Science of Discworld III

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 bebigger 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)