This is going to be rather a philosophical post, so hang onto your hats. For those not used to this sort of thing, let me assure you that in a larger frame, the following is of practical importance. Very often, I think that people mainly just don’t take infinity seriously enough. If you don’t take infinity seriously enough you can end up losing track of the limitations involved in being finite, which is a source of delusion in all sorts of ways.

Infinity is something we have a concept of, but cannot experience or even imagine. That concept is thus one that leaps beyond experience, and makes sense to us only as a negation of what is finite. Not surprisingly, then, it defies all our attempts to tame it conceptually and treat it as though it was another finite thing we experience. Metaphysical beliefs, of the kind avoided by the Middle Way, all depend on infinity in one way or another, and a lot of confusion about them seems to arise from not considering the full implications of making an infinite claim.

The problems arise when people use *probability*, or any other kind of assessment based on experience, and apply it to infinity. Probability is based on the number of occasions we have observed a particular phenomenon in the past, taking into account the conditions we have observed it in, and estimating from that how likely that phenomenon is in the future. However, it makes no difference how many times you’ve observed something in the past when you apply it to infinity: the probability will be zero. That’s because whenever you put a finite thing in a context of infinity, whatever measurable features it has become infinitesimal by comparison – they keep on being infinitely dwarfed by the infinite quantity.

So, let’s take something for which there are a great many supporting observations – the operation of the laws of physics. Also, let us assume the universe is infinite.

*What is the probability of the laws of physics operating in the whole universe? Zero.*

Yes, it’s so easy to forget what a small part we play in this. Such a result obviously shows the complete inappropriateness of applying probability at all to infinity.

This seems to offer a pretty decisive response to determinism. If we define determinism as the belief that all events in the universe are sufficiently caused, and let’s assume that we have observed a very great many causal relationships which appear to be regular and theoretically predictable. However, it doesn’t matter how many we have observed…

*What is the probability of determinism being correct? Zero.*

Of course, we don’t know for sure whether the universe is infinite, but we certainly haven’t established that it isn’t. Whenever scientists come up with an explanation of observed phenomena that suggests the universe is finite, there seems still to be an infinite possibility of other things they haven’t observed that would undermine this claim. The same point applies both to infinity in space and in time – forwards and backwards.

But I don’t want to just have a go at excessive claims associated with science. Religion, too, is full of problems created by a lack of respect for infinity. One of the basic characteristics of God is that he is infinite. This means, of course, that there can be no such thing as ‘evidence’ for the existence of God, or indeed for the non-existence of God, whether you’re talking about signs of design, causal explanations, or arguments from the presence of morality or beauty.

*What is the probability of evidence establishing the existence of an infinite being? Zero.*

The same point applies just as much to attempts to disprove the existence of God. Whatever finite evidence you apply to this disproof will have no effect on an infinite being.

*What is the probability of evidence establishing the non-existence of an infinite being? Zero.*

The Dawkins Foundation recently put an advert on London buses saying “There’s probably no God.” Sorry Dawkins, but there’s a basic confusion in that statement. Unless you’re talking about a God that’s not infinite – but that would have nothing to do with the God believed in by Christians, Muslims and Jews.

But the problems raised by infinity are by no means confined to these ‘big’ topics. In the purely conceptual terms in which infinity works, any space or time is infinitely divisible. Take a piece of paper and a pair of scissors and there is, in practice, a limitation to the number of times you can cut up the paper within the physical capacities of the scissors. However, *conceptually* no such limitations exist. Even when you get down to sub-atomic particles in the paper you can *theoretically *carry on chopping those infinitely into smaller and smaller pieces.

One of the implications of infinite divisibility is that there are an infinite number of possible ways of ‘chopping up’ the world using concepts. So, imagine that you’re chopping up a map of the world with your scissors. You could chop it up by cutting along the national boundaries, so that you had China, Russia, Mongolia etc as separate pieces. But alternatively you could combine a bit or Russia with a bit of China, merge Mongolia into another bit of China, and so on. In fact, there are an infinite number of ways you could chop up that map. In the same way we could conceptually rearrange the boundaries of the world in an infinite number of possible ways. The same applies to any ways we chop up phenomena into conceptual groups – let’s say the division of animals into species, genera etc. Since the qualities of animals could be infinitely chopped up and recombined, there are an infinite number of possible conceptual schemes (i.e. basic ways of organising our experience in terms of ideas) for organising species. The same applies to all the other boundaries between objects or qualities in our experience. If you take the greyscale between black and white, you can have not just fifty shades of grey, but an infinite number.

