THIRTEEN
INFINITY IS A BIT TRICKY
IT WAS JUST GONE HALF five in the morning, too late for Nibbles and yet not time for Early Breakfast. Jogging through the grey mist, Archchancellor Ridcully saw the lights on in the Great Hall. Steeling himself in case Ponder had students in there, he pushed open the door.
There were a few students around. One of them was asleep under the coffee spigot.
Ponder Stibbons was still on the stepladder, waving his hands through the timelines.
‘Getting anywhere, Stibbons?’ said Ridcully, running on the spot.
Ponder managed to steady himself just in time.
‘Er … general progress, sir,’ he said, and climbed down.
‘Bit of a big job, eh?’ said Ridcully.
‘Rather taxing, sir, yes. We’ve done the instructions, though. We’re nearly ready.’
‘Hit ’em hard, that’s the style,’ said Ridcully, punching the air.
‘Quite probably, sir,’ said Ponder, yawning.
‘I was thinking while I was running, Stibbons, as is my wont,’ said Ridcully.
It’s going to be about the eyeball, isn’t it, Ponder thought. I’m pretty good on the eyeball now, but then he’ll ask about the parasitic wasp and that’s a puzzler, and then he’ll ask how exactly is evolution passed on and there’s a god-space right there. And then he’ll ask how do you get from a blob in the ocean to people by adding nothing but sunlight and time? And he’ll probably say: people know they’re people, did blobs know they were blobs? What bit of a blob knows that? Where did consciousness come from, then? Did the big lizards have it? What’s it for? What about imagination? And even if I can think up some kind of answers to all those, he’ll say: look, Stibbons, what you’ve got there is a lot of clockwork answers, and if I ask you how you can get from a big bang to turtles and spoons and Darwin, all you’ll be able to come up with is more clockwork. How did all this happen? Who wound it up? How can nothing explode? Theology of Species makes so much sense when—
‘Are you all right, Stibbons?’ He was aware of the Archchancellor looking at him with uncharacteristic concern.
‘Yes, sir, Just a bit tired.’
‘Only, your lips were moving.’
Ponder sighed. ‘What was it you were thinking about, sir?’
‘Lots of Darwins get through this voyage, right?’
‘Yes. An infinite number.’
‘Well, in that case—’ the Archchancellor began.
‘But Hex did say it’s a much smaller infinite number that the number that don’t,’ said Ponder. ‘And that’s an even smaller number than the very large infinity when he never goes on the voyage. And the number of infinities where he’s never even born is—’
‘Infinite?’ Ridcully asked.
‘At least,’ said Ponder. ‘However, there is a positive side to this.’
‘Do tell, Stibbons.’
‘Well, sir, once Origin is published, the number of universes in which it is published will also become infinite in an infinitely small space of time. So even though the book may only be written once, it will, by human standards, immediately have been written in untold billions of adjacent universes.’
‘An infinite number, I suspect?’ said Ridcully.
‘Yes, sir. Sorry about that. Infinity is a bit tricky.’
‘You can’t imagine half of it, for one thing.’
‘That’s true. It’s not really a number at all. You can’t get to it starting from one. And that’s the problem, sir. Hex is right, the oddest number in the multiverse isn’t infinity, it’s one. Just one Charles Darwin writing The Origin of Species … it’s impossible.’
Ridcully sat down. ‘I’ll be damn glad when he finishes the book,’ he said. ‘We’ll get all those nody things sorted and get him back and I personally will hand him the pen.’
‘Er … that doesn’t happen immediately, sir,’ said Ponder. ‘He didn’t write it until he was back home.’
‘Fair enough,’ said Ridcully. ‘Probably a bit tricky, writin’ on a boat.’
‘He thought about it a lot first, sir,’ said Ponder. ‘I did mention that.’
‘How long?’ said Ridcully.
‘About twenty-five years, sir.’
‘What?’
‘He wanted to be sure, sir. He researched and wrote letters, lots of letters. He wanted to know everything about, well, everything – silk worms, sheep, jaguars … He wanted to be sure he was right.’ Ponder thumbed through the papers on his clipboard. ‘This interested me. It was from a letter he wrote in 1857, and he says “what a jump it is from a well-marked variety, produced by a natural cause, to a species produced by the separate act of the Hand of God”.
