I was watching an old episode of *Horizon* on BBC4 (available to watch on the iPlayer here), which was on the subject of infinity; it was mainly about the mathematical concept, although it touched on the implications it has for physics. I remember watching it before, but I can’t have being paying proper attention, because I don’t recall my mind bending in the way it did this last weekend. I seem to have remembered the pure maths parts more accurately than the parts relating to physics. Certainly, I was able to greet the mathematical proof that there must be more than one infinity, and that some infinities are bigger than others, as an old friend.

If I understood correctly, the point is that, if you start with a number, you can then write an infinite list of other numbers starting from the same root point. For example:

1

11

111

1111

11111

111111

1111111 etc

You can keep doing this forever – you can add infinitely many 1s to the end of the number, meaning that the series (the list of numbers) is infinite. But, even though this series is infinite, it’s obvious that there are numbers it doesn’t contain – 22, say. So it’s therefore possible to imagine a second series of numbers that contains all the numbers from the first series, but also the other number we just thought of. This means that, even though the first series was infinite, the second series must have more numbers in it than the first. Therefore, and no matter how weird it instinctively feels, it follows that there must be more than one infinity, and that some of those infinities are larger than others.

In fact, because the original series is infinitely long, and every number in that infinite series can be altered, there must be an infinite number of infinities, each one nested inside another infinity larger than itself – like a Russian Doll that never stops. It’s tempting to think there must be an ‘infinite infinity’ that contains every other infinity – but, no matter how large the series of numbers gets, you can always include another number and, hey presto!, you just created another, larger, infinity.

This is, on its own, a pleasingly mind-bending idea. It’s a logically illogical (or do I mean illogically logical?) concept, and deeply inimical to common sense: if infinity is endless, how can there be something larger than infinity? And if infinity is the point at which counting breaks down – because there are more things than could ever be counted, even if time were infinite – then how is it possible to count the number of things in different infinities and say “This one over here’s bigger”? But then I quite like things that are inimical to common sense, and I also like using logic to stretch abstract concepts to the point where they break, as people who’ve had the misfortune to discuss philosophy with me know to their cost (it must be the deconstructionist in me). Although in this case, of course, the concept hasn’t actually broken; it’s been stretched to the point where it looks thoroughly illogical, but it keeps stolidly working: you can always include another number in an infinite series of numbers, and create thereby a larger infinity.*

The part of the programme which I didn’t recall as clearly – and ended up seriously bending my mind – was the application of infinity to the physical universe. I got the impression from the programme that physicists don’t much like the concept of infinity, even as they rely on it to make their equations work. I think it’s perhaps one of those things that they think is an approximation of reality, rather than reality itself, because it’s complex and inelegant; in other words, an area that has yet to be adequately explained. As a complete and utter layman, I think I see their point: I don’t have a huge amount of difficulty getting my head round the idea that something abstract can be infinite, but the universe is a real thing, made out of real stuff. Accepting that something like that can be infinite is a much harder stretch.

I mean, the universe has, ever since the big bang, been expanding out from a single point. I can see that, if time is infinite, the universe might keep expanding until it’s infinitely large, but is that the same thing as saying that at this present moment – a measurably finite time after the big bang – the universe is already infinite? But then again, if it isn’t currently infinite, when will it ever become so? The universe might keep growing forever, but will there ever come a point when it isn’t theoretically capable of being measured? That seems to leave me with only two options for describing the universe’s relationship with infinity, neither of which seems very satisfying. I can say that a constantly expanding universe will never be infinite, even though anything that expands forever must eventually become infinite; or I can say that the universe always has been and always will be infinite, even though currently it is measurably finite: both seem like essentially meaningless statements.

There was quite a lot of talk in the programme about ‘the paradoxes of infinity’, so perhaps this is one of them. Or perhaps I just misunderstood something fundamental. Either way round, my brain hurts.

Anyway, one of the consequences of applying infinity to the physical universe is that, if existence is infinite, it becomes necessary to conclude that anything that *can* exist *must* exist. As a teenage sci fi geek I got quite familiar with this idea. Although the novels I read and TV shows I watched were borrowing far more from the many-worlds interpretation of quantum mechanics than they were engaging directly with the idea of infinity, they nonetheless suggested there were an infinite number of versions of me somewhere out in the multiverse. This was always explained, ultimately, in terms of a decision-tree: that, at every point where I (or any one of my ancestors, or any third party or impersonal event that impacted on me or my ancestors) did something, somewhere else an alternative version of me (or them, or it) did something different. Some of those versions of reality were radically different – in many, the planet Earth never formed – and some were almost identical – perhaps I turned on the TV at 21:58:57 on the 27^{th} June 2008 instead of 21:58:58.

