The Science of Discworld Revised Edition

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:
    • S IMPLICITY : 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.’
    • C HAOS : 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.
    • E MERGENT O RDER : 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