[MUSIC PLAYING] EUGENIA CHENG:
Thank you very much. Well, first of all, thank you
to Google for inviting me here to speak to you. And thank you for
coming in what I suppose is your lunchtime to listen
to maths during a lunch break, which is pretty amazing. So I am going to talk
about logic and about how to make sense in a
world that doesn't, because a lot of the world does
not make sense at the moment. In particular, many people
don't seem to make sense in the world at the moment. And I have previously
written books to show how maths is fun,
and delicious, and amazing. And this is less about how it's
fun and more about how it's important, because the previous
books were about how it's important, but recent events
in the last couple of years have driven me to
feeling the need to say slightly more urgent
and more, dare I say it, political things about
how maths can help us. And this might be surprising,
because usually pure maths is thought of as
being pure and remote. I am a pure mathematician,
I am a category theorist, that is my research. And category theory is sometimes
thought of as the most abstract type of mathematics. Well, I say that pure
maths is a framework for agreeing on things. But in general,
pure maths is often thought of as really only being
relevant to applied maths. And applied maths is then
helpful to science, possibly, and science is helpful to
engineering and medicine. And engineering and medicine
are helpful to the world. And maths is helpful to
kind of the numerical part of the world. And we might think of
differential equations, or solving equations, and
calculating things in finance, and science, and business. And those are the ways
in which maths is useful. But that enables people
to say, well, that's fine, I'm glad somebody does that,
but it doesn't have to be me, it's not relevant to me,
someone else can do it. And I'm sure many
of you do do it. And there are many people
in the world who just ignore it and hope for the best. Well, I say that, actually,
because pure maths is a framework for
agreeing on things, it's a framework for
how to think well. And because it's
about how to think, it's actually about the entire
human world, or at least the parts of the human
world that thinks, which might not be all
of it at the moment. [LAUGHTER] But wouldn't it be
great if everyone were able to think better? That's what I say. And I truly believe
that pure mathematics is a discipline
for thinking better and for how to have better
arguments, because I'm not claiming to tell
everyone what to think, but I wish we could
have better arguments. Because at the moment, we
seem to be mired in conflict, divisive arguments, fake news,
yelling, bigotry, prejudice, hate, exploitation, victimhood. And it can sometimes seem like
there is not a productive way to have a discussion. But I think there
is a productive way to have discussion, it just
might not be on social media, necessarily. But we can try. And so this is what
my book is about, it's about how can we have more
productive conversations using lessons from
abstract mathematics? So I'm going to talk
about analogies and how pure maths uses analogies. I'm going to talk about the
interconnectedness of things and how we should
always think about how things are interconnected. I'm going to talk about
relationships of various kinds. I'm going to talk about pivoting
between different contexts so that we can help ourselves
understand other people from their point of view. And finally, I'm going to talk
about what I think intelligence really actually is. So first of all, analogies-- I say that pure
maths is a framework for agreeing on things. It's also a theory of analogies. Because fundamentally,
what it says is, there are lots of
different situations that have something
in common if we just forget some of the pesky
details about them. And then, that takes us from
the real world of things into the abstract world of
ideas where more things become the same by analogy. But what pure maths does
that we don't necessarily do in normal life
is, we make very precise what the analogy is. We don't just say this
thing is analogous to that. We say, what is the analogy
that we're thinking about here? And that removes a
lot of ambiguity. And having a framework
for agreeing on things is partly about removing
possible ambiguities. So here is an
example of how this works at a very basic level. If you think about two
apples and two bananas, then you can go, oh,
these things have something in common,
they're both two things. And that's how we
get the number two. It's an analogy between a lot
of different sets of two things. And when we first teach
small children how to count, we have to wait for them
to make the abstract leap from the objects in front
of them to the concept. And you can't really do
that for them, you just have to sort of wait, show them
over and over again, and wait until they make that leap. It's a leap of abstraction. Now, there are other ways
we could have done this. We could instead
have said that they are both two fruits,
which is still true, it just encompasses less stuff. So if we had instead made
the analogy via two fruits, then we would not be able to
encompass two chairs in this because that is not two fruits. This doesn't mean
it's a wrong analogy, it just means it's an analogy
at a different level that encompasses fewer things. Whereas now if we go all
the way up to two things, then we get to encompass
the two chairs as well. And a lot of pure maths is
about finding the right level for the situation
that you're in. It's not that there is one
absolutely right level, maths, as it turns out, is not
about getting the right answer. It's about different
points of view that illuminates situations
in different ways. And I think arguments
in life are often not really about one
person being right and one person being wrong,
but a sense in which something is right. So here is the sense in
which two chairs are not analogous to two bananas along
with a sense in which they are. And then once we've made precise
those two different levels, we can then have a
discussion about which one is a better level to use
for the current discussion. And this happens at
a more profound level in maths when we think
about, say, 1 plus 2 and we compare that to 2 plus 3. And this is where
that thing happens that makes some
people very scared, where their numbers
turn into letters. And these are both
examples of a plus b. But the point of doing that
is not to terrify everyone, although that can sometimes
be what seems like the point. The point is to encompass
more examples, that's always the idea of abstraction, to
