Terry Pratchett - The Science of Discworld II - The Globe

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 'quantum wave-function' of every single subatomic particle in the cat. Having done this, you apply a mathematical rule known as Schrodinger's equation, which physicists tell us will predict the future state of the cat [70] Schrodinger pointed out that quantum mechanics often gives silly answers like 'the cat is half alive and half dead'. His intention was to dramatise the gap between a quantum-level description of reality and the world we actually live in, but most physicists missed the point and derived complicated explanations of why cats really are like that. And why the universe needs conscious observers to ensure that it continues to exist. Only recently did they twig what Schrodinger was on about, and come up with the concept of 'decoherence', which shows that superpositions of quantum states rapidly change into single states unless they are protected from interaction with the surrounding environment. And the universe doesn't need us to make it hold together, sorry. See The Science of Discworld, with a cameo appearance of Nanny Ogg's cat Greebo. .

However, no sensible physicist would attempt any such thing, because the wavefunction is far too complicated. The number of subatomic particles in a cat is enormous; even if you could measure their states precisely -which of course you can't do anyway -the universe does not contain a sheet of paper big enough to list all the numbers. So the calculation can't even get started, because in practical terms the present state of the cat is indescribable in the language of quantum wavefunctions. As for plugging the wavefunction into Schrodinger's equation, well, forget it.

Agreed, this is not a sensible way to model the behaviour of a cat. But it does make it clear that the usual physicists' rhetoric about quantum mechanics being 'fundamental' is at best true in a philosophical sense. It's not fundamental to our understanding of the cat, although it might be fundamental to the cat.

Despite these difficulties, cats generally manage to behave like cats, and in particular they discover their own futures by living them. Down on the philosophical level, again, this may be because the universe is a lot better at solving Schrodinger's equation than we are, and because it doesn't need a description of the quantum wavefunction of the cat: it's already got the cat, which is its own quantum wavefunction from this point of view.

Let's accept that, even though it's rather likely that the universe doesn't propagate a cat into its future by applying anything that corresponds to Schrodinger's equation. The equation is a human model, not the reality. But even if Schrodinger's equation is what the universe 'really' does -and more so if it's not -there's no way that we limited humans can follow the 'calculation' step by step. There are too many steps. What interests us about cats occurs on the system level: things like purring, catching mice, drinking milk, getting stuck in the catflap. Schrodinger's equation doesn't help us understand those phenomena.

When the logical chain that leads from an entity-level description of a complex system to system-level behaviour is far too complicated for any human being to follow it, that behaviour is said to be an emergent property of the complex system, or just to be 'emergent'. A cat drinking milk is an emergent property of Schrodinger's equation applied to the subatomic particles that make up the cat. And the milk, and the saucer ... and the kitchen floor, and ...