Darwin's Watch

by Terry Pratchett

  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 GoogleTM

  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 that 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 nearimmortal 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.

  As the philosophers would say: counting is an instance of potential infinity. It is a process that can go on for ever (or at least, so it seems to our naive pattern-recognising brains) but never gets to `for

  ever'.

  The development of new mathematical ideas tends to follow a pattern. If mathematicians were building a house, they would start with the downstairs walls, hovering unsupported a foot or so above the damp-proof course ... or where the damp-proof course ought to be. There would be no doors or windows, just holes of the right shape. By the time the second floor was added, the quality of the brickwork would have improved dramatically, the interior walls would be plastered, the doors and windows would all be in place, and the floor

  would be strong enough to walk on. The third floor would be vast, elaborate, fully carpeted, with pictures on the walls, huge quantities of furniture of impressive but inconsistent design, six types of wallpaper in every room ... The attic, in contrast, would be sparse but elegant - minimalist design, nothing out of place, everything there for a reason. Then, and only then, would they go back to ground level, dig the foundations, fill them with concrete, stick in a dampproof course, and extend the walls downwards until they met the foundations.

  At the end of it all you'd have a house that would stand up. Along the way, it would have spent a lot of its existence looking wildly improbable. But the builders, in their excitement to push the walls skywards and fill the rooms with interior decor, would have been too busy to notice until the building inspectors rubbed their noses in the structural faults.

  When new mathematical ideas first arise, no one understands them terribly well, which is only natural because they're new. And no one is going to make a great deal of effort to sort out all the logical refinements and make sense of those ideas unless they're convinced it's all going to be worthwhile. So the main thrust of research goes into developing those ideas and seeing if they lead anywhere interesting. `Interesting', to a mathematician, mostly means `can I see ways to push this stuff further?', but the acid test i
s `what problems does it solve?' Only after getting a satisfactory answer to these questions do a few hardy and pedantic souls descend into the basement and sort out decent foundations.

  So mathematicians were using infinity long before they had a clue what it was or how to handle it safely. In 500 Bc Archimedes, the greatest of the Greek mathematicians and a serious contender for a place in the all-time top three, worked out the volume of a sphere by (conceptually) slicing it into infinitely many infinitely thin discs, like an ultra-thin sliced loaf, and hanging all the slices from a balance, to compare their total volume with that of a suitable shape

  whose volume he already knew. Once he'd worked out the answer by this astonishing method, he started again and found a logically acceptable way to prove he was right. But without all that faffing around with infinity, he wouldn't have known where to start and his logical proof wouldn't have got off the ground.

  By the time of Leonhard Euler, an author so prolific that we might consider him to be the Terry Pratchett of eighteenth-century mathematics, many of the leading mathematicians were dabbling in `infinite series' - the school child's nightmare of a sum that never ends. Here's one:

  1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + . -

  where the `. . .' means `keep going'. Mathematicians have concluded that if this infinite sum adds up to anything sensible, then what it adds up to must be exactly two. [1] If you stop at any finite stage, though, what you reach is slightly less than two. But the amount by which it is less than two keeps shrinking. The sum sort of sneaks up on the correct answer, without actually getting there; but the amount by which it fails to get there can be made as small as you please, by adding up enough terms.

  Remind you of anything? It looks suspiciously similar to one of Zeno/Xeno's paradoxes. This is how the arrow sneaks up on its victim, how Achilles sneaks up on the tortoise. It is how you can do infinitely many things in a finite time. Do the first thing; do the second thing one minute later; do the third thing half a minute after that; then the fourth thing a quarter of a minute after that ... and so on. After two minutes, you've done infinitely many things.

  The realisation that infinite sums can have a sensible meaning is only the start. It doesn't dispel all of the paradoxes. Mostly, it just

  [1] To see why, double it: the result now is 2 + 1 + V2 + '/ + y8 + 'A+ . . . , which is 2 more than the original sum. What number increases by 2 when you double it? There's only one such number, and it's 2.

  sharpens them. Mathematicians worked out that some infinities are harmless, others are not.

  The only problem left after that brilliant insight was: how do you tell? The answer is that if your concept of infinity does not lead to logical contradictions, then it's safe to use, but if it does, then it isn't. Your task is to give a sensible meaning to whatever `infinity' intrigues you. You can't just assume that it automatically makes sense.

  Throughout the eighteenth and early nineteenth centuries, mathematics developed many notions of `infinity', all of them potential. In projective geometry, the `point at infinity' was where two parallel lines met: the trick was to draw them in perspective, like railway lines heading off towards the horizon, in which case they appear to meet on the horizon. But if the trains are running on a plane, the horizon is infinitely far away and it isn't actually part of the plane at all - it's an optical illusion. So the point `at' infinity is determined by the process of travelling along the train tracks indefinitely. The train never actually gets there. In algebraic geometry a circle ended up being defined as `a conic section that passes through the two imaginary circular points at infinity', which sure puts a pair of compasses in their place.

  There was an overall consensus among mathematicians, and it boiled down to this. Whenever you use the term `infinity' you are really thinking about a process. If that process generates some well determined result, by however convoluted an interpretation you wish, then that result gives meaning to your use of the word `infinity', in that particular context.

  Infinity is a context-dependent process. It is potential.

