The Science of Discworld III - Darwin's Watch

14. 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.

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 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[35]. 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.

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 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[36].

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 trans-finite 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 Violet[37].

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:

SundayBerilia

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[38]. 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 infinities, 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 10^aleph-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: n ('pi'). To one decimal place, n 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 n. 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.

The pattern is clear. If we work to umpty decimal places, there are 10^umpty possibilities. That is, umptyplex.

If the decimal places go on `for ever', we first must ask `what kind of for ever?' And the answer is `Cantor's aleph-zero', because there is a first decimal place, a second, a third ... the places match the whole numbers. So if we set 'umpty' equal to 'aleph-zero', we find that the cardinal of the set of all real numbers (ignoring anything before the decimal point) is aleph-zeroplex. The same is true, for slightly more complicated reasons, if we include the bit before the decimal point.'[39]

All very well, but presumably aleph-zeroplex is going to turn out to be aleph-zero in heavy disguise, since all infinities surely must be equal? No. They're not. Cantor proved that you can't match the real numbers with the whole numbers. So aleph-zeroplex is a bigger infinity than aleph-zero.

He went further. Much further. He proved[40] that if umpty is any infinite cardinal, the umptyplex is a bigger one. So aleph-zeroplexplex is bigger still, and aleph-zeroplexplexplex is bigger than that, and ...

There is no end to the list of Cantorian infinities. There is no 'hyperinfinity' that is bigger than all other infinities.

The idea of infinity as `the biggest possible number' is taking some hard knocks here. And this is the sensible way to set up infinite arithmetic.

If you start with any infinite cardinal aleph-umpty, then alephumptyplex is bigger. It is natural to suppose that what you get must be aleph-(umpty+1), a statement dubbed the Generalised Continuum Hypothesis. In 1963 Paul Cohen (no known relation either to Jack or the Barbarian) proved that ... well, it depends. In some versions of set theory it's true, in others it's false.

The foundations of mathematics are like that, which is why it's best to construct the house first and put the foundations in later. That way, if you don't like them, you can take them out again and put something else in instead. Without disturbing the house.

This, then, is Cantor's Paradise: an entirely new number system of alephs, of infinities beyond measure, never-ending - in a very strong sense of `never'. It arises entirely naturally from one simple principle: that the technique of `matching' is all you need to set up the logical foundations of arithmetic. Most working mathematicians now agree with Hilbert, and Cantor's initially astonishing ideas have been woven into the very fabric of mathematics.

The wizards don't just have the mathematics of infinity to contend with. They are also getting tangled up in the physics. Here, entirely new questions about the infinite arise. Is the universe finite or infinite? What kind of finite or infinite? And what about all those parallel universes that the cosmologists and quantum theorists are always talking about? Even if each universe is finite, could there be infinitely many parallel ones?

According to current cosmology, what we normally think of as the universe is finite. It started as a single point in the Big Bang, and then expanded at a finite rate for about 13 billion years, so it has to be finite. Of course, it could be infinitely finely divisible, with no lower limit to the sizes of things, just like the mathematician's line or plane - but quantum-mechanically speaking there is a definite graininess down at the Planck length, so the universe has a very large but finite number of possible quantum states.

The `many worlds' version of quantum theory was invented by the physicist Hugh Everett as a way to link the quantum view of the world to our everyday `sensible' view. It contends that whenever a choice can be made - for example, whether an electron spin is up or down, or a cat is alive or dead - the universe does not simply make a choice and abandon all the alternatives. That's what it looks like to us, but really the universe makes all possible choices. Innumerable `alternative' or `parallel' worlds branch off from the one that we perceive. In those worlds, things happen that did not happen here. In one of them, Adolf Hitler won the Second World War. In another, you ate one extra olive at dinner last night.

Narratively speaking, the many worlds description of the quantum realm is a delight. No author in search of impressive scientific gobbledegook that can justify hurling characters into alternative storylines - we plead guilty - can possibly resist.

The trouble is that, as science, the many-worlds interpretation is rather overrated. Certainly, the usual way that it is described is misleading. In fact, rather too much of the physics of multiple universes is usually explained in a misleading way. This is a pity, because it trivialises a profound and beautiful set of ideas. The suggestion that there exists a real universe, somehow adjacent to ours, in which Hitler defeated the Allies, is a big turn-off for a lot of people. It sounds too absurd even to be worth considering. `If that's what modern physics is about, I'd prefer my tax dollars to go towards something useful, like reflexology.'

