The Science of Discworld

by Terry Pratchett

  For a more homely example, think of a stone rolling down a bumpy hillside, skidding on a clump of grass, bouncing wildly off bigger rocks, splashing through muddy puddles, and eventually coming to rest against the trunk of a tree. If fundamentalist reductionism is right, then every aspect of the stone's movement, right down to how the blades of grass get crushed, what pattern the mud makes when it splatters, and why the tree is growing where it is anyway, are consequences of one set of rules, that Theory of Everything. The stone 'knows' how to roll, skid, bounce, splash, and stop because the Theory of Everything tells it what to do. More than that: because the Theory of Everything is true, the stone itself is tracking through the logical consequences of those rules as it skitters down the hillside. In principle you could predict that the stone would hit that particular tree, just by working out necessary consequences of the Theory of Everything.

  The picture of causality that this viewpoint evokes is one in which the only reasons for things to happen are because the Theory of Everything says so. The alternative is that the universe is doing whatever the universe does, and the stone is in a sense exploring the consequences of what the universe does. It doesn't 'know' that it will skid on grass until it hits some grass and finds itself skidding. It doesn't 'know' how to splash mud all over the place, but when it hits the puddle, that's what happens. And so on. Then we humans come along and look at what the stone does, and start finding patterns. 'Yes, the reason it skids is because friction works like this ...' 'And the laws of fluid dynamics tell us that the mud must scatter like that...'

  We know that these human-level rules are approximate descriptions, because that's why we invented them. Mud is lumpy, but the rules of fluid dynamics don't take account of lumps. Friction is something rather complicated involving molecules sticking together and pulling apart again, but we can capture a lot of what it does by thinking of it as a force that opposes moving bodies when in contact with surfaces. Because our human-level theories are approximations, we get very excited when some more general principle leads to more accurate results. We then, unless we are careful, confuse 'the new theory gives results that are closer to reality than the old' with 'the new theory's rules are closer to the real rules of the universe than the old one's rules were'. But that doesn't follow: we might be getting a more accurate description even though our rules differ from whatever the universe 'really' does. What it really does may not involve following neat, tidy rules at all.

  There is a big gap between writing down a Theory of Everything and understanding its consequences. There are mathematical systems that demonstrate this point, and one of the simplest is Langton's Ant, now the small star of a computer program. The Ant wanders around on an infinite square grid. Every time it comes to a square, the square changes colour from black to white or from white to black, and if it lands on a white square then it turns right, but if it lands on a black square then it turns left. So we know the Theory of Everything for the Ant's universe — the rule that governs its complete behaviour by fixing what can happen on the small scale — and everything that happens in that universe is 'explained' by that rule.

  When you set the Ant in motion, what you actually see is three separate modes of behaviour. Everybody — mathematician or not — immediately spots them. Something in our minds makes us sensitive to the difference, and it's got nothing to do with the rule. It's the same rule all the time, but we see three distinct phases:

  SIMPLICITY: During the first two or three hundred moves of the Ant, starting on a completely white grid, it creates tiny little patterns which are very simple and often very symmetric. And you sit there thinking 'Of course, we've got a simple rule, so that will give simple patterns, and we ought to be able to describe everything that happens in a simple way.'

  CHAOS: Then, suddenly, you notice it's not like that any more. You've got a big irregular patch of black and white squares, and the Ant is wandering around in some sort of random walk, and you can't see any structure at all. For Langton's Ant this kind of pseudo-random motion happens for about the next 10,000 steps. So if your computer is not very fast you can sit there for a long time saying 'Nothing interesting is going to happen, it's going to go on like this forever, it's just random.' No, it's obeying the same rule as before. It's just that to us it looks random.

  EMERGENT ORDER: Finally the Ant locks into a particular kind of repetitive behaviour, and it builds a 'highway'. It goes through a cycle of 104 steps, after which it has moved out two squares diagonally and the shape and the colours along the edge are the same as they were at the beginning of that cycle. So that cycle repeats forever, and the Ant just builds a diagonal highway — forever.

  Those three modes of activity are all consequences of the same rule, but they are on different levels from the rule itself. There are no rules that talk about highways. The highway is clearly a simple thing, but a 104-step cycle isn't a terribly obvious consequence of the rule. In fact the only way mathematicians can prove that the Ant really does build its highway is to track through those 10,000 steps. At that point you could say 'Now we understand why Langton's Ant builds a highway.' But no sooner.

  However, if we ask a slightly more general question, we realize that we don't understand Langton's Ant at all. Suppose that before the Ant starts we give it an environment — we paint a few squares black. Now let's ask a simple question: does the Ant always end up building a highway? Nobody knows. All of the experiments on computers suggest that it does. On the other hand, nobody can prove that it does. There might be some very strange configuration of squares, and when you start it off on that it gets triggered into some totally different behaviour. Or it could just be a much bigger highway. Perhaps there is a cycle of 1,349,772,115,998 steps that builds a different kind of highway, if only you start from the right thing. We don't know. So for this very simple mathematical system, with one simple rule, and a very simple question, where we know the Theory of Everything ... it doesn't tell us the answer.

