The Science of Discworld II

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 abillion 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 particles 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, isbeyond 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