TEDxLondon
Eugenia Cheng: An unexpected tool for understanding inequality: abstract math
尤金尼娅·陈: 用一种意想不到的工具理解不平等现象：抽象数学
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在这个疯狂的世界，我们怎样才能做到理智？数学家尤金尼娅·陈告诉我们，应该着眼于一些意想不到的地方。她解释了如何通过将抽象的数学理论运用到日常生活中，能帮我们更深入理解愤怒的根源，以及特权的作用。更多的了解这个不可思议的工具如何帮助我们以同理心对待他人。
Eugenia Cheng - Mathematician, pianist
Eugenia Cheng devotes her life to mathematics, the piano and helping people. Full bio
Eugenia Cheng devotes her life to mathematics, the piano and helping people. Full bio
Double-click the English transcript below to play the video.
00:13
The world is awash
with divisive arguments,
with divisive arguments,
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这个世界充斥着引发分歧的观点，
冲突，
00:18
conflict,
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00:20
fake news,
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虚假新闻，
00:22
victimhood,
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受害者情绪，
00:25
exploitation, prejudice,
bigotry, blame, shouting
bigotry, blame, shouting
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剥削， 偏见，偏执，责怪，喊叫
00:30
and minuscule attention spans.
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和极短的注意力时间。
00:34
It can sometimes seem
that we are doomed to take sides,
that we are doomed to take sides,
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有时候似乎我们注定要选边站队，
00:40
be stuck in echo chambers
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固执己见，
再也无法与人达成共识。
00:42
and never agree again.
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00:45
It can sometimes seem
like a race to the bottom,
like a race to the bottom,
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有时候似乎我们在比惨，
00:48
where everyone is calling out
somebody else's privilege
somebody else's privilege
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每个人都在说别人有特权，
00:52
and vying to show that they
are the most hard-done-by person
are the most hard-done-by person
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然后争先恐后地怨天尤人，
00:57
in the conversation.
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都说自己最惨。
01:01
How can we make sense
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在这个疯狂的世界，
01:02
in a world that doesn't?
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我们怎样才能做到理智？
01:07
I have a tool for understanding
this confusing world of ours,
this confusing world of ours,
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我有一样工具，
可以帮助理解这个费解的世界，
可以帮助理解这个费解的世界，
01:12
a tool that you might not expect:
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一样你们可能意想不到的工具：
01:16
abstract mathematics.
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抽象数学。
01:19
I am a pure mathematician.
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我是个纯粹的数学家。
01:22
Traditionally, pure maths
is like the theory of maths,
is like the theory of maths,
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传统意义上，纯数学更多是
研究数学理论，
研究数学理论，
01:26
where applied maths is applied
to real problems like building bridges
to real problems like building bridges
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而应用数学是解决实际问题，
比如建造大桥，
比如建造大桥，
01:31
and flying planes
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开飞机，
01:32
and controlling traffic flow.
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控制交通流量。
01:35
But I'm going to talk about a way
that pure maths applies directly
that pure maths applies directly
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但我打算讲一种
将纯数学作为一种思维方式
将纯数学作为一种思维方式
01:40
to our daily lives
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直接运用到
01:42
as a way of thinking.
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我们日常生活的情况。
01:44
I don't solve quadratic equations
to help me with my daily life,
to help me with my daily life,
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我解二次方程并不是为了
方便我的日常生活，
方便我的日常生活，
01:49
but I do use mathematical thinking
to help me understand arguments
to help me understand arguments
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但我确实运用数学思维来理解争论，
与别人产生共鸣。
01:54
and to empathize with other people.
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01:57
And so pure maths helps me
with the entire human world.
with the entire human world.
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所以纯数学可以帮助我
理解整个人类世界。
理解整个人类世界。
02:04
But before I talk about
the entire human world,
the entire human world,
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但在谈整个人类世界之前，
02:07
I need to talk about something
that you might think of
that you might think of
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我必须和你们谈一些看起来可能
02:10
as irrelevant schools maths:
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无关的学校里教的数学：
02:13
factors of numbers.
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数字的因数。
我们先想一下30的因数。
02:16
We're going to start
by thinking about the factors of 30.
by thinking about the factors of 30.
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02:19
Now, if this makes you shudder
with bad memories of school maths lessons,
with bad memories of school maths lessons,
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如果这令你回想起数学课的糟糕回忆，
02:24
I sympathize, because I found
school maths lessons boring, too.
school maths lessons boring, too.
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我深表同情，因为我也觉得数学课无聊。
02:29
But I'm pretty sure we are going
to take this in a direction
to take this in a direction
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但我很确定，接下来发生的事情
02:33
that is very different
from what happened at school.
from what happened at school.
