The Science of Discworld II - The Globe tsod-2

by Terry Pratchett

  'We'll find them and bring them back!' said Ponder. 'How hard can it be?'

  'It can be incredibly hard! There's elves there! You know elves! They are dangerous! Drop your guard for a moment and they can control your mind!'

  'They chased me through some woods once,' said Ponder. 'They are very frightening. I remember writing that down in my diary.'

  'You wrote down in your diary that you were scared?'

  Yes. Why not? Don't you?'

  'I haven't got a big enough diary. But it makes no sense! There's nothing on the Roundworld that elves would be interested in! They like to have ... slaves. And we've never seen anything evolve that's bright enough to be a slave.'

  'You might have missed something,' said Ponder.

  'No, I say you, you say we,' said Rincewind.

  They both stared at the globe.

  'Look, it's like having a pot plant,' said Ponder. 'If it has greenfly, you try to squash them.'

  'I never do that,' said Rincewind. 'Greenfly may be small, but there's a lot of them ...'

  'It was a metaphor, Rincewind,' said Ponder, wearily.

  '... I mean, supposing they decide to gang up?'

  'Rincewind, you are the only other person here who knows anything at all about Roundworld.

  You will come with us or ... or ... I'll tell the Archchancellor about the seven buckets.'

  'How do you know about the seven buckets?'

  'And I'll explain to him how all of your jobs could easily be done by a simple set of instructions for Hex, too. It'd take me about, oh, thirty seconds. Let's see ...

  # Rincewind SUB WAIT

  WAIT

  RETURN

  Or possibly RUN RINCEWIND'

  'You wouldn't do that!' said Rincewind. 'Would you?'

  'Certainly. Now, are you coming? Oh, and bring the Luggage.'

  Knowledge = power = energy = matter = mass, and on that simple equation rests the whole of L- space. It is via L-space that all books are connected (quoting the ones before them, and influencing the ones that come after). But there is no time in L-space. Nor is there, strictly speaking, any space. Nevertheless, L-space is infinitely large and connects all libraries, everywhere and everywhen. It's never further than the other side of the bookshelf, yet only the most senior and respected librarians know the way in.

  From inside, L-space looked to Rincewind like a library designed by someone who did not have to worry about time, budget, strength of materials or physics. There are some laws, though, that are coded into the very nature of the universe, and one is: There Is Never Enough Shelf Space[11].

  He turned and looked back. They'd entered L-space by walking through what had looked like a solid wall of books. He knew it was a solid wall, he'd taken books off those shelves before now.

  You had to be a very senior Librarian indeed to know in what precise circumstances you could step straight through it.

  He could still see the library through the gap, but it faded from view as he watched. What remained was books. Mountains of books. Hills and valleys of books. Perilous precipices of books. Even in what passed for the sky, which was a sort of blue grey, there was a distant suggestion of books. There is never enough shelf space, anywhere.

  Ponder was carrying a considerable amount of magical equipment. Rincewind, being a more experienced traveller, was carrying as little weight as possible. Everything else was being carried by the Luggage, which looked like a sea chest but with a number of pink, human-like and fully operational feet.

  'Under the rules of the Roundworld, magic can't work,' said Ponder, as they followed the Librarian. 'Won't the Luggage stop existing?'

  'It's worth a try,' said Rincewind, who felt that owning a semi-sapient and occasionally homicidal box on legs reduced his opportunities to make live friends, 'but it doesn't usually worry about rules. They bend round it. Anyway, it's already been there before, for a very long time, without any damage. To the Luggage, anyway.'

  The walls of books shifted as the wizards approached; in fact, each step radically changed the nature of the bookscape which was in any case, said Ponder, a mere metaphorical depiction created by their brains to allow them to deal with the unimaginable reality. The shifting perspective would have given most people a serious headache at least, but Unseen University had rooms where the gravity moved around during the course of a day, one corridor of infinite length and several windows that only existed on one side of their walls. Life at UU reduced your capacity for surprise by quite a lot.