The key thing, anyway, is that, even with all this, I still retained my uniqueness; none of the other versions of ‘me’ – even the ones that were very, very similar – was precisely *me*; at the very least, they lived in a universe that was different to mine. And, if you’re following the many-worlds theory then that works (I think: this is way over my head, except at the most basic level); a ‘new universe’ is only produced at each moment of decoherence, so each universe is subtly different to every other. But what if the universe, or the multiverse, is infinite: what then?

One of the contributors to the programme explained it by using pool balls; I’ve adapted this for something that will work in a blog post. So, let’s imagine a universe that is made up of two constituent parts, **A** and **B**. There is a limited number of ways these parts can be arranged:

**AA**

**AB**

**BA**

**BB**

Because there are only four possible permutations, once I have written each of the four permutations down, any other time I write a permutation it must be an exact duplicate of one of the earlier permutations: whether I write **AA**, **AB**, **BA** or **BB**, I am replicating something that already exists.

This is, of course, an incredibly simple universe, but the principle holds for our vastly more complex one. There may be an unimaginable number of possible permutations, but given enough time and space the permutations will eventually start to repeat. This means that, if the universe or multiverse is infinite, then everything that appears in it once – the Earth, you, me – must appear more than once. In fact, if the multi-/universe is infinite, everything must repeat an infinite number of times. There aren’t just two precisely identical versions of you and me – there are an infinite number of precisely identical versions of you and me.

The more you think about it, the more bizarre this idea gets. It’s already quite odd to think that now, at the precise moment you sit reading this sentence, there are an infinite number of precisely identical other yous all sitting and reading this sentence – and, more, that they have done everything at exactly the same time as you up until this point, and will continue to do everything at exactly the same time until all of the separate-but-identical yous cease to exist. We’re used to thinking of, say, Usain Bolt’s attainments as deeply impressive personal achievements, but don’t they start to look rather less impressive when you realise that they’re not unique at all, but have been replicated an infinite number of times? In fact, what’s the point of you, personally, doing anything at all, if an infinite number of other yous are going to do it anyway? So what if you don’t do the washing up? Let one of the other yous do it instead, dozy sap that s/he is.

Also, what happens to the concept of free will? That idea is based on the assumption that I take independent decisions on my own behalf – but if there are infinitely many people, all taking precisely the same decisions at precisely the same time, how can any of my decisions be said to belong to me personally? Aren’t I just following the herd? What makes this me the ‘original’, and all the others copy-cats? Mightn’t it be the other way around, and it’s me who’s slavishly imitating the actions of someone else, rather than existing as an individual in my own right? Do the words ‘I’ and ‘me’ even retain any useful content when I have to use them to refer to people who are not me, but simultaneously are me? In an infinite universe can I even make the claim that ‘I’ (the unique, individual me) exists at all?

This is all good, mind-bending stuff, and I recommend watching the documentary if this is the kind of thing that floats your boat too. There are irritating aspects to the documentary – the inclusion of the actor Steven Berkoff seemed particularly gratuitous to me, since his sections added nothing to the ideas that were being discussed. I’d guess they wanted him there to make things seem weird and wibbly-wobbly but, truthfully, the panel of mathematicians and physicists they’d assembled for the programme were already resplendently weird (as you’d expect and hope), and given they were shot in the semi-dark and the editing left in the kind of hesitations and tics that are normally left out, the idea that the weirdness quotient needed a boost is frankly almost insulting. Still these are minor quibbles – the documentary as a whole is interesting, and well worth watching. But be aware that it may make your brain hurt – especially if your brain is as slow and imperfect as mine.

—

* – This is one of the things that attracts me towards the study of Maths, even though I have fairly limited mathematical ability – I find the idea that there may be abstract concepts that can’t be deconstructed fascinating. You see, there ought to be, because if every abstract idea can be deconstructed – that is shown to ultimately fall apart as the result of trying to reconcile things that can’t be reconciled – then it follows that the abstract concept of deconstruction should, itself, eventually do the same thing: if deconstruction is an accurate description of the way abstract ideas work, then there must be abstract concepts that cannot be deconstructed. It’s already clear to me (and anyone familiar with the Sokal hoax) that deconstruction falls catastrophically apart in the face of bodies of knowledge based on empirical observation – i.e. science – but those are not fully abstract ideas, because even the most theoretical of physicists ultimately bases her equations in observable reality (assuming I understand correctly; don’t forget I’m a complete layman when it comes to all scientific matters). Mathematical concepts, on the other hand, are fully abstract, and it seems they are likely candidates for the kinds of concepts that are impervious to deconstruction.**

