The world is awash
with divisive arguments,

conflict, fake news,

victimhood,

exploitation, prejudice,
bigotry, blame, shouting,

and minuscule attention spans.

It can sometimes seem

that we are doomed to take sides,

be stuck in echo chambers,

and never agree again.

It can sometimes seem
like a race to the bottom,

where everyone is calling out
somebody else's privilege

and vying to show that they

are the most hard-done-by person
in the conversation.

How can we make sense

in a world that doesn't?

I have a tool for understanding
this confusing world of ours,

a tool that you might not expect:

abstract mathematics.

I am a pure mathematician.

Traditionally, pure maths
is like the theory of maths,

where applied maths is applied
to real problems like building bridges

and flying planes

and controlling traffic flow.

But I'm going to talk about a way
that pure maths applies directly

to our a daily lives as a way of thinking.

I don't solve quadratic equations
to help me with my daily life,

but I do use mathematical thinking
to help me understand arguments

and to empathize with other people.

And so pure maths helps me
with the entire human world.

But before I talk about
the entire human world,

I need to talk about something
that you might think of

as irrelevant schools maths:

factors of numbers.

We're going to start by thinking
about the factors of 30.

Now, if this makes you shudder
with bad memories of school maths lessons,

I sympathize,

because I found school
maths lessons boring too.

But I'm pretty sure we are going
to take this in a direction

that is very different
from what happened at school.

So what are the factors of 30?

Maybe you can remember them.
We'll work them out.

It's one, two, three,

five, six,

10, 15, and 30.

It's not very interesting.

It's a bunch of numbers
in a straight line.

We can make it more interesting
by thinking about which of these numbers

are also factors of each other
and drawing a picture,

a bit like a family tree

to show those relationships.

So 30 is going to be at the top
like a kind of great grandparent.

Six, 10, and 15 go into 30.

Five goes into 10 and 15.

Two goes in six and 10.

Three goes into six and 15.

And one goes into two, three, and five.

So now we see that 10
is not divisible by three,

but that is this the corners of a cube,

which is I think a bit more interesting

than a bunch of numbers
in a straight line.

We can see something more here.
There's a hierarchy going on.

At the bottom level is the number one,

then there's the numbers
two, three, and five,

and nothing goes into those
except one and themselves.

You might remember
this means they're prime.

At the next level up,
we have six, 10, and 15,

and each of those is a product
of two prime factors.

So six is two times three,

10 is two times five,

15 is three times five,

and then at the top, we have 30,

which is a product of three prime numbers,

two times three times five.

So I could redraw this diagram
using those numbers instead.

So we see that we've got
two, three, and five at the top,

we have pairs of numbers
at the next level,

and we have single elements
at the next level,

and then the empty set at the bottom.

And each of those arrows shows
losing one of your numbers in the set.

Now maybe it can be clear

that it doesn't really matter
what those numbers are.

In fact it doesn't matter what they are.

So we could replace them with
something like A, B, and C instead

and we get the same picture.

So now this has become very abstract.

The numbers have turned into letters.

But there is a point to this abstraction,

which is that it now suddenly
becomes very widely applicable,

because A, B, and C could be anything.

For example, they could be
three types of privilege:

rich, white, and male.

So then at the next level,
we have rich white people.

Here we have rich male people.

Here we have white male people.

Then we have rich, white, and male.

And finally people with
none of those types of privilege.