study a lot of different things at the same time to
save your brain power and be more efficient. And this is a good thing to
do if you are lazy like me and you don't want to do the
same thing over and over again. And I'd say a lot of pure
maths comes from that. You do the same
thing a few times, and you go, oh, I don't want
to do that any more, I'm going to make a theory
that will do it for me. So you're kind of
front load your effort, and then you never have to
make the effort ever again, like getting a dishwasher. So this is very similar
to 1 times 2 and 2 times 3, which can be
abstracted to a times b. But now, we can say that
whole left-hand part is analogous to you the
entire right-hand part. And these are both an
example of a something b. They're a way of combining
some a and some b. And so we can go even
one level more abstract, and say this is a
bleh b, and it's a way of combining two things. This is called a binary
operation in mathematics. And it's one level more abstract
that you probably only see if you do an undergraduate
degree in maths and you do something
like group theory where you look at things
with binary operations. Now, sometimes the top level
is a really good place to look, and sometimes the middle level
is a better place to look. It just depends what
you're trying to do. So if you're thinking
about something like 1 plus 2 equals 2 plus 1,
which you might remember is the commutativity
of addition. There's also commutativity
of multiplication, which says that 1 times
2 equals 2 times 1, and a times b equals b times a. And if we go up
to the top level, we can study both of those
phenomena at the same time. However, if we want to look
at undoing the process, we can't go up to the
top level, because at that level, addition and
multiplication aren't the same, because you can't always undo
the process of multiplication, because if you
multiply by 0, you can't undo that, you're done. Whereas whatever you
add, you can always subtract it to
get it back again. You can divide by any
number apart from 0. And so, at that level,
addition and multiplication are not analogous. So we have to pick carefully. And the main point
of talking about this is to show that making precise
which level we're talking about helps us to be clear about
what's going to be useful now. And what isn't, isn't
true in that context. So a more real-life
example, and my book is full of actual life
examples of things that people argue about on
the internet all the time, is when a straight marriage
and gay marriage are analogous. So some people say
that they do not believe that gay people
should be allowed to marry each other because
they think that marriage should be between a man and a woman. Now, that is not exactly
an illogical argument. But it's also not a
very profound argument, because all they've really done
is said the same thing twice. So if I say, I like orange juice
because I like orange juice, that's not illogical. It just hasn't got
anywhere at all, you haven't achieved anything. And so just being
logical versus illogical is not the be all and end all
of what an intelligent person should try and do. What we should try and do is
actually make progress and gain understanding, not just say the
same thing over and over again with slightly different words. So the sense in which these
two things are not analogous is if you do think
that marriage is about an unrelated
man and woman, and then it is true that
same-sex marriage does not come under that. However, if you think that it
is about two unrelated adults, then same-sex marriage
does come under that. And the question shouldn't
be whether those things are the same or not, the
question should be, which level is a good
level for us to consider? Now, the next thing
that happens if you go as far as this is that
somebody will get really upset and say, oh, well, the
next thing you know, we'll be allowing incest. And so what they've done is,
they've gone up a further level and gone up to the
level of two adults. And then, the next
thing you know, you could go up to the
level of just two humans. And then, the next
thing you know, you could go up to the level
of two living creatures. And then, you could just go up
to the level of two creatures. This is the slide that could
have been redacted had there been any minors in the room. Now, all of these levels
are logically valid. I would argue that some of them
are not very morally valid. But that's a different question. And what is certainly
not logically valid is to claim that just
because somebody has accepted this level, some people
claim that means that you've accepted all the other levels. And that is not logically valid. So once we've made precise
what all these levels are, we can now have a discussion
about which level anyone thinks that we should go up to. But we should not conflate
all the different levels. And the trouble with not making
those levels precise is that somebody can hallucinate,
willfully or otherwise, that somebody else has shot all
the way up to the top when they haven't. So this is one of the
advantages, I think, of the precision of
abstract mathematics. I'd like to move on to talking
about the interconnectedness of things, because
everything is interconnected. Unfortunately, many
people often like to blame exactly one thing,
or exactly one person, or one group of people. And that is a way to
simplify situations, but I don't think
it's a very good way. And yes, we are
all trying to make sense of a complicated world. And that requires somehow
simplifying the world. But just ignoring
huge quantities of it is not a great way
of doing it, I think. I think that a better
way of doing it is to become more intelligent,
because then the world becomes simpler relative to
your intelligence. And then, you
simplify the world. So here is one of my favorite
diagrams of interconnectedness. We all know this picture. It's not a geographically
accurate picture. It doesn't show
geographical location. What it shows is the
interconnectedness of stations by different tube lines. And this is all we need to know
if we're trying to get from one tube station to another. Once we're in the
train, we don't really need to know exactly where
the tunnel goes, necessarily. We just need to know where
it's going to pop out at. And this does mean that
hapless tourists sometimes find themselves taking a
tube ridiculous distance because they didn't realize
that actually if they just went above ground, it was
two minutes' walk that way. This is an abstract
picture because it's just showing the
interconnectedness of things. This is the geographical
tube map picture. It is much harder to