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跟学校里学到的会非常不一样。
02:37
So what are the factors of 30?
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那么30的因数有哪些？
02:39
Well, they're the numbers that go into 30.
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它们是30能整除的数字。
你们或许还记得，我们来算一下。
02:42
Maybe you can remember them.
We'll work them out.
We'll work them out.
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有1、2、3、
02:45
It's one, two, three,
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02:48
five, six,
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5、6、
02:51
10, 15 and 30.
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10、15和30。
02:53
It's not very interesting.
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这没什么意思。
02:55
It's a bunch of numbers
in a straight line.
in a straight line.
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就是一串数字。
02:58
We can make it more interesting
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我们能让它变得有趣一些，
03:00
by thinking about which of these numbers
are also factors of each other
are also factors of each other
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想一下，这些数字中，
有哪些互为因数，
有哪些互为因数，
03:04
and drawing a picture,
a bit like a family tree,
a bit like a family tree,
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然后画一张像家谱图的图，
03:06
to show those relationships.
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来展示这些关系。
03:08
So 30 is going to be at the top
like a kind of great-grandparent.
like a kind of great-grandparent.
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30在最上方，像是曾祖父母。
03:12
Six, 10 and 15 go into 30.
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6、10、15连上30。
03:15
Five goes into 10 and 15.
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5连上10和15。
03:18
Two goes into six and 10.
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2连上6和10。
03:21
Three goes into six and 15.
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3连上6和15。
03:24
And one goes into two, three and five.
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1连上2、3和5。
03:29
So now we see that 10
is not divisible by three,
is not divisible by three,
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我们看到10不能被3整除，
03:32
but that this is the corners of a cube,
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但这里是一个立方体的8个角，
03:36
which is, I think, a bit more interesting
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我觉得这样的关系
相比于一串数字要有趣得多。
03:38
than a bunch of numbers
in a straight line.
in a straight line.
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03:41
We can see something more here.
There's a hierarchy going on.
There's a hierarchy going on.
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我们能发现更多东西。
它是分层级的。
它是分层级的。
03:44
At the bottom level is the number one,
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最底层是数字1，
03:46
then there's the numbers
two, three and five,
two, three and five,
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然后是 2、3、5，
这些数字能被1和它们自己整除。
03:48
and nothing goes into those
except one and themselves.
except one and themselves.
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03:51
You might remember
this means they're prime.
this means they're prime.
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你或许记得，这表示他们是质数。
03:54
At the next level up,
we have six, 10 and 15,
we have six, 10 and 15,
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再往上一层，是6、10和15，
它们都是两个质数的乘积。
03:57
and each of those is a product
of two prime factors.
of two prime factors.
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04:00
So six is two times three,
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6 = 2 × 3
04:02
10 is two times five,
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10 = 2 × 5，
15 = 3 × 5。
04:04
15 is three times five.
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04:06
And then at the top, we have 30,
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最顶层的30，
04:08
which is a product
of three prime numbers --
of three prime numbers --
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是三个质数的乘积，
04:10
two times three times five.
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2 x 3 x 5。
04:12
So I could redraw this diagram
using those numbers instead.
using those numbers instead.
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所以我可以重新画这个图，
用数字来替代。
用数字来替代。
04:18
We see that we've got
two, three and five at the top,
two, three and five at the top,
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我们看到2、3、5在最顶层，
成对的数字在第二层，
04:21
we have pairs of numbers
at the next level,
at the next level,
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04:24
and we have single elements
at the next level
at the next level
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单个的数字在下一层
04:26
and then the empty set at the bottom.
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最底层是空集。
04:29
And each of those arrows shows
losing one of your numbers in the set.
losing one of your numbers in the set.
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每个箭头表示在集合中少一个数字。
04:34
Now maybe it can be clear
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现在或许清楚了，
04:37
that it doesn't really matter
what those numbers are.
what those numbers are.
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这些数字是多少并不重要。
04:40
In fact, it doesn't matter what they are.
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实际上，它们是什么都不重要。
04:42
So we could replace them with
something like A, B and C instead,
something like A, B and C instead,
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所以我们可以用A, B, C替代它们，
04:46
and we get the same picture.
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也会得到同样的图。
04:49
So now this has become very abstract.
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这样就变得抽象了。
04:51
The numbers have turned into letters.
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数字变成了字母。
04:54
But there is a point to this abstraction,
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但这个抽象化是有意义的，
04:57
which is that it now suddenly
becomes very widely applicable,
becomes very widely applicable,
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因为这张图能被广泛应用了，
05:02
because A, B and C could be anything.
