18 June 2007

first post

You've stumbled across "God plays dice".

What is this? It's a blog. I plan to use it to talk about probability. Probability's everywhere around us, and in fact the world is basically governed by it. As you probably know, the ultimate laws of physics don't reflect what you see around you every day. The state-of-the-art theories in physics involve quantum mechanics. In our world, you can't walk through walls. But if you were a subatomic particle, maybe you could -- it's called quantum tunnelling. The probability of it happening would be very low -- but it could happen. And in fact, there's a chance that you, too, could end up on the other side of the wall next to you, or that your chair could suddenly disintegrate, or something like that. It's just that that chance is so small that it would take many times the lifetime of the universe for that to happen. Nevertheless, I had a friend in college who claimed to be afraid of quantum tunnelling. He was very smart and knew more physics than I do, so I hope he was joking.

Some people -- people who know a little bit about physics -- think, "oh, quantum mechanics says that things are random" and assume that's it. But that's not true. There's something called a wavefunction which tells us just how likely it is for a particle to be at a certain point in space. And physicists can compute these likelihoods very precisely. (Sometimes the computations get hard, but in principle there's always an answer even if they can't write down the number.) You might not like this. Einstein didn't like this, and Einstein was a very smart man. Einstein once wrote in a letter: "I, at any rate, am convinced that He does not throw dice." Niels Bohr is said to have said "Albert, stop telling God what to do."

(By the way, Einstein also didn't wear socks, or so they say. I tried this for a little while once. It's not such a good idea -- your shoes end up smelling horrible if you don't wear socks. So he didn't know everything.)

That's where the name of this blog comes from. But this isn't a blog about quantum mechanics. It's a blog about probability in all its guises. For example, do you worry that all the air in your room is going to go over to where you aren't, and suffocate you? Probably not. If you have money in the stock market, you might worry that the stock market goes up and down -- but you keep your money there because the "experts" tell you that the market won't collapse. The experts might not know what they're talking about, though -- from what I've heard a lot of them use models that assume the way prices vary follow something called the normal distribution. This isn't because they actually do -- it's just that that makes the math easier.

And that's why, at least in this blog, I will strive to be a "probabilist" and not a mathematician who does probability. What's the difference? Well, I'm a graduate student in mathematics. In mathematics there's a very strict way of doing things. You start out with certain axioms, which are things that you just assume are true, and then using the laws of logic you can show lots and lots of things are true. This goes back as far as Euclid's Elements, over two thousand years ago. Euclid started from a few pages of axioms about points, lines, and planes, and from that it's possible to prove a wide variety of statements about things you can draw in the plane. But here's something important -- he never defines what a point, a line, or a plane is. It's expected that you just know these things from being a human being. And it turns out that Euclid's geometry is not the geometry that applies, say, on the surface of the earth. On the plane, if you stand at the intersection of two lines, they never meet. But on the surface of the earth, if you are standing at the intersection of two "lines", then those lines also intersect at a point that is literally half a world away. Probability has a similar batch of axioms -- the probability axioms. And you can reason about these all you want and come up with nice results -- say, the central limit theorems, or the Lovasz local lemma. But when you apply these things to the "real world", you get things like the Doomsday argument. This says, among other things, that we don't know how long the human species is likely to survive, so we ought to act as if the human species will continue to exist for exactly as long as it already has existed. Or does it say that the number of people who are yet to be born is the same as the number who have already been born? Since population has grown a lot lately, these aren't the same things. We get ourselves into trouble when we apply probability to the real world -- but that's what I hope to do here.

I hope to convince you, my readers, that randomness is everywhere. (I admit I have not done that in this post.) And I aim to use this blog to teach you about probability, and to try to learn more of it for myself. As I said, I'm a graduate student in math at a Big Fancy University. But we don't talk about things like this. We just assume that Someone Else has figured out how to translate what we're doing into mathematical symbols, and we take it from there. We lose track of "reality", whatever that is. But probability theory was first developed not because a bunch of mathematicians were sitting around wondering how to torture their students -- no, it came into being because people wanted to analyze games of chance. And the world is a game of chance.

23 comments:

Mary Pat said...

Howdy there -- I'm a big probability fan (as you might know), so I wish you well on your new blog! -meep

Max said...

The doomsday argument is neat. And then I learned about Bertran's Paradox....

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