  It couldn't stay that way.

  avid Hilbert was one of the top two mathematicians in the world at the end of the nineteenth century, and he was one of the great enthusiasts for a new approach to the infinite, in which - contrary to what we've just told you - infinity is treated as a thing, not as a process. The new approach was the brainchild of Georg Cantor, a German mathematician whose work led him into territory that was fraught with logical snares. The whole area was a confused mess for about a century (nothing new there, then). Eventually he decided to sort it out for good and all by burrowing downwards rather than building ever upwards, and putting in those previously non-existent foundations. He wasn't the only person doing this, but he was among the more radical ones. He succeeded in sorting out the area that drove him to these lengths, but only at the expense of causing considerable trouble elsewhere.

  Many mathematicians detested Cantor's ideas, but Hilbert loved them, and defended them vigorously. `No one,' he declaimed, `shall expel us from the paradise that Cantor has created.' It is, to be sure, as much paradox as paradise. Hilbert explained some of the paradoxical properties of infinity a la Cantor in terms of a fictitious hotel, now known as Hilbert's Hotel.

  Hilbert's Hotel has infinitely many rooms. They are numbered 1, 2, 3, 4 and so on indefinitely. It is an instance of actual infinity - every room exists now, they're not still building room umpty-ump gazillion and one. And when you arrive there, on Sunday morning, every room is occupied.

  In a finite hotel, even with umpty-ump gazillion and one rooms, you're in trouble. No amount of moving people around can create an extra room. (To keep it simple, assume no sharing: each room has exactly one occupant, and health and safety regulations forbid more than that.)

  In Hilbert's Hotel, however, there is always room for an extra guest. Not in room infinity, though, for there is no such room. In room one.

  But what about the poor unfortunate in room one? He gets moved to room two. The person in room two is moved to room three. And so on. The person in room umpty-ump gazillion is moved to room umpty-ump gazillion and one. The person in room umpty-umpgazillion and one is moved to room umpty-ump gazillion and two. The person in room n is moved to room n+1, for every number n.

  In a finite hotel with umpty-ump gazillion and one rooms, this procedure hits a snag. There is no room umpty-ump gazillion and two into which to move its inhabitant. In Hilbert's Hotel, there is no end to the rooms, and everyone can move one place up. Once this move is completed,' the hotel is once again full.

  That's not all. On Monday, a coachload of 50 people arrives at the completely full Hilbert Hotel. No worries: the manager moves everybody up 50 places - room 1 to 51, room 2 to 52, and so on - which leaves rooms 1-50 vacant for the people off the coach.

  On Tuesday, an Infinity Tours coach arrives containing infinitely many people, helpfully numbered A1, A2, A3, .... Surely there won't be room now? But there is. The existing guests are moved into the even-numbered rooms: room 1 moves to room 2, room 2 to room 4, room 3 to room 6, and so on. Then the odd-numbered rooms are free, and person A1 goes into room 1, A2 into room 3, A3 into room 5 ... Nothing to it.

  By Wednesday, the manager is really tearing his hair out, because infinitely many Infinity Tours coaches turn up. The coaches are labelled A, B, C, ... from an infinitely long alphabet, and the people in them are A1, A2, A3, ... , B1, B2, B3, ... C1, C2, C3, .

  .. and so on. But the manager has a brainwave. In an infinitely large corner of the infinitely large hotel parking lot, he arranges all the new guests into an infinitely large square:

  Al A2 A3 A4 A5 ...

  B1 B2 B3 B4 B5 ...

  C1 C2 C3 C4 C5 ...

  D1 D2 D3 D4 D5 ...

  E1 E2 E3 E4 E5 .. .

  Then he rearranges them into a single infinitely long line, in the order

  A1 - A2 B1 - A3 B2 C1 - A4 B3 C2 D1 - A5 B4 C3 D2 El ...

  (To see the pattern, look along success
ive diagonals running from top right to lower left. We've inserted hyphens to separate these.) What most people would now do is move all the existing guests into the even-numbered rooms, and then fill up the odd rooms with new guests, in the order of the infinitely long line. That works, but there is a more elegant method, and the manager, being a mathematician, spots it immediately. He loads everybody back into a single Infinity Tours coach, filling the seats in the order of the infinitely long line. This reduces the problem to one that has already been solved.[1]

  Hilbert's Hotel tells us to be careful when making assumptions about infinity. It may not behave like a traditional finite number. If you add one to infinity, it doesn't get bigger. If you multiply infinity by infinity, it still doesn't get bigger. Infinity is like that. In fact, it's easy to conclude that any sum involving infinity works out as infinity, because you can't get anything bigger than infinity.

  That's what everybody thought, which is fair enough if the only infinities you've ever encountered are potential ones, approached as a sequence of finite steps, but in principle going on for as long as you wish. But in the 1880s Cantor was thinking about actual

  [1] If you've never encountered the mathematical joke, here it is. Problem 1: a kettle is hanging on a peg. Describe the sequence of events needed to make a pot of tea. Answer: take the kettle off the peg, put it in the sink, turn on the tap, wait till the kettle fills with water, turn the tap off ... and so on. Problem 2: a kettle is sitting in the sink. Describe the sequence of events needed to make a pot of tea. Answer: not `turn on the tap, wait till the kettle fills with water, turn the tap off ... and so on'. Instead: take the kettle out of the sink and hang it on the peg, then proceed as before. This reduces the problem to one that has already been solved. (Of course the first step puts it back in the sink - that's why it's a joke.)