  Langton's Ant will be our icon for a very important idea: emergence, Simple rules may lead to large, complex patterns. The issue here is not what the universe 'really does'. It is how we understand things and how we structure them in our minds. The simple Ant and its tiled universe are technically a 'complex system' (it consists of a large number of entities that interact with each other, even though most of those entities are simply squares that change colour when an Ant walks on them).

  We can create a system, and give it simple rules which 'common sense' suggests should lead to a rather dull future, and we will often find that quite complex features will result. And they will be 'emergent' — that is, we have no practical way of working out what they are going to be apart from ... well, watching. The Ant must dance. There are no short cuts.

  Emergent phenomena, which you can't predict ahead of time, are just as causal as the non-emergent ones: they are logical consequences of the rules. And you have no idea what they are going to be. A computer will not help — all it will do is run the Ant very fast.

  A 'geographical' image is useful here. The 'phase space' of a system is the space of all possible states or behaviours — all of the things that the system could do, not just what it does do. The phase space of Langton's Ant consists of all possible ways to put black and white squares on a grid — not just the ones that the Ant puts there when it follows its rules. The phase space for evolution is all conceivable organisms, not just the ones that have existed so far. Discworld is one 'point' in the phase space of consistent universes. Phase spaces deal with everything that might be, not what is.

  In this imagery, the features of a system are structures in phase space that give it a well-defined 'geography'. The phase space of an emergent system is indescribably complicated: a generic term for such phase spaces is 'Ant Country', which you can think of as a computational form of infinite suburbia. To understand an emergent feature you would have to find it without traversing Ant Country step by step. The same problem arises when you try to start from a Theory of
Everything and work out what it implies. You may have pinned down the micro-rules, but that doesn't mean that you understand their macro-consequences. A Theory of Everything would tell you what the problem is, in precise language, but that might not help you solve it.

  Suppose, for instance, that we had very accurate rules for fundamental particles, rules that really do govern everything about them. Despite that, it's pretty clear that those rules would not greatly help our understanding of something like economics. We want to understand someone who goes into a supermarket, buys some bananas, and pays over some money. How do we approach that from the particle rules? We have to write down an equation for every particle in the customer's body, in the bananas, in the note that passes from customer to cashier. Our description of the transaction — money for bananas — and our explanation of it is in terms of an incredibly complicated equation about fundamental particles.

  Solving that equation is even harder. And it might not even be the only fruit they buy.

  We're not saying that the universe hasn't done it that way. We're saying that even if it has, that won't help us understand anything. So there's a big, emergent gap between the Theory of Everything and its consequences.

  A lot of philosophers seem to have got the idea that in an emergent phenomenon the chain of causality is broken. If our thoughts are emergent properties of our brain, then to many philosophers they are not physically caused by the nerve cells, the electrical currents, and the chemicals in the brain. We don't mean that. We think it's confused nonsense. We're perfectly happy that our thoughts are caused by those physical entities, but you can't describe someone's perceptions or memory in terms of electrical currents and chemicals.

  Human beings never understand things that way. They understand things by keeping them simple — in Archchancellor Ridcully's case, the simpler the better. A little narrativium goes a long way: the simpler the story, the better you understand it. Storytelling is the opposite of reductionism; 26 letters and some rules of grammar are no story at all.

  One set of modern physical rules poses more philosophical questions than all the others combined: Quantum Mechanics. Newton's rules explained the universe in terms of force, position, speed, and the like — things that make intuitive sense to human beings and let us tell good stories. A century or so ago, however, it became clear that the universe's hidden wiring has other, less intuitive layers. Concepts such as position and speed not only ceased to be fundamental — they ceased to have a well defined meaning at all.

  This new layer of explanation, quantum theory, tells us that on small scales the rules are random. Instead of something happening or not, it may do a bit of both. Empty space is a seething mass of potentialities, and time is something you can borrow and pay back again if you do it quickly enough for the universe not to notice. And the Heisenberg Uncertainty Principle says that if you know where something is then you can't also know how fast it's going. Ponder Stibbons would consider himself lucky if he did not have to explain this to his Archchancellor.

  A thorough discussion of the quantum world would need a book all to itself, but there's one topic that benefits from some Discworld insights. This is the notorious case of the cat in the box. Quantum objects obey Schrödinger's Equation, a rule named after Erwin Schrödinger which describes how 'wave functions' — waves of quantum existence, propagate through space and time. Atoms and their sub-atomic components aren't really particles: they're quantum wave functions.

  The early pioneers of quantum mechanics had enough problems solving Schrödinger's equation: they didn't want to worry about what it meant. So they spatchcocked together a cop-out clause, the 'Copenhagen interpretation' of quantum observations. This says that whenever you try to observe a quantum wave function it immediately 'collapses' to give a single particle-like answer. This seems to promote the human mind to a special status, it has even been suggested that our purpose in the universe is to observe it — thereby ensuring its existence, an idea that the wizards of UU consider to be simple common sense.