‘That’s the author of The Origin? Sounds more like the author of The Ology.’
‘It was a big thing he was going to do, sir. It worried him.’
‘I’ve read The Ology,’ said Ridcully. ‘Well, some of it. Makes a lot of sense.’
‘Yes, sir.’
‘I mean, if we hadn’t watched the world all happen from Day One, we’d have thought—’
‘I know what you mean, sir. I think that’s why The Ology was so popular.’
‘Darwin – I mean our Darwin – thought that no god would make so many kinds of barnacle. It’s so wasteful. A perfect being wouldn’t do it, he thought. But the other Darwins, the religious ones, said that was the whole point. They said that just as mankind had to strive for perfection, so must the whole animal kingdom. Plants, too. Survival of the Worthiest, they called it. Things weren’t made perfect, but had an inbuilt, er, striving to achieve perfection, as if part of the Plan was inside them. They could evolve. In fact, that was a good thing. It meant they were getting better.’
‘Seems logical,’ said Ridcully. ‘By god logic, at least.’
‘And there’s the whole thing about the Garden of Eden and the end of the world,’ said Ponder.
‘I must’ve missed that chapter,’ said Ridcully.
‘Well, sir, it’s your basic myth of a golden age at the start of the world and terrible destruction at the end of it, but codified in some very interesting language. Darwin suggested that the early chroniclers had got things mixed up. Like trolls, you know? They think the past is ahead of them because they can see it? The terrible destruction was in fact the birth of the world—’
‘Oh, you mean the red hot rocks, planets smacking together, that sort of thing?’
‘Exactly. And the end of the world, well, as experienced, would be the assembly of perfect creatures and plants in a perfect garden, belonging to the god.’
‘To get congratulated, and so on? Prizes handed out, marks awarded?’
‘Could be, sir.’
‘Like an everlasting picnic?’
‘He didn’t put it like that, but I suppose so.’
‘What about the perfect wasps?’ said Ridcully. ‘You always get them, you know. And ants.’
Ponder had been ready for this.
‘There was a lot of debate about that sort of thing,’ he said.
‘And it concluded how?’ said the Archchancellor.
‘It was decided that it was the kind of subject on which there could be a lot of debate, and that earthly considerations would not apply.’
‘Hah! And Darwin got all this past the priests?’
‘Oh, yes. Most of them, anyway,’ said Ponder.
‘But he was turning their whole world upside down!’
‘Um, that was happening anyway, sir. But this way, the god didn’t drop out of the bottom. People were poking around and proving that the world really was very old, that seabeds had become mountain tops, that all kinds of strange animals had lived a long, long time ago. Lots of people already accepted the idea of evolution. The idea of natural selection, as Darwin called it, of life just evolving itself, was hovering in the air. It was a big threat. But Theology of Species said there was a Plan. A huge, divine Plan, unfolding across millions of years! It even included the
planet itself! All that turmoil and volcanoes and drowning lands, that was a world evolving, you see? A world that would end up with topsoil, and the right kind of atmosphere, and minerals that were easily accessible, and seas full of fish—’
‘A world for humans, in other words.’
‘Got it in one, sir,’ said Ponder. ‘Humans. The top of the tree. A creature that knew what it was, that gave things names, that had a concept of epiphany. That Darwin later wrote another book, called The Ascent of Man. Oddly enough, our Darwin is going to write a similar book called The Descent of Man—’
‘Ah, I can see a bad choice of words right there,’ said Ridcully.
‘Quite,’ said Ponder. ‘The Ology Darwin was considered daring but … acceptable. And there was so much evidence that this was a planet made for humans. The religion changed quite a lot, but so did the technomancy. The god was still in charge.’
‘All very neat,’ said the Archchancellor. ‘So … what about the dinosaurs?’
‘Sorry, sir?’
