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It's certainly true that for a some mathematical problems memorizing the steps to the solution is often the most practical way forward (especially if you're pla
by Traster 5y ago
It's certainly true that for a some mathematical problems memorizing the steps to the solution is often the most practical way forward (especially if you're planning on passing an exam), but there is definitely more to it. Most of the work proving things is actually exactly what you say - going down a different route. That's why mathematicians have careers, they spend days/months/years going down different routes and taking different appraoches to solve something. Naturally, they don't explain all the ways they didn't solve the problem when they finally do solve a problem. You don't need to know the answer to get to the solution. This is just like saying "The problem with a maze is you need to know the route through it before you can get through the maze"- no, what you do is you search for the route through the maze and once you've finished searching then you know the answer and suddenly you can walk through the maze repeatedly very easily.
And similarly, whilst one proof might have limited practical applications, it's very difficult to know which proofs do have practical applications and it's almost impossible to know which results might later yeild further proofs that do have practical applications.
None of this is really a matter of intelligence, it's a matter of putting in work (although obviously there is intelligence involved to find particularly neat ways of appraoching problems).
And why do mathematicians learn proofs? Often, to see the approaches that you can use to solve other problems.
- antonfire 5y ago> Naturally, they don't explain all the ways they didn't solve the problem when they finally do solve a problem. I do think something related to this is less obvious than you're suggesting. A proof generally presents a train of thought gets you from point A to point B. It's almost never a representation of the train of thought that the author actually followed to get from point A to point B. This is obvious to people with experience in mathematics, but it's much less obvious before you have that experience. Even if you know it's true in principle, appreciating it viscerally and adapting the way you interact with mathematics to account for it is basically a lifelong task. A big component of good mathematical communication is explaining what motivates a solution to a problem (or even what motivates the problems in the first place). Sure, you don't have to do that every time, but it's fair criticize a broad lack of this. A lot of mathematical communication is... not good in this sense. Notably, the problem/solution in the OP isn't really motivated by the author. A skilled reader will think about it themselves, but they need to learn that skill somewhere. More or less on topic, here's an example where an experienced mathematician tries to reproduce one of Sylow's theorems without looking it up, and writes down a more-honest account of the thought process involved: https://gowers.wordpress.com/2011/12/10/group-actions-iv-intrinsic-actions/ https://gowers.wordpress.com/2011/12/10/group-actions-iv-int...
- JackC 5y agoI like the 3blue1brown youtube series for explaining mathematical problem solving, like this example explaining a difficult problem from the Putnam: https://www.youtube.com/watch?v=OkmNXy7er84 https://www.youtube.com/watch?v=OkmNXy7er84 "And those of you who follow the channel know that rather than just jumping straight to the solution, which in this case will be surprisingly short, when possible I prefer to take the time to walk through how you might stumble upon the solution yourself. That is, make the video more about the problem-solving process than the particular problem used to exemplify it." I bailed out of math around calculus, and these videos helped me at least appreciate what I was missing.