  Schrödinger, however, thought this was silly, and in support he introduced a thought experiment now called Schrödinger's Cat. Imagine a box, with a lid that can be sealed so tightly that nothing, not even the barest hint of a quantum wavelet, can leak out. The box contains a radioactive atom, which at some random moment will decay and emit a particle, and a particle detector that releases poison gas when it detects the atom decaying. Put the cat in the box and close the lid. Wait a bit.

  Is the cat alive or dead?

  If the atom has decayed, then the cat's dead. If not, it's alive. However, the box is sealed, so you can't observe what's inside. Since unobserved quantum systems are waves, the quantum rules tell us that the atom must be in a 'mixed' state — half decayed and half not. Therefore the cat, which is a collection of atoms and so can be considered as a gigantic quantum system, is also in a mixed state: half alive, half dead. In 1935 Schrödinger pointed out that cats aren't like that. Cats are macroscopic systems with classical yes/no physics. His point was that the Copenhagen interpretation does not explain — or even address — the link from microscopic quantum physics to macroscopic classical physics. The Copenhagen interpretation replaces a complex physical process (which we don't understand) by a piece of magic: the wave collapses as soon as you try to observe it.

  Most of the time this problem is discussed, physicists manage to turn Schrödinger's point on its head. 'No, quantum waves really are like that!' And they've done lots of experiments to prove they're right. Except... those experiments have no box, no poison gas, no alive, no dead, and no cat. What they have is quantum-scale analogues — an electron for a cat, positive spin for alive and negative for dead, and a box with Chinese walls, through which anything can be observed, but you take great care not to notice.

  These discussions and experiments are lies-to-children: their aim is to convince the next generation of physicists that quantum-level systems do actually behave in the bizarre way that they do. Fine ... but it's got nothing to do with cats. The wizards of Unseen University, who know nothing about electrons but have an intimate familiarity with cats, wouldn't be fooled for an instant. Neither would the witch Gytha Ogg, whose cat Greebo is shut in a box in Lords and Ladies. Greebo is the sort of cat that would take on a ferocious wolf and eat it.* In Witches Abroad he eats a vampire by accident, and the witches can't understand why the local villagers are so ecstatic.

  Greebo has his own way of handling quantum paradoxes: 'Greebo had spent an irritating two minutes in that box. Technically, a cat locked in a box may be alive or it may be dead. You never know until you look. In fact, the mere act of opening the box will determine the state of the cat, although in this case there were three determinate states the cat could be in: these being Alive, Dead, and Bloody Furious.'

  Schrödinger would have applauded. He wasn't talking about quantum states: he wanted to know how they led to ordinary, classical physics in the large, and he could see that the Copenhagen interpretation didn't have anything to say about that. So how do classical yes/no answers emerge from quantum Ant Country? The closest we have to an answer is something called 'decoherence', which has been studied by a number of physicists, among them Anthony Leggett, Roland Omnés, Serge Haroche and Luis Davidovich. If you have a big collection of quantum waves and you leave it to its own devices, then the component waves get out of step and fuzz out. This is what a classical object is 'really' like from the quantum standpoint, and it means that cats do, in fact, behave like cats. Experiments show that the same is true even when the role of the detector is played by a microscopic quantum object: a photon's wave function can collapse without any observers being aware, at the time, that it has done so. Even with a quantum cat, death occurs at the instant that the detector notices that the atom has decayed. It doesn't require a mind.

  In short, Archchancellor, the universe always notices the cat. And a tree in a forest does make a sound when it falls, even if no one is around. The forest is always there.

>   This rule does require some special assumptions, such as the chronic and irreversible stupidity of humanity.

  As Nanny Ogg always says, 'He’s just a big softy. '

  THIRTEEN

  NO, IT CAN'T DO THAT

  ARCHCHANCELLOR RIDCULLY LOOKED AROUND at his colleagues. They'd chosen the long table in the Great Hall for the meeting, since the HEM was getting too crowded.

  'All here? Good,' he said. 'Carry on, Mister Stibbons.'

  Ponder sifted through his papers.

  'I've, er, asked for this meeting,' he said, 'because I'm afraid we're doing things wrong.'

  'How can that be?' said the Dean. 'It's our universe!'

  'Yes, Dean. And, er, no. It's made up its own rules.'

  'No, no, it can't do that,' said the Archchancellor. 'We're intelligent creatures. We make the rules. Lumps of rock don't make rules,'

  'Not exactly, sir,' said Ponder, employing the phrase in its traditional sense of 'absolutely wrong'. 'There are some rules in the Project.'

  'How? Is someone else meddling with it?' the Dean demanded. 'Has a Creator turned up?'

  'An interesting thought, sir. I'm not qualified to answer that one. The point I'm trying to make is that if we want to do anything constructive, we've got to obey the rules.'

  The Lecturer in Recent Runes looked down at the table in front of him. It had been laid for lunch.

  'I don't see why,' he said. 'This knife and fork don't tell me how to eat.'

  'Er ... in fact, sir, they do. In a roundabout way.'

  'Are you trying to tell us that the rules are built in?' said Ridcully.