The science of `the' multiverse - there are numerous alternatives, which is only appropriate - is fascinating. Some of it is even useful.

And some - not necessarily the useful bit - might even be true. Though not, we will try to convince you, the bit about Hitler.

It all started with the discovery that quantum behaviour can be represented mathematically as a Big Sum. What actually happens is the sum of all of the things that might have happened. Richard Feynman explained this with his usual extreme clarity in his book QED (Quantum Electro Dynamics, not Euclid). Imagine a photon, a particle of light, bouncing off a mirror. You can work out the path that the photon follows by `adding up' all possible paths that it might have taken. What you really add is the levels of brightness, the light intensities, not the paths. A path is a concentrated strip of brightness, and here that strip hits the mirror and bounces back at the same angle.

This `sum-over-histories' technique is a direct mathematical consequence of the rules of quantum mechanics, and there's nothing objectionable or even terribly surprising about it. It works because all of the `wrong' paths interfere with each other, and between them they contribute virtually nothing to the overall sum. All that survives, as the totals come in, is the `right' path. You can take this unobjectionable mathematical fact and dress it up with a physical interpretation. Namely: light really takes all possible paths, but what we observe is the sum, so we just see the one path in which the light `ray' hits the mirror and bounces off again at the same angle.

That interpretation is also not terribly objectionable, philosophically speaking, but it verges into territory that is. Physicists have a habit of taking mathematical descriptions literally - not just the conclusions, but the steps employed to get them. They call this `thinking physically', but actually it's the reverse: it amounts to projecting mathematical features on to the real world - `reifying' abstractions, endowing them with reality.

We're not saying it doesn't work - often it does. But reification tends to make physicists bad philosophers, because they forget they're doing it.

One problem with `thinking physically' is that there are sometimes several mathematically equivalent ways to describe something - different ways to say exactly the same thing in mathematical language. If one of them is true, they all are. But, their natural physical interpretations can be inconsistent.

A good example arises in classical (non-quantum) mechanics. A moving particle can be described using (one of) Newton's laws of motion: the particle's acceleration is proportional to the forces that act on it. Alternatively, the motion can be described in terms of a `variational principle': associated with each possible particle path there is a quantity called the `action'. The actual path that the particle follows is the one that makes the action as small as possible.

The logical equivalence of Newton's laws and the principle of least action is a mathematical theorem. You cannot accept one without accepting the other, on a mathematical level. Don't worry what `action' is. It doesn't matter here. What matters is the difference between the natural interpretations of these two logically identical descriptions.

Newton's laws of motion are local rules. What the particle does next, here and now, is entirely determined by the forces that act on it, here and now. No foresight or intelligence is needed; just keep on obeying the local rules.

The principle of least action has a different style: it is global. It tells us that in order to move from A to B, the particle must somehow contemplate the totality of all possible paths between those points. It must work out the action associated with each path, and find whichever one of them has the smallest action. This `computation' is non-local, because it involves the entire path(s), and in some sense it has to be carried out before the particle knows where to go. So in this natural interpretation of the mathematics, the particle appears to be endowed with miraculous foresight and the ability to choose, a rudimentary kind of intelligence.

So which is it? A mindless lump of matter which obeys the local rules as it goes along? Or a quasi-intelligent entity with vast computational powers, which has the foresight to choose, among all the possible paths that it could have taken, precisely the unique one that minimises the action?

We know which interpretation we'd choose.

Interestingly, the principle of least action is a mechanical analogue of Feynman's sum-over-histories method in optics. The two really are extremely close. Yes, you can formulate the mathematics of quantum mechanics in a way that seems to imply that light follows all possible paths and adds them up. But you are not obliged to buy that description as the real physics of the real world, even if the mathematics works.

The many-worlds enthusiasts do buy that description: in fact, they take it much further. Not the history of a single photon bouncing off a mirror, but the history of the entire universe. That, too, is a sum of all possibilities - using the universe's quantum wave function in place of the light intensity due to the photon - so by the same token, we can interpret the mathematics in a similarly dramatic way. Namely: the universe really does do all possible things. What we observe is what happens when you add all those possibilities up.