understand if you're trying to use it
to get from A to B, but that doesn't
mean it's worthless. It's quite interesting to see
where the tube lines actually go and to see where the tube
stations really are in London, I think. So it's not that one of these is
right and one of them is wrong, it's just that it depends
what you're trying to achieve and that depending on what
you're trying to achieve, you should look
at the things that are relevant to the
thing you're trying to achieve or understand,
and then be able to switch between thinking about relevant
these things in this situation and those things
in that situation. And that's one of the things
that abstract mathematics does a lot, is that it
doesn't say, this is actually the full
explanation of this situation. It says, here is an explanation
of an aspect of this situation. And then on another
day when we're worrying about
something else, we can think about another aspect. Relationship
breakdown is something that involves a lot
of interconnectedness and also a lot of blame. And often when
relationships break down, everyone blames one person, and
then the other people all blame the other person. So often people take sides. And often the people in
the relationship breakdown take sides. Now, there are situations
where it might really be one person's fault in cases
of extreme domestic abuse. But usually, it's about
a breakdown of things going in both directions. And here's an example of how
I think about it sometimes. Supposing Alex is
someone who doesn't like feeling disrespected. And if Alex feels
disrespected, then Alex is unable to show love. And then, maybe Alex's
partner Sam is someone who really likes to feel loved. And if Sam feels
unloved, then Sam is unable to show
respect, in which case Alex feels disrespected. And we get in this cycle. And it can get worse,
and worse, and worse. Now, I can classify these
arrows in different ways. So these arrows are
arrows about action that people take in
response to something, and these arrows are
about feelings that people have in response to something. And they can get into a
very bad vicious cycle. Once we've seen it's
this vicious cycle by the interconnectedness
of these arrows, we can think about how
to break the cycle. We can think about
which of these arrows is the weakest one
that we could break. We can think about whether
it's more difficult to change your feelings or your actions. I think it's harder
to change feelings because feelings just are,
you just respond to things, but you don't have to
act on those feelings. But some other people think
that they can't stop themselves acting on those
feelings, so they have to change their feelings. But at least we can then have a
clearer discussion about what's going on in this situation. Now, there's an
analogous situation, that is to say it makes the
same picture, that's possibly more urgent, which
is about police violence against black people. This is a particular
issue in America, where we have another
vicious circle that looks like this that involves,
maybe, police who feel threatened by black people. And that means that they
aggressively defend themselves against black people. But in turn, that
means that black people feel very threatened
by the police, which means that black people
aggressively defend themselves against the police, which means
that police feel threatened by black people. And again, this separates
out into feeling responses and action responses. And once we've
observed that there is this interconnectedness
of causes, then we can have, I think,
a more productive argument. Instead of half the people
on the internet yelling, well, they should just do
what the police tell them, and then they won't
get shot, and then other half saying,
well, the police shouldn't shoot
them, that's unfair. I think that we can look at this
interconnectedness in saying, there are many things going on. Now, whose responsibility
is it to change something? I think in this situation and
in a relationship breakdown, whoever has more power
in the situation should take the responsibility
for changing it. But maybe some people
disagree with me. At least then we can
talk about really what we disagree with rather
than having futile shouting matches across the internet. I use the
interconnectedness of things to help me understand
difficult situations that have arisen when
people are yelling about exactly who is to blame. So one case where I
drew a diagram, that looks like a category
theory diagram to me, was in the egregious
United incident, or rather after the
egregious United incident, when the flight was overbooked,
and they picked someone to kick off the plane,
and he didn't want to go. And so they called security,
they hauled him off extremely violently, and he sustained
quite serious injuries on the way out. At which point,
half the internet went, well, he should have
just left when he was told, and then wouldn't get injured. And the other half
of the internet said, well, you shouldn't
just beat people up because they don't
want to leave a plane. And some people said it was
just one person's fault, some people said it was
just something else. But it was really
a lot of things. And there was even an editorial
by a British journalist whom I probably shouldn't
name, since this is going on the
internet, but who wrote an editorial saying
it's all of your faults, yours, because you're
sometimes late for flights. And that means that they
overbook flights in case people are late. And it's because of that
overbooking that they had to get someone off the plane. So it's actually your fault
for being late for a flight. I thought, well, that's
pushing it a bit far. [LAUGHTER] Maybe there is some
kind of a point. So what kind of
a point is there? Well, the end result was
that some injury was caused. Why was the injury caused? Well, the guy refused
to leave the plane, and also the
security used force. But the security
wouldn't even have been there unless the
airline had called security in the first place. Now, he refused to leave because
he needed to get to work. But why did the airline even
choose him in the first place? You could say, he was
a doctor at a hospital, he had a very valid reason
to want to get to work. Why did they pick him? There are questions
about whether there was racial profiling,
why did they pick those particular people? Then, there was, why did they
even need to kick people off in the first place? Whose fault was that? Well, nobody volunteered. Why did nobody volunteer? The airline offered
them too little money. Were they being
stingy, or why did they really want to get home? Why did they even