  Occasionally the Librarian would stop, and sniff at the books nearest to him. At last he said 'ook', quietly, and pointed to another stack of books. There were, drawn gently on the spine of an old leather-bound volume, some chalk marks.

  'Librarian-sign,' said Rincewind. 'He's been here before. We're close to Roundworld book-space.'

  'How could he—' Ponder began, and then said: 'Oh. I see. Er ... Roundworld exists in L-space even before we created it? I mean yes, obviously I know that's true, but even so—'

  Rincewind took a book from a pile near him. The cover was brightly coloured and made of paper, suggesting an absence of cows on the originating world, and had the title: Sleep Well My Lovely Falcon. The words inside made even less sense.

  'It might not have been worth our trouble,' he said.

  The Librarian said 'ook', which Rincewind understood as 'I'm going to get into real trouble with the Secret Masters of the Library for this day's work'.

  Then the ape appeared to triangulate on the bookscape around him and knuckled forward, and vanished.

  Ponder looked at Rincewind. 'Did you see how he did that?' he said and then a hairy red arm appeared out of the air and jerked him off his feet. A moment later the same thing happened to Rincewind.

  It wasn't much of a library, but Rincewind knew how this worked. Two books were a library - for a lot of people, two books were an enormous library. But even one book could be a library, if it was a book that made a big enough dimple in L-space. A book with a title like 100 Ways with Broccoli was unlikely to be one such, whereas A Relationship Between Capital and labour might be, especially if it has an appendix on making explosives. The deeply magical and interminably ancient volumes in the Library of UU strained the fabric L-space like a baby elephant on a worn- out trampoline, leaving it so thin that the Library was a potent and easy portal.

  Sometimes, though, even one book could do that. Even one line. Even one word, in the right place and the right time.

  The room was large, panelled and sparsely furnished. Quite a lot of paperwork was strewn on a desk. Quill pens lay by an inkwell. A window looked out on to broad gardens, where it was raining. A skull lent a homely touch.

  Rincewind leaned down and tapped it.

  'Hello?' he said. He looked up at the others.

  'Well, the one in the Dean's office can sing comic songs,' he said defensively. He stared at the paperwork on the desk. It was covered in symbols which had a magical look, although he didn't recognise any of them. On the other side of the room, the Librarian was leafing through one of the books. Strangely, they weren't on shelves. Some were neatly piled, others locked in boxes, or at least in boxes that were locked until the Librarian tried to lift the lid.

  Occasionally he pursed his lips and blew a disdainful raspberry.

  'Ook,' he muttered.

  'Alchemy?' said Rincewind. 'Oh dear. That stuff never works.' He lifted up what looked like a small leather hatbox, and removed the lid. 'This is more like it!' he said, and pulled out a ball of smoky quartz. 'Our man is definitely a wizard!'

  'This is very bad,' said Ponder, staring at a device in his hand. "Very, very bad indeed.'

  'What is?' said Rincewind, turning around quickly.

  'I'm reading a very high glamour quotient,' said Ponder.

  'There's elves here?'

  'Here? The place is practically elvish!' said Ponder. 'The Archchancellor was right.'

  All three explorers stood quietly. The Librarian's nostrils flared. Rincew
ind sniffed, very cautiously.

  'Seems okay to me,' he said, at last.

  Then a man in black entered the room. He came in quickly, opening the door no more than necessary, in a kind of aggressive sidle, and stopped in astonishment. Then his hand flew to his belt and he drew a thin, businesslike sword.

  He saw the Librarian. He stopped. And then it was really all over, because the Librarian could unfold his arm very fast and, importantly, there was a fist like a sledgehammer on the end of it.

  As the dark figure slid down the wall, the crystal sphere in Rincewind's hand said: 'I believe I now have enough information. I advise departure from this place at a convenient opportunity and in any case before this gentleman awakes.' Hex?' said Ponder.

  'Yes. Let me repeat my advice. Lack of absence from this place will undoubtedly result in metal entering the body.'

  But you're talking via a crystal ball! Magic doesn't work here!'

