ABOUT THE SPEAKER
Eugenia Cheng - Mathematician, pianist
Eugenia 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 TimesNational GeographicScientific American, and she was interviewed around the world including on the BBCNPR 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.

More profile about the speaker
Eugenia Cheng | Speaker | TED.com
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

Filmed:
478,298 views

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。
- Mathematician, pianist
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世界 is awash充斥着
with divisive分裂 arguments参数,
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这个世界充斥着引发分歧的观点,
冲突,
00:18
conflict冲突,
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00:20
fake news新闻,
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虚假新闻,
00:22
victimhood受害者,
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受害者情绪,
00:25
exploitation开发, prejudice偏见,
bigotry偏执, blame, shouting叫喊
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剥削, 偏见,偏执,责怪,喊叫
00:30
and minuscule微不足道 attention注意 spans跨度.
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和极短的注意力时间。
00:34
It can sometimes有时 seem似乎
that we are doomed注定 to take sides双方,
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有时候似乎我们注定要选边站队,
00:40
be stuck卡住 in echo回声 chambers
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固执己见,
再也无法与人达成共识。
00:42
and never agree同意 again.
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00:45
It can sometimes有时 seem似乎
like a race种族 to the bottom底部,
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有时候似乎我们在比惨,
00:48
where everyone大家 is calling调用 out
somebody else's别人的 privilege特权
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每个人都在说别人有特权,
00:52
and vying百舸争流 to show显示 that they
are the most hard-done-by硬地比 person
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然后争先恐后地怨天尤人,
00:57
in the conversation会话.
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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世界 that doesn't?
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我们怎样才能做到理智?
01:07
I have a tool工具 for understanding理解
this confusing扑朔迷离 world世界 of ours我们的,
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我有一样工具,
可以帮助理解这个费解的世界,
01:12
a tool工具 that you might威力 not expect期望:
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一样你们可能意想不到的工具:
01:16
abstract抽象 mathematics数学.
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抽象数学。
01:19
I am a pure mathematician数学家.
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我是个纯粹的数学家。
01:22
Traditionally传统, pure maths数学
is like the theory理论 of maths数学,
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传统意义上,纯数学更多是
研究数学理论,
01:26
where applied应用的 maths数学 is applied应用的
to real真实 problems问题 like building建造 bridges桥梁
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而应用数学是解决实际问题,
比如建造大桥,
01:31
and flying飞行 planes飞机
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开飞机,
01:32
and controlling控制 traffic交通 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数学 applies适用 directly
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但我打算讲一种
将纯数学作为一种思维方式
01:40
to our daily日常 lives生活
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直接运用到
01:42
as a way of thinking思维.
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我们日常生活的情况。
01:44
I don't solve解决 quadratic二次 equations方程
to help me with my daily日常 life,
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我解二次方程并不是为了
方便我的日常生活,
01:49
but I do use mathematical数学的 thinking思维
to help me understand理解 arguments参数
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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同情 with other people.
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01:57
And so pure maths数学 helps帮助 me
with the entire整个 human人的 world世界.
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所以纯数学可以帮助我
理解整个人类世界。
02:04
But before I talk about
the entire整个 human人的 world世界,
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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威力 think of
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我必须和你们谈一些看起来可能
02:10
as irrelevant不相干 schools学校 maths数学:
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无关的学校里教的数学:
02:13
factors因素 of numbers数字.
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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开始
by thinking思维 about the factors因素 of 30.
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02:19
Now, if this makes品牌 you shudder不寒而栗
with bad memories回忆 of school学校 maths数学 lessons教训,
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如果这令你回想起数学课的糟糕回忆,
02:24
I sympathize同情, because I found发现
school学校 maths数学 lessons教训 boring无聊, 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漂亮 sure we are going
to take this in a direction方向
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但我很确定,接下来发生的事情
02:33
that is very different不同
from what happened发生 at school学校.
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跟学校里学到的会非常不一样。
02:37
So what are the factors因素 of 30?
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那么30的因数有哪些?
02:39
Well, they're the numbers数字 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记得 them.
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有趣.
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这没什么意思。
02:55
It's a bunch of numbers数字
in a straight直行 line线.