decide to remove people in the first place? That was because the
flight was too full, the flight was too
full and they needed to get some crew to Lewisville. Why did they need to get some
crew to Lewisville so badly, was it some scheduling issue? And why was the flight too full? Partly because it
was overbooked, and partly because there
was not enough no-shows. Now whose fault is that,
is it the airline's fault for overbooking, or is
it the people's fault for not showing up? It's both of those things. And then finally,
there's that people often miss flights, which is a
very tenuous contribution to this situation. But I think that changing
any one of those factors would have changed
the outcome, but it doesn't mean that any one of
those factors by themselves is individually to blame. It's really the
interconnectedness of the whole system. So another situation
where people often go, oh, it's really simple-- whenever people say
that, it really usually isn't that simple. [LAUGHTER] It might be simple
in a different way, but it's rarely simple in the
way that people are saying. And one is when they
talk about losing weight. So I used to be quite
fat, and I lost 50 pounds, and I would like to
leave it that way. And so I think quite
a lot about this because it's quite an effort. And then, periodically
someone will go, oh, well, it's not
rocket science, you just have to eat
less and exercise more. And I say, well, even rocket
science is just applied maths. [LAUGHTER] So why do I put on weight? Well, I put on weight
because I take in more energy than I burn. And that's because
I eat too much and because I
exercise too little. But it's also because
of my metabolism. And what affects my metabolism? Well, my metabolism slows
down if I eat too little and I exercise too much. So already, we have the
fact that I gain weight because I eat too
much and too little, and I exercise too
much and too little. So yes, it's a very
simple situation, together with the fact that why is
my metabolism like that, it's also some genetics,
the amount of sleep I do and don't get. And why do I eat too much? I really like food-- but also because emotions,
I emotionally eat. And why do I eat too much? Well, why do I eat too
much because of emotions? Some people don't. It's partly to do
with my genetics and partly to do
with my upbringing where we definitely had an
emotional connection to food when I was growing up. And there's also social
pressure to eat too much. And some people say, you can't
blame everyone else all time. But let's look at
all the factors. There is social pressure to
eat too much, because people get bored with you if you don't
join in stuffing your face when they are. The social norms
also cause emotions that cause me to eat too much. Then there's also time
pressure-- if I'm very busy, I get stressed and
I also sleep less. And there's also life
that causes emotions. And then, there's also
the entire food industry. So there's a multibillion
pound food industry that is trying to get us to
eat too much because that's how they make money. And they're also trying to
get us to want to lose weight, and to try and lose
weight and fail, so that they can carry on with
a diet industry that doesn't really achieve anything. And so advertising and so on
causes us to eat too much, and then they make
money out of us. So it's kind of money,
maybe it's money that's at the root
of everything. Then, there's the fact
that if I do gain weight, then I try to eat less,
and exercise too much, and I get stressed about it. And so I get all of
these vicious cycles, and interconnectedness, and
a conglomeration of factors. So now try and tell me that
it really is that simple. I don't think it
really is that simple, but it really is this simple. And the point is,
if you get used to understanding interconnected
networks of factors, then this actually is
simple, the whole thing. You don't have to forget
huge parts of the diagram in order to make it simple. Your brain can comprehend
this as a single unit. And that's what I
mean by becoming more intelligent in order
to make things simpler. I did draw a diagram
for the 2016 US election because it's a
classic case of people yelling at each other
about whose fault it is. Here's my diagram. [LAUGHTER] I'm not going to leave
it up for very long, but I did find it
very helpful for me to put all of these
different factors in and understand why
people are yelling at-- there are some
people who just say, it's the entire fault of
the third party voters. But what they don't realize
is that the people voting third party wouldn't have made
a difference if the voting system were different. So it's people voting
third party and the fact that the voting system is
the way it is, and so on. So this to me, is the
logic inside the situation. Working out why
each of these things was there in the first
place is more the politics or the sociology of it. But understanding the logic of
the interconnectedness right at the outset can help us
start to think about all of the other issues. It's definitely not
the full understanding, but it makes it so
much clearer to me, and it means that
I can see what's going on with other
people's arguments about it. I'd now like to talk about
relationships between things. I mentioned relationships
between people just now. But a lot of what
I've been talking about in the
previous section was about how things
are interconnected by different relationships. And actually, my
field of research, which is category
theory, is all about studying relationships
between things instead of their
intrinsic characteristics. And so I've learned
a lot from this. And my brain is very
geared towards thinking about relationships
between things a lot. And I realize that it can
apply to many important things in the world around us, but
starting from something quite banal like the factors of 30. So this is obviously
mathematical, maybe we can remember what
the factors of 30 are-- they are 1, and 2, 3,
5, 6, 10, 15, and 30. Very good. It's not very
interesting, it's a bunch of numbers in a straight line. And we live in a
three-dimensional world, so we write on
two-dimensional pieces of paper in one-dimensional
straight lines, which means that we stuff
all our thoughts into a one-dimensional
straight line all the time. And sometimes they
don't want to be in a one-dimensional
straight line, they have natural geometry
in higher dimensions. And I like to say
this is why I don't tidy the papers on
my desk, because they have natural geometry
in higher dimensions. [LAUGHTER] They don't want to be stuffed
into a one-dimensional pile. So I'm going to exhibit
