Darwin's Watch by Terry Pratchett

Author:Terry Pratchett
Language: eng
Format: epub
Publisher: Knopf Doubleday Publishing Group
Published: 2015-06-09T04:00:00+00:00

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

10000000​00000000​00000000​00000000​00000000​00000000​000000000​0000000000​00000000​00000000​00000000​00000000​00

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