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因为A，B，C可以是任何东西。
05:06
For example, they could be
three types of privilege:
three types of privilege:
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比如，它们可以是3种特权：
05:10
rich, white and male.
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有钱的，白人，男性。
05:14
So then at the next level,
we have rich white people.
we have rich white people.
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所以下一层，我们得到“有钱的”“白人”。
05:18
Here we have rich male people.
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这里是“有钱的”“男性”。
05:20
Here we have white male people.
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这里是“白人”“男性”。
05:22
Then we have rich, white and male.
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然后是“有钱的”、“白人”、“男性”。
05:27
And finally, people with none
of those types of privilege.
of those types of privilege.
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最后，是没有任何特权的人。
05:30
And I'm going to put back in
the rest of the adjectives for emphasis.
the rest of the adjectives for emphasis.
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我把剩下的形容词补上，用来强调。
05:33
So here we have rich, white
non-male people,
non-male people,
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所以这里是“有钱的”、“白人”、“非男性”，
05:36
to remind us that there are
nonbinary people we need to include.
nonbinary people we need to include.
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别忘了还有人既不是男性也不是女性，
05:39
Here we have rich, nonwhite male people.
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这里是“有钱的”、“非白人”、“男性”，
这里是“非有钱的”、“白人”、“男性”，
05:42
Here we have non-rich, white male people,
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05:45
rich, nonwhite, non-male,
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“有钱的”、“非白人”、“非男性”，
05:48
non-rich, white, non-male
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“非有钱的”、“白人”、“非男性”，
05:51
and non-rich, nonwhite, male.
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以及“非有钱的”、“非白人”、“男性”，
05:53
And at the bottom,
with the least privilege,
with the least privilege,
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以及在最底层，特权最少的
05:55
non-rich, nonwhite, non-male people.
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“非有钱的”、“非白人”、“非男性”。
05:59
We have gone from a diagram
of factors of 30
of factors of 30
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我们从一个30的因数图表
06:03
to a diagram of interaction
of different types of privilege.
of different types of privilege.
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到了一个不同特权的交叉图表。
06:08
And there are many things
we can learn from this diagram, I think.
we can learn from this diagram, I think.
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我们能从这张图表中学到很多。
06:11
The first is that each arrow represents
a direct loss of one type of privilege.
a direct loss of one type of privilege.
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首先，每个箭头表示失去一种特权。
06:19
Sometimes people mistakenly think
that white privilege means
that white privilege means
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有时候人们错误地以为，
白人特权意味着
白人特权意味着
06:23
all white people are better off
than all nonwhite people.
than all nonwhite people.
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所有的白人都比非白人过得更好。
06:28
Some people point at superrich
black sports stars and say,
black sports stars and say,
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有些人指着超级有钱的
黑人运动明星说，
黑人运动明星说，
06:32
"See? They're really rich.
White privilege doesn't exist."
White privilege doesn't exist."
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“看到没？他们超有钱，
白人特权不存在。”
白人特权不存在。”
06:36
But that's not what the theory
of white privilege says.
of white privilege says.
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但是这不是白人特权的内涵。
白人特权是指，如果其他特征
跟那个超有钱的运动明星一样，
跟那个超有钱的运动明星一样，
06:39
It says that if that superrich sports star
had all the same characteristics
had all the same characteristics
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06:44
but they were also white,
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同时还是白人，
06:45
we would expect them
to be better off in society.
to be better off in society.
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我们会认为他们在社会上混得更好。
06:51
There is something else
we can understand from this diagram
we can understand from this diagram
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我们从这张图表中还能学到更多
如果我们沿着一个箭头看。
06:54
if we look along a row.
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06:56
If we look along the second-to-top row,
where people have two types of privilege,
where people have two types of privilege,
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沿最顶层到第二层的箭头看，
拥有两种特权的人，
拥有两种特权的人，
07:00
we might be able to see
that they're not all particularly equal.
that they're not all particularly equal.
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他们并不是特别平等。
07:04
For example, rich white women
are probably much better off in society
are probably much better off in society
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比如，有钱的白人女性
或许比贫穷的白人男性
或许比贫穷的白人男性
07:10
than poor white men,
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混得更好，
07:12
and rich black men are probably
somewhere in between.
somewhere in between.
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而有钱的黑人男性或许介于两者之间。
07:15
So it's really more skewed like this,
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所以其实这张图应该更加倾斜，
07:18
and the same on the bottom level.
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最下面一层也是一样。
07:20
But we can actually take it further
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我们可以更仔细地
07:23
and look at the interactions
between those two middle levels.
between those two middle levels.