  'Don't argue with a voice saying "run away"!' said Rincewind. 'That's good advice! You don't question it! Let's get out of here!'

  He looked at the Librarian, who was sniffing along the bookshelves with a puzzled expression.

  Rincewind had a sense for the universe's tendency to go wrong. He didn't leap to conclusions, he plunged headlong towards them.

  'You've brought us out through a one-way door, haven't you ...' he said.

  'Oook!'

  'Well, how long will it take to find the way in?'

  The Librarian shrugged and returned his attention to the shelves.

  'Leave now,' said the crystal Hex. 'Return later. The owner of the house will be useful. But leave before Sir Francis Walsingham wakes up, because otherwise he will kill you. Steal his purse from him first. You will need money. For one thing, you will need to pay someone to give the Librarian a shave.'

  'Oook?'

  4. THE ADJACENT POSSIBLE

  The concept of L-space, short for 'Library-space', occurs in several of the Discworld novels. An early example occurs in Lords and Ladies, a story that is mostly about elvish evil. We are told that Ponder Stibbons is Reader in Invisible Writings, and this phrase deserves (and gets) an explanation: The study of invisible writings was a new discipline made available by the discovery of the bidirectional nature of Library-space. The thaumic mathematics are complex, but boil down to the fact that all books, everywhere, affect all other books. This is obvious: books inspire other books written in the future, and cite books written in the past. But the General Theory[12] of L-space suggests that, in that case, the contents of books as yet unwritten can be deduced from books now in existence.

  L-space is a typical example of the Discworld habit of taking a metaphorical concept and making it real. The concept here is known as 'phase space', and it was introduced by the French mathematician Henri Poincare about a hundred years ago to open up the possibility of applying geometrical reasoning to dynamics. Poincare's metaphor has now invaded the whole of science, if not beyond, and we will make good use of it in our discussion of the role of narrativium in evolution of the mind.

  Poincare was the archetypal absent-minded academic -no, come to think of it he was 'presentminded somewhere else', namely in his mathematics, and it's easy to understand why. He was probably the most naturally gifted mathematician of the nineteenth century. If you had a mind like his, you'd spend most of your time somewhere else too, revelling in the beauty of the mathiverse.

  Poincare ranged over almost all of mathematics, and he wrote several best-selling popular science books, too. In one piece of research which single-handedly created a new 'qualitative'

  way of thinking about dynamics, he pointed out that when you are studying some physical system that can exist in a variety of different states, then it may be a good idea to consider the states that it could be in, but isn’t as well as the particular state in which it is. By doing that, you set up a context that lets you understand what the system is doing, and why. This context is the

  'phase space' of the system. Each possible state can be thought of as a point in that phase space.

  As time passes, the state changes, so this representative point traces out a curve, the trajectory of the system. The rule that determines the successive steps in the trajectory is the dynamic of the system. In most areas of physics, the dynamic is completely determined, once and for all, but we can extend, this terminology to cases where the rule involves possible choices. A good example is a game. Now the phase space is the space of possible positions, the dynamic is the rules of the game and a trajectory is a legal sequence of moves by the players.

  The formal setting and terminology for phase spaces is not as important, for us, as the viewpoint that they encourage. For example, you might wonder why the surface of a pool of water, in the absence of' wind or other disturbances, is flat. It just sits there, flat; it isn't even doing anything.

  But you start to make progress immediately if you ask the question 'what would happen if it wasn't flat?' For instance, why can't the water be piled up into a hump in the middle of the pond?

  Well, imagine that it was. Imagine that you can control the position of every molecule of water, and that you pile it up in this way, miraculously keeping every molecule just where you've placed it. Then, you let go. What would happen? The heap of water would collapse, and waves would slosh across the pool until everything settled down to that nice, flat surface that we've learned to expect. Again, suppose you arranged the water so that there was a big dip in the middle. Then as soon as you let go, water would move in from the sides to fill the dip.