Of course there's also a less dramatic interpretation: the universe trundles along obeying the local laws of quantum mechanics, and does exactly one thing ... which just happens, for purely mathematical reasons, to equal the sum of all the things that it might have done.

Which interpretation do you buy?

Mathematically, if one is `right' then so is the other. Physically, though, they carry very different implications about how the world works. Our point is that, as for the classical particle, their mathematical equivalence does not require you to accept their physical truth as descriptions of reality. Any more than the equivalence of Newton's laws with the principle of least action obliges you to believe in intelligent particles that can predict the future.

The many-worlds interpretation of quantum mechanics, then, is resting on dodgy ground even though its mathematical foundations are impeccable. But the usual presentation of that interpretation goes further, by adding a hefty dose of narrativium. This is precisely what appeals to SF authors, but it's a pity that it stretches the interpretation well past breaking-point.

What we are usually told is this. At every instant of time, whenever a choice has to be made, the universe splits into a series of `parallel worlds' in which each of the choices happens. Yes, in this world you got up, had cornflakes for breakfast, and walked to work. But somewhere `out there' in the vastness of the multiverse, there is another universe in which you had kippers for breakfast, which made you leave the house a minute later, so that when you walked across the road you had an argument with a bus, and lost, fatally.

What's wrong here is not, strangely enough, the contention that this world is `really' a sum of many others. Perhaps it is, on a quantum level of description. Why not? But it is wrong to describe those alternative worlds in human terms, as scenarios where everything follows a narrative that makes sense to the human mind. As worlds where `bus' or `kipper' have any meaning at all. And it is even less justifiable to pretend that every single one of those parallel worlds is a minor variation on this one, in which some human-level choice happens differently.

If those parallel worlds exist at all, they are described by changing various components of a quantum wave function whose complexity is beyond human comprehension. The results need not resemble humanly comprehensible scenarios. Just as the sound of a clarinet can be decomposed into pure tones, but most combinations of those tones do not correspond to any clarinet.

The natural components of the human world are buses and kippers. The natural components of the quantum wave function of the world are not the quantum wave functions of buses and kippers. They are altogether different, and they carve up reality in a different way. They flip electron spins, rotate polarisations, shift quantum phases.

They do not turn cornflakes into kippers.

It's like taking a story and making random changes to the letters, shifting words around, probably changing the instructions that the printer uses to make the letters, so that they correspond to no alphabet known to humanity. Instead of starting with the Ankh-Morpork national anthem and getting the Hedgehog Song, you just get a meaningless jumble. Which is perhaps as well.

According to Max Tegmark, writing in the May 2003 issue of Scientific American, physicists currently recognise four distinct levels of parallel universes. At the first level, some distant region of the universe replicates, almost exactly, what is going on in our own region. The second level involves more or less isolated `bubbles', baby universes, in which various attributes of the physical laws, such as the speed of light, are different, though the basic laws are the same. The third level is Everett's many-worlds quantum parallelism. The fourth includes universes with radically different physical laws - not mere variations on the theme of our own universe, but totally distinct systems described by every conceivable mathematical structure.

Tegmark makes a heroic attempt to convince us that all of these levels really do exist - that they make testable predictions, are scientifically falsifiable if wrong, and so on. He even manages to reinterpret Occam's razor, the philosophical principle that explanations should be kept as simple as possible, to support his view.

All of this, speculative as it may seem, is good frontier cosmology and physics. It's exactly the kind of theorising that a Science of Discworld book ought to discuss: imaginative, mind-boggling, cutting-edge. We've come to the reluctant conclusion, though, that the arguments have serious flaws. This is a pity, because the concept of parallel worlds is dripping with enough narrativium to make any SF author out-salivate Pavlov's dogs.

We'll summarise Tegmark's main points, describe some of the evidence that he cites in their favour, offer a few criticisms, and leave you to form your own opinions.

Level 1 parallel worlds arise if - because - space is infinite. Not so far back we told you it is finite, because the Big Bang happened a finite time ago so it's not had time to expand to an infinite extent[41]. Apparently, though, data on the cosmic microwave background do not support a finite universe. Even though a very large finite one would generate the same data.

