TEDxLondon
Eugenia Cheng: An unexpected tool for understanding inequality: abstract math
尤吉妮亞 · 程: 誰想得到可以用這個工具來了解不平等：抽象數學
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在這個不合理的世界上，我們要如何找出意義？數學家尤吉妮亞 · 程說，我們可以去試著看未預期的地方。她解釋如何把抽象數學的概念應用到日常生活中，引導我們更深入了解一些事物，比如憤怒的根源以及特權怎麼運作。來看看這場演說，進一步了解這項驚人的工具如何協助我們同理他人。
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.
（掌聲）
00:13
The world is awash
with divisive arguments,
with divisive arguments,
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這個世界被淹沒了，
充斥著造成不和的爭論、
00:18
conflict,
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衝突、
00:20
fake news,
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假新聞、
00:22
victimhood,
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受害、
00:25
exploitation, prejudice,
bigotry, blame, shouting
bigotry, blame, shouting
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剝削、偏見、不容忍、責怪、叫喊，
00:30
and minuscule attention spans.
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以及短暫注意力。
00:34
It can sometimes seem
that we are doomed to take sides,
that we are doomed to take sides,
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有時，似乎我們注定得要選邊站，
00:40
be stuck in echo chambers
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被困在回聲室裡，
00:42
and never agree again.
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永遠不會再有共識。
00:45
It can sometimes seem
like a race to the bottom,
like a race to the bottom,
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有時，感覺就像是逐底競爭，
00:48
where everyone is calling out
somebody else's privilege
somebody else's privilege
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每個人都在大聲指出別人的特權，
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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在談話中搶著要展現出
他們是最受委屈的人。
00:57
in the conversation.
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01:01
How can we make sense
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在這個不合理的世界上，
我們要如何找出意義？
我們要如何找出意義？
01:02
in a world that doesn't?
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01:07
I have a tool for understanding
this confusing world of ours,
this confusing world of ours,
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我有一個工具，能協助了解
我們這個讓人困惑的世界，
我們這個讓人困惑的世界，
01:12
a tool that you might not expect:
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這個工具可能是
你們未曾預期過的：
你們未曾預期過的：
01:16
abstract mathematics.
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抽象數學。
01:19
I am a pure mathematician.
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我是個純數學家。
01:22
Traditionally, pure maths
is like the theory of maths,
is like the theory of maths,
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傳統上，純數學就像是數學理論，
01:26
where applied maths is applied
to real problems like building bridges
to real problems like building bridges
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而應用數學是被用在真實問題上，
比如建造橋樑、讓飛機飛行，
01:31
and flying planes
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01:32
and controlling traffic flow.
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以及控制交通車流。
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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但我要談的，是將純數學
做為一種思考方式，
01:40
to our daily lives
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01:42
as a way of thinking.
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直接應用到我們的日常生活中。
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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我不用解二次方程來協助日常生活，
01:49
but I do use mathematical thinking
to help me understand arguments
to help me understand arguments
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但我會用數學思维來幫我理解論點，
01:54
and to empathize with other people.
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及去同理他人。
01:57
And so pure maths helps me
with the entire human world.
with the entire human world.
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所以，純數學幫我
處理整個人類世界。
處理整個人類世界。
02:04
But before I talk about
the entire human world,
the entire human world,
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但在我談整個人類世界之前，
02:07
I need to talk about something
that you might think of
that you might think of
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我得要先談談你可能認為
02:10
as irrelevant schools maths:
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學了沒用的學校數學：
02:13
factors of numbers.
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數字的因數。
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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我們從思考 30 的因數開始。
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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如果這會引起你過去不好的
學校數學課回憶而讓你發抖，
學校數學課回憶而讓你發抖，
02:24
I sympathize, because I found
school maths lessons boring, too.
school maths lessons boring, too.
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我感同身受，因為我也覺得
學校數學課很無聊。
學校數學課很無聊。
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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但我很確定，我們今天
會走不同的方向，
會走不同的方向，
02:33
that is very different
from what happened at school.
from what happened at school.
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絕對不是學校的那一套。
02:37
So what are the factors of 30?
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所以，30 的因數是什麼？
02:39
Well, they're the numbers that go into 30.