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就是一串数字。
02:58
We can make it more interesting有趣
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我们能让它变得有趣一些,
03:00
by thinking思维 about which哪一个 of these numbers数字
are also factors因素 of each other
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想一下,这些数字中,
有哪些互为因数,
03:04
and drawing画画 a picture图片,
a bit like a family家庭 tree,
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然后画一张像家谱图的图,
03:06
to show显示 those relationships关系.
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来展示这些关系。
03:08
So 30 is going to be at the top最佳
like a kind of great-grandparent曾祖.
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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整除 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角落 of a cube立方体,
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但这里是一个立方体的8个角,
03:36
which哪一个 is, I think, a bit more interesting有趣
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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数字
in a straight直行 line线.
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03:41
We can see something more here.
There's a hierarchy等级制度 going on.
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我们能发现更多东西。
它是分层级的。
03:44
At the bottom底部 level水平 is the number one,
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最底层是数字1,
03:46
then there's the numbers数字
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他们自己.
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03:51
You might威力 remember记得
this means手段 they're prime主要.
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你或许记得,这表示他们是质数。
03:54
At the next下一个 level水平 up,
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产品
of two prime主要 factors因素.
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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最佳, we have 30,
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最顶层的30,
04:08
which哪一个 is a product产品
of three prime主要 numbers数字 --
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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重绘 this diagram
using运用 those numbers数字 instead代替.
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所以我可以重新画这个图,
用数字来替代。
04:18
We see that we've我们已经 got
two, three and five at the top最佳,
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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数字
at the next下一个 level水平,
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04:24
and we have single elements分子
at the next下一个 level水平
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单个的数字在下一层
04:26
and then the empty set at the bottom底部.
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最底层是空集。
04:29
And each of those arrows箭头 shows节目
losing失去 one of your numbers数字 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明确
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现在或许清楚了,
04:37
that it doesn't really matter
what those numbers数字 are.
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这些数字是多少并不重要。
04:40
In fact事实, 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更换 them with
something like A, B and C instead代替,
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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相同 picture图片.
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也会得到同样的图。
04:49
So now this has become成为 very abstract抽象.
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这样就变得抽象了。
04:51
The numbers数字 have turned转身 into letters.
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数字变成了字母。
04:54
But there is a point to this abstraction抽象化,
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但这个抽象化是有意义的,
04:57
which哪一个 is that it now suddenly突然
becomes very widely广泛 applicable适用,
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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类型 of privilege特权:
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比如,它们可以是3种特权:
05:10
rich丰富, white白色 and male.
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有钱的,白人,男性。
05:14
So then at the next下一个 level水平,
we have rich丰富 white白色 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丰富 male people.
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这里是“有钱的”“男性”。
05:20
Here we have white白色 male people.
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这里是“白人”“男性”。
05:22
Then we have rich丰富, white白色 and male.
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然后是“有钱的”、“白人”、“男性”。
05:27
And finally最后, people with none没有
of those types类型 of privilege特权.
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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休息 of the adjectives形容词 for emphasis重点.
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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丰富, white白色
non-male非男 people,
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所以这里是“有钱的”、“白人”、“非男性”,
05:36
to remind提醒 us that there are
nonbinary非二进制 people we need to include包括.
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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丰富, nonwhite非白人 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非富, white白色 male people,
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05:45
rich丰富, nonwhite非白人, non-male非男,
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“有钱的”、“非白人”、“非男性”,
05:48
non-rich非富, white白色, non-male非男
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“非有钱的”、“白人”、“非男性”,
05:51
and non-rich非富, nonwhite非白人, male.
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以及“非有钱的”、“非白人”、“男性”,
05:53
And at the bottom底部,
with the least最小 privilege特权,
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以及在最底层,特权最少的
05:55
non-rich非富, nonwhite非白人, non-male非男 people.
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“非有钱的”、“非白人”、“非男性”。
05:59
We have gone走了 from a diagram
of factors因素 of 30
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我们从一个30的因数图表
06:03
to a diagram of interaction相互作用
of different不同 types类型 of privilege特权.
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到了一个不同特权的交叉图表。
06:08
And there are many许多 things
we can learn学习 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箭头 represents代表
a direct直接 loss失利 of one type类型 of privilege特权.
