Math Girls (『数学ガール』の英訳)で気になった部分 (まだ途中)
[2012/1/16追記] There was a problem connecting to Twitter. There was a problem connecting to Twitter.
Mathematics is timeless.
When I think back on those days, equations seem to pop into my head and fresh ideas flow like a spring. Equations don’t fade with the passage of time. Even today they reveal to us the insights of giants: Euclid, Gauss, Euler.
Mathematics is ageless.
Through equations, I can share the experience of mathematicians from ages past. They might have worked their proofs hundreds of years ago, but when I trace the path of their logic, the thrill that fills me is mine.
Mathematics leads me into deep forests and reveals hidden treasures. It’s a competition of intellect, a thrilling game where finding the most powerful soution to a problems is the goal. It is drama. It is battle.
Math isn’t about dredging up half-remembered formulas. It’s about making new discoveries. Sure, there are some things that require rote memorization: the names of people and places, words, the symbols of the elements. But math isn’t like that. With a math problem, you have a set of rules. You have tools and aterials, laid out on the table in front of you. Math’s not about memorizing, it’s about thinking. Or at least, that’s what it is to me.
If you only saw the 1,2,3,4, then you’d expect the next number to be 5. But you’d be wrong. Rules won’t always reveal themselves in a small sample.
Mathematicians are always the lookout for useful concepts to help build the world of mathematics. When they find something really good, they give it a name. That’s what a definition is. So you could
define the primes to include 1 if you wanted to. But there’s a difference between a possible definition and a useful one.
Well, it’s not like you have to get a PhD, but when you’re reading math, you do have to make an effort to get into it. Don’t just read it, write it out. That’s the only way to be sure you really understand.
Seeing something from higher up can make your description simpler, and a simpler description is a sign of deeper understanding. When you look for the next number in a fragment of a sequence, you’re just solving a puzzle. When you generalize the seuence, you’re uncovering it’s hidden form. And that’s when the magic happens.
Circles are a more natural source of repetition. People who can only see real numbers on a number line would probably call it an oscillation, but once you can see it on the complex plane you notice that it’s a revolution. More hidden structure!
When most people hear the word ‘formula,’ I think they imagine something that’s written in stone, something to be memorized but never questioned. Once you start playing around with math, you see them in a different light. They’re more like a clay than stone: the more you work with them, the softer they get.
Testing thing with concrete examples like that is really important. Examples are the key to understanding, right.
When you keep plugging away at something you love, you learn to tell the difference between what’s important and what’s fluff. I’ve had math teachers that try to impress their students by talking really loud, or by acting smarter than they are. I guess they want to be the center of attention, like it’s an ego thing, or something. But someone who’s used to thinking about things, someone who knows what’s real, doesn’t act like that. You can’t figure out a recurrence relation by shouting at it. You can’t solve an equation by pretending you’re a genius. So no matter what people say to you, no matter what people think about you, just do your own thing until you figure it out. I think that’s really important. you have to follow what you love. You have to chase after —