the natural geometry of this situation or find some
by looking at which numbers are also factors of each other. And I'm going to draw something
a bit like the family tree to show those relationships. So like in a family
tree, I don't need to show relationships
across two generations because we can deduce those. So we have 30 at the top like
a kind of great-grandparent. And then, we have 6, 10,
and 15 that go into 30. We have 5 goes into 10 and 15. 2 goes into 6 and 10. 3 goes into 6 and 15. And 1 goes into 2, 3, and 5. So now, we see that it's
the corners of a cube. We can also see a
hierarchy going on here, because you might notice
that one is at the bottom because it's the smallest,
and then 2, 3, and 5 are directly above it
because nothing else goes into them apart from 1, which
is that they are prime numbers. And then at the
level above that, we have products of
two prime numbers. And then at the top, we have a
product of three prime numbers. So we have a hierarchy according
to the number of prime factors that each number has. And if I draw out a picture
of the prime factors, instead I get this. So at the bottom,
it's the empty set, there are no prime factors. And then, one prime factor,
pairs of prime factors, and then three. And we can get back to
the left-hand diagram by just multiplying together
all the numbers at each corner. But now, maybe we can
see it didn't really matter what those numbers were. In fact, it didn't really
matter what they were at all. They could have
been a, b, and c. So we've turned our
numbers into letters, ooh. And now we have arrows
that show forgetting one thing from each
set, so this is also called subset interaction. But this is now,
it's very powerful because we can try it
with something else. So we could switch to
doing the factors of 42. And we can draw a
diagram similarly where we start with 1 and then
prime factors 2, 3, and 7, then the products of
two prime factors, and the product of
[INAUDIBLE] prime factors. We can redraw the
other diagrams, and you can see that the
right-hand one is just a, b, and c. It's the same. The middle one
looks very similar, except that 7 has been
put in place of every 5. So once we get as far abstract
as the right-hand side, it's become the same picture. It looks less and less
the same towards the left, and this is the
point of abstraction that many things
become the same. And there is a
particular feature about the left-hand diagram
that I want to point out, which is that 6 is less than 7. Now that might not
sound very profound, but notice that 6 is higher
than 7 in the diagram, but 6 is less than 7
in the normal hierarchy of size of numbers. So there is a difference,
there is a tension between the hierarchy
in this diagram and the just sheer
size of numbers. And that is something
that's going to be crucial in the next
couple of minutes as we move on, because now I'm going to
show that this has become very powerful at the
abstract level, because a, b, and c can be anything. They don't have to be
numbers, they can be anything. So for example, they can be
three types of privilege-- rich, white, and male. And then at the next
level down, we'll have people with two types
of privilege, rich and white, rich and male, and white male,
and then at the next level down, just rich, white,
or male, and then at the bottom, people with none
of those types of privilege. And if I put back in the
other words for emphasis, we have rich, white, non-men,
we have rich non-white men, we have poor white men. I should have said non-rich,
but it doesn't fit sideways, so I said poor. [LAUGHTER] We have rich non-white
non-men, poor white non-men, and poor non-white men. And at the bottom,
poor non-white non-men. So 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 I think we can learn from this diagram. The first is that each arrow
represents a direct loss of one type of privilege. And it doesn't mean that
all the people who are white are better off than all the
people who are not white, because you can see
along the second level that there are some people
with no arrows in between them. So there is no direct loss of
privilege along that level. Now, this is
something that people make a mistake
about when they're getting angry about the
theory of privilege. Because people sometimes
point at the super rich black sportstar, and say, look at
that really rich black person. See, white privilege
doesn't exist. But that's not what
white privilege means. White privilege, in the
theory of white privilege, means that if that super
rich black sportstar had all the same features
but was also white, then we would expect them
to be better off in society. And so it's not trying to say
that every white person is better off than every
non-white person, it's about losing
one type of privilege along one of these
arrows and how that affects people's experiences. Now, there's something else I
think we can learn from this. If we just look
along the second row of people who are level, having
two types of privilege each. Because I don't think
that they are really level in terms of absolute
privilege in society. If we think about
rich white women, I think we can probably
agree that they're better off in society than,
say, rich black men, who are in turn better off
than poor white men. Because it turns
out that money can mitigate for an awful
lot of other problems. So actually, it's
more skewed like this. And the same is true
on the bottom level. Although actually, I think
it's skewed further than that, because if we look at the
interaction between the two levels in the middle,
I think we might be able to agree that rich
non-white non-men are probably better off in society
than poor white men. Because again, riches
mitigate for many things. And if we think about
some extreme cases, like Michelle Obama
or Oprah Winfrey, I'm sure we'll all agree
that they're much better off than poor, white,
unemployed, homeless men in the middle of nowhere
in America, for example. So actually, it's
skewed more like this. And it's just like the
fact that 6 is less than 7. So 6 was at a higher level
in terms of the cube, but it was lower in
absolute terms than 7. And poor white men have
two types of privilege, so they're quite high up
in terms of privilege. But when it comes to
absolute privilege, they're lower down than
people who are considered less privileged than them. And this has helped
me understand why some poor white men are so
angry in society at the moment, because they are considered
to be privileged, but they don't feel any actual
manifestations or benefits of that privilege. And I think that it's
