Episode 09 - Laplace's demon and the butterfly

Tomlin Hawthorne

Imagine the weather forecasters get everything they've ever wanted. A sensor on every lamp post on Earth. Every ship, every aircraft, every square mile of ocean reporting temperature and wind and pressure, every second. And a computer a billion times faster than anything that exists. With all that, how far ahead could they tell you, reliably, whether it will rain on a particular afternoon? A month? A year? Ten years?

Tomlin Hawthorne

The Answer: About two weeks. And if you made the computer a billion times faster again, and put a sensor on every blade of grass, you might get to three. This isn't a complaint about meteorologists. It's a discovery, one of the most important of the twentieth century, and it was made more or less by accident in the winter of 1961 by a man who rounded off a number. Today we find out what he found, and what it did to the grandest dream in science.

Tomlin Hawthorne

Last time, a coin in a mug. You backed away until the rim hid it, somebody poured in water, and the coin floated up into view without moving. Light, bending at the surface to take the quickest way to your eye. I find it slightly eerie every time. We've spent four episodes admiring the Newtonian world. Conserved energy, deep symmetry, laws you can write two ways. It is a magnificent machine. Today we ask the question the machine invites. If the laws are fixed, is the future fixed? And can we know it?

Tomlin Hawthorne

I want to start in the theatre, because a playwright got to the heart of this before most physicists did. Tom Stoppard's Arcadia opened in London in 1993. It's set in one room of an English country house, in two periods at once: 1809, and the present day. In the 1809 scenes there's a girl of thirteen called Thomasina Coverly, who is a mathematical prodigy, and her tutor, Septimus Hodge, who is clever and idle and friends with Byron. The play opens with Thomasina asking her tutor what carnal embrace means, which he handles badly. But a few pages later she asks him a better question. She says: when you stir your rice pudding, the spoonful of jam spreads out in red trails. But if you stir backwards, the jam doesn't come together again. Why not? And a bit further on she has an idea. Suppose, she says, you could freeze every atom in the universe where it is, and note which way each one was heading. And suppose you were extremely good at algebra. Then you could write down a formula for the whole of the future. Nobody could really be that clever, she admits. But the formula must exist all the same. Septimus is rather impressed. He says, yes, as far as I know you are the first person to have thought of this. She wasn't quite. But Stoppard has the date right. She's saying it in 1809, and the man who said it for real published it in 1814. His name was Pierre-Simon Laplace. He was the greatest mathematical physicist in France, he had survived the Revolution by being politically agile, shall we say, and he'd spent his career showing that Newton's laws could account for every wobble of every planet. There's a story that Napoleon, leafing through his enormous book on the heavens, remarked that he could find no mention of God in it. And Laplace is supposed to have said: Sire, I had no need of that hypothesis. And here's what Laplace wrote. Picture an intellect, he said, that at one given instant knows all the forces acting in nature and the position of every single thing in the universe. And suppose it's vast enough to do the calculation. For such an intellect, nothing would be uncertain. The future, just like the past, would be present before its eyes. This imaginary intellect became known as Laplace's demon. And for a century and a half it was the unspoken creed of science. Think about what it claims. The world is a set of billiard balls. Know where they all are and how they're moving, and you know everything that will ever happen. The fall of Rome. What you'll have for lunch. This sentence. It was all settled at the beginning, and the only reason we can't read it off is that we aren't clever enough. I want you to feel how reasonable that is, given what we've covered. Newton's laws really are deterministic. The same start always gives the same finish. Halley's comet turned up on schedule. Neptune was exactly where the pen said. Why should it stop there?

