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.

(Laughter)

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.

(Laughter)

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.

(Applause)