The Abstract Mathematical Universe

Hi,

My name is Kevin Buzzard, and I'm a professor of pure mathematics

at Imperial College London,

and I'm going to talk to you about numbers.

So here's an easy question about numbers to get started with. What's to add to?

I think you will know the answer to that.

But if I ask you to prove what to add to is, how are you going to do it?

How are you going to show me? The to add to is for

I guess you could look around in your bag and maybe you've

got four pens in there so you could pull out some pens.

You could put two pens in one hand

to pens on the other hand

and then put them together. Count them and you've got four pens.

Or maybe you've got two books in your bag and the person

sitting next to you they've got two books in their bag,

so you put your books on the table.

The other person puts their books on the table.

You count the books and you've got four books

and you want you want to tell me then that two and two is four, but

I'm going to say to you that maybe you haven't proved that to add to his four.

Maybe you've been doing some physics experiments there.

You've been doing experimental science. You haven't shown that to add to his four.

You've shown that two pens at two pens is four pens

or that two books at two books is four books.

Let's take a more exotic object. Let's take elephants, for example.

Maybe many of you have never even seen an elephant.

How do you actually know for sure that two elephants and two

more elephants if you put them together again to make four elephants?

Maybe elephants are different. Maybe there'd be five.

Elephants are like 37 elephants.

How do we really know that there's some fundamental property

of books and elephants and whatever exists in the universe

that makes to add to always come out to before?

This is one of the questions are going to be talking about in these lectures.

So I'll tell you,

what's going on really is that when we talk

about two pens or two elephants were using to,

that's kind of an objective.

This is this is not This is not a noun. It's an adjective.

I was talking about the abstract number two, which is a different thing

when we start talking about properties of things in the universe like

we could talk about a happy child or a happy baby,

a happy dog you can.

You can look at these things and they have some kind of properties in common.

They're all happy. They have some abstract property of happiness.

But happiness is kind of an abstract now.

It's a strange thing, happy as an objective and happiness is an abstract noun.

And that's what's going on with numbers as well.

We've got four pens,

but with defining a property of that collection of

pens for is being used as an objective there.

And what I want to step away from is the objective.

And I want to talk about the abstract noun for

and so this is This is a kind of an abstraction.

This is different from a concrete property that we can measure in the real world.

When we talk about a red pen or a red toy, we have some abstract concept of redness,

and physicists can come along and try and start X,

explaining that in terms of wavelengths and frequencies of light.

Or biologists can come and try and tell us about redness,

using the rods and cones in our eyes.

Similarly,

mathematicians can come along and trying to explain

to you about the fondness of things.

Being four ish is some kind of property of a collection of things

very early on in our lives.

We learn very quickly about this property because we start learning to count

and we begin by count on our fingers we count 1234,

and we learned that this is somehow four fingers.

And then we abstract this concept onto other things where we've got pens,

we want to count the pens.

We start matching up the pens with our fingers,

one pen for one finger, 234 pens or four fingers.

And now the pens and the fingers have got some kind of abstract property in common

four NUS.

And this is what mathematicians study the abstract concept of four.

So where does this concept of foreignness live?

It's kind of an interesting question because, actually,

uh,

once you learn about counting, you start realising that you can always add one.

You've got four. Then there's another one. You've got five and then another one.

You've got six,

and this goes on and on and on.

And the abstract concept of numbers, This seems to be an infinite idea.

There seems to be infinitely many numbers, and of course, mathematicians can prove

that there are infinitely many numbers.

And yet physicists tell us that the universe is finite.

The universe has got a finite size,

and it's only actually got about 10 to the 70 atoms in.

So this is quite strange,

because our abstraction doesn't seem to fit into the universe at all.

So what is what is what? Our numbers?

If we can always add one, then numbers are going to be too big to fit into our universe.

At school,

you learn about the number line and the

number line is this very innocent looking thing.

Just a straight line starts at one keeps going up, maybe starts at zero, depending on

depending on how sophisticated your model is.

It keeps going up. Just add one. Add one, add one.

It keeps moving on to the right, but yet this innocent little picture

there's something very strange about it, because that line goes on forever.

The number line

that line is too big to fit in our universe,

the number I lives in a more abstract place

called the Mathematical universe.

And that's where mathematicians do their work. That's where pure mathematicians

do their work.

I'd like to finish by suggesting that if

the mathematical universe is not the real universe,

then maybe there's no point to it.

And maybe there's no point to doing mathematics at all.

But in practise, this turns out not to be the case.

In practise, mathematicians have discovered that by creating

abstract stuff in this abstract universe and working with it and

constructing abstract constructs,

they can use those things to model reality.

And so, for example, applied mathematicians

and study sound waves

create abstract models of these things using

numbers which maybe don't really exist.

In our real universe, at least, big numbers might not exist,

and then they can use these things to design.

For example, materials that will stop certain waves,

sound waves of a certain frequency,

getting through and making industrial machinery quieter.

Or statisticians can use abstract stuff that lives in the math magical universe and

use it to do model things like like virus spreading through the population,

for example,

and computer scientists can use abstract stuff created in the

mathematical universe and use them to create codes complex codes,

which are basically unbreakable.

And we can use those to transfer data across networks like the Internet.

So the abstract mathematical universe, which is what I'm going to talk about,

contains these strange,

infinite collection of numbers and yet seems to have a lot of importance

in our real world universe today.