Saint Petersburg Mathematical Society

Transcription

MARK JAGO: We're going to talk about four paradoxes that touch on the topic of infinity. First one, we've got the Hilbert Hotel. This is a hotel with infinitely many rooms. And at the moment, there's somebody in every single room. So it's a full hotel, and then a new customer rolls up at the reception desk. Now, you might think they're going to turn him away because the hotel's full. But the manager's clever. And here's what he does. He shifts the person in room 1 to room 2, and the person in room 2 to room 3, and the person in room 3 to room 4, and so on. Everyone gets shifted forward one room. And because there's infinitely many rooms, you never run out of rooms to put people in. And when you've done all of this, room 1's free. So the new guest gets put in room number 1. So you can fit in a new guest. You can fit in two new guests. You could fit in 10 new guests. And even if an infinite bus of new customers rolls up, you can fit them all in. I guess it's that when I first heard the description, I thought, well, that hotel's full. You can't fit anybody in it. That's how full hotels work. But then once you're shown that you can fit as many new people in the hotel as you like, you learn something new about infinity. So your intuitions change. And the trick is because it's an infinite hotel, there is no final room. If there were a final room, you could count all the rooms, and there wouldn't be infinitely many of them. I think he's using the idea to say, infinity is interesting, it's philosophically interesting because you might have thought things went like this. But they don't, and we can show mathematically that they don't. Paradox number two-- this is the paradox of Gabriel's trumpet. So the idea is we have this mathematical shape, this mathematical object. And it's shaped like a trumpet. It'll start off here, and it'll taper and get thinner and thinner and thinner and thinner and thinner. And it tends off to infinity, getting thinner and thinner all the way. So that mathematical object has an infinite surface area. But the volume it encloses, the air in the center of the trumpet, is only a finite volume. Suppose you were told you had to paint the inside of that trumpet. Well, on the one hand, you've got to paint an infinitely big surface. So you're going to need an infinite bucket of paint. But on the other hand, you could just take a finite, big enough can of paint, tip it into the trumpet, and let it filter its way down. So it looks like you only need a finite amount of paint to paint an infinite surface area. That's really, really strange. If we're thinking about real physical paint, you just can't paint the whole surface. Because it gets so thin at one end that you're just not going to fit the paint down there. So I guess the clash arises when we think about the difference between the mathematical idea of a surface and the physical idea of paint, which is real thick stuff. If we had mathematical paint where the molecules had no size whatsoever, looks like you'd need an infinite amount of that to cover that infinite surface area. He thought there was something really strange going on here. He gave lots of proofs of this idea because I think at first, he thought he'd made a mistake. And lots of mathematicians thought that there was something wrong with the idea of infinity, that maybe somehow we should banish infinity. Because puzzles like this clash. They show there's something wrong with the idea, so we should somehow ban it from our mathematics. But in fact, infinity works just fine in mathematics. But we have to sometimes change our ideas about how the world works to fit it in. Number three-- so this is the puzzle of the dartboard. So we suppose we've got a dartboard, and we've got a dart. And we're going to throw the dart at the dartboard. And let's just suppose we're guaranteed 100% chance to hit the dartboard. And then we ask the question, think about the exact center point of the dart. And we think about an exact point, a mathematical point on the dartboard. And we ask the question, what's the chance our dart is going to hit that exact point? And it turns out we can't give a sensible answer. So suppose on the one hand that we say, it's got a chance greater than 0 of hitting that point. Well, there's a problem with that. If it's got a greater than 0 chance of hitting that point, the same goes for every other point on the dartboard. And there's infinitely many of them. In fact, there's a big infinity of those mathematical points. When we add all those chances together to give us the chance that the dart's going to hit the dartboard at all, we end up with an infinite probability. But we can't have an infinite probability. You can't get a probability bigger than 1 that something will happen. OK, so we've got a problem with the idea that the chance of the dart hitting that point is bigger than 0. So on the other hand, what happened to the chance of 0? Well, that's also strange. Because if the chance of hitting that point is 0, we could say the same for every point. So the chance of hitting any point on the dart board is 0. It's not going to happen. But that's really strange because we're sure the dart's going to hit the dartboard somewhere. Again, one thing philosophers say about this is the idea of banning infinity. So we say, don't think about the exact mathematical point of the dart. We think about an area-- any real physical dart has an area at its center. And we think about what's the chance that it hits an area on the dartboard. And then we divide these up in a finite way. And the problem goes away. As the area gets bigger, the probability that the dart hits there is bigger. As it gets smaller, the probability the dart hits there gets smaller. BRADY HARAN: So it's this kind of granular nature of our existence that gets us out of trouble? MARK JAGO: Yeah, that's right. So if everything that we think about is an area rather than an exact point, the problem goes away. But that's not completely satisfying. After all, we can think of an area getting smaller and smaller and smaller and tending to infinity. Shouldn't we have a decent answer to this problem when we get to the infinite case? And maybe we don't have a very good answer to it. OK, so here's the fourth paradox. This is perhaps the most interesting one of them all because it relates to what rational humans should do in betting games. You go to a casino, and they put a pound in the pot. And they say, we're going to flip a fair coin. And when it lands on a tail, you get whatever's in the pot. If we flip a head, the casino doubles what's in the pot, and we play again. If it gets a tail, you get what's in the pot. If it's a head, they double the pot, and we carry on playing. But then they ask you the question, how much would you pay to get into this game? Name your price. You've got to pay something to play. What would you pay? And I think most people think about this a bit and say something like, maybe a few pounds? Maybe if you're rich, you'd say 20 pounds. The mathematical theory, on the other hand, tells you, bet whatever you've got. Stake any amount of money you can get your hands on on this game, because the expected winnings is infinite. Your expected take home is infinite if you play the game. So the way we work out the expected take-home winnings is we look at each case. If you flip a tail on the first go, how much would you win? Add it to if you flip a tail on the second go, how much would you win? Add it to if you flip a tail on the third go, how much would you win? And we sum all of these. But because the game hasn't got a fixed end point, we sum infinitely many of these. And in fact, the expected value of each play of the game is 1/2. So we're adding 1/2 to 1/2 to 1/2 to 1/2 infinitely many times. That's why the expected value of what you will take home is infinitely big. BRADY HARAN: But we wouldn't bet our house, would we? We wouldn't bet everything. MARK JAGO: I wouldn't. And I don't think you would. And-- BRADY HARAN: What does that say? MARK JAGO: I think it says something interesting perhaps about us. Perhaps it says that we're more risk averse than the mathematical theory would have us believe. Perhaps it says that the value of money changes the more you accrue. Perhaps it says something else that we haven't quite worked yet, that we should figure in to what makes us rational reasoners that we haven't yet taken account of in the maths. BRADY HARAN: You're sort of saying that as if humans are wrong to not bet everything on it. But surely if humans did that, casinos would set that game up and own all our houses by now. MARK JAGO: Well, good point. To make the maths come out, the casino has to guarantee, so you have to know in advance that they will bet as much as they need. So there would have to be a casino with an infinite amount of money, potentially, to put into this game. So one way out of it is to say, look, there's no casino with that amount of money. And if we say after x amount of money has been put in the pot, that's it, that's all the casino can do. Then the numbers change, and it turns out you would be irrational to put lots of money into it. We have this mathematical theory of rational decision making. Most mathematical theories don't crumble when you put infinity in. They work just fine. So why is it that theories about us, how we should or shouldn't behave, go really weird when you put infinity in? So suppose we take all the natural numbers together. None of those are infinitely big. Each one is a finite number. But there's infinitely many of them. So if we were asked to count them, we'd say there's an infinite connection.