`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.'

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

10000000000000000000000 0000000000000000000000000000 000000000000000000000000000000 00000000000000000000 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 GoogleTM

Even though a googol is very big, it is definitely not infinite. It is easy to write down a bigger number:

10000000000000000000000 0000000000000000000000000000 0000000000000000000000000000000 0000000000000000001

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.