Tomlin Hawthorne

The first hairline crack appears in 1889, and it involves a king, a prize, and an expensive mistake. King Oscar the Second of Sweden, to mark his sixtieth birthday, offered a prize for the solution to a famous problem. Take three bodies, say the Sun, the Earth and the Moon, each pulling on the others by Newton's gravity. Work out their motion for all time. Two bodies is easy. Newton did it. You get a neat ellipse, for ever. Three, everyone assumed, was just harder. The prize went to Henri Poincaré, the leading French mathematician of the day. His essay was brilliant. It was printed. And then, as it was being prepared for wider publication, an editor queried a step, and Poincaré, going back over it, discovered that his argument contained a serious error. He had to have the entire print run recalled and destroyed, and he paid for a corrected one out of his own pocket. It cost him more than the prize was worth. But the corrected version is the one that matters. Because what he'd found, when he fixed the mistake, was that the three bodies don't settle into any tidy pattern. Their paths wind around each other in a tangle so intricate that he wrote he could not even attempt to draw it. Tiny differences in how you start them lead to utterly different motions. He'd seen chaos. He understood what he'd seen. And then almost nobody followed it up for seventy years, because without a computer there was nothing much you could do with it. Which brings us to the coffee. It's the winter of 1961, at the Massachusetts Institute of Technology. Edward Lorenz is a meteorologist, a quiet man who liked hiking and had once wanted to be a mathematician. He has a computer in his office. It's the size of a large desk, it's full of vacuum tubes, it makes a tremendous racket, and it breaks down about once a week. On it he's built a toy weather. Not real weather. Twelve equations, describing a simplified atmosphere. He types in some starting numbers, and the machine grinds away and prints out a row of figures for each day of pretend weather. Winds shifting, pressure rising and falling. It never quite repeats itself, which he rather likes. One day he wants to look at a stretch of it again, in more detail. Rather than start from the very beginning, which would take hours, he takes a shortcut. He picks up an earlier printout, finds a line partway through, and types those numbers back in as his starting point. Then he goes down the hall to get a cup of coffee. When he comes back an hour later, the new weather has nothing to do with the old one. At first they track each other. Then they drift apart. Within a couple of pretend months, they have no resemblance whatsoever. One has storms where the other has calm. His first thought is that a vacuum tube has blown again. But it hasn't. And then he works out what happened. Inside the computer, the numbers were stored to six decimal places. One of them was 0.506127. But to save paper, the printout only showed three. So what he'd typed in was 0.506. A difference of about one part in four thousand. In real-weather terms, a puff of wind. He had assumed, as anyone would, that a tiny difference at the start would make a tiny difference at the end. That's what we mean when we say something is approximately right. And in his little weather, it wasn't true. The tiny difference grew. It doubled, and doubled again, and kept doubling, until it had swallowed everything.