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首先,每个箭头表示失去一种特权。
06:19
Sometimes有时 people mistakenly think
that white白色 privilege特权 means手段
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有时候人们错误地以为,
白人特权意味着
06:23
all white白色 people are better off
than all nonwhite非白人 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超级富豪
black黑色 sports体育 stars明星 and say,
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有些人指着超级有钱的
黑人运动明星说,
06:32
"See? They're really rich丰富.
White白色 privilege特权 doesn't exist存在."
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“看到没?他们超有钱,
白人特权不存在。”
06:36
But that's not what the theory理论
of white白色 privilege特权 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,
06:39
It says that if that superrich超级富豪 sports体育 star
had all the same相同 characteristics特点
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06:44
but they were also white白色,
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同时还是白人,
06:45
we would expect期望 them
to be better off in society社会.
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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其他
we can understand理解 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沿 a row.
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06:56
If we look along沿 the second-to-top从第二到顶 row,
where people have two types类型 of privilege特权,
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沿最顶层到第二层的箭头看,
拥有两种特权的人,
07:00
we might威力 be able能够 to see
that they're not all particularly尤其 equal等于.
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他们并不是特别平等。
07:04
For example, rich丰富 white白色 women妇女
are probably大概 much better off in society社会
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比如,有钱的白人女性
或许比贫穷的白人男性
07:10
than poor较差的 white白色 men男人,
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混得更好,
07:12
and rich丰富 black黑色 men男人 are probably大概
somewhere某处 in between之间.
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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偏斜 like this,
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所以其实这张图应该更加倾斜,
07:18
and the same相同 on the bottom底部 level水平.
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最下面一层也是一样。
07:20
But we can actually其实 take it further进一步
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我们可以更仔细地
07:23
and look at the interactions互动
between之间 those two middle中间 levels水平.
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看中间两层的相互关系。
07:27
Because rich丰富, nonwhite非白人 non-men非男性
might威力 well be better off in society社会
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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
07:33
than poor较差的 white白色 men男人.
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过得更好。
07:35
Think about some extreme极端
examples例子, like Michelle米歇尔 Obama奥巴马,
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举几个极端的例子,比如米歇尔·奥巴马
07:39
Oprah奥普拉 Winfrey温弗瑞.
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奥普拉·温弗瑞。
07:40
They're definitely无疑 better off
than poor较差的, white白色, unemployed失业的 homeless无家可归 men男人.
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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。
07:46
So actually其实, the diagram
is more skewed偏斜 like this.
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所以这张图更像是这样倾斜。
07:49
And that tension张力 exists存在
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图表中各个层级的人
07:52
between之间 the layers
of privilege特权 in the diagram
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在生活中体验到的特权,
07:55
and the absolute绝对 privilege特权
that people experience经验 in society社会.
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与在图表中所处的位置存在差异。
07:59
And this has helped帮助 me to understand理解
why some poor较差的 white白色 men男人
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这帮助我理解了,
为什么有些贫穷的白人男性
08:02
are so angry愤怒 in society社会 at the moment时刻.
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在社会中如此愤怒。
08:06
Because they are considered考虑 to be high up
in this cuboid长方体 of privilege特权,
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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条款 of absolute绝对 privilege特权,
they don't actually其实 feel the effect影响 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。
08:15
And I believe that understanding理解
the root of that anger愤怒
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我认为理解愤怒的根源
08:19
is much more productive生产的
than just being存在 angry愤怒 at them in return返回.
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比反过来对他们感到愤怒
更有实际帮助。
08:25
Seeing眼见 these abstract抽象 structures结构
can also help us switch开关 contexts上下文
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这种抽象的结构
还能帮我们转换情境,
08:29
and see that different不同 people
are at the top最佳 in different不同 contexts上下文.
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看看如果不同的人
位于顶端,会有什么不同。
08:33
In our original原版的 diagram,
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在我们最初的图表里,
08:35
rich丰富 white白色 men男人 were at the top最佳,
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有钱的白人男性在顶层,
08:37
but if we restricted限制
our attention注意 to non-men非男性,
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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。
08:42
and now the rich丰富, white白色
non-men非男性 are at the top最佳.