** – Yes, yes I know I talked about things like ‘empirical observation’ and ‘observable reality’, and people who know a little bit about philosophy (i.e. deconstructionists) think these are controversial terms for philosophers, but they aren’t, really, any more: most philosophers accept that the real world really exists, even if they can only *infer* rather than *prove* its existence. As the old joke has it: I’ll be happy to debate the non-existence of the physical world with anyone who’s first jumped out of an aeroplane at 30,000 feet without a parachute…

I haven’t seen the programme, but it sounds like the BBC mashing a lot of distantly related ideas together. Which it does quite a lot in its pop science.

The most useful thing I ever read about infinity was an article by Ian Stewart on how the word has several meanings which produce confusion when conflated. 10/0=infinity…in the sense of ‘not defined within the system of mathematics’. There’s the notion of an infinite (neverending, limitless) expanse of space, but as Stephen Hawking pointed out in

Brief History of Time, the universe could also be infinite in the same way as a circle is infinite – you can trace the line as much as you want, but you’ll never come to the end.The business with the infinite lists of numbers sounds like the Diagonal Slash Argument, which I first came across in Roger Penrose’s

Emporor’s New Mine– a book which I’d be willing to bet is on your shelf somewhere.Zeno’s Arrow paradox is a (pseudo)paradox of infinity, which I’ve been heatedly discussion on a bulletin board recently. It’s the idea that, having divided a quantity of space traveled in a duration of time into infinitely small pieces, you can’t then reassemble them into a finite quantity because…being infinitely small, they can’t ever add up to anything that isn’t also infinitely small.

And the reason we should care about an ancient Greek thinker (and his elder boyfriend Parmenides) getting confused about infinity? Georg Hegel came up with a pseudosolution to the pseudoparadox, and made it a centerpiece of his system. Which was taken up by Frederich Engels, from where it was taken up by Vladimir Lenin…who used it to win a leadership contest thinly disguised as a philosophical debate in the Bolshevik party.

So if Zeno had been more careful, 20th century European history might have been completely different! :-).

As for the many worlds notion, it’s not the only interpretation of quantum mechanics. There’s another one which states that there’s only one universe, but it’s fuzzy and indeterminate, existing in many states simultaneously – but the fuzzyness is only significant at the quantum level.

At least, that’s my understanding. Which may be no better than your average BBC documentary maker.

Brill, shared to facebook :-)

It sounds a lot like Nietzsche and eternal return! He thought there was a sort of solace in the mystical idea that time repeated and looped back on itself so that everything repeated.

He also thought this was something that could be proved on a similar basis to your example with the permutations above: if there are only finite objects, in the course of infinite time the same arrangements of objects would have to recur. However, I read a disproof (which I now can’t find) of this argument: it was along the lines that you could have a clock with three hands, which all start pointing at 12 but move at different speeds; no matter how long the clock ran, the hands would never point at 12 at the same time again.

Oh actually turns out not to be that obscure, it’s on the Wikipedia entry for Eternal Return!

“Even if there were exceedingly few things in a finite space in an infinite time, they would not have to repeat in the same configurations. Suppose there were three wheels of equal size, rotating on the same axis, one point marked on the circumference of each wheel, and these three points lined up in one straight line. If the second wheel rotated twice as fast as the first, and if the speed of the third wheel was 1/π of the speed of the first, the initial line-up would never recur”

@Kapitano Thanks for the fascinating extra info. You’re right about the mathematical proof of endless – and endlessly larger – infinities; I remembered it was called the diagonal something-or-other from the programme, but I was feeling lazy, and didn’t bother looking it up. So thanks for doing it for me. :o)

@blackberryjuniper Thanks, and thanks!

@Andrew Godfrey Thanks for the comment – it’s fascinating stuff. :o)

Off the top of my head, one apparent problem with the disproof is that it doesn’t engage with the idea that there is an infinite array of infinities, each one larger than its predecessor. While there may be an infinite series of locations for the three hands – meaning that the same location never recurs within that series – that doesn’t preclude the existence of an even larger infinity which contains the whole of the first series, and the point at which the hands occupy the same location again. In the same way, an infinite series of numbers built from the digit 1 can never contain the digit 2 – but this doesn’t preclude the existence of a larger infinity which contains both the whole of the first series and the digit 2.

But, as always, I may have completely missed the point!