That was the voice of the Senior Wrangler. It sounded more petulant than usual.
Ponder went to see what was going on.
TWELVE
WHERE DO RULES COME FROM?
SOMETHING IS MAKING Roundworld do strange things …
It seems to be obeying rules.
Or maybe it’s just making them up as it goes along.
Isaac Newton taught us that our universe runs on rules, and they are mathematical. In his day they were called ‘laws of nature’, but ‘law’ is too strong a word, too final, too arrogant. But it does seem that there are more or less deep patterns in how the universe works. Human beings can formulate those patterns as mathematical rules, and use the resulting descriptions to work out some aspects of nature that would otherwise be totally mysterious, and even exploit them to make tools, vehicles, technology.
Thomas Malthus changed a lot of people’s minds when he found a mathematical rule for social behaviour. He said that food grows arithmetically (1-2-3-4-5), but populations grow geometrically (1-2-4-8-16). Whatever the growth rates, eventually population will outstrip food supply: there are limits to growth.1 Malthus’s law shows that there are rules Down Here as well as Up There, and it tells us that poverty is not the result of evil or sin. Rules can have deep implications.
What are rules? Do they tell us how the universe ‘really’ works, or do our pattern-seeking brains invent or select them?
There are two main viewpoints here. One is fundamentalist at heart, as fundamentalist as the Taliban and Southern Baptists – indeed, as fundamentalist as the exquisitor Vorbis in Small Gods who states his position thus: ‘… that which appears to our senses is not the fundamental truth. Things that are seen and heard and done by the flesh are mere shadows of a deeper reality.’
Scientific fundamentalism holds that there is one set of rules, the Theory of Everything, which doesn’t just describe nature rather well, but is nature. For about three centuries science seems to have been converging on just such a system: the deeper our theories of nature become, the simpler they become too. The philosophy behind this view is known as reductionism, and it proceeds by taking things to bits, seeing what the bits are and how they fit together, and using the bits to explain the whole. It’s a very effective research strategy, and it’s served us well for a long time. We’ve now managed to reduce our deepest theories to just two: quantum mechanics and relativity.
Quantum mechanics set out to describe the universe on very small scales, subatomic scales, but then became involved in the largest scales of all, the origin of the universe in the Big Bang. Relativity set out to describe the universe on very large scales, supergalactic ones, but then became involved in the smallest scales of all, the quantum effects of gravity. Despite this, the two theories disagree in fundamental ways about the nature of the universe and what rules it obeys. The Theory of Everything, it is hoped, will subtly modify both theories in such a way that they fit seamlessly together into a unified whole, while continuing to work well in their respective domains. With everything reduced to one Ultimate Rule, reductionism will have reached the end of its quest, and the universe will be completely explained.
The extreme version of the alternative view is that there are no ultimate rules, indeed that there are no totally accurate rules either. What we call laws of nature are human approximations to regularities that crop up in certain specialized regions of the universe – chemical molecules, galaxy dynamics, whatever. There is no reason why our formulations of regularities in molecules and regularities in galaxies should be part of some deeper set of regularities that explains both, any more than chess and soccer should somehow be aspects of the same greater game. The universe could perfectly well be patterned on all levels, without there being an ultimate pattern from which all the others must logically follow. In this view, each set of rules is accompanied by a statement of which areas it can safely be used to describe – ‘use these rules for molecules with fewer than a hundred atoms’ or ‘this rule works for galaxies provided you don’t ask about the stars that make them up’. Many such rules are contextual rather than reductionist: they explain why things work the way they do in terms of what is outside them.
Evolution, especially before it was interpreted through the eyes of DNA, is one of the clearest examples of this style of reasoning. Animals evolve because of the environment in which they live, including other animals. A curious feature of this viewpoint is that to a great extent the system builds its own rules, as well as obeying them. It is rather like a game of chess played with tiles that can be used to build new bits of board, upon which new kinds of chess piece can move in new ways.
Could the entire universe sometimes build its own rules as it proceeds? We’ve suggested as much a couple of times: here’s a sense in which it might happen. It’s hard to see how rules for matter could meaningfully ‘exist’ when there is no matter, only radiation – as there was at an early stage of the Big Bang. Fundamentalists would maintain that the rules for matter were always implicit in the Theory of Everything, and became explicit when matter appeared. We wonder whether the same ‘phase transition’ that created matter might also have created its rules. Physics might not be like that, but biology surely is. Before organisms appeared, there couldn’t have been any rules for evolution.
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 s
mall 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 pseudorandom 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 – for ever.
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.
The Science of Discworld Revised Edition Page 11