On "Difficulty"

At last I'm on vacation! Time to try and get some work done - but first some long-postponed blogging...

One thing that always annoyed me is to be told that something is difficult. "Mathematics is very hard," some people say. Even teachers fall into this trap.

Why is something hard? Who says it's hard? I don't find it easy, but that doesn't mean it's hard for everyone. Some people even find math easy; I wish I were one of those folks. The

How many times have you heard a story about somebody who was interested in something, went ahead and figured it out, and then heard after the fact that he wasn't supposed to be able to do it? This is worth considering.

What makes something difficult? Not the material itself. It's entirely relative to the individual, or culture, or species, depending on what the nature of the challenge is. And then it's largely a matter of practice.

For human beings, it's very hard to walk on two legs - at first. Once they get it, though, they're off and running before you know it. For other critters, they're off and running once they hit the ground.

Answering fundamental questions about the world - now that's pretty hard, always has been. Just when you think you've got it, something or somebody comes along and messes up your tidy little theory. And it's back to square one.

For some cultures, it's hard to imagine an impersonal God. I'm not talking about believing in one, just the idea of one. And the reason for that is they're used to conceiving of a personal God with a face (sometimes literally).

For some individuals, math is hard. Maybe it'll always be that way, but maybe they'll get it and even be pretty good at it. It all depends on the person and the teachers they get - blockheads or angels.

Some things are supposed to be impossible for anybody. There's even proofs to show you that the difficulty isn't just in your head. Take trisection, for example. The task is simple: for any angle, divide it into three equal angles. Easy, right?

Wrong. People have beat their heads against the wall for centuries in an effort to figure out a way to do this. But in 1836 a proof was published by a mathematician named Wantzel that this task is not only difficult but impossible. Mathematicians have been grinning smugly ever since. (You can read all about it here and here. Archimedes cheated - see here.)

The only trouble is, there seems to be a method that works. I found one in an obscure journal years ago, and it obeys the rules of the game. Now I'm not a mathematician, but I know what I see, and this method looks very convincing to me. I've seen many other attempts, but they are either obviously wrong or inapplicable beyond a certain angle. But this method apparently lacks both these shortcomings.

(Yes, I do have a copy of it. Just not at the time of this writing. I will try and post it later.)

What have I learned from this?

1. I need to learn math.
2. I need to learn math to do, among other things,
a. understand that damn proof,
b. see what's wrong with it,
c. prove that this method works.

Is there any practical benefit to this exercise? No, not if you're really asking, "Can it fix your car?" or "Will it make you money?" But those aren't the only benefits around. Consider these:

1. A hidden assumption will be exposed.
2. We will have yet another example showing that we can do more than we think.
3. How we think greatly influences our abilities.

One thing about inquiry is that, to a large degree, it concerns finding limitations. That's what world records are for. Kids do it all the time: How far can I jump? How many pieces of chewing gum can I stuff into my mouth? How much can I get away with before Teacher smacks me? Grown-ups too: What happens if I mix these chemicals together? How long can I ride a unicycle? If I leave this number out, can I pay less on my income tax? The basic question underlying all these is, How far can you go?

If you don't try, you'll never find out. Even when the proofs say it can't be done, it's worth asking, Why not?

(Image joyfully pirated from http://mathworld.wolfram.com/AngleTrisection.html)