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November 20, 2007

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I make it a habit to learn as little as possible by rote, and just derive what I need when I need it. This means my knowledge is already heavily compressed, so if you start plucking out pieces of it at random, it becomes unrecoverable fairly quickly. As near as I can tell, my knowledge rarely vanishes for no good reason, though, so I have not really found this to be a handicap.

Reminds me of the time that my daughter asked me how to solve a polynomial equation. Many moons removed from basic algebra I had to start from scratch and quickly ended up with the quadratic equation without realizing where I was going until the end. It was a satisfying experience although there's no way to tell how much the work was guided by faint memories.

A valuable method of learning math is to start at the beginning of recorded history and read the math-related texts that were produced by the people who made important contributions to the progression of mathematical understanding.

By the time you get to Newton, you understand the basic concepts of everything and where it all comes from much better than if you had just seen them in a textbook or heard a lecture.

Of course, speaking from experience, reading page after page of Euclid's proofs can be exhausting to continue to pay enough mental attention to actual understand them before moving on to the next one. :)

Still, it does help tremendously to be able to place the knowledge in the mental context of people who actually needed and made the advances.

@Sharper: There's actually a school that teaches math (and other things) that way, St John's College in the US (http://en.wikipedia.org/wiki/St._John's_College%2C_U.S). Fascinating place.

I make it a habit to learn as little as possible by rote, and just derive what I need when I need it. This means my knowledge is already heavily compressed, so if you start plucking out pieces of it at random, it becomes unrecoverable fairly quickly.

This is why I find learning a foreign language to be extremely difficult. There's no way to derive the word for "desk" in another language from anything other than the word itself. There's no algorithm for an English-Spanish dictionary that's significantly simpler than a huge lookup table. (There's a reason it takes babies years to learn to talk!)

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.

So, what about the notion of mathematical proof? Anyone want to give a shot at explaining how that can be regenerated?

Dynamically_Linked: On that one, I'm having a hard time understanding what exactly is being regenerated. If it's just a matter of "systematizing the process of deducing from assumptions", then it doesn't sound hard. The question is just -- what knowledge do I have before that, on which I'm supposed to come up with the concept? What's the "the sides of this triangle are 3 and 4 and this angle is right, and the hypotenuse is calculable"?

Very good post -- I think it'd be helpful to have a series of examples of knowledge being regenerated. Then people could really get your idea and use it.

Life is full of contradictions. Your boss wants you to work more, you want to spend more time with your family. On the one hand you need the salary to support your family and on the other hand you need a private life to enjoy yourself, recharge and be ready again to work some more. Do you work to live, or do you live to work? Can the question even be answered with a simple 'yes' or 'no'? Assuming you do not live to work - then why do you work? And the other way around: if you do not work to live, then why do you live? That is a contradiction.

But life is not a matter of yes or no questions. Or is life a matter of yes and no questions? This is a clear 'yes or no question' and clearly a matter concerning life. Assuming life is, then it would not be a matter of yes or no questions and the statement 'Life is not a matter of yes or no questions' would be false, assuming on the other hand that life is a matter of yes and no questions then the statement would be false as well. No matter how you approach it the statement is always false but you nevertheless agree with it. Another contradiction - how can this be?

The answer is of course the middle ground. You do not only work just to life and you do not only life just to work. Being the smart person that you are you look at you options, understand the consequences and strike a compromise. Work some so you can life some so you can work some more... A part of your salary is flowing back into your next salary by allowing you to recharge and a part of your life supported by your salary is the cause that lets you recharge in order to earn more salary. It is a recursive self referencing feedback loop - like a Moebius snail.

How to understand this recursive self-referencing feedback loop - let us call it the Moebius effect - to know what you have to do, is what I want to help you realize.

Those "meaningless" tokens aren't only used in one place, however. If you had a bunch of other facts including the tokens involved, like "waves produce interference patterns when they interact" and "light produces interference patterns when it interacts", then you can regenerate "light is waves" if it is lost.

Similarly, while "happiness is a state of mind" is not enough to define happiness, a lot of other facts about it certainly would. The fact that it is a state of mind would also let us apply facts we know about states of mind, giving us even more information about happiness.

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