of creating them in the first place, which usually involves large amounts of trying out ideas that worked on similar-looking problems you have seen before but which turn out to be dead ends in this case, and so on. The intimidating "neatness" of a written proof lies about how meandering and messy the path to get there was. | |
| Aug 7, 2020 at 13:16 | comment | added | Yakk | @user28324 You might be running into the fact that your physics "proofs" are closer to "just so stories" than to a solid arguments, while the mathematical "proofs" are closer to an actual solid argument. (Note: formal and provably correct proofs are much longer than even technical proofs in mathematics. See Coq) | |
| Aug 6, 2020 at 22:21 | comment | added | Mars | I really, really disagree with this answer. I don't see proofs as there just for the sake of "obligation", nor do I think they are not there for learning. I learn a great deal about the subject matter from studying proofs. I learn things that are not in the statement of the theorem nor in the surrounding text. I think authors write proofs so that they will be understood by their intended audience, perhaps with some challenge to a student to work at it a bit. (It's not that I find reading proofs to be easy. That depends, but I have to work hard at it, often.) | |
| Aug 6, 2020 at 18:42 | comment | added | user28324 | i think there is a problem with proofs in all sciences. Physics ( specially electromagnetism) has some nasty proofs too. But I've always preferred the intuitive way. What was really frustrating for me that i never really suffered with proofs in physics even the ones involving advanced calculus but i couldn't get through one assignment of induction without help | |
| Aug 6, 2020 at 18:03 | comment | added | user114966 | Maybe we learned from different books (which is likely, since I've mostly learned from local books), since my experience is always "formal $\gg$ intuitive". But I don't suggest to get rid of the formal part altogether: it's still there, at the bottom level of the tree. When you concat the derivations there into the string, you'll get the original formal proof. | |
| Aug 6, 2020 at 17:43 | comment | added | apen | @Dmitry That's exactly what happens, right? When a proof is so formal and detailed, you get lost in the woods. Hence, proofs are presented in short, intuitive forms. But the only problem is that my intuition is different from yours, and if that gap exists, it is sometimes insurmountable; I can't get inside your brain. With a formal proof, at least most readers can go through it if they could tolerate the grind, which I suspect is the intention of Intro to Alg. However, I dislike that approach pretty much the same. | |
| Aug 6, 2020 at 16:57 | comment | added | user114966 | My point is yes, they write proofs as an obligation. And as obligation, their proofs are correct, but hard to understand. So a different approach is needed: something like a tree: you have a main idea at the top level (and explanation why this idea is natural!), which can be split into subproblems (children nodes). At the leaves of this tree we have a hardcore math. This way will allow a reader to predict what happens next (this is what happened to me with my awesome math analysis teacher), not just follow the proof. | |
| Aug 6, 2020 at 16:54 | comment | added | user114966 | While I agree that "proofs are not written for the purpose of teaching", I think "writing them in exhausting details" is exactly what happens. Take "Introduction to algorithms" for example: it's one of the most famous books, and I believe that their way to present proofs is terrible. They prove the simplest things, like DFS and BFS, in very formal, but so much hard to understand way. I remember I couldn't understand dynamic programming for weeks using this book; and I understood it instantly after someone explained Fibonacci example to me. | |
| Aug 6, 2020 at 15:32 | history | answered | apen | CC BY-SA 4.0 |