An unexpected tool for understanding inequality abstract math Eugenia Cheng
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 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?
Well, they’re the numbers that go into 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 into 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 this is 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.
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.
And I’m going to put back in
the rest of the adjectives for emphasis.
So here we have rich, white
non-male people,
to remind us that there are
nonbinary people we need to include.
Here we have rich, nonwhite male people.
Here we have non-rich, white male people,
rich, nonwhite, non-male,
non-rich, white, non-male
and non-rich, nonwhite, male.
And at the bottom,
with the least privilege,
non-rich, nonwhite, non-male people.
We have gone from a diagram
of factors of 30
to a diagram of interaction
of different types of privilege.
And there are many things
we can learn from this diagram, I think.
The first is that each arrow represents
a direct loss of one type of privilege.
Sometimes people mistakenly think
that white privilege means
all white people are better off
than all nonwhite people.
Some people point at superrich
black sports stars and say,
“See? They’re really rich.
White privilege doesn’t exist.”
But that’s not what the theory
of white privilege says.
It says that if that superrich sports star
had all the same characteristics
but they were also white,
we would expect them
to be better off in society.
There is something else
we can understand from this diagram
if we look along a row.
If we look along the second-to-top row,
where people have two types of privilege,
we might be able to see
that they’re not all particularly equal.
For example, rich white women
are probably much better off in society
than poor white men,
and rich black men are probably
somewhere in between.
So it’s really more skewed like this,
and the same on the bottom level.
But we can actually take it further
and look at the interactions
between those two middle levels.
Because rich, nonwhite non-men
might well be better off in society
than poor white men.
Think about some extreme
examples, like Michelle Obama,
Oprah Winfrey.
They’re definitely better off
than poor, white, unemployed homeless men.
So actually, the diagram
is more skewed like this.
And that tension exists
between the layers
of privilege in the diagram
and the absolute privilege
that people experience in society.
And this has helped me to understand
why some poor white men
are so angry in society at the moment.
Because they are considered to be high up
in this cuboid of privilege,
but in terms of absolute privilege,
they don’t actually feel the effect of it.
And I believe that understanding
the root of that anger
is much more productive
than just being angry at them in return.
Seeing these abstract structures
can also help us switch contexts
and see that different people
are at the top in different contexts.
In our original diagram,
rich white men were at the top,
but if we restricted
our attention to non-men,
we would see that they are here,
and now the rich, white
non-men are at the top.
So we could move to
a whole context of women,
and our three types of privilege
could now be rich, white and cisgendered.
Remember that “cisgendered” means
that your gender identity does match
the gender you were assigned at birth.
So now we see that rich, white cis women
occupy the analogous situation
that rich white men did
in broader society.
And this has helped me understand
why there is so much anger
towards rich white women,
especially in some parts
of the feminist movement at the moment,
because perhaps they’re prone
to seeing themselves as underprivileged
relative to white men,
and they forget how overprivileged
they are relative to nonwhite women.
We can all use these abstract structures
to help us pivot between situations
in which we are more privileged
and less privileged.
We are all more privileged than somebody
and less privileged than somebody else.
For example, I know and I feel
that as an Asian person,
I am less privileged than white people
because of white privilege.
But I also understand
that I am probably among
the most privileged of nonwhite people,
and this helps me pivot
between those two contexts.
And in terms of wealth,
I don’t think I’m super rich.
I’m not as rich as the kind of people
who don’t have to work.
But I am doing fine,
and that’s a much better
situation to be in
than people who are really struggling,
maybe are unemployed
or working at minimum wage.
I perform these pivots in my head
to help me understand experiences
from other people’s points of view,
which brings me to this
possibly surprising conclusion:
that abstract mathematics
is highly relevant to our daily lives
and can even help us to understand
and empathize with other people.
My wish is that everybody would try
to understand other people more
and work with them together,
rather than competing with them
and trying to show that they’re wrong.
And I believe that abstract
mathematical thinking
can help us achieve that.
Thank you.
(Applause)