`Is there a copy of you reading this article?' Tegmark asks. Assuming the universe is infinite, he tells us that `even the most unlikely events must take place somewhere'. A copy of you is likelier than many, so it must happen. Where? A straightforward calculation indicates that `you have a twin in a galaxy about 10 to the power 10^21 metres from here'. Not 10^21 metres, which is already 25 times the size of the currently observable universe, but 1 followed by 1028 zeros. Not only that: a complete copy of (the observable part of) our universe should exist about 10 to the power 10^118 metres away. And beyond that ...

We need a good way to talk about very big numbers. Symbols like 10^118 are too formal. Writing out all the zeros is pointless, and usually impossible. The universe is big, and the multiverse is substantially bigger. Putting numbers to how big is not entirely straightforward, and finding something that can also be typeset is even harder.

Fortunately, we've already solved that problem with our earlier convention: if `umpty' is any number, then `umptyplex' will mean 10^umpty, which is 1 followed by umpty zeros.

When umpty = 118 we get 118plex, which is roughly the number of protons in the universe. When umpty is 118plex we get 118plexplex, which is the number that Tegmark is asking us to think about, 10 to the power 10 to the power 118. Those numbers arise because a `Hubble volume' of space - one the size of the observable universe - has a large but finite number of possible quantum states.

The quantum world is grainy, with a lower limit to how far space and time can be divided. So a sufficiently large region of space will contain such a vast number of Hubble volumes that every one of those quantum states can be accommodated. Specifically, a Hubble volume contains 118plex protons. Each has two possible quantum states. That means there are 2 to the power 118plex possible configurations of quantum states of protons. One of the useful rules in this type of mega-arithmetic is that the `lowest' number in the plexified stack - here 2 - can be changed to something more convenient, such as 10, without greatly affecting the top number. So, in round numbers, a region 118plexplex metres across can contain one copy of each Hubble volume.

Level 2 worlds arise on the assumption that spacetime is a kind of foam, in which each bubble constitutes a universe. The main reason for believing this is `inflation', a theory that explains why our universe is relativistically flat. In a period of inflation, space rapidly stretches, and it can stretch so far that the two ends of the stretched bit become independent of each other because light can't get from one to the other fast enough to connect them causally. So spacetime ends up as a foam, and each bubble probably has its own variant of the laws of physics - with the same basic mathematical form, but different constants.

Level 3 parallel worlds are those that appear in the many-worlds interpretation of quantum mechanics, which we've already tackled.

Everything described so far pales into insignificance when we come to level 4. Here, the various universes involved can have radically different laws of physics from each other. All conceivable mathematical structures, Tegmark tells us, exist here: How about a universe that obeys the laws of classical physics, with no quantum effects? How about time that comes in discrete steps, as for computers, instead of being continuous? How about a universe that is simply an empty dodecahedron? In the level N multiverse, all these alternative realities actually exist.

But do they?

In science, you get evidence from observations or from experiments.

Direct observational tests of Tegmark's hypothesis are completely out of the question, at least until some remarkable spacefaring technology comes into being. The observable universe extends no more than 27plex metres from the Earth. An object (even the size of our visible universe) that is 118plexplex metres away cannot be observed now, and no conceivable improvement on technology can get round that. It would be easier for a bacterium to observe the entire known universe than for a human to observe an object 118plexplex metres away.

We are sympathetic to the argument that the impossibility of direct experimental tests does not make the theory unscientific. There is no direct way to test the previous existence of dinosaurs, or the timing (or occurrence) of the Big Bang. We infer these things from indirect evidence. So what indirect evidence is there for infinite space and distant copies of our own world?

Space is infinite, Tegmark says, because the cosmic microwave background tells us so. If space were finite, then traces of that finitude would show up in the statistical properties of the cosmic background and the various frequencies of radiation that make it up.

This is a curious argument. Only a year or so ago, some mathematicians used certain statistical features of the cosmic microwave background to deduce that not only is the universe finite, but that it is shaped a bit like a football.* There is a paucity of very long-wavelength radiation, and the best reason for not finding it is that the universe is too small to accommodate such wavelengths. Just as a guitar string a metre long cannot support a vibration with a wavelength of 100 metres - there isn't room to fit the wave into the available space.

