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The link about research and life partners moved
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!
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 ?
In response to
Circular Altruism
I am truly confused. This post does not endorse either side.
I just would like to note something about my cognitive process here: in the "step by step" argument, what I seem to be thinking is "rigorously the same torture" and "for more people". The argument may be sound, but it does not seem to be hitting my brain in a sound way
In response to
Circular Altruism
Can someone please post a link to a paper on mathematics, philosophy, anything, that explains why there's this huge disconnect between "one-off choices" and "choices over repeated trials"? Lee?
Here's the way across the philosophical "chasm": write down the utility of the possible outcomes of your action. Use probability to find the expected utility. Do it for all your actions. Notice that if you have incoherent preferences, after a while, you expect your utility to be lower than if you do not have incoherent preferences.
You might have a point if there existed a preference effector with incoherent preferences that could only ever effect one preference. I haven't thought a lot about that one. But since your incoherent preferences will show up in lots of decisions, I don't care if this specific decision will be "repeated" (note: none are ever really repeated exactly) or not. The point is that you'll just keep losing those pennies every time you make a decision.
1. Save 400 lives, with certainty. 2. Save 500 lives, with 90% probability; save no lives, 10% probability. What are the outcomes? U(400 alive, 100 dead, I chose choice 1) = A, U(500 alive, 0 dead, I chose choice 2) = B, and U(0 alive, 500 dead, I chose choice 2) = C.
Remember that probability is a measure of what we don't know. The plausibility that a given situation is (will be) the case. If 1.0*A > 0.9*B + 0.1*C, then I prefer choice 1. Otherwise 2. Can you tell me what's left out here, or thrown in that shouldn't be? Which part of this do you have a disagreement with?
http://en.wikipedia.org/wiki/Prisoner%27s_dilemma#The_iterated_prisoners.27_dilemma
(just an example of such a disconnect, not a general defence of disconects)
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The link about research and life partners moved
http://waitbutwhy.com/2014/02/pick-life-partner.html
Also, although I know this is not the point of your post, could you please share why you like that page/that blog. i.e.: why is it trustworthy?
I mean to read the citations (and probably their citations) but maybe you already have convincing evidence that this guy(gal?) is methotical enough to be trusted