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)