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08:45
So we could move移动 to
a whole整个 context上下文 of women妇女,
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我们可以把整个情境转换到女性,
08:48
and our three types类型 of privilege特权
could now be rich丰富, white白色 and cisgendered顺性别.
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那么我们的三种特权变成了
“有钱的”、“白人“、“本性别”。
08:53
Remember记得 that "cisgendered顺性别" means手段
that your gender性别 identity身分 does match比赛
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“本性别”是指自我认同的性别
08:57
the gender性别 you were assigned分配 at birth分娩.
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和生理性别一致。
09:00
So now we see that rich丰富, white白色 cisCis women妇女
occupy占据 the analogous类似 situation情况
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现在“有钱的”、“白人”、“本性别女性”
与“有钱的”、“白人”、“男性”
09:06
that rich丰富 white白色 men男人 did
in broader更广泛 society社会.
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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帮助 me understand理解
why there is so much anger愤怒
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这让我理解了,为什么会有那么多人
09:12
towards rich丰富 white白色 women妇女,
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讨厌“有钱的”、“白人”、“女性”,
09:14
especially特别 in some parts部分
of the feminist女权主义者 movement运动 at the moment时刻,
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尤其在最近的很多女权主义活动中,
09:17
because perhaps也许 they're prone易于
to seeing眼看 themselves他们自己 as underprivileged弱势
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因为她们倾向于认为自己是弱势群体,
09:21
relative相对的 to white白色 men男人,
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如果跟白人男性比的话,
09:23
and they forget忘记 how overprivileged过度特权
they are relative相对的 to nonwhite非白人 women妇女.
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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。
09:30
We can all use these abstract抽象 structures结构
to help us pivot between之间 situations情况
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我们能利用这些抽象的结构
在情境之间转换
09:36
in which哪一个 we are more privileged特权
and less privileged特权.
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我们有时占优势,有时占劣势。
09:38
We are all more privileged特权 than somebody
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我们总会比一些人占优势,
09:41
and less privileged特权 than somebody else其他.
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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亚洲 person,
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比如,我知道并且感觉到
作为一个亚洲人,
09:49
I am less privileged特权 than white白色 people
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比白人更弱势,
09:52
because of white白色 privilege特权.
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因为白人特权的存在。
09:53
But I also understand理解
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但是我也知道,
09:55
that I am probably大概 among其中
the most privileged特权 of nonwhite非白人 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帮助 me pivot
between之间 those two contexts上下文.
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这能让我在不同情境下转换。
10:03
And in terms条款 of wealth财富,
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说到财富,
10:05
I don't think I'm super rich丰富.
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我不觉得自己超级有钱。
10:07
I'm not as rich丰富 as the kind of people
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情况 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奋斗的,
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那些失业的,或者拿最低工资的人,
10:15
maybe are unemployed失业的
or working加工 at minimum最低限度 wage工资.
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我已经过得很好了。
10:20
I perform演出 these pivots支点 in my head
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我在脑中进行这些转换
10:24
to help me understand理解 experiences经验
from other people's人们 points of view视图,
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来帮助我理解别人的处境,
10:30
which哪一个 brings带来 me to this
possibly或者 surprising奇怪 conclusion结论:
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这让我得出了一个意外的结论:
10:35
that abstract抽象 mathematics数学
is highly高度 relevant相应 to our daily日常 lives生活
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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理解
and empathize同情 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希望 is that everybody每个人 would try
to understand理解 other people more
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我希望每个人都能
尝试更多去理解他人,
10:56
and work with them together一起,
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共同努力,
10:58
rather than competing竞争 with them
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而不是相互竞争,
11:00
and trying to show显示 that they're wrong错误.
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说对方错了。
11:04
And I believe that abstract抽象
mathematical数学的 thinking思维
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我相信,抽象的数学思维
11:08
can help us achieve实现 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掌声)
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(掌声)
Translated by jiayi jiang
Reviewed by Alvin Lee

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ABOUT THE SPEAKER
Eugenia Cheng - Mathematician, pianist
Eugenia 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 TimesNational GeographicScientific American, and she was interviewed around the world including on the BBCNPR 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.

More profile about the speaker
Eugenia Cheng | Speaker | TED.com