more productive for us to understand this
root of that anger rather than simply to be
angry with them in return. Seeing these
abstract structures, I think, helps us pivot,
or can help us pivot, between different
situations that are analogous in which we play
different roles so that we can understand what it's like to
be in different situations in those abstract structures. And we can look at analogous
people in different structures. So for example, we've seen that
in these three cubes, 30, 42, and rich white men all
occupy an analogous position. And if we look at just one
type of power relationship over another one, we can see
some more analogous things, such as that male people
over female people are analogous in a way to
white people's relationships to black people, which is
analogous to rich people's power over poor people,
because each of these involves a power
difference in society. We can then have a
discussion about how exactly that power difference
manifests itself, but there is an analogous
power difference in the sense of this diagram. However, there is not an
analogous power difference between these two situations. So the interaction of male
people to female people is not analogous to the
interaction of female people to male people because
of that power difference. And that is why that gives
us a sense in which men are being sexist
against women is not exactly the same as women
being sexist against men. And it gives us a sense in
which white people being racist against black people is
different from black people being racist against
white people, and a sense in which straight
people being prejudiced against gay people is not
the same as gay people being prejudiced against
straight people because of power differences. And I think that's
an important thing that these abstract structures
can help us remember together with the fact that
something could be not at the top
in one context, but at the top in a
different context. So in our original
diagram of privilege, rich, white men are at the top. But if we restrict our
attention to non-men, then we get these people. And we find that rich, white,
non-men are now at the top. And if, in fact, we restrict our
entire context to, say, women, and we think of three types
of privilege among women, then we can think about
rich, white, and cisgendered as being our three types
of privilege, for example. And now, we'll see that
rich, white, cis women occupy the analogous position
that rich, white men do in broader society. And this has helped
me to understand why there is so much
anger against white women, especially in some parts
of the feminist movement at the moment, because they can
be prone to seeing themselves as under-privileged
relative to white men, because we women do all feel
everyday sexism every day. And that is a very
big part of our lives. But then as a result,
they can maybe forget how privileged they are
relative to non-white women who have it even worse than them. And understanding those
underprivileged and over-privileged aspects
of the same person can help us to
remember what it's like to be somewhere
else in the cube. We're all
over-privileged relative to someone and underprivileged
relative to somebody else. And so we could all pivot
between a situation in which we are one thing and a
situation in which we're other to understand it from
someone else's point of view. And in that way, these
abstract structures can help us empathize
with other people. And here are some pivots that
I do or that we could do. So one is that as
an Asian person, I've realized that I
understand something of what it's like
to be non-white, because I'm not white. But also, I am probably
among the most privileged of non-white people, because
Asians occupy, in many ways, a much better position than
other non-white people in most of the world. And so I can pivot between
understanding things from the underprivileged and the
over-privileged point of view from a racial perspective. In terms of wealth,
I can also pivot, because I don't tend to think
of myself as extremely rich. I'm not so rich that I don't
have to work ever again, I'm not that kind of rich. But I'm doing perfectly fine,
and that is a much better situation to be than people
who are really struggling, who are maybe unemployed
or working at minimum wage. And so I think most people,
especially in this country, we tend to view rich
people as other people. And however, we are all
richer than somebody. And there is somebody who is
viewing us as rich in turn. And so we can pivot
between those situations. And here is the pivot
that white women could do, because white women are less
privileged than white men, but more privileged
than non-white women. And any time you take two
adjectives where one of them has privilege and one doesn't
and put them together, you can have this
situation where if you only see yourself
as under-privileged and everyone else sees
you as over-privileged, then antagonism will
result from that. So finally, I'd like to
talk about, sum up, what I think intelligence is, really. And because I've been
drawing all these diagrams of interconnectedness, I thought
I'd draw one about intelligence to show what I think
are the contributing factors to intelligence. So I do think that intelligence
involves being reasonable, being not just logical,
but powerfully logical, and also being helpful. So if you just sit in
a cave and use logic, I don't think that's
particularly intelligent, because you're not
really helping anything. I mean, people can disagree. You can disagree
with me about this, but I don't think
that's very intelligent. So what do these things mean? I think that being reasonable
means you can be reasoned with. And what that means is not
just that you use logic, but you have some kind
of framework for it, a framework for telling when
you should change your mind. I think that's really important. You can tell when
someone isn't reasonable if there's no possible situation
in which they would change their mind, if there is
no possible evidence that would convince them that
the moon landings actually happened, then that isn't
terribly reasonable. So what about being
powerfully logical? Well, I gave an
example of saying, if you say something twice,
so if you don't believe in gay marriage because
you think marriage should be between a man
and a woman, you've just said the same thing twice. That's not powerfully logical,
even though it's not illogical. So I think being
powerfully logical involves not just using logic,
but also using techniques to build up logical
arguments and unravel logical arguments so that you
