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cata comments on Truly Part Of You - Less Wrong

59 Post author: Eliezer_Yudkowsky 21 November 2007 02:18AM

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Comment author: cata 23 December 2010 11:27:47PM *  2 points [-]

Well, if you like reading things, I know of one extremely good book about the different methods and heuristics that are useful in problem-solving: George Polya's How to Solve It. I strongly recommend it. Hell, I'll mail it to you if you like.

However, it feels to me personally that every single drop of the problem-solving and figuring-things-out ability I have comes purely from active experience solving problems and figuring things out, and not from reading books.

Comment author: giambolvoe 02 January 2011 06:08:15AM *  0 points [-]

Well, here's my background:
I taught myself math from Algebra to Calculus (by "taught myself" I mean went through the Saxxon Math books and learned everything without a teacher, except for the few times when I really didn't understand something, when I would go to a math teacher and ask).
I made sure I tried to understand every single proof I read. I found that when I understood the proofs of why things worked, I would always know how to solve the problems. However, I remember thinking, every time I came across a new proof, that I wouldn't have been able to come up with it on my own, without someone teaching it to me. Or, at least, I may have been able to come up with one or two by accident, as a byproduct of something I was working on, but I really don't think I'd be able to sit down and try to figure out the differentiation, for example, on purpose, if someone asked me to figure out a method to find the slope of a function.
That's what I meant when I said that I'm intimidated by this. It's not impossible that I wouldn't ever figure out one of the theorems on accident, by working on something else, I just can't see myself sitting down to figure out the basic theorems of mathematics. If you think it'll help, I'll have to pick up "How to Solve It" from a library. Thanks for the advice!

Comment author: josinalvo 20 December 2014 10:27:08PM 1 point [-]

One true thing that might be applicable: Usually math textbooks have 'neat' proofs. That is, proofs that, after being discovered (often quite some time ago) where cleaned up repeatedly, removing the previous (intuitive) abstractions and adding abstractions that allow for simpler proofs (sometimes easier to understand, sometimes just shorter)

Rather than trying to prove a theorem straight, a good intermediary step is to try to find some particular case that makes sense. Say, instead of proving the formula for the infinite sum of geometric progressions, try the infinite sum of the progression 1, 1/2, 1/4. Instead of proving a theorem for all integers, it it easier for powers of two ?

Also, you can try the "dual problem". Try to violate the theorem. What is holding you back ?