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看中间两层的相互关系。
07:27
Because rich, nonwhite non-men
might well be better off in society
might well be better off in society
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因为有钱的、非白人、非男性
可能比贫穷的白人男性
可能比贫穷的白人男性
07:33
than poor white men.
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过得更好。
07:35
Think about some extreme
examples, like Michelle Obama,
examples, like Michelle Obama,
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举几个极端的例子，比如米歇尔·奥巴马
07:39
Oprah Winfrey.
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奥普拉·温弗瑞。
07:40
They're definitely better off
than poor, white, unemployed homeless men.
than poor, white, unemployed homeless men.
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她们绝对比贫穷的、失业的、
无家可归的白人男性过得好。
无家可归的白人男性过得好。
07:46
So actually, the diagram
is more skewed like this.
is more skewed like this.
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所以这张图更像是这样倾斜。
07:49
And that tension exists
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图表中各个层级的人
07:52
between the layers
of privilege in the diagram
of privilege in the diagram
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在生活中体验到的特权，
07:55
and the absolute privilege
that people experience in society.
that people experience in society.
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与在图表中所处的位置存在差异。
07:59
And this has helped me to understand
why some poor white men
why some poor white men
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这帮助我理解了，
为什么有些贫穷的白人男性
为什么有些贫穷的白人男性
08:02
are so angry in society at the moment.
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在社会中如此愤怒。
08:06
Because they are considered to be high up
in this cuboid of privilege,
in this cuboid of privilege,
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因为他们被认为处于特权阶级的上层，
08:10
but in terms of absolute privilege,
they don't actually feel the effect of it.
they don't actually feel the effect of it.
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而现实生活中，他们这种
享有特权的感受并不明显。
享有特权的感受并不明显。
08:15
And I believe that understanding
the root of that anger
the root of that anger
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我认为理解愤怒的根源
08:19
is much more productive
than just being angry at them in return.
than just being angry at them in return.
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比反过来对他们感到愤怒
更有实际帮助。
更有实际帮助。
08:25
Seeing these abstract structures
can also help us switch contexts
can also help us switch contexts
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这种抽象的结构
还能帮我们转换情境,
还能帮我们转换情境,
08:29
and see that different people
are at the top in different contexts.
are at the top in different contexts.
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看看如果不同的人
位于顶端，会有什么不同。
位于顶端，会有什么不同。
08:33
In our original diagram,
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在我们最初的图表里，
08:35
rich white men were at the top,
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有钱的白人男性在顶层，
08:37
but if we restricted
our attention to non-men,
our attention to non-men,
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但如果我们只看非男性，
08:41
we would see that they are here,
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他们集中在这个区域，
现在“有钱的”、“白人”、
“非男性”在顶层了。
“非男性”在顶层了。
08:42
and now the rich, white
non-men are at the top.
non-men are at the top.
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08:45
So we could move to
a whole context of women,
a whole context of women,
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我们可以把整个情境转换到女性，
08:48
and our three types of privilege
could now be rich, white and cisgendered.
could now be rich, white and cisgendered.
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那么我们的三种特权变成了
“有钱的”、“白人“、“本性别”。
“有钱的”、“白人“、“本性别”。
08:53
Remember that "cisgendered" means
that your gender identity does match
that your gender identity does match
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“本性别”是指自我认同的性别
08:57
the gender you were assigned at birth.
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和生理性别一致。
09:00
So now we see that rich, white cis women
occupy the analogous situation
occupy the analogous situation
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现在“有钱的”、“白人”、“本性别女性”
与“有钱的”、“白人”、“男性”
与“有钱的”、“白人”、“男性”
09:06
that rich white men did
in broader society.
in broader society.
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在更宽泛的社会中拥有了类似的地位。
09:09
And this has helped me understand
why there is so much anger
why there is so much anger
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这让我理解了，为什么会有那么多人
09:12
towards rich white women,
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讨厌“有钱的”、“白人”、“女性”，
09:14
especially in some parts
of the feminist movement at the moment,
of the feminist movement at the moment,
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尤其在最近的很多女权主义活动中，
09:17
because perhaps they're prone
to seeing themselves as underprivileged
to seeing themselves as underprivileged
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因为她们倾向于认为自己是弱势群体，
09:21
relative to white men,
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如果跟白人男性比的话，
09:23
and they forget how overprivileged
they are relative to nonwhite women.
they are relative to nonwhite women.
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但她们忘记了，跟非白人女性相比，
自己享受了多少特权。
自己享受了多少特权。
09:30
We can all use these abstract structures
to help us pivot between situations
to help us pivot between situations
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我们能利用这些抽象的结构
在情境之间转换
在情境之间转换
09:36
in which we are more privileged
and less privileged.
and less privileged.
