To simplify the description, imagine someone playing solo snakes and ladders, so that there is only one counter on the board. Then at any instant, the `state' of the game is determined by a single square: whichever one is currently occupied by the counter. In this analogy, the board itself becomes the phase space, our analogue of Platonia. The counter represents the entire universe. As the counter hops around, according to the rules of the game, the state of the `universe' changes. The path that the counter follows - the list of squares that it successively occupies - is analogous to the world-line of the universe. In this interpretation, time does exist, because each successive move of the counter corresponds to one tick of the cosmic clock.
Quantum snakes and ladders is very different. The board is the same, but now all that matters is the probability with which the counter occupies any given square - not just at one stage of the game, but overall. For instance, the probability of being on the first square, at some stage in the game, is 1, because you always start there. The probability of being on the second square is 1/6, because the only way to get there is to throw a 1 with the dice on your first throw. And so on. Once we have calculated all these probabilities, we can forget about the rules of the game and the concept of a `move'. Now only the probabilities remain. This is the quantum version of the game, and it has no explicit moves, only probabilities. Since there are no moves, there is no notion of the `next' move, and no sensible concept of time.
Our universe, Barbour tells us, is a quantum one, so it is like quantum snakes and ladders, and `time' is a meaningless concept. So why do we naive humans imagine that time flows; that the universe (at least, the bit near us) passes through a linear sequence of changes?
To Barbour, the apparent flow of time is an illusion. He suggests that Platonian configurations which have high probability must contain within them `an appearance of history'. They will look as though they had a past. It's a bit like the philosophers' old chestnut: maybe the universe is being created anew every instant (as in Thief of Time), but at each moment, it is created along with apparent records of a lengthy past history. Such apparently historical clouds in Platonia are called time capsules. Now, among those high-probability configurations we find the arrangement of neurons in a conscious brain. In other words, the universe itself is timeless, but our brains are time capsules, high-probability configurations, and these automatically come along with the illusion that they have had a past history.
It's a neat idea, if you like that sort of thing. But it hinges on Barbour's claim that Platonia must be timeless because `there can only be once-and-for-all probabilities for each possible configuration'. This statement is remarkably reminiscent of one of Xeno's - sorry, Zeno's - paradoxes: the Arrow. Which, you recall, says that at each instant an arrow has a specific location, so it can't be moving. Analogously, Barbour tells us that at each instant (if such a thing could exist) Platonia must have a specific probability mist, and deduces that this mist can't change (so it doesn't).
What we have in mind as an alternative to Barbour's timeless probability mist is not a mist that changes as time passes, however. That would fall foul of the non-Newtonian relation between space and time; different parts of the mist would correspond to different times depending on who observed them. No, we're thinking of the mathematical resolution of the Arrow paradox, via Hamiltonian mechanics. Here, the state of a body is given by two quantities, position and momentum, instead of just position. Momentum is a `hidden variable', observable only through its effect on subsequent positions, whereas position is something we can observe directly. We said: `a body in a given position with zero momentum is not moving at that instant, whereas one in the same position with non-zero momentum is moving, even though instantaneously it stays in the same place'. Momentum encodes the next change of position, and it encodes it now. Its value now is not observable now, but it is (will be) observable. You just have to wait to find out what it was. Momentum is a `hidden variable' that encodes transitions from one position to another.
Can we find an analogue of momentum in quantum snakes and ladders? Yes, we can. It is the overall probability of going from any given square to any other. These `transition probabilities' depend only on the squares concerned, not on the time at which the move is made, so in Barbour's sense they are `timeless'. But when you are on some given square, the transition probabilities tell you where your next move can lead, so you can reconstruct the possible sequences of moves, thereby putting time back into the physics.
For exactly the same reason, a single fixed probability mist is not the only statistical structure with which Platonia can be endowed. Platonia can also be equipped with transition probabilities between pairs of states. The result is to convert Platonia into what statisticians call a `Markov chain', which is just like the list of transition probabilities for snakes and ladders, but more general. If Platonia is made into a Markov chain, each sequence of configurations gets its own probability. The most probable sequences are those that contain large numbers of highly probable states - these look oddly like Barbour's time capsules. So instead of single-state Platonia we get sequentialstate Markovia, where the universe makes transitions through whole sequences of configurations, and the most likely transitions are the ones that provide a coherent history - narrativium.
This Markovian approach offers the prospect of bringing time back into existence in a Platonian universe. In fact, it's very similar to how Susan Sto Helit and Ronnie Soak managed to operate in the cracks between the instants, in Thief of Time.
Tick.
THE FISH IS OFF
Two HOURS LATER A SINGLE sheet of paper slid off Hex's writing table. Ponder picked it up.
`There are about ten points where we must intervene to ensure that The Origin is written,' he said.
'well, that doesn't seem too bad,' said Ridcully. `We got Shakespeare born, didn't we?[1] We just have to tinker.'
`These look a little more complicated,' said Ponder, doubtfully.
`But Hex can move us around,' said Ridcully. `It could be fun, especially if something or someone is playing les buggeurs risibles. It could be educational, Mr Stibbons.'
`And they do really good beer, ` said the Dean. `And the food was excellent. Remember that goose we had last time? I've seldom eaten better.'
`We will be setting out to save the world,' said Ridcully, severely. `We will have other things on our minds!'
`But there will be mealtimes, yes?' said the Dean Second Lunch and Mid-afternoon Snack went past almost unnoticed. Perhaps the wizards were already leaving space for goose ...
[1] Yes, they did - in The Science of Discworld II.
It was turning out to be a long day. Easels had been set up around Hex. Paper was strewn across every table. The Librarian had practically built up a branch library in one corner, and was still fetching books from the distant reaches of L-Space.
And the wizards had changed their clothes, ready for hands-on intervention. There had barely been a discussion about it, not after the Dean had mentioned the goose. Hex had a great deal of control over the Globe, but when it came to the fine detail you needed to be hands-on, especially hands on cutlery. Hex had no hands. Besides, he'd explained at length, there was no such thing as absolute control, not in a fully functioning universe. There was just a variable amount of lack of control. In fact, Ponder thought, Hex was a Great Big Thing as far as Roundworld was concerned. Almost ... godlike. But he still couldn't control everything. Even if you knew where every tiny spinning particle of stuff was, you couldn't know what it'd do next.
The wizards would have to go in. They could do that. They'd done it before. No trouble is too much if it saves some excellent chefs from extinction.