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 damp-proof 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 is ‘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 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.
David 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 à 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 ro
om umpty-ump gazillion and one. The person in room umpty-ump gazillion 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 Al, 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:
A1 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 E1 …
(To see the pattern, look along successive 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.2
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 infinities, and he opened up a veritable Pandora’s box of ever-larger infinities. He called them transfinite numbers, and he stumbled across them when he was working in a hallowed, traditional area of analysis. It was really hard, technical stuff, and it led him into previously uncharted byways. Musing deeply on the nature of these things, Cantor became diverted from his work in his entirely respectable area of analysis, and started thinking about something much more difficult.
Counting.
The usual way that we introduce numbers is by teaching children to count. They learn that numbers are ‘things you use for counting’. For instance, ‘seven’ is where you get to if you start counting with ‘one’ for Sunday and stop on Saturday. So the number of days in the week is seven. But what manner of beast is seven? A word? No, because you could use the symbol 7 instead. A symbol? But then, there’s the word … anyway, in Japanese, the symbol for 7 is different. So what is seven? It’s easy to say what seven days, or seven sheep, or seven colours of the spectrum are … but what about the number itself? You never encounter a naked ‘seven’, it always seems to be attached to some collection of things.
Cantor decided to make a virtue of necessity, and declared that a number was something associated with a set, or collection, of things. You can put together a set from any collection of things whatsoever. Intuitively, the number you get by counting tells you how many things belong to that set. The set of days of the week determines the number ‘seven’. The wonderful feature of Cantor’s approach is this: you can decide whether any other set has seven members without counting anything. To do this, you just have to try to match the members of the sets, so that each member of one set is matched to precisely one of the other. If, for instance, the second set is the set of colours of the spectrum, then you might match the sets like this:
Sunday Red
Monday Orange
Tuesday Yellow
Wednesday Green
Thursday Blue
Friday Violet3
Saturday Octarine
The order in which the items are listed does not matter. But you’re not allowed to match Tuesday with both Violet and Green, or Green with both Tuesday and Sunday, in the same matching. Or to miss any members of the sets out.
In contrast, if you try to match the days of the week with the elephants that support the Disc, you run into trouble:
Sunday Berilia
Monday Tubul
Tuesday Great T’Phon
Wednesday Jerakeen
Thursday ?
More precisely, you run out of elephants. Even the legendary fifth elephant fails to take you past Thursday.
Why the difference? Well, there are seven days in the week, and seven colours of the spectrum, so you can match those sets. But there are only four (perhaps once five) elephants, and you can’t match four or five with seven.
The deep philosophical point here is that you don’t need to know about the numbers four, five or seven, to discover that there’s no way to match the sets up. Talking about the numbers amounts to being wise after the event. Matching is logically primary to counting.4 But now, all sets that match each other can be assigned a common symbol, or ‘cardinal’, which effectively is the corresponding number. The cardinal of the set of days of the week is the symbol 7, for instance, and the same symbol applies to any set that matches the days of the week. So we can base our concept of number on the simpler one of matching.
So far, then, nothing new. But ‘matching’ makes sense for infinite sets, not just finite ones. You can match the even numbers with all numbers:
2 1
4 2
6 3
8 4
10 5
…
and so on. Matchings like this explain the goings-on in Hilbert’s Hotel. That’s where Hilbert got the idea (roof before foundations, remember).
What is the cardinal of the set of all whole numbers (and hence of any set that can be matched to it)? The traditional name is ‘infinity’. Cantor, being cautious, preferred something with fewer mental associations, and in 1883 he named it ‘aleph’, the first letter of the Hebrew alphabet. And he put a small zero underneath it, for reasons that will shortly transpire: aleph-zero.
He knew what he was starting: ‘I am well aware that by adopting such a procedure I am putting myself in opposition to widespread views regarding infinity in mathematics and to current opinions on the nature of number.’ He got what he expected: a lot of hostility, especially from Leopold Kronecker. ‘God created the integers: all else is the work of Man,’ Kronecker declared.
Nowadays, most of us think that Man created the integers too.
Why introduce a new symbol (and Hebrew at that?). If there had been only one infinity in Cantor’s sense, he might as well have named it ‘infinity’ like everyone else, and used the traditional symbol of a figure 8 lying on its side. But he quickly saw that from his point of view, there might well be other infiniti
es, and he was reserving the right to name those aleph-one, aleph-two, aleph-three, and so on.
How can there be other infinities? This was the big unexpected consequence of that simple, childish idea of matching. To describe how it comes about, we need some way to talk about really big numbers. Finite ones and infinite ones. To lull you into the belief that everything is warm and friendly, we’ll introduce a simple convention.
If ‘umpty’ is any number, of whatever size, then ‘umptyplex’ will mean 10umpty, which is 1 followed by umpty zeros. So 2plex is 100, a hundred; 6plex is 1000000, a million; 9plex is a billion. When umpty = 100 we get a googol, so googol = 100plex. A googolplex is therefore also describable as 100plexplex.
In Cantorian mode, we idly start to muse about infinityplex. But let’s be precise: what about aleph-zeroplex? What is 10aleph-zero?
Remarkably, it has an entirely sensible meaning. It is the cardinal of the set of all real numbers – all numbers that can be represented as an infinitely long decimal. Recall the Ephebian philosopher Pthagonal, who is recorded as saying, ‘The diameter divides into the circumference … It ought to be three times. But does it? No. Three point one four and lots of other figures. There’s no end to the buggers.’ This, of course, is a reference to the most famous real number, one that really does need infinitely many decimal places to capture it exactly: π (‘pi’). To one decimal place, π is 3.1. To two places, it is 3.14. To three places, it is 3.141. And so on, ad infinitum.
There are plenty of real numbers other than π. How big is the phase space of all real numbers?
Think about the bit after the decimal point. If we work to one decimal place, there are 10 possibilities: any of the digits 0, 1, 2, …, 9. If we work to two decimal places, there are 100 possibilities: 00 up to 99. If we work to three decimal places, there are 1000 possibilities: 000 up to 999.
Science of Discworld III Page 18