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The world is awash
with divisive arguments,

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conflict, fake news,

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victimhood,

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exploitation, prejudice,
bigotry, blame, shouting,

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and minuscule attention spans.

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It can sometimes seem

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that we are doomed to take sides,

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be stuck in echo chambers,

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and never agree again.

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It can sometimes seem
like a race to the bottom,

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where everyone is calling out
somebody else's privilege

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and vying to show that they

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

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How can we make sense

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in a world that doesn't?

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

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a tool that you might not expect:

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abstract mathematics.

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I am a pure mathematician.

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Traditionally, pure maths
is like the theory of maths,

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where applied maths is applied
to real problems like building bridges

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and flying planes

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and controlling traffic flow.

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

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to our a daily lives as a way of thinking.

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

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but I do use mathematical thinking
to help me understand arguments

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and to empathize with other people.

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

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But before I talk about
the entire human world,

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I need to talk about something
that you might think of

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as irrelevant schools maths:

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factors of numbers.

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

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

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I sympathize,

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because I found school
maths lessons boring too.

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

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that is very different
from what happened at school.

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So what are the factors of 30?

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

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It's one, two, three,

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five, six,

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10, 15, and 30.

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It's not very interesting.

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

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

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are also factors of each other
and drawing a picture,

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a bit like a family tree

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to show those relationships.

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

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Six, 10, and 15 go into 30.

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Five goes into 10 and 15.

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Two goes in six and 10.

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Three goes into six and 15.

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And one goes into two, three, and five.

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

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but that is this the corners of a cube,

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which is I think a bit more interesting

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than a bunch of numbers
in a straight line.

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

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At the bottom level is the number one,

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

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and nothing goes into those
except one and themselves.

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You might remember
this means they're prime.

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

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

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So six is two times three,

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10 is two times five,

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15 is three times five,

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and then at the top, we have 30,

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which is a product of three prime numbers,

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two times three times five.

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So I could redraw this diagram
using those numbers instead.

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

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we have pairs of numbers
at the next level,

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and we have single elements
at the next level,

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and then the empty set at the bottom.

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

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Now maybe it can be clear

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

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In fact it doesn't matter what they are.

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

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and we get the same picture.

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So now this has become very abstract.

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The numbers have turned into letters.

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But there is a point to this abstraction,

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which is that it now suddenly
becomes very widely applicable,

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because A, B, and C could be anything.

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For example, they could be
three types of privilege:

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rich, white, and male.

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

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Here we have rich male people.

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Here we have white male people.

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Then we have rich, white, and male.

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And finally people with
none of those types of privilege.