The main other item of evidence is of a very different nature - not an observation as such, but an observation about how we interpret observations. Cosmologists who analyse the microwave background to work out the shape and size of the universe habitually report their findings in the form `there is a probability of one in a thousand that such and such a shape and size could be consistent with the data'. Meaning that with 99.9 per cent probability we rule out that size and shape. Tegmark tells us that one way to interpret this is that at most one Hubble volume in a thousand, of that size and shape, would exhibit the observed data. `The lesson is that the multiverse theory can be tested and falsified even when we cannot see the other universes. The key is to predict what the ensemble of parallel universes is and to specify a probability distribution over that ensemble.'

This is a remarkable argument. Fatally, it confuses actual Hubble volumes with potential ones. For example, if the size and shape under consideration is `a football about 27plex metres across[42] - a fair guess for our own Hubble volume - then the `one in a thousand' probability is a calculation based on a potential array of one thousand footballs of that size. These are not part of a single infinite universe: they are distinct conceptual `points' in a phase space of big footballs. If you lived in such a football and made such observations, then you'd expect to get the observed data on about one occasion in a thousand.

There is nothing in this statement that compels us to infer the actual existence of those thousand footballs - let alone to embed the lot in a single, bigger space, which is what we are being asked to do. In effect, Tegmark is asking us to accept a general principle: that whenever you have a phase space (statisticians would say a sample space) with a well-defined probability distribution, then everything in that phase space must be real.

This is plain wrong.

A simple example shows why. Suppose that you toss a coin a hundred times. You get a series of tosses something like HHTTTHH ... THH. The phase space of all possible such tosses contains precisely 2100 such sequences. Assuming the coin is fair, there is a sensible way to assign a probability to each such sequence - namely the chance of getting it is one in 2100. And you can test that `distribution' of probabilities in various indirect ways. For instance, you can carry out a million experiments, each yielding a series of 100 tosses, and count what proportion has 50 heads and 50 tails, or 49 heads and 51 tails, whatever. Such an experiment is entirely feasible.

If Tegmark's principle is right, it now tells us that the entire phase space of coin-tossing sequences really does exist. Not as a mathematical concept, but as physical reality.

However, coins do not toss themselves. Someone has to toss them.

If you could toss 100 coins every second, it would take about 24plex years to generate 2100 experiments. That is roughly 100 trillion times the age of the universe. Coins have been in existence for only a few thousand years. The phase space of all sequences of 100 coin tosses is not real. It exists only as potential.

Since Tegmark's principle doesn't work for coins, it makes no sense to suppose that it works for universes.

The evidence advanced in favour of level 4 parallel worlds is even thinner. It amounts to a mystical appeal to Eugene Wigner's famous remark about `the unusual effectiveness of mathematics' as a description of physical reality. In effect, Tegmark tells us that if we can imagine something, then it has to exist.

We can imagine a purple hippopotamus riding a bicycle along the edge of the Milky Way while singing Monteverdi. It would be lovely if that meant it had to exist, but at some point a reality check is in order. We don't want to leave you with the impression that we enjoy pouring cold water over every imaginative attempt to convey a feeling for some of the remarkable concepts of modern cosmology and physics. So we'll end with a very recent addition to the stable of parallel worlds, which has quite a few things going for it. Perhaps unsurprisingly, the main thing not currently going for it is a shred of experimental evidence.

The new theory on the block is string theory. It provides a philosophically sensible answer to the age-old question: why are we here? And it does so by invoking gigantic numbers of parallel universes.

It is just much more careful how it handles them.

Our source is an article, `The String Theory Landscape' by Raphael Bousso and Joseph Polchinski, in the September 2004 issue of Scientific American - a special issue on the theme of Albert Einstein.

If there is a single problem that occupies the core of modern physics, it is that of unifying quantum mechanics with relativity. This search for a `theory of everything' is needed because although both of those theories are extraordinarily successful in helping us to understand and predict various aspects of the natural world, they are not totally consistent with each other. Finding a consistent, unified theory is hard, and we don't yet have one. But there's one mathematically attractive attempt, string theory, which is conceptually appealing even though there's no observational evidence for it.

