The Science of Discworld

by Terry Pratchett

  The Lecturer in Recent Runes had designed a creature.

  'Really, all we need is a basic limpet or whelk, to begin with,' he said, as they looked at the blackboard. 'We bring it back here where proper magic works, try a few growth spells, and then let Nature take its course. And, since these extinctions seem to be wiping out everything, it'll gradually become the dominant feature.'

  'What's the scale again?' said Ridcully, critically.

  'About two miles to the tip of the cone,' said the Lecturer. 'About four miles across the base.'

  'Not very mobile, then,' said the Dean.

  'The weight of the shell will certainly hamper it, but I imagine it should be able to move its own length in a year, perhaps two.'

  'What'll it eat, then?'

  'Everything else.'

  'Such as...'

  'Everything. I'd advise suction holes around the base here so that it can filter seawater for useful things like plankton.'

  'Plankton being —?'

  'Oh, whales, shoals of fish and so on.'

  The wizards looked long and hard at the huge cone-shaped object.

  'Intelligence?' said Ridcully.

  'What for?' said the Lecturer in Recent Runes.

  'Ah.'

  'It will withstand anything except a direct hit with a comet, and I estimate it'll have a lifespan of about 500,000 years.'

  'And then it'll die?' said Ridcully.

  'Yes. I estimate it will, by then, take it twenty-four hours and one second to absorb enough food to last it for twenty-four hours.'

  'So after that it will be dead?'

  'Yes.'

  'Will it know?'

  'Probably not.'

  'Back to the drawing board, Senior Lecturer.'

  Ponder sighed.

  'It's no good ducking,' he said. 'That won't help. We're paying special attention to comets. We'll let you know in plenty of time.'

  'You've got no idea what it was like!' said Rincewind, creeping along the beach. 'And the noise!'

  'Have you seen the Luggage?'

  'It certainly made my ears ring, I can tell you!'

  'And the Luggage?'

  'What? Oh ... gone. Have you looked at that side of the planet? There's a whole new set of mountain ranges!'

  The wizards had let time run forward for a while after the strike. It made such a depressing mess of everything. Now, drawing on its bottomless reserves of bloodimindium, life was returning in strength. Crabs were already back although, here, at least, they didn't seem inclined to make even simple structures. Perhaps something in their souls told them it'd be a waste of time in the long run.

  Rincewind mentally crossed them off the list. Look for signs of intelligence, the Archchancellor had said. As far as Rincewind was concerned, anything really intelligent would be keeping out of the way of the wizards. If you saw a wizard looking at you, Rincewind would advise, then you should walk into a tree or say 'dur?'.

  All along the beaches, and out below the surf, everything was acting with commendable stupidity.

  A soft sound made him look down. He'd almost stepped on a fish.

  It was some way from the water line, and squirming across the mud towards a pool of brackish water.

  A kind man by nature, Rincewind picked it up gingerly and carried it back to the sea. It flopped around in the shallows for a while and then, to his amazement, inched its way back on to the mud.

  He put it back again, in deeper water this time.

  Thirty seconds kter, it was back on the beach.

  Rincewind crouched down, as the thing wiggled determinedly onwards.

  'Would it help to talk to someone?' he said. 'I mean, you've got a good life out there in the sea, no sense in throwing it all away, is there? There's always a silver lining if you know where to look. Okay, okay, life's a beach. And you're a pretty ugly fish. But, you know, beauty is only sk— scale deep, and —’

  'What's happening?' said Ponder's voice in his ear.

  'I was talking to this fish,' said Rincewind.

  'Why?'

  'It keeps coming out of the water. It seems to want to go for whatever is the opposite of a paddle.'

  'Well?'

  'You told me to keep a look out for anything interesting.'

  'The consensus here is that fish aren't interesting,' said Ponder. 'Fish are dull.'

  'I can see bigger fish in the shallows,' said Rincewind. Terhaps it's trying to keep away from them?'

  'Rincewind, fish are designed for living in water That's why they're fish. Go and find some crabs. And put the poor freakish thing back in the sea, for goodness' sake.'

  'Perhaps a rethink is in order here,' said Ridcully.