So, whenever we make a claim about the essential features of the world, the probabilities of those essential features being chopped up correctly are infinitesimal.

*What are the chances that any given conceptual scheme is correct? Zero.*

*What are the chances that any given concept truly identifies features of the universe? Zero.*

*What are the chances of any given claim, using those concepts, being true? Zero.*

*What are the chances of (for example) it being true that a particular dog is black? Zero.*

The boundaries of blackness could be drawn in an infinite number of possible places, and you have no way of telling whether you have drawn it in the right place to include the colour of the dog’s coat. Thus it is that even everyday claims about things being *true* rely on a confusion about infinity. The whole idea of a set of concepts arranged in a claim lining up with a truth out there (or with a falsehood out there) is just infinitely wrong.

But don’t groan or panic. This is scepticism – in the sense used by Pyrrho in ancient Greece, casting doubt on both positive and negative claims – and scepticism is not a problem, but a tool for liberation.

Firstly, there’s a much better way of explaining why our concepts mean what they do for us, without them having to line up with essential truth or falsity, and it’s called the embodied meaning thesis. I won’t attempt to explain that here, but you can see it explained elsewhere. Secondly, our confidence in the correctness of what we believe doesn’t have to be based on delusions about how it lines up with truths out there. We can start in a different place, respecting the infinity of infinity rather than trespassing on it, but also enjoying the finiteness of our finiteness.

Our bodies are the basis of our finiteness, and it is our bodies that give us a sense of justifiable confidence in beliefs that have served us well in practice. You don’t need to start disbelieving in your coffee-mug on the grounds that the concept you are using to understand it has been randomly selected out of an infinite range of possibilities. Instead, you need to accept that your life has been sculpted out of finite experience, and your coffee mug is part of that too. We do have plenty of grounds to doubt claims of ‘truth’, which do not take into account the infinite space they have been selected from, but no grounds to worry about a lack of *justification* as it impacts our lives.

Picture by tdadamemd (Wikimedia Commons – Creative Commons licence)

I’d not heard you make these claims about probability before, Robert, so this was interesting to read. One immediate concern is that the probability of the negation of a proposition is equal to 1 minus the probability of that proposition (this is a straightforward corollary of the Additivity axiom of probability theory). So it follows from what you say here that the probability of indeterminism being true is 1, that the probability of the laws of physics being somewhere transgressed is 1, and that the probability that Bo (the black Portuguese Water Dog owned by the Obama family) is not black is 1. These conclusions are indefensible, though. (Bo does have some white fur on his paws and tummy (see [A]), but there’s no way it’s *maximally probable* that he’s not black!)

References:

[A] http://en.wikipedia.org/wiki/Bo_%28dog%29

Hi Mike,

Thanks for this. It sounds as though the probability theory may have been designed with an inappropriate application of the Aristotelian dichotomy. As usual agnosticism is not being seriously considered as an option, perhaps? It also needs to be clear here that I am talking about the chances of a precise lineup between a proposition and a state of affairs. If that chance is nil when the claim is positive, it is also nil when the chance is negative in the sense of the precise boundary drawn being right but the claim being on the other side of it. The chances of it being negative in the sense of the boundary being unknown are, I agree, 1, but that is just confirming that we do not know where the boundary is.

Hi Robert,

I don’t see any reason to charge probability theory with neglecting agnosticism. Agnosticism about a proposition can be understood as the assignment of equal probability to that proposition and its negation: given Additivity, that means assigning probability 0.5 to each. If you wish to maintain that determinism and indeterminism each have probability zero, you must fundamentally depart from the standard mathematical treatment of probability, according to which the probability of , where P and Q are mutually exclusive, is equal to the sum of the probabilities of P and Q. Personally, I would need to actually see the details of any such non-standard treatment of probability, and have some assurance that it didn’t yield wildly implausible results about probability, before I could consider it an option. (For example, what probability do you assign to the disjunction of determinism and indeterminism? That disjunction is a logical truth, so it ought to be probability 1. But in that case you’re committed to saying that although it is maximally probable that either determinism or indeterminism is true, determinism and indeterminism are each maximally likely not to be true. This is implausible.) And, of course, if you reject standard probability theory, you can’t appeal to it to justify your claims about infinity and zero probability above.

I’m afraid it doesn’t make things any clearer for me when you say that you are ‘talking about the chances of a precise lineup between a proposition and a state of affairs’. Consider the proposition that water contains hydrogen. What more is it for that proposition to ‘precisely line up with a state of affairs’ than for water to contain hydrogen?