String theory holds that what we usually consider to be individual points of spacetime, dimensionless dots with no interesting structure of their own, are actually very, very tiny multidimensional surfaces with complicated shapes. The standard analogy is a garden hose. Seen from some way off, a hose looks like a line, which is a onedimensional space - the dimension being distance along the hose. Look more closely, though, and you see that the hose has two extra dimensions, at right angles to that line, and that its shape in those directions is a circular band.

Maybe our own universe is a bit like that hosepipe. Unless we look very closely, all we see is three dimensions of space plus one of time - relativity. An awful lot of physics is observed in those dimensions alone, so phenomena of that type have a nice four-dimensional description - relativity again. But other things might happen along extra `hidden' dimensions, like the thickness of the hose. For instance, suppose that at each point of the apparent four-dimensional spacetime, what seems to be a point is actually a tiny circle, sticking out at right angles to spacetime itself. That circle could vibrate. If so, then it would resemble the quantum description of a particle. Particles have various `quantum numbers' such as spin. These numbers occur as whole number multiples of some basic amount. So do vibrations of a circle: either one wave fits into the circle, or two, or three ... but not two and a quarter, say.

This is why it's called `string theory'. Each point of spacetime is replaced by a tiny loop of string.

In order to reconstruct something that agrees with quantum theory, however, we can't actually use a circular string. There are too many distinct quantum numbers, and plenty of other problems that have to be overcome. The suggestion is that instead of a circle, we have to use a more complicated, higher-dimensional shape, known as a `bran'[43]. Think of this as a surface, only more so. There are many distinct topological types of surface: a sphere, a doughnut, two doughnuts joined together, three doughnuts ... and in more dimensions than two, there are more exotic possibilities.

Particles correspond to tiny closed strings that loop around the brane. There are lots of different ways to loop a string round a doughnut - once through the hole, twice, three times ... The physical laws depend on the shape of the brane and the paths followed by these loops.

The current favourite brane has six dimensions, making ten in all. The extra dimensions are thought to be curled up very tightly, smaller than the Planck length, which is the size at which the universe becomes grainy. It is virtually impossible to observe anything that small, because the graininess blurs everything and the fine detail cannot be seen. So there's no hope of observing any extra dimensions directly. However, there are several ways to infer their presence indirectly. In fact, the recently discovered acceleration in the rate of expansion of the universe can be explained in that manner. Of course, this explanation may not be correct: we need more evidence.

The ideas here change almost by the day, so we don't have to commit ourselves to the currently favoured six-dimensional set-up. We can contemplate any number of different branes and differently arranged loops. Each choice - call it a loopy brane - has a particular energy, related to the shape of the brane, how tightly it is curled up, and how tightly the loops wind round it. This energy is the 'vacuum energy' of the associated physical theory. In quantum mechanics, a vacuum is a seething mass of particles and antiparticles coming into existence for a brief instant before they collide and annihilate each other again. The vacuum energy measures how violently they seethe. We can use the vacuum energy to infer which loopy brane corresponds to our own universe, whose vacuum energy is extraordinarily small. Until recently it was thought to be zero, but it's now thought to be about 1/120plex units, where a unit is one Planck mass per cubic Planck length, which is a googol grammes per cubic metre.

We now encounter a cosmic `three bears' story. Macho Daddy Bear prefers a vacuum energy larger than +1/118plex units, but such a spacetime would be subject to local expansions far more energetic than a supernova. Wimpy Mummy Bear prefers a vacuum energy smaller than -1/120plex units (note the minus sign), but then spacetime contracts in a cosmic crunch and disappears. Baby Bear and Goldilocks like their vacuum energy to be `just right': somewhere in the incredibly tiny range between +1/118plex and -1/120plex units. That is the Goldilocks zone in which life as we know it might possibly exist.

It is no coincidence that we inhabit a universe whose vacuum energy lies in the Goldilocks zone, because we are life as we know it. If we lived in any other kind of universe, we would be life as we don't know it. Not impossible, but not us.

This is our old friend the anthropic principle, employed in an entirely sensible way to relate the way we function to the kind of universe that we need to function in. The deep question here is not `why do we live in a universe like that?', but `why does there exist a universe like that, for us to live in?' This is the vexed issue of cosmological fine-tuning, and the improbability of a random universe hitting just the right numbers is often used to prove that something - they always say `We don't know, could be an alien,' but what they're all thinking is: 'God'- must have set our universe up to be just right for us.