  'About the newts,' said Ponder.

  'Newts is going far too far,' said the Dean. 'I've seen more shapely things in the privy.'

  'I want the person who put the newts on this continent to own up right now,' said Ridcully.

  'No one could,' said the Senior Wrangler. 'No one's seen the Luggage since the last comet. We couldn't get anything in there.'

  'I know, because I had a tank of thaumically treated whelks all ready to go,' said the Lecturer in Recent Runes. 'And what, pray, am I supposed to do with them?'

  'Some sort of chowder would appear to be in order,' said the Dean.

  'Evolution makes things better,' said Ridcully. 'It can't make them different. All right, some rather dull amphibians seem to have turned up. But, and this is important, those fish Rincewind reported are still around. Now, if they were going to turn into things with legs, why are they still here?'

  'Tadpoles are fish,' said the Bursar.

  'But a tadpole knows it's going to be a frog,' said Ridcully patiently. 'There's no narrativium on this world. That fish couldn't be saying to itself "Ah, a new life beckons on dry land, walking around on things I haven't yet got a name for."' No, either the planet is somehow generating new life, or we're back to the old "hidden gods" theory.'

  'It's all gone wrong, you know,' said the Dean. 'It's the bloodi-mindium. Even gods couldn't control this place. Once there's life, there's complete and utter chaos. Remember that book the Librarian brought back? It's a complete fantasy! Nothing seems to happen like that at all! Everything just does what it likes!'

  'Progress is being made,' said Ponder.

  'Big amphibians?' sneered the Senior Wrangler. 'And things were going so well in the sea. Remember those jellyfish that made nets? And the crabs even had a flourishing land civilization! They had practically got a culture!'

  'They ate captured enemies alive,' said the Lecturer in Recent Runes, patiently.

  'Well... yes. But with a certain amount of etiquette, at least,' the Senior Wrangler admitted. 'And in front of their sand statue of the Great Big Crab. They were obviously attempting to control their world. And what good did it do them? A million tons of white hot ice smack between the eyestalks. It's so upsetting.'

  'Perhaps they should have eaten more enemies,' said the Dean.

  'Perhaps sooner or later the planet will get the message,' said Ridcully.

  'Time for the giant whelks, perhaps?' said the Lecturer in Recent Runes, hopefully.

  'Big newts is what we've got right now,' said Ridcully. He glanced at the Dean and Senior Wrangler. Ridcully hadn't maintained his position atop the boiling heap of UU wizardry without a little political savvy. 'And newts, gentlemen, might be the way to go. Amphibians? At home in the water and on land? The best of both worlds, I fancy.'

  The two wizards exchanged sheepish glances.

  'Well ... I suppose ...' said the Senior Wrangler.

  'Could be,' the Dean said grudgingly. 'Could be.'

  'There we are then,' said Ridcully happily. 'The future is newt.'

  THIRTY-FOUR

  NINE TIMES OUT OF TEN

  THERE'S NO NARRATIVIUM ON THIS WORLD.'

  Let's take a step away from the unfolding ancestral tale of The Fish That Came Out From The Sea and look at a more philosophical issue.

  The w
izards are puzzled. On Discworld, things happen because narrative imperative makes them happen. There is no choice about ends, only about means. The Lecturer in Recent Runes is trying to make a sustainable lifeform happen. He thinks that the obstacle to sustainability is the fragility of life — so the only way he can see to achieve this is the two-mile limpet, proof against everything the sky can drop on it.

  It never occurs to him that lifeforms might achieve sustainability by other, less direct methods, despite the evidence of his eyes that suggests that a dogged tenacity appears to allow life to arise in the most inhospitable environment, effectively re-creating itself over and over again. The wizards are torn between evidence that a planet is the last place you'd choose to create life, and evidence that life doesn't agree.

  On Discworld, it is clearly recognized that million-to-one chances happen nine times out of ten.* The reason is that every Discworld character lives out a story, and the demands of the story determine how their lives unfold. If a million to one chance is required to keep that story on track, then that's what will happen, appalling odds notwithstanding. On Discworld, abstractions generally show up as things, so there is even a thing — narrativium — that ensures that everybody obeys the narrative imperative. Another personification of the abstract, Death, also makes sure that each individual's story comes to an end exactly when it's supposed to. Even if a character tries to behave contrary to the story in which they find themselves, narrativium makes sure that the end result is consistent with the story anyway.