  Mathematically, this idea can be formalised in terms of the space of all possible shapes for the water's surface. 'Possible' here doesn't mean physically possible: the only shape you'll ever see in the real world, barring disturbances, is a flat surface. 'Possible' means 'conceptually possible'. So we can set up this space of all possible shapes for the surface as a simple mathematical construct, and this is the phase space for the problem. Each 'point' -location -in phase space represents a conceivable shape for the surface. Just one of those points, one state, represents 'flat'.

  Having defined the appropriate phase space, the next step is to understand the dynamic: the way that the natural flow of water under gravity affects the possible surfaces of the pool. In this case, there is a simple principle that solves the whole problem: the idea that water flows so as to make its total energy as small as possible. If you put the water into some particular state, like that piled-up hump, and then let go, the surface will follow the 'energy gradient' downhill, until it finds the lowest possible energy. Then (after some sloshing around which slowly subsides because of friction) it will remain at rest in this lowest-energy state.

  The energy in this problem is 'potential energy', determined by gravity. The potential energy of a mass of water is equal to its height above some arbitrary reference level, multiplied by the mass concerned. Suppose that the water is not flat. Then some parts are higher up than others. So we can transfer some water from the high level to the lower one, by flattening a hump and filling a dip. When we do that, the water involved moves downwards, so the total energy decreases.

  Conclusion: if the surface is not flat, then the energy is not as small as possible. Or, to put it the other way round: the minimum energy configuration occurs when the surface is flat.

  The shape of a soap bubble is another example. Why is it round? The way to answer that question is to compare the actual round shape with a hypothetical non-round shape. What's different? Yes, the alternative isn't round, but is there some less obvious difference? According to Greek legend, Dido was offered as much land (in northern Africa) as she could enclose with a bull's hide. She cut it into a very long thin strip and enclosed a circle. There she founded the city of Carthage. Why did she choose a circle? Because the circle is the shape with greatest area, for a given perimeter. In the same way, a sphere is the shape with greatest volume, for a given surface area; or, to put it another way, it is the shape with the smallest surface ar
ea that contains a given volume. A soap bubble contains a fixed volume of air and its surface area gives the energy of the soap film due to surface tension. In the space of all possible shapes for bubbles, the one with the least energy is a sphere. All other shapes have larger energy, and are therefore ruled out.

  You may not feel that bubbles are important. But the same principle explains why Roundworld

  (the planet not the universe, but maybe that, too) is round. When it was molten rock, it settled into a spherical shape, because that had the least energy. For the same reason, the heavy materials like iron sank into the core, and the lighter ones, like continents and air, floated up to the top. Actually, Roundworld isn't exactly a sphere, because it rotates, so centrifugal forces cause it to bulge at the equator. But the amount of bulge is only one-third of one per cent. And that bulging shape is the minimum-energy configuration for a mass of liquid spinning at the same speed as the Earth's rotation when it was just starting to solidify.

  The physics here isn't important for the message of this book. What is important is the 'Worlds of If' point of view involved in the application of phase spaces. When we discussed the shape of water in a pond, we pretty much ignored the flat surface, the thing we were trying to explain. The entire argument hinged upon non-flat surfaces, humps and dips, and hypothetical transfers of water from one to the other. Almost all of the explanation involved thinking about things that don't actually happen. Only at the end, having ruled out all non-flat surfaces, did we observe that the only possibility left was therefore what the water would actually do. The same goes for the bubble.

  At first sight, this might seem to be a very oblique way of doing physics. It takes the stance that the way to understand the real world is to ignore it, and focus instead on all the possible alternative unreal worlds. Then we find some principle (in this case, minimum energy) to rule out nearly all of the unreal worlds, and see what's left. Wouldn't it be easier to start with the real world, and focus solely on that? No, it wouldn't. As we've just seen, the real world alone is too limited to offer a convincing explanation. What you get from the real world alone is 'the world is like it is, and there's nothing more to be said'. However, if you take the imaginative leap of considering unreal worlds, too, you can compare the real world with all of those unreal worlds, and maybe find a principle that picks out the real one from all the others. Then you have answered the question 'Why is the world the way it is, rather than something else?'