On the Question of Infinity

Something a little different today. The politics news is a total horrorshow, but it’s the same horrorshow we’ve been seeing for months, so nothing new to say about that, and I haven’t quite finished the next Batpost or my post on Smile. So for now you’re getting a thought about infinity. This may be gibberish — I’m very, very, sick this week.

This was partly inspired by Plok, who in an email that started about a song we’re co-writing, but as is the way with him touched on a million other subjects, said:
“By the way, I really do believe that an system of infinite universes is an Aleph-1 thing? So what is the fucking point in giving them numbers, I ask you. I know no one cares about this, but it bugs me: the number-line simply ISN’T INFINITE ENOUGH to number all the worlds, and if I were Grant Morrison I would employ this fact in a big cosmic crossover story. Worlds unnumbered because they are un-numerable…all control is an illusion…”

And this got me thinking about how the concept of infinity is really the central unanswered question of our times, and how unsure we are of whether the concept even has a meaning. Because the question of whether it does or not should hold an importance equivalent to the question of whether God exists — and in fact the two may be different ways of phrasing the same question.

Because it is an open question as to whether the word “infinity” actually has a meaning at all. If the universe is of finite extent, and contains a finite number of discrete particles, then the word “infinity” is literally meaningless. There is a finite (though incomprehensibly large) number of ways in which that finite number of particles can be arranged. That means that there exists some number past which it is not, even in principle, possible to represent in the universe. A last meaningful number.

If that is the case, then it is literally meaningless to talk about “infinity”, or a number higher than that number. And most of what we know about low-level physics seems to suggest that — that reality is fundamentally granular, and that time, space, and matter come in lumps beyond which there is no “smaller”. If this is correct, then there are only a finite number of possible states of the universe, and thus only a finite number of numbers.

However, everything we know about large-scale physics suggest we live in a universe that is continuous, rather than discrete — an analogue, rather than a digital, universe (a “classical” rather than “quantum” universe, to use the terms physicists use). Such a universe makes infinity as a concept make sense.

One of the main tasks facing physicists is to resolve this, and to unify these two views to find what’s really there. It’s the consensus among physicists that when they do resolve the classical and quantum views of the universe, though, they will show that the quantum view is right.

But there’s a problem with that — because the maths that we use to figure out the laws of physics is the same maths that requires infinity in order to work. If the universe is finite, bounded, and granular, for example, the proof that there is no highest prime number is trivially false — if there’s a highest number, then the highest prime number must be less than or equal to that.

So there are three options, none of which are especially palatable. The first is that our physics is wrong. This doesn’t seem to be the case. The second is that our mathematics — including the mathematics we used to develop the physics — is wrong. If this is the case, then we need a very good explanation as to how this wrong mathematics manages to get us to the right answer when we use it in the real world.

The only other option is one that several mathematicians, and some theologians and philosophers, but no-one else, think might be the case — that maths is true, but not in this universe. In other words, there is some other, greater, universe outside the physical universe, which contains truths which are necessarily true, but which have no physical nature, and which we can access with our minds.

This, also, seems on the face of it to be not the most likely of options.

So either everything we know about the universe is wrong, or it’s right but we somehow got to the right answer using the wrong tools (tools which are the basis of things like the computer on which I’m typing this and the information network through which you’re reading it, somehow, despite being wrong), or we can magically know stuff that has no basis in physical reality but which is somehow true on some deeper level.

Of course, if infinity *is* a coherent concept, then that also leads to a whole lot of other, even more incomprehensible, conclusions of the kind that philosophers speculate about (for example that we’re probably computer simulations, or that we should all expect to find ourselves, though not anyone else we know, to be immortal). The point here is that there’s a major flaw in the most basic ways we understand the universe, and that it’s a flaw no-one seems to have the first idea how to begin addressing in a sensible way.

Anyway, that’s your blog post for today. This may not be the most coherent blog post you ever read — as I said, I’m extremely ill at the moment — but its incoherence at least has the advantage of being about something that is a genuine mystery, and a genuinely confusing problem. Which puts it ahead of all the political incoherence in the world at the moment, which seems to boil down to “should [the US/the UK/Hungary/name your country]” utterly destroy itself in order to hurt foreign people because stupid, angry, people want that?”

The answer to that one seems pretty obvious to me, even in this state.

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10 Responses to On the Question of Infinity

  1. Paul Barker says:

    Personally I find the idea of a Finite Universe a bit claustrophobic & slightly UKIPpy.

  2. Not incoherent at all. Very interesting,

  3. Andrew Hickey says:

    Thanks — given the way current politics is exacerbating my stress-related illnesses, I’m probably going to be writing more in the semi-delirious state I was in yesterday, so I’m glad it has interesting results ;)

  4. Brian says:

    I’m not grasping why the concept of infinite numbers is tied to the concept of infinite space. Numbers have utility beyond the limit of things that can be counted, don’t they?

    • Andrew Hickey says:

      Utility, perhaps, but meaning? There is, of course, a debate about whether one can meaningfully talk about concepts that have no referent in the real world ( http://plato.stanford.edu/entries/nonexistent-objects/#ConMeiTheNonAbs ) but a truly finite, bounded, granular universe would also knock out half the arguments that we can talk meaningfully about those things (such as the other worlds strategy listed in that link).
      My own view is that it is only meaningful to talk about things which have a referent in the universe — certainly when talking about things that affect things which do have such referents.

      • Quantum says:

        I have to raise a couple of points here, sorry, it’s a compulsion. First off, the idea of words having a referent is usually blamed on Frege and falls apart quite quickly IMHO. For example abstract concepts (courage, absurdity, comedy) do not have referents in the world, yet they have no trouble being meaningful, and more importantly when we use words for things that don’t exist yet and may never exist (Matryoshka brains, computronium, a LibDem govt.) those words indubitably have meaning *right now*, whether or not a ‘real’ thing ever exists to be the word’s referent.
        Secondly, the meaningfulness of propositions being derived from their relation to the real world *also* falls apart in a similar way. For example the Verification Principle espoused by the logical positivists of the Vienna Circle says only statements that are empirically verifiable are meaningful, but that principle bites itself on the arse because it isn’t empirically verifiable.
        All of which is a long winded way of saying in a finite universe the concept of infinity is still valid. In a universe of only 100 things, the number 101 is still meaningful.

  5. plok says:

    Hmm, well there is a “greater universe” out there where math is real, though it isn’t really the one you’re talking about…it’s the universe beyond the particle horizon! All the other parts of “Big Bang Space” that got blown up by cosmological inflation, that we’ll never again interact with, no never, unless somehow there’s a Big Crunch. So “if the universe is of finite extent”, hmm, that’s a tricky one…perhaps it’s like a number-line itself, starts at 1 but then never stops expanding?

  6. Mike Taylor says:

    For a mathematician (which I used to be, back in my undergraduate days) there is no question about whether infinity is real. It is just as real a concept as the square root of minus one, which also has no direct physical representation in our physical universe. To a mathematician, the physical universe is an irrelevance at best, a distraction at worse: concepts exist independently of it.

    To put it another way, if you struggle with the idea of whether infinity “exists”, then you should similarly be struggling with the idea of whether three exists. It’s just as abstract. You can count three things, but you can’t point to a thing that is three.

    It’s all in Plato, all in Plato: Bless me, what do they teach them at these schools?

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