can actually get somewhere with your logic. And finally, being helpful
crucially involves, well, some techniques
for being helpful, but it really involves
using our emotions, because people are
emotional beings. And if we don't use emotions to
communicate with other people, then we'll just be
throwing cold logic at them all the time instead
of understanding what it is that they are feeling
and why they are feeling it. So I really like Carlo
Cipolla's theory of stupidity. It's a slightly tongue-in-cheek
but quite serious theory. He starts by saying he
thinks that in any given group of people, there's
the same proportion of stupid people, whether
those are scientists, people who work at Google,
maybe more in politicians-- [LAUGHTER] --professors, Nobel Prize
winners, prisoners, criminals, street sweepers-- he thinks that there's
exactly the same proportion of stupid people
everywhere, and that it's more than you are
ever expecting, even after you've taken
that into account. So OK, what does
he mean by stupid? Well, he has a
definition, and it involves a graph with two axes. And this axis is how much
you benefit yourself, and this axis is how much
you benefit other people. So if you benefit yourself
and hurt other people, then you are a bandit. If you benefit other
people but hurt yourself, then you are what he
calls unfortunate. I sometimes think of
this as being a martyr, where you sacrifice
yourself all the time for the good of others. And I used to do
this a lot, and I think that quite
a lot of women do, because we think
that it's noble, and we're supposed
to efface ourselves for the good of our people. I've eventually realized
that that's not terribly productive in the long run. If you hurt others and hurt
yourself at the same time, you are stupid. [LAUGHTER] And that's how he
defines stupid. Whereas, if you
benefit other people and you benefit yourself at the
same time, that's intelligent. And I think this is
what intelligence is. I don't think it has anything
to do with qualifications, degrees, education, money,
status, job, achievements, prizes, accolades,
any of those things. I think it's about this. And I think that we should use
logic and emotions together so that we can benefit
ourselves and other people at the same time, and in that
way, create a virtuous circle, because logic and abstract
mathematical thinking, I think, can actually help
us to understand how other people think
from their point of view rather than from our point
of view, which helps us to empathize with other people. And if we can empathize
with other people, then that will help us act
in a logical way for them and for us all
together so that we can create this virtuous circle. And I think it might
be a surprising idea that abstract mathematics
can do that for us, but I truly believe
that it can, and I hope that I can persuade
many people to join in with this lofty aim. Thank you. [APPLAUSE] SPEAKER 1: Thank you very much,
we do have time for questions. Please raise your hand. AUDIENCE: Thanks for the
talk, that was really amazing. I also have a background
in mathematics and have always wished that I
could help other people not be as afraid of it. And I see that you're writing
books to accomplish this. And I had always planned to
eventually create a website where I could
offer free tutoring and then end up with this
kind of database of questions that are well-answered. But now after
seeing your chart, I feel like that's
the martyr route. [LAUGHTER] So I'm just wondering if you
have any suggestions for what I could potentially do or
if this is something you've thought about at length, maybe. EUGENIA CHENG: I did start out
by just making videos of me explaining things to people. And it wasn't too much
martyrdom-ish at the time, because I did also feel that-- I mean, there are many ways
that you can benefit yourself. And one is that if you get
a feeling of self-worth from having helped someone
understand something, then you have
benefited yourself. And personally, I
do derive my self worth from how much
I help other people. So maybe it gets a bit cyclic. And if I have
helped other people, then I have helped myself. But what I had to stop doing
was actively hurting myself. So I was I was quite
unhappy in a job that I had, but I thought that
I should carry on going because I
was helping people, even though I was
completely miserable. And that was what was
particularly unproductive about it. So I think that the way in
which you benefit yourself can be measured in many ways. And if you feel like
it's benefiting you, then it's not
necessarily martyrdom. If it feels like
it's hurting you and that it's making
your life miserable, then I think it
limits your ability to help people in the long run. So you help people tons for a
week, and then you're burnt out and you hate everybody, and then
you can't help anyone anymore. Or you help people a
bit less, and you also look after yourself or
do things for yourself, and then you have a
much longer run of being able to benefit other people. So that's some sort of vague
answer to that, I think. AUDIENCE: Hi. EUGENIA CHENG: Hi. AUDIENCE: Amazing talk,
and what a fantastic way to look at the world. One question though, when
you're starting to break down challenges in life, and
particularly looking at the weight loss slide
as one that jumped out, I start to see a building
series of contributing factors to why this becomes a
challenge to overcome. How'd you get over that? Do you find that when you
start breaking things out into all the different reasons
why something is one way, mentally it becomes
harder to then make that change, how do you
get to that next step? EUGENIA CHENG: What I do
is, well, first of all, I'm so used to dealing
with these huge diagrams that I'm less daunted
by them, maybe. And the more you deal with them,
the less daunted you become. And it actually becomes
clearer, it's reassuring to me. And that's not the
same for everybody, but because I've been trained
in this way of thinking for a long time, it has
become very reassuring to me. And I think that
what I can then do is look at all the
individual parts and see which is the
arrow that would be the best one to try and break. And once you've
actually made clear what all the connections are-- this is a bit like a
cognitive behavioral therapy technique, where you try and
see which arrow you can break-- but it is about seeing
which one is the weakest or which one would be the
easiest one to try and change. And then, you can try it. And if it doesn't
work that well, at least you might
have weakened it a bit. And then, you can try
weakening another one. And then eventually,