‘Mr Stibbons, you know what I’m talking about. We saw them, remember? Not the big ones, the little ones who painted their bodies and herded animals? And the octopuses building cities under the sea? Not to mention the crabs! Oh, yes, the crabs. They were really doing well, the crabs. They were building rafts with sails and enslaving other crab nations. That’s practically civilisation! But they all got wiped out. Was that part of a divine plan?’
Ponder hesitated.
‘They did worship a crab-shaped god,’ he said, as a holding statement until actual thought happened.
‘Well, they would, wouldn’t they?’ said Ridcully. ‘They were crabs.’
‘Um. Perhaps they just weren’t … satisfactory?’ said Ponder. ‘In some way?’
‘They were pretty clever,’ said Ridcully.
Ponder squirmed. ‘Darwin didn’t know about them,’ he said. ‘They didn’t build anything that lasted. I suppose the Darwin who wrote The Ology would have taken the view that they simply failed, or were wicked in some way. One of the major religious texts does mention a divine flood that drowned everything in the world except one family and a boatload of animals.’
‘Why?’
‘Because they were all wicked, I believe.’
‘How can animals be wicked? How can a crab be wicked, for that matter?’
‘I don’t know, Archchancellor!’ Ponder burst out. ‘Maybe if they eat forbidden seaweed? Dig a burrow on the wrong day? I’m not a theologian!’
They sat in despondent silence.
‘It’s a bit of mess, isn’t it?,’ said Ridcully.
‘Yes, sir.’
‘We’ve really got to see to it that The Origin gets written.’
‘We have indeed, sir.’
‘But you’d like to think there’s someone in charge, yes?’ said Ridcully, gently. ‘Of everything, I mean.’
‘Yes! Yes, I would, sir! Not a big beard in the sky, but … something! Some kind of frame, some sense that good and bad have real meanings! I can see why The Ology was so popular. It wrapped everything up! But how does evolution get passed on? Where does order come from? If you start with a lot of exploding firmament, how do you end up with butterflies? Were butterflies built in from the start? How? What bit of burning hydrogen carried the plans for people? Even the Darwin who wrote The Origin called on a god to start life. It’s be nice to know that underneath it all is some kind of … sense.’
‘You didn’t used to talk like this, Mr Stibbons.’
Ponder sagged. ‘Sorry, sir. It’s all getting me down, I think.’
‘Well, I can see why,’ said Ridcully. ‘Surely there must be some Deitium here. Some things can’t just happen. Now, the eyeball –’
Ponder gave a little yelp.
‘– is easy,’ said Ridcully. ‘Are you all right, Stibbons?’
‘Er, fine, fine, sir. I’m fine. Easy, is it?’
‘Seeing keeps you alive,’ said Ridcully. ‘Any kind of seeing is better than nothing. I can see, ha, what the Origin Darwin is getting at there. You don’t have to have a god. But there’s a kind of wasp that’s parasitical on a spider … unless I’m thinking about a kind of spider that is parasitical on a wasp … anyway, what it does is, it waits until—’
‘Ah,’ said Ponder brightly, ‘wasn’t that the gong for Early Breakfast?’
‘I didn’t hear anything,’ said Ridcully.
‘I’m positive,’ said Ponder, edging towards the door. ‘I’ll tell you what, sir, I’ll just go and check.’
FOURTEEN
ALEPH-UMPTYPLEX
THE WIZARDS ARE NOT ONLY grappling with the apparent absurdities of ‘quantum’, their catch-all phrase for advanced physics and cosmology, but with the explosive philosophical/mathematical concept of infinity. In their own way, they have rediscovered one of the great insights of nineteenth-century mathematics: that there can be many infinities, some of them bigger than others.
If this sounds ridiculous, it is. Nonetheless, there is an entirely natural sense in which it turns out to be true.
There are two important things to understand about infinity. Although the infinite is often compared with numbers like 1, 2, 3, infinity is not itself a number in any conventional sense. As Ponder Stibbons says, you can’t get there from 1. The other is that, even within mathematics, there are many distinct notions that all bear the same label ‘infinity’. If you mix up their meanings, all you’ll get is nonsense.
And then – sorry, three important things – you have to appreciate that infinity is often a process, not a thing.
But – oh, four important things – mathematics has a habit of turning processes into things.