…oops, that should be “…the probability of ‘P or Q’, where P and Q are mutually exclusive,..”

Hi Mike,

I think the problem is that the established philosophical and mathematical tradition interprets probability in relation to actual events (which are just assumed to be representable), rather than our chances of having an accurate representation of events. In other words, they neglect the human perspective and implicitly jump to a God’s eye view. Agnosticism is not a position of assigning equal probabilities, and I think that involves a complete misunderstanding of agnosticism. Rather, as I’ve been arguing, absolute or infinite claims are just not a matter of probability – and neither is the rejection of such claims.

I am using probability theory here only to show its absurdities when applied to infinity, to show that probability doesn’t apply usefully. The accusation of circularity applies just as much to the ways you have just demonstrated that such theory rules out the very possibility of an agnostic position a priori. I do reject any version of probability theory that makes such assumptions, but that doesn’t mean that I reject the whole useful concept of probability. For me probability is based merely on the Humean approach – a way of conceptualising the implications of past experience.

In discussing determinism and indeterminism there, you are still thinking about the possibilities of those positions being true, aside from the limitations of our representations of them, rather than the chances of our representations being correct given that they are applied to the whole universe. By the chances of the representation being correct, I mean the chances for the whole conceptual scheme that talks in terms of determinism and indeterminism being correct, not the chances of either of them being true in opposition to each other and in their own terms.

Thanks –

I guess I’m really struggling to understand what your position is. You say that “agnosticism is not a position of assigning equal probabilities”, and yet, if I’ve understood you, you *do* wish to assign equal probabilities to determinism and its negation – namely, zero probability. You say that you are concerned with “the chances for the whole conceptual scheme that talks in terms of determinism and indeterminism being correct, not the chances of either of them being true in opposition to each other and in their own terms”, but in your post you *do* make a claim about chance of determinism being true: you claim that the chance is nil.

My objection is that it is dubiously coherent to assign zero probability both to a proposition and its negation. Although you’ve raised general misgivings about the established tradition on probability (which, I’m afraid, I’m also not sure I’ve understood), and have said that you favour a Humean approach, I don’t think you’ve responded to this objection. And I continue to wonder what probability you can coherently assign to the disjunction of determinism and indeterminism.

Hi Mike,

As far as I’m concerned, the assignation of zero probability to events that we would normally ascribe a large or at least substantial probability to is a way of showing that the notion of probability is absurd when applied to infinity. I’m learning from this that I would probably have done better to draw out that point, to avoid people such as yourself getting stuck on the attribution of zero probability. Read it like this – ‘If one is going to assign probability to such a thing, which is absurd, it would be zero.’ I took it for granted that zero probability would be read as absurd, but obviously I was wrong to assume this.

As for ‘not understanding’ – I’m routinely told by analytic philosophers that what I’m saying is ‘unintelligible’. After many fruitless attempts to give them the ‘detail’ they apparently wanted but that made no difference to their assumptions, I have tended to draw the conclusion that they mean ‘unintelligible’ within the framework of assumptions they are willing to consider seriously, and that if what I’m saying challenges that framework then they are not willing to consider it, or at least not willing to devote the time they would need to understanding it, given how deeply their neural channels are etched in incompatible directions. It does not seem to be unintelligible to other people – particularly people with a more interdisciplinary background, who can think in a variety of ways, and consider synthetically and practically how they might relate to each other.

It doesn’t matter so much whether you understand or agree with this particular argument as whether you can understand the more basic perspective it comes from. Although this argument about infinity, along with whatever flaws it has, is my responsibility, it’s more broadly just another version of Pyrrhonist sceptical argument, of a kind that you’ll also find reflected in some Buddhist material, and more recently in the embodied meaning thesis (which is based on neuroscientific evidence) and in Nassim Nicholas Taleb. It amounts to a new paradigm, or perhaps an old one that never entirely went away. As I suggested, it requires a human’s-eye view perspective, rather than the implicit God’s eye view that seems to be routine in analytic philosophy – and moving from the latter to the former can involve a paradigm shift a bit like a shift in perception as you reinterpret an ambiguous picture.

I have added in an extra sentence “Such a result obviously shows the complete inappropriateness of applying probability at all to infinity. ” to try to avoid any further such literalistic readings as Mike’s. I’ve also been reflecting on the limitations of the reduction ad absurdum form that this argument takes (when you take a way of thinking to extremes to show it is not viable). When people are so used to thinking in ways that bear no relationship to experience, and these have been enshrined in conventions that they take for granted, they may not even notice that a reduction to absurdity is intended.