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 [1]
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
[1]Yes, traditionally `Indigo' goes here, but that's silly - Indigo is just another shade of blue. You could equally well insert `Turquoise' between Green and Blue. Indigo was just included because seven is more mystical than six. Rewriting history, we find that we have left a place for Octarine, the Discworld's eighth colour. Well, seventh, actually. Septarine, anyone?
to counting.[1] 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
[1] This is why, even today when the lustre of `the new mathematics' has all but worn to dust, small children in mathematics classes spend hours drawing squiggly lines between circles containing pictures of cats to circles containing pictures of flowers, busily `matching' the two sets. Neither the children nor their teachers have the foggiest idea why they are doing this. In fact they re doing it because, decades ago, a bunch of demented educators couldn't understand that just because something is logically prior to another, it may not be sensible to teach them in that order. Real mathematicians, who knew that you always put the roof on the house before you dug the foundation trench, looked on in bemused horror.
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.' [1]
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 [2] that if umpty is any infinite cardinal, the umptyplex is a bigger one. So aleph-zeroplexplex is
[1] Briefly: since the bit before the decimal point is a whole number, taking that into account multiplies the answer by aleph-zero. Now aleph-zero x aleph-zeroplex is less than or equal to aleph-zeroplex x aleph-zeroplex, which is (2 x aleph-zeroplex, which is aleph-zeroplex. OK?
[2] The proof isn't hard, but it's sophisticated. If you want to see it, consult a textbook on the foundations of mathematics.
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 descript
ions.
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.
Darwin's Watch Page 18