there will be enough that has become weaker
that the end result doesn't happen quite so strongly. AUDIENCE: Thank you
for this amazing talk, I really enjoyed it. EUGENIA CHENG: Thank you. AUDIENCE: I was
wondering about-- so obviously, we would
all be better off if we all used such logical
mathematical constructs, but I think it's clear that,
especially on the internet, a lot of people do not. I was wondering,
are you optimistic, or do you have any tips for how
we can engage people via logic if they're not, even at
all, willing to acknowledge that maybe that's a
helpful thing to do? EUGENIA CHENG: I am optimistic,
and that's why I'm here, and that's why I write
books, and that's why I'm in education, because
I am optimistic that we can change things. I think that there are some very
extreme people who we might not be able to reach immediately,
and that that's OK, and we shouldn't get too
upset about those people. But there are a whole
load of other people who aren't so far gone. We shouldn't start with the most
absolutely difficult people. But there are other people
who are, first of all, there are people who really,
really want to do better but don't know how. And those are people we can
definitely help to do better. Then, are the people who
aren't really terrible people, they just don't have the
techniques or don't realize. And then once you
explain it to them, just like with
things like sexually inappropriate comments, some
men, especially older ones, have just got used to the fact
that they can say those things and they don't understand. But they're not really
terrible people, and once you explain it
to them, then they get it, and then they can change. Whereas the ones who are
really absolutely morons, maybe we can't change
them yet, but eventually, if we sway enough people
in that direction. So I see it as being
a kind of continuum, and that these are
the people that we can most easily convince,
and then once we convinced these people, then
everything will have shifted a bit further
this way so that then we can convince these people. And then everything
will shift this way, and we can convince
these people. And eventually,
everything will shift, and then maybe we can
get back to a situation where it is not acceptable
to be those people anymore. And one of the problems
with the current climate is that it has become acceptable
again to be those people. And they probably
were always there, but maybe they felt like they
weren't supposed to show it or that they was not supposed
to be really like that. And if we can persuade everyone
to come a bit further back that way, I don't think we have
to just flip everybody over overnight. But if we can shift the
balance a bit further, then I think that could
just flip it back again. And yes, I am optimistic. The people who are
unreasonable on the internet are often the loudest
and the most annoying, but I don't think that they're
the majority of people. And I've seen enough reasonable
conversations on Twitter, even, that I believe it is possible
if we all slow down a bit and try to understand what
other people are thinking, rather than immediately trying
to show that they're wrong. AUDIENCE: You are an
accomplished artist as well. And your title of your
book [? says ?] about art. So where does the art play a
role in making sense of things? EUGENIA CHENG: Oh, thank you. I think that anything is an art. Well, it's interesting,
because I teach art students at the school of the Art
Institute of Chicago, so we've had many
discussions about what art is and what science is. And what we have collectively
decided one semester is that science is about
trying to understand the world around us,
and that art is about trying to interpret
the world around us, and that in order to
understand the world, you have to interpret it. And in order to
interpret the world, you have to understand it. So they're really
joined together, they're not completely separate. And to me, something
becomes an art if it's not an algorithmic
process that you just follow step-by-step. And that's why there is
an art to using logic. Because if you were
just doing, say, computer-aided proofs
of something, then maybe the computer wouldn't be
using an art, because it would be using an algorithm. But when we're
using logic, we have to understand the
people around us, and we have to engage
emotions, and we have to pick ways
of speaking, and we have to use techniques to
understand what's going on. And that, to me, is
what makes it an art. Hi. AUDIENCE: Hi. I also enjoyed your
talk very much. EUGENIA CHENG: Thank you. AUDIENCE: You claim that
in any group of people, there is roughly the
same ratio of stupid-- EUGENIA CHENG: That's
Carlo Cipolla's claim. AUDIENCE: But you seem
to be agreed with that. EUGENIA CHENG: I
was neutral about whether I agreed with that. I agreed with his
definition of intelligence. AUDIENCE: OK. Because it is somehow-- how to understand
that one would feel that in a group of more
intelligent people, are were less martyrs. EUGENIA CHENG: Well, only if you
define intelligence like this. If you define intelligence
by who has really prestigious degrees from prestigious
universities, I don't know that that's-- I've worked in some
of those places, and I don't know that that's-- [LAUGHTER] I don't know. AUDIENCE: Thank you. AUDIENCE: I was struck right at
the beginning when you started talking about the
abstraction of 2 by the fact that that's maybe
straightforward in English, and there are several
languages where there are particles you
put after the number to talk about what kind it is. So they've got more
levels of abstraction built into the language. Does that change anything
in practice or anything an implication in the process
you're thinking about? I guess the child development
aspects is way outside scope, but-- EUGENIA CHENG: I think that
it means, like you say, that there are some more
intermediate levels. But I think at the level of 2,
there's still the concept of 2, then there are some
other refinements of 2 bleh, and 2 bleh. And I know what you
mean, because there is this in Cantonese, and
it's terribly confusing when you're trying to
speak it and it's not your first language. But I think at the
level of 2, there is still a concept
of 2 that sits kind of above all of those things. And so yes, at different
levels of abstraction, you get different subtle
nuances in between things. And I don't know what that means
for we native English speakers and whether we have a
less subtle distinction between different
types of objects. SPEAKER 1: Thank you very much. Again, Dr. Eugenia Cheng. EUGENIA CHENG:
Thank you very much. [APPLAUSE]
