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I think you are making a philosophical mistake here.
The whole comment? Or just one part? I don't necessarily see any mistakes (possibly because I don't see any conclusions he makes, other than the pithy quote at the end*)
*which I like, regardless of its pithiness
In response to
Original Seeing
This happens to me every time someone asks me to explain what I believe. I say "uhhh..."
I try to ask people to be more specific (what do you believe about this particular topic)
If they don't, I just tell them I believe human rationality can come to an understanding of everything, and can at least attempt to account for the things it doesn't understand. I may be wrong, but it's so damn hard to start from nothing, even if you do know everything (which I don't).
In response to
Truly Part Of You
I've always been intimidated by this. I'm quite positive I couldn't regenerate the Pythagorian Theorem, but I know that I should be able to. I certainly wouldn't be able to figure out basic calculus on my own. I wish that I could, but I know that I wouldn't be able to. Are there any things we've learned from mathematicians in the past that make figuring out such things easier? Anything I can learn to make learning easier?
In response to
Truly Part Of You
I make it a habit to learn as little as possible by rote, and just derive what I need when I need it.
Do realize that you're trading efficiency (as in speed of access in normal use) for that space saving in your brain. Memorizing stuff allows you to move on and save your mental deducing cycles for really new stuff.
Back when I was memorizing the multiplication tables, I noticed that
9 x N = 10 x (N-1) + (9 - (N-1))
That is, 9 x 8 = 70 + 2
So, I never memorized the 9's the same way I did all the other single digit multiplications. To this day I'm slightly slower doing math with the digit 9. The space/effort saving was worth it when I was 8 years old, but definitely not today.
I always do my 9x multiplications like this! We were taught this, though. I can't say I figured it out on my own.
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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.
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!