Oh, and – all right, five important things – one kind of infinity is a number, though a slightly unconventional one.
As well as the mathematics of infinity, the wizards are also contending with its physics. Is the Roundworld universe finite or infinite? Is it true that in any infinite universe, not only can anything happen, but everything must? Could there be an infinite universe consisting entirely of chairs … immobile, unchanging, wildly unexciting? The world of the infinite is paradoxical, or so it seems at first, but we shouldn’t let the apparent paradoxes put us off. If we keep a clear head, we can steer our way through the paradoxes, and turn the infinite into a reliable thinking aid.
Philosophers generally distinguish two different ‘flavours’ of infinity, which they call ‘actual’ and ‘potential’. Actual infinity is a thing that is infinitely big, and that’s such a mouthful to swallow that until recently it was rather disreputable. The more respectable flavour is potential infinity, which arises whenever some process gives us the distinct impression that it could be continued for as long as we like. The most basic process of this kind is counting: 1, 2, 3, 4, 5 … Do we ever reach ‘the biggest possible number’ and then stop? Children often ask that question, and at first they think that the biggest number whose name they know must be the biggest number there is. So for a while they think that the biggest number is six, then they think it’s a hundred, then they think it’s a thousand. Shortly after, they realise that if you can count to a thousand, then a thousand and one is only a single step further.
In their 1949 book Mathematics and the Imagination, Edward Kasner and James Newman introduced the world to the googol – the digit 1 followed by a hundred zeros. Bear in mind that a billion has a mere nine zeros: 1000000000. A googol is
100000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
and it’s so big we had to split it in two to fit the page. The name was invented by Kasner’s nine-year-old nephew, and is the inspiration for the internet search engine Google™.
Even though a googol is very big, it is definitely not infinite. It is easy to write down a bigger number:
100000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000001
Just
add 1. A more spectacular way to find a bigger number than a googol is to form a googolplex (name also courtesy of the nephew), which is 1 followed by a googol of zeros. Do not attempt to write this number down: the universe is too small unless you use subatomic-sized digits, and its lifetime is too short, let alone yours.
Even though a googolplex is extraordinarily big, it is a precisely defined number. There is nothing vague about it. And it is definitely not infinite (just add 1). It is, however, big enough for most purposes, including most numbers that turn up in astronomy. Kasner and Newman observe that ‘as soon as people talk about large numbers, they run amuck. They seem to be under the impression that since zero equals nothing, they can add as many zeros to a number as they please with practically no serious consequences,’ a sentence the Mustrum Ridcully himself might have uttered. As an example, they report that in the late 1940s a distinguished scientific publication announced that the number of snow crystals needed to start an ice age is a billion to the billionth power. ‘This,’ they tell us, ‘is very startling and also very silly.’ A billion to the billionth power is 1 followed by nine billion zeros. A sensible figure is around 1 followed by 30 zeros, which is fantastically smaller, though still bigger than Bill Gates’s bank balance.
Whatever infinity may be, it’s not a conventional ‘counting’ number. If the biggest number possible were, say, umpty-ump gazillion, then by the same token umpty-ump gazillion and one would be bigger still. And even if it were more complicated, so that (say) the biggest number possible were umpty-ump gazillion, two million, nine hundred and sixty-four thousand, seven hundred and fifty-eight … then what about umpty-ump gazillion, two million, nine hundred and sixty-four thousand, seven hundred and fifty-nine?
Given any number, you can always add one, and then you get a number that is (slightly, but distinguishably) bigger.
The counting process only stops if you run out of breath; it does not stop because you’ve run out of numbers. Though a near-immortal might perhaps run out of universe in which to write the numbers down, or time in which to utter them.
In short: there exist infinitely many numbers.
The wonderful thing about that statement is that it does not imply that there is some number called ‘infinity’, which is bigger than any of the others. Quite the reverse: the whole point is that there isn’t a number that is bigger than any of the others. So although the process of counting can in principle go on for ever, the number you have reached at any particular stage is finite. ‘Finite’ means that you can count up to that number and then stop.
Science of Discworld III Page 17
