'That simple, eh?'

'And now I shall test it,' said Wen. He moved it a little less this time.

That simple, eh?'

'And now I shall test it,' said Wen. This time he twisted it gently to and fro.

That si-si-si That simple-pie, eh eheh simple, eh?' said Clodpool.

'And I have tested it,' said Wen.

On Roundworld we don't have history monks -or, at least, we've never caught anyone playing that role, but could we ever do so? -but we do have a kind of historical narrativium. We have a saying that 'history repeats itself -the first time as comedy, the second time as tragedy', because the one thing we learn from history is that we never learn from history.

Roundworld history is like biological evolution: it obeys rules, but even so, it seems to make itself up as it goes along. In fact, it seems to make up its rules as it goes along. At first sight, that seems incompatible with the existence of a dynamic, because a dynamic is a rule that takes the system from its present state to the next one, a tiny instant into the future. Nonetheless, there must be a dynamic, otherwise historians would not be able to make sense of history, even after the event. Ditto evolutionary biology.

The solution to this conundrum lies in the strange nature of the historical dynamic. It is emergent. Emergence is one of the most important, but also the most puzzling, features of complex systems. And it is important for this book, because it is the existence of emergent dynamics that leads humans to tell stories. Briefly: if the dynamic wasn't emergent, then we wouldn't need to tell stories about the system, because we'd all be able to understand the system on its own terms. But when the dynamic is emergent, a simplified but evocative story is the best description that we can hope to find ...

But now we're getting ahead of our own story, so let's back up a little and explain what we're talking about.

A conventional dynamical system has an explicit, pre-stated phase space. That is, there exists a simple, precise description of everything that the system can possibly do, and in some sense this description is known in advance. In addition, there is a fixed rule, or rules, that takes the current state of the system and transforms it into the next state. For example, if we are trying to understand the solar system, from a classical point of view, then the phase space comprises all possible positions and velocities for the planets, moons, and other bodies, and the rules are a combination of Newton's law of gravity and Newton's laws of motion.

Such a system is deterministic: in principle, the future is entirely determined by the present. The reasoning is straightforward. Start with the present state and work out what it will be one time- step into the future by applying the rules. But we can now consider that state as the new 'present'

state, and apply the rule again to find out what the system will be doing two time-steps into the future. Repeat again, and we know what will happen after three time-steps. Repeat a billion times, and the future is determined for the next billion time-steps.

This mathematical phenomenon led the eighteenth-century mathematician Pierre Simon de Laplace to a vivid image of a 'vast intellect that could predict the entire future of every particle in the universe once it was furnished with an exact description of all those particle at one instant.

Laplace was aware that performing such a computation was far too difficult to be practical, and he was also aware of the difficulty, indeed the impossibility, of observing the state of every particle at the same moment. Despite these problems, his image helped to create an optimistic attitude about the predictability of the universe. Or, more accurately, of small enough bits of it.

And for several centuries, science made huge inroads into making such predictions feasible.

Today, we can predict the motion of the solar system billions of years in advance, and we can even predict the weather (fairly accurately) three whole days in advance, which is amazing.

Seriously. Weather is a lot less predictable than the solar system.

Laplace's hypothetical intellect was lampooned in Douglas Adams's The Hitchhiker's Guide to the Galaxy as Deep Thought, the supercomputer which took five million years to calculate the answer to the great question of life, the universe, and everything. The answer it got was 42.

'Deep Thought' is not so far away from 'Vast Intellect', although the name originates in the pornographic movie Deep Throat, whose title was the cover-name of a clandestine source in the Watergate scandal in which the presidency of Richard Nixon self-destructed (how soon people forget ...).

One reason why Adams was able to poke fun at Laplace's dream is that about forty years ago we learned that predicting the future of the universe, or even a small part of it, requires more than just a vast intellect. It requires absolutely exact initial data, correct to infinitely many decimal places. No error, however minuscule, can be tolerated. None. No marks for trying. Thanks to the phenomenon known as 'chaos', even the smallest error in determining the initial state of the universe can blow up exponentially fast, so that the predicted future quickly becomes wildly inaccurate. In practice, though, measuring anything to an accuracy of more than one part in a trillion, 12 decimal digits, is beyond the abilities of today's science. So, for instance, although we can indeed predict the motion of the solar system billions of years in advance, we can't predict it correctly. In fact, we have very little idea where Pluto will be, a hundred million years from now.

Ten million, on the other hand, is a cinch.

Chaos is just one of the practical reasons why it's generally impossible to predict the future (and get it right). Here we'll examine a rather different one: complexity. Chaos afflicts the prediction method, but complexity afflicts the rules.. Chaos occurs because it is impossible to say in practice what the state of the system is, exactly. In a complex system, it may be impossible to say what the range of possible states of the system is, even approximately. Chaos throws a spanner in the works of the scientific prediction machine, but complexity turns that machine into a small cube of crumpled scrap metal.

We've already discussed the limitations of the Laplacian world-picture in the context of Kauffman's theory of autonomous agents expanding into the space of the adjacent possible. Now we'll take a closer look at how such expansions occur. We'll see that the Laplacian picture still has a role to play, but a less ambitious one.

A complex system consists of a number (usually large) of entities or agents, which interact with each other according to specific rules. This description makes it sound as though a complex system is just a dynamical system whose phase space has a huge number of dimensions, one or more per entity. This is correct, but the word 'just' is misleadingly dismissive. Dynamical systems with big phase spaces can do remarkable things, far more remarkable than what the solar system can do.

The new ingredient in complex systems is that the rules are 'local', stated on the level of the entities. In contrast, the interesting features of the system itself are global, stated on the level of the entire system. Even if we know the local rules for entities, it may not be possible -either in practice, or in principle -to deduce the dynamical rules of the system as a whole. The problem here is that the calculations involved may be intractable, either in the weak sense that they would take far too long to do, or in the strong sense that you can't actually do them at all.

Suppose, for example, that you wanted to use the laws of quantum mechanics to predict the behaviour of a cat. If you take the problem seriously, the way to do this is to write down the