  What's puzzling the wizards is that our world isn't like that ...

  Or is it?

  After all, people live on our world too, and it's people that drive stories.

  As case in point, a story about people who drive. The setting is Jerez Grand Prix circuit, last race of the 1997-98 Formula One motor racing season ... Ace driver Michael Schumacher is one Championship point ahead of arch-rival Jacques Villeneuve. Villeneuve's team-mate Heinz-Harold Frentzen may well play a crucial tactical role. The drivers are competing for 'pole position' on the starting grid, which goes to whoever produces the fastest lap in the qualifying sessions. So what happens? Unprecedentedly, Villeneuve, Schumacher, and Frentzen all lap in 1 minute 21.072 seconds, the same time to a thousandth of a second. An amazing coincidence.

  Well: 'coincidence' it surely was — the lap times coincided. But was it truly amazing?

  Questions like this arise in science, too, and they're important. How significant is a statistical cluster of leukaemia cases near a nuclear installation? Does a strong correlation between lung cancer and having a smoker in the family really indicate that secondary smoking is dangerous? Are sexually abnormal fish a sign of oestrogen-like chemicals in our water supply?

  A case in point. It is said that 84% of the children of Israeli fighter pilots are girls. What is it about the life of a fighter pilot that produces such a predominance of daughters? Could an answer lead to a breakthrough in choosing the sex of your children? Or is it just a statistical freak? It's not so easy to decide. Gut feelings are worse than useless, because human beings have a rather poor intuition for random events. Many people believe that lottery numbers that have so far been neglected are more likely to come up in future. But the lottery machine has no 'memory' — its future is independent of its past. Those coloured plastic balls do not know how often they have come up in previous draws, and they have no tendency to compensate for past imbalances.

  Our intuition goes even further astray when it comes to coincidences. You go to the swimming baths, and the guy behind the counter pulls a key at random from a drawer. You arrive in the changing room and are relieved to find that very few lockers are in use ... and then it turns out that three people have been given lockers next to yours, and it's all 'sorry!' and banging locker doors together. Or you are in Hawaii, for the only time in your life ... and you bump into the Hungarian you worked with at Harvard. Or you're on honeymoon camping in a remote part of Ireland ... and you and your new wife meet your Head of Department and his new wife, walking the other way along an otherwise deserted beach. All of these have happened to Jack.

  Why do we find coincidences so striking? Because we expect random events to be evenly distributed, so statistical clumps surprise us. We think that a 'typical' lottery draw is something like 5, 14, 27, 36, 39,45, but that 1,2, 3,19,20,21 is far less likely. Actually, these two sets of numbers have exactly the same probability, which for the UK lottery is: 1 in 13,983,816. A typical lottery draw often includes several numbers close together, because sequences of six random numbers between 1 and 49 are more likely to be clumpy than not.

  How do we know this? Probability theorists tackle such questions using 'sample spaces' — their name for what we earlier called a 'phase space', a conceptual 'space' that organizes all the possibilities. A sample space contains not just the event that concerns us, but all possible alternatives. If we are rolling a die, for instance, then the sample space is 1, 2, 3, 4, 5, 6. For the lottery, the sample space is the set of all sequences of six different numbers between 1 and 49. A numerical value is assigned to each event in the sample space, called its 'probability', and this corresponds to how likely that event is to happen. For fair dice each value is equally likely, with a probability of 1/6. Ditto for the lottery, but now with a probability of 1/13,983,816.

  We can use a sample space approach to get a ball-park estimate of how amazing the Formula One coincidence was. Top drivers all lap at very nearly the same speed, so the three fastest times can easily fall inside the same tenth-of-a-second period. At intervals of a thousandth of a second, there are one hundred possible lap times for each to 'choose' from: this list determines the sample space, The probability of the coincidence turns out to be one chance in ten thousand. Unlikely enough to be striking, but not so unlikely that we ought to feel amazed.