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我们有时占优势，有时占劣势。
09:38
We are all more privileged than somebody
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我们总会比一些人占优势，
09:41
and less privileged than somebody else.
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也总会比另一些人更吃亏。
09:44
For example, I know and I feel
that as an Asian person,
that as an Asian person,
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比如，我知道并且感觉到
作为一个亚洲人，
作为一个亚洲人，
09:49
I am less privileged than white people
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比白人更弱势，
09:52
because of white privilege.
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因为白人特权的存在。
09:53
But I also understand
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但是我也知道，
09:55
that I am probably among
the most privileged of nonwhite people,
the most privileged of nonwhite people,
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我可能是非白人人群中最有特权的，
09:59
and this helps me pivot
between those two contexts.
between those two contexts.
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这能让我在不同情境下转换。
10:03
And in terms of wealth,
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说到财富，
10:05
I don't think I'm super rich.
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我不觉得自己超级有钱。
10:07
I'm not as rich as the kind of people
who don't have to work.
who don't have to work.
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我比不上那些甚至不需要工作的人。
10:10
But I am doing fine,
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但我过得也很滋润，
10:11
and that's a much better
situation to be in
situation to be in
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比起在温饱线上挣扎的人，
10:13
than people who are really struggling,
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那些失业的，或者拿最低工资的人，
10:15
maybe are unemployed
or working at minimum wage.
or working at minimum wage.
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我已经过得很好了。
10:20
I perform these pivots in my head
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我在脑中进行这些转换
10:24
to help me understand experiences
from other people's points of view,
from other people's points of view,
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来帮助我理解别人的处境，
10:30
which brings me to this
possibly surprising conclusion:
possibly surprising conclusion:
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这让我得出了一个意外的结论：
10:35
that abstract mathematics
is highly relevant to our daily lives
is highly relevant to our daily lives
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抽象数学和我们的日常生活息息相关
10:42
and can even help us to understand
and empathize with other people.
and empathize with other people.
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甚至能帮我们理解他人并产生共情。
10:50
My wish is that everybody would try
to understand other people more
to understand other people more
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我希望每个人都能
尝试更多去理解他人，
尝试更多去理解他人，
10:56
and work with them together,
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共同努力，
10:58
rather than competing with them
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而不是相互竞争，
11:00
and trying to show that they're wrong.
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说对方错了。
11:04
And I believe that abstract
mathematical thinking
mathematical thinking
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我相信，抽象的数学思维
11:08
can help us achieve that.
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能帮助我们实现这些。
11:12
Thank you.
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谢谢大家。
11:13
(Applause)
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（掌声）
ABOUT THE SPEAKER
Eugenia Cheng - Mathematician, pianistEugenia Cheng devotes her life to mathematics, the piano and helping people.
Why you should listen
Dr. Eugenia Cheng quit her tenured academic job for a portfolio career as a research mathematician, educator, author, columnist, public speaker, artist and pianist. Her aim is to rid the world of math phobia and develop, demonstrate and advocate for the role of mathematics in addressing issues of social justice.
Her first popular math book, How to Bake Pi, was published by Basic Books in 2015 to widespread acclaim including from the New York Times, National Geographic, Scientific American, and she was interviewed around the world including on the BBC, NPR and The Late Show with Stephen Colbert. Her second book, Beyond Infinity was published in 2017 and was shortlisted for the Royal Society Insight Investment ScienceBook Prize. Her most recent book, The Art of Logic in an Illogical World, was published in 2018 and was praised in the Guardian.
Cheng was an early pioneer of math on YouTube, and her most viewed video, about math and bagels, has been viewed more than 18 million times to date. She has also assisted with mathematics in elementary schools and high schools for 20 years. Cheng writes the "Everyday Math" column for the Wall Street Journal, is a concert pianist and founded the Liederstube, a not-for-profit organization in Chicago bringing classical music to a wider audience. In 2017 she completed her first mathematical art commission, for Hotel EMC2 in Chicago; her second was installed in 2018 in the Living Architecture exhibit at 6018 North.
Cheng is Scientist In Residence at the School of the Art Institute of Chicago and won tenure in Pure Mathematics at the University of Sheffield, UK. She is now Honorary Fellow at the University of Sheffield and Honorary Visiting Fellow at City University, London. She has previously taught at the universities of Cambridge, Chicago and Nice and holds a PhD in pure mathematics from the University of Cambridge. Her research is in the field of Category Theory, and to date she has published 16 research papers in international journals.
You can learn more about her in this in-depth biographic interview on the BBC's Life Scientific.
Eugenia Cheng | Speaker | TED.com