Tomlin Hawthorne

Let me tell you what that means, because the popular version usually gets it slightly wrong. It does not mean the weather is random. There wasn't a single random thing in Lorenz's machine. It was twelve equations, pure Newtonian clockwork. Type in exactly the same numbers and you'd get exactly the same weather, every time. Laplace would have been perfectly at home. What it means is that determined and predictable are not the same thing. We'd always assumed they were. A system can follow strict laws, with no chance in it at all, and still be impossible to forecast, because to forecast it you'd need to know the starting point with perfect, infinite precision. And you never do. There's always a seventh decimal place. And in a system like this, the seventh decimal place eventually decides everything. That's why two weeks. Every time you want one more day of forecast, you need to know today's weather several times more precisely. It's a doubling game, and a doubling game always beats you. A billion times more computing power buys you a few more days. In 1972 Lorenz gave a talk about this, and the organiser, who needed a title for the programme, came up with one: Does the flap of a butterfly's wings in Brazil set off a tornado in Texas? In earlier versions Lorenz had used a seagull. I think we can agree the butterfly was the better choice. The field got a name, chaos, and once people knew what to look for, they found it everywhere. A dripping tap. A pendulum with a second pendulum hanging from its end. The populations of animals from one year to the next. The rhythm of a heart going into fibrillation. My favourite example is a game of pool. Surely that's the very picture of predictability? But there's a well-known estimate of what it would take to predict a long chain of collisions, ball hitting ball hitting ball. Each collision magnifies any uncertainty. And the answer is that after only about nine collisions, your prediction is thrown off by something as small as the gravitational pull of the people standing around the table. Even the solar system isn't safe. That was the original clockwork, Laplace's own life's work. In 1989 a French astronomer, Jacques Laskar, showed that the orbits of the inner planets are chaotic. Not on any timescale that matters to us. But ask where the Earth will be in its orbit a hundred million years from now, and the question has no answer. The demon's own masterpiece turns out to have a horizon. Now, I don't want to leave you thinking chaos just means mess. Because the second half of the discovery is, if anything, more surprising. Chaos has structure. When Lorenz plotted his weather, not as a list of numbers but as a shape, tracing how the system moved, he got a figure that looks like a pair of butterfly wings. The system loops round one wing, then flips to the other, then back, and it never repeats and never leaves the shape. You can't say which wing it'll be on next month. You can say for certain that it'll be on the butterfly. That's the difference between weather and climate, by the way. We can't predict the rain on a day in July ten years from now. We can say a great deal about what Julys will be like. And the shapes that turn up in chaos have a peculiar property. Zoom in on any piece and you find the same kind of intricacy again, and again inside that. They're called fractals. A coastline is like this. Seen from space it's ragged. Walk along it and it's ragged at the scale of bays. Kneel down and it's ragged at the scale of pebbles. Ferns are like this, and river deltas, and the branching in your lungs. Which is what Thomasina is doing in Arcadia. She's bored with her geometry, all those circles and triangles. She points out that nothing in nature looks like that. And she sets out to find the equation of a leaf, by taking a simple rule and feeding the answer back in, over and over. She fills her notebook and runs out of margin. A century and a half later, one of the modern characters puts her rule into a laptop and watches the leaf appear. Simple rules, repeated, make endless complexity. That's the hopeful half of chaos. The world can be intricate without the laws being complicated.

Tomlin Hawthorne

The myth of the week comes in two parts.

Tomlin Hawthorne

First: that chaos means randomness. It doesn't. It's the opposite, really. It's perfectly rule-bound behaviour that only looks random because we can't measure finely enough.

Tomlin Hawthorne

Second: that the butterfly causes the tornado. As if you could find the culprit. The point is that every butterfly matters, and every seagull, and every time anyone opens a door. The tornado has no single cause smaller than the whole state of the atmosphere. If you're hoping to prevent tornadoes by disciplining butterflies, I have bad news. And the question to take away. You'll have seen it coming. Suppose Laplace was right about the laws. Everything is determined. But suppose Lorenz is right too, and no mind that fits inside the universe could ever know the starting point well enough to predict it. Then the future is fixed, and unknowable in principle, by anyone, ever. Is that freedom? It certainly feels like an open future. Nobody can tell you what you'll do, including you. But is an unpredictable fate any less a fate? And for the historians among you. If a puff of wind in the atmosphere can decide whether there's a storm in a month, what about a puff of wind in human affairs? The driver who took a wrong turn in Sarajevo. Is history the kind of system where small things grow without limit? Or does it have a shape it returns to, like the butterfly? When you argue about whether great events have great causes, you're arguing about whether history is chaotic. I don't think physics can settle that for you. But it can tell you both kinds of system exist. Your challenge this week needs a calculator. The one on your phone is fine. Type in 0.3. Now do this: take one minus your number, multiply that by your number, and multiply by four. So for 0.3, that's 0.7 times 0.3 times 4, which gives 0.84. Now do the same thing to 0.84. And again to the result. Do it fifteen times and write down the numbers as you go. Then start again, but with 0.3001 in place of 0.3. A difference of one part in three thousand, a rounding error. Do the same fifteen steps. Compare the two lists. For the first few steps they'll be twins. By about the twelfth, they won't be on speaking terms. You'll have done, on your phone, in five minutes, what Lorenz did in 1961. And if you'd like the visual version, search for a video of a double pendulum. Better, two of them, started side by side. Next time, the last piece of classical physics we need, and it's a pleasure. What do a violin string, a stadium crowd, an ancient Greek theatre and a pair of noise-cancelling headphones have in common? And how can adding one sound to another give you silence?