Mathematics

Does infinity exist?

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Well, mathematicians all the time talk about infinitary structures. To start with the very simplest examples, they talk about the set of all natural numbers, {0, 1, 2, 3, ...}: and they also talk about the set of all subsets of the natural numbers. And they introduce "infinite cardinal numbers" which indicate the size of such infinite sets. There's a beautiful theorem by Cantor which shows that, on a very natural understanding of "number of members", the set of all numbers has a smaller number of members than the set of all subsets of the set of all numbers. So there are different infinite cardinal numbers, which we can order in size. Indeed, again on natural assumptions, there's an infinity of them.

Is that little reminder about what mathematicians get up to enough to settle the question whether "infinity exists"? Well, perhaps not. There remain a number of questions here. Here's one (so to speak) about the pure mathematics, and one about applied mathematics.

First, then, we might say "Sure, mathematicians talk about infinite sets, and infinite numbers: but then Conan Doyle talked about Sherlock Holmes. But Holmes doesn't really exist: he's just in a story. And maybe the mathematicians are just telling a story, and there aren't really infinite sets and infinite numbers." But to get to grips with that suggestion, we'd have to tackle the whole question of whether "fictionalism" about mathematics is a viable position -- and that's far too much to take on here!

Second, even if we do allow the existence of infinite structures as entities populating the mathematician's abstract realm (denizens of "Plato's heaven"), we might still wonder whether such structures are exemplified in the physical world. We do use heavily infinitary mathematics in our best going physical theories (they presume, e.g., that space is infinitely divisible); but is that really essential? Could it be that, e.g., the world is ultimately grainy at the small scale, and is finite too at the big scale, and there are no physically existing infinite structures? Again, that's far too big a question to take on now.

So all I've been able to do here is make some preliminary clarifications, and suggest two different ways of sharpening up the original question into something we might get our teeth into. But then a lot of philosophy is like that!

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