The pattern is clear. If we work to umpty decimal places, there are 10umpty 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.5
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 proved6 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 aleph-umptyplex 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.
&n
bsp; 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.7 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 1028 metres from here’. Not 1028 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 10118 metres away. And beyond that …
We need a good way to talk about very big numbers. Symbols like 10118 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 some
thing 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 10umpty, 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.
Science of Discworld III Page 19