The string theorists are made of sterner stuff, and they have a more sensible answer.

In 2000 Bousso and Polchinski combined string theory with an earlier idea of Steven Weinberg to explain why we shouldn't be surprised that a universe with the right level of vacuum energy exists. Their basic idea is that the phase space of possible universes is absolutely gigantic. It is bigger than, say, 500plex. Those 500plex universes distribute their vacuum energies densely in the range -1 to +1 units. The resulting numbers are much more closely packed than the 1/118plex units that determine the scale of the `acceptable' range of vacuum energies for life as we know it. Although only a very tiny proportion of those 500plex universes fall inside that range, there are so many of them that that a tiny proportion is still absolutely gigantic - here, around 382plex. So a whacking great 382plex universes, from a phase space of 500plex loopy branes, are capable of supporting our kind of life.

However, that's still a very small proportion. If you pick a loopy brane at random, the odds are overwhelmingly great that it won't fall inside the Goldilocks range.

Not a problem. The string theorists have an answer to that. If you wait long enough, such a universe will necessarily come into being. In fact, all universes in the phase space of loopy branes will eventually become the `real' universe. And when the real universe's loopy brane gets into the Goldilocks range, the inhabitants of that universe will not know about all that waiting. Their sense of time will start from the instant when that particular loopy brane first occurred.

String theory not only tells us that we're here because we're here - it explains why a suitable `here' must exist.

The reason why all of those 500plex or so universes can legitimately be considered `real' in string theory stems from two features of that theory. The first is a systematic way to describe all the possible loopy branes that might occur. The second invokes a bit of quantum to explain why, in the long run, they will occur. Briefly: the phase space of loopy branes can be represented as an `energy landscape', which we'll name the branescape. Each position in the landscape corresponds to one possible choice of loopy brane; the height at that point corresponds to the associated vacuum energy.

Peaks of the branescape represent loopy branes with high vacuum energy, valleys represent loopy branes with low vacuum energy. Stable loopy branes lie in the valleys. Universes whose hidden dimensions look like those particular loopy branes are themselves stable ... so these are the ones that can exist, physically, for more than a split second.

In hilly districts of the world, the landscape is rugged, meaning that it has a lot of peaks and valleys. They get closer together than elsewhere, but they are still generally isolated from each other. The branescape is very rugged indeed, and it has a huge number of valleys. But all of the valleys' vacuum energies have to fit inside the range from -1 units to +1 units. With so many numbers to pack in, they get squashed very close together.

In order for a universe to support life as we know it, the vacuum energy has to lie in the Goldilocks zone where everything is just right. And there are so many loopy branes that a huge number of them must have vacuum energies that fall inside it.

Vastly more will fall outside that range, but never mind.

The theory has one major advantage: it explains why our universe has such a small vacuum energy, without requiring it to be zero - which, we now know, it isn't.

The upshot of all the maths, then, is that every stable universe sits in some valley of the branescape, and an awful lot of them (though a tiny proportion of the whole) lie in the Goldilocks range. But all of those universes are potential, not actual. There is only one real universe. So if we merely pick a loopy brane at random, the chance of hitting the Goldilocks zone is pretty much zero. You wouldn't bet on a horse at those odds, let alone a universe.

Fortunately, good old quantum gallops to our rescue. Quantum systems can, and do, `tunnel' from one energy valley to another. The uncertainty principle lets them borrow enough energy to do that, and then pay it back so quickly that the corresponding uncertainty about timing prevents anyone noticing. So, if you wait long enough - umptyplexplexplex years, perhaps, or umptyplexplexplexplex if that's too short - then a single quantum universe will explore every valley in the entire branescape. Along the way, at some stage it finds itself in a Goldilocks valley. Life like ours then arises, and wonders why it's there.

It's not aware of the umptyplexplexplexplex years that have already passed in the multiverse: just of the few billion that have passed since the wandering universe tunnelled its way into the Goldilocks range. Now, and only now, do its human-like inhabitants start to ask why it's possible for them to exist, given such ridiculous odds to the contrary. Eventually, if they're bright enough, they work out that thanks to the branescape and quantum, the true odds are a dead certainty.

It's a beautiful story, even if it turns out to be wrong.