  Estimates like this help to explain astounding coincidences reported in newspapers, such as a bridge player getting a 'perfect hand' — all thirteen cards in one suit. The number of games of bridge played every week worldwide is huge — so huge that every few weeks the actual events explore the entire sample space. So occasionally a perfect hand actually does turn up — with the frequency that its small but non-zero probability predicts. The probability of all four players getting a perfect hand at the same time, though, is so micoscopic that even if every planet in the galaxy had a billion inhabitants, all playing bridge every day for a billion years, you wouldn't expect it to happen.

  Nevertheless, every so often the newspapers report a four-way perfect hand. The sensible conclusion is not that a miracle happened, but that something changed the odds. Possibly the players got close to a four-way perfect hand, and the tale grew in the telling, so that when the journalist arrived with a photographer, another kind of narrative imperative ensured that their story fitted what the journalist had been told. Possibly they deliberately cheated to get their names in the papers. Scientists, especially, tend to underestimate the propensity of people to lie. More than one scientist has been fooled into accepting apparent evidence of extrasensory perception or other 'supernatural' events, which can actually be traced to deliberate trickery.

  Many other apparent coincidences, on close investigation, slither into a grey area in which trickery is strongly suspected, but may never be proved — either because sufficient evidence is unobtainable, or because it's not worth the trouble. Another way to be fooled about a coincidence is to be unaware of hidden constraints that limit the sample space. That 'perfect hand' could perhaps be explained by the way bridge players often shuffle cards for the next deal, which can be summed up as: poorly. If a pack of cards is arranged so that the top four cards consist of one from each suit, and thereafter every fourth card is in the same suit, then you can cut (but not shuffle, admittedly) the pack as many times as you like, and it will deal out a four-way perfect hand. At the end of a game, the cards lie on the table in a fairly ordered manner, not a random one — so it's not so surprising if they possess a degree of s
tructure after they've been picked up.

  So even with a mathematically tidy example like bridge, the choice of the 'right' sample space is not entirely straightforward. The actual sample space is 'packs of cards of the kind that bridge players habitually assemble after concluding a game', not 'all possible packs of cards'. That changes the odds.

  Unfortunately, statisticians tend to work with the 'obvious' sample space. For that question about Israeli fighter pilots, for instance, they would naturally take the sample space to be all children of Israeli fighter pilots. But that might well be the wrong choice, as the next tale illustrates.

  According to Scandinavian folklore, King Olaf of Norway was in dispute with the King of Sweden about ownership of an island, and they agreed to throw dice for it: two dice, highest total wins. The Swedish king threw a double-six. 'You may as well give up now,' he declared in triumph. Undeterred, Olaf threw the dice ... One turned up six ... the other split in half, one face showing a six and the other a one. 'Thirteen, I win,' said Olaf.'*

  Something similar occurs in The Colour of Magic, where several gods are playing dice to decide certain events on the Discworld:

  The Lady nodded slightly. She picked up the dice-cup and held it steady as a rock, yet all the Gods could hear the three cubes rattling about inside. And then she sent them bouncing across the table.

  A six. A three. A five.

  Something was happening to the five, however. Battered by the chance collision of several billion molecules, the die flipped onto a point, spun gently and came down a seven.

  Blind lo picked up the cube and counted the sides.

  'Come on,' he said wearily. 'Play fair.'

  Nature's sample space is often bigger than a conventional statistician would expect. Sample spaces are a human way to model reality: they do not capture all of it. And when it comes to estimating significance, a different choice of sample space can completely change our estimates of probabilities. The reason for this is an extremely important factor — 'selective reporting', which is a type of narra-tivium in action. This factor tends to be ignored in most conventional statistics. That perfect hand at bridge, for instance, is far more likely to make it to the local or even national press than an imperfect one. How often do you see the headline BRIDGE PLAYER GETS ENTIRELY ORDINARY HAND, for instance? The human brain is an irrepressible pattern-seeking device, and it seizes on certain events that it considers significant, whether or not they really are. In so doing, it ignores all the 'neighbouring' events that would help it judge how likely or unlikely the perceived coincidence actually is.