*Kit Fine – Philosopher*

Numbers are strange. They are not physical objects. No one has bumped into the number two or tripped over the number three; not even your crazy math professor. They are not mental objects either. The thought of your beloved isn’t your beloved no matter how much you might want it to be. And no more is the thought of the number three, the number three. Nor do numbers exist in space or time. You don’t expect to find the number three in the kitchen cupboard, and you don’t need to worry that numbers once didn’t exist or might one day cease to exist.

But even though numbers are far removed from the familiar world of thoughts and things, they’re intimately connected to that world because we do things with numbers. We count with them, we measure with them, we formulate our scientific theories with them. So this makes it all the stranger what they are.

How can they be so far removed from the familiar world and yet so intimately connected to it? In this talk I want to consider three views about the nature of number that were developed by mathematicians and philosophers around the end of the 19th century and the beginning of the 20th century. All of these views presuppose that strictly speaking what we count are not things, but sets of things.

A set is just many things, any things you like, considered as one. So for example, we have the set of beer bottles that you drank last night. The bottles are put in these braces to indicate that the six bottles are being considered as one object. Then we have the set consisting of your two favorite pets, Fido and Felix. Or we have a set consisting of all the natural numbers, so they’re put together in this very big set: 0, 1, 2, 3, 4 and so on. And so what we do when we count is associate a number with a set.

In the case of the beer bottles, the number six, assuming you’re not too drunk to count them. In the case of your pets, the number two. And in the case of the natural numbers, when we put them into one big set, is going to be some infinite number.

The first view I want to consider about the nature of numbers was developed independently by two great philosopher mathematicians, Gottlob Frege and Bertrand Russell. These two individuals were very different from one another. Russell came from the English aristocracy; Frege from the comfortable German middle class.

Russell was a crusading liberal; Frege, I’m sorry to say, was a proto-Nazi. Russell had four wives, and innumerable mistresses; Frege had a single wife, and as far as I know, enjoyed a happy, staid marital existence. But despite these differences, they had more or less the same view about the nature of number. So what was it? Well let’s take the number two as an example. Two can be used to number any two-membered set or pair. So it can be used to number the set whose members are Frege and Russell. Or it can be used to number the set consisting of your favorite pets, Fido and Felix. Or it can be used to number Dickens’ famous two cities, London and Paris. I insisted that London be placed first there.

Now the idea of Russell and Frege was to put all of these pairs into one big set. We pile them all into one big set, and that would be the number two. So the number two would be a set of sets, and these sets would just be all the pairs that could be counted by the number two.

Similarly, for all other numbers, the number three would be the set of all triples, the number four the set of all quadruples, and so on. A simple and beautiful theory. Unfortunately, it led to contradiction. I can’t give a demonstration of the contradiction here, but I can give you a feel for how it arose. You’ll recall that the number two was the set of all pairs, all pairs of whatever. So in particular, it would include pairs that themselves contain the number two.

So let’s look at the particular such pair, the pair consisting of the number two and the number one. Then that pair, the pair {1, 2}, would itself be inside the number two. So the number two would contain itself, and that looks as if it’s impossible. So here’s an analogy: imagine a very hungry serpent that tries to eat its own tail.

Now, it could succeed in doing this – this is the best we could do by way of illustration – This is gross, but still possible. But imagine now that the serpent is so ravenous that it attempts to eat itself in its entirety. That’s not even possible because then, the serpent’s stomach would have to be inside its stomach. And that’s what happens with the number two. The number two, as you see, has itself inside of its very own stomach. What was to be done?

The mathematician John von Neumann came up with a brilliant solution; von Neumann was perhaps one of the most versatile mathematicians who ever existed. He helped invent game theory and the modern computer. He was a prodigy and had the most amazing computational skills. So what was his solution? There he is. He said: “Well look, rather than take the number two to be the set of all pairs, take it to be a particular pair.”

Well, which pair would it be? He suggested that the number two should be the set of its predecessors. Two has two predecessors, zero and one. We take two to be the set whose members are zero and one. But we still have numbers; we have zero and one. Well, zero is the set of its predecessors. Zero has no predecessors, so it’s what’s called the ‘null set,’ the set without any members. And one has one predecessor, which is zero. So one is the set whose sole member is zero. So there we have two defined, one defined and zero defined.

If we put these definitions together, we get the set. The number two is the set whose two members are the null set, which is the number zero, and the set whose sole member is the null set, which is the number one. So that’s according to von Neumann what the number two is; it’s sets all the way down – sets, not turtles – And you actually hit rock bottom, too.

And similarly for all other numbers, the number three would be an even more complicated thing, and so on. Remember, the Frege-Russell view gave birth to monsters. Here, we no longer have a monster; the monster has turned into an angel, because although the number two contains other numbers, it doesn’t contain itself. The monster is always eating a smaller monster, so to speak. It doesn’t get in the way of itself. This view is generally accepted by philosophers and mathematicians today, but it also has its difficulties.

One difficulty that especially bothers me is there’s nothing special about the number two. We want the number two to be what is common to all pairs, but von Neumann’s number two is just one pair among many, and there’s no special way in which that pair is what’s common to all pairs. So it doesn’t make the number two special anyway; it’s just one pair among many.

We come now to the final view, and the one I like most of all. It’s a view that’s generally dismissed or ignored by philosophers and mathematicians of today. It was developed by Georg Cantor in the late 19th century. Cantor was a multi-talented individual, a brilliant violinist with wide-ranging interests ranging from religion to literature. But he’s best known for his theory of infinite number. Cantor wanted to count not only finite collections – I know there are a lot of people here, but it’s still a finite number – so not just the finite collections, like the number of people here, or the number of stars in the Milky Way, he also wanted to count infinite collections, like the collection of all natural numbers or the collection of all points in space.

And to this end, he attempted to develop a general theory of number. So what was his view? Again, let’s consider the number two. Let’s take two objects, Fido and Felix. Now Cantor said: “Look, let’s deprive these two objects of all of their individuating features beyond their being distinct from one another.” So we remove their fur, we remove their flesh and blood, until we’re simply left with two bare objects – what he called units without any differentiating features. I hope there are no animal lovers amongst you. But anyway, this is what happens to pets when Cantor gets hold of them.

So what are these units? Well, take the two dollars in your bank account – I hope you still have two dollars left after paying the admission fee. These two dollars aren’t any particular dollars, but when you go to the ATM machine, you can redeem them for two particular dollars. So they aren’t those particular dollars, but they can be redeemed for any two particular dollars. This is what Cantor’s units are like.

But when you go to the Cantorian ATM machine to redeem your units, you get back any two objects or whatever. It’s the ultimate lucky dip. Cantor’s idea was this: we take the number two to be the set of these two units. So we take these two units, which could be derived from any two objects, and the number two is the set of those two units. And similarly, for all other numbers; the number three would be the set of three units, and so on and so forth.

So we have three views on the table. The Frege-Russell view according to which the number two is the set of all pairs; the view of von Neumann, according to which the number two is the set whose members are zero and one; and the Cantorian view, according to which two is the set of two units. The Frege-Russell view breeds monsters, so we can’t have it.

The von Neumann view doesn’t properly account for why the number two is common to all pairs. The Cantorian view suffers from neither of those difficulties. It doesn’t breed monsters because the number two only contains units; it doesn’t itself contain the number two. And it’s, in an obvious sense, common to all pairs, because it’s derived by this process of abstraction, or stripping away, from each pair. So thanks to Cantor, we now know what numbers are. Thank you.