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就是能夠整除 30 的數字。
02:42
Maybe you can remember them.
We'll work them out.
We'll work them out.
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也許你能想起來。
我們來把它們解出來。
我們來把它們解出來。
02:45
It's one, two, three,
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答案是 1、2、3、
02:48
five, six,
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5、6、
02:51
10, 15 and 30.
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10、15，及 30。
02:53
It's not very interesting.
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這沒什麼樂趣。
02:55
It's a bunch of numbers
in a straight line.
in a straight line.
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就只是一連串的數字。
02:58
We can make it more interesting
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我們可以讓它更有趣，
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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想想看這些數字中，
有哪些也是彼此的因數？
有哪些也是彼此的因數？
03:04
and drawing a picture,
a bit like a family tree,
a bit like a family tree,
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畫一張圖，有點像是家族樹狀圖，
03:06
to show those relationships.
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把那些關係畫出來。
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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所以，最頂端是 30，
有點像是曾祖父。
有點像是曾祖父。
03:12
Six, 10 and 15 go into 30.
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30 可以分出 6、10，和 15。
03:15
Five goes into 10 and 15.
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10 和 15 可以分出 5。
03:18
Two goes into six and 10.
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6 和 10 則可以分出 2。
03:21
Three goes into six and 15.
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6 和 15 能分出 3。
03:24
And one goes into two, three and five.
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2、3、5 則能分出 1。
03:29
So now we see that 10
is not divisible by three,
is not divisible by three,
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現在我們就能看到，
10 無法被 3 整除，
10 無法被 3 整除，
03:32
but that this is the corners of a cube,
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但，這是立方體的一個角，
03:36
which is, I think, a bit more interesting
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我認為，這樣會比
一連串的數字更有趣些。
一連串的數字更有趣些。
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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這裡還可以看出更多資訊。
有一種階層存在。
有一種階層存在。
03:44
At the bottom level is the number one,
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在最底層是數字 1，
03:46
then there's the numbers
two, three and five,
two, three and five,
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接著是數字 2、3、5。
除了 1 和它們本身之外，
無法分出其他數字。
無法分出其他數字。
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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你們可能記得，
這就表示它們是質數。
這就表示它們是質數。
03:54
At the next level up,
we have six, 10 and 15,
we have six, 10 and 15,
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再向上一層，是 6、10、15，
03:57
and each of those is a product
of two prime factors.
of two prime factors.
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它們每一個都是
兩個質數因數的乘積。
兩個質數因數的乘積。
04:00
So six is two times three,
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6 就是 2 乘以 3，
04:02
10 is two times five,
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10 就是 2 乘以 5，
04:04
15 is three times five.
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15 就是 3 乘以 5。
04:06
And then at the top, we have 30,
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接著，最頂層是 30，
04:08
which is a product
of three prime numbers --
of three prime numbers --
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它是三個質數的乘積——
04:10
two times three times five.
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2 乘以 3 乘以 5。
04:12
So I could redraw this diagram
using those numbers instead.
using those numbers instead.
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所以，我用那些數字
把這張圖重新畫過。
把這張圖重新畫過。
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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我們可以看見，
最頂層是 2、3、5，
最頂層是 2、3、5，
04:21
we have pairs of numbers
at the next level,
at the next level,
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下面一層則是成對的數字，
04:24
and we have single elements
at the next level
at the next level
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再下面一層是單一個元素，
04:26
and then the empty set at the bottom.
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最底下則是一個空集合。
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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每一個箭頭表示
從集合中拿掉一個數字。
從集合中拿掉一個數字。
04:34
Now maybe it can be clear
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也許這樣更清楚顯示，
04:37
that it doesn't really matter
what those numbers are.
what those numbers are.
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那些數字是什麼其實無所謂。
04:40
In fact, it doesn't matter what they are.
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事實上，它們是什麼都無所謂。
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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我們可以用 a、b、c 來取代它們，
04:46
and we get the same picture.
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我們還是會有同樣的圖。
04:49
So now this has become very abstract.
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現在，它就變得非常抽象了。
04:51
The numbers have turned into letters.
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數字被轉成英文字。
04:54
But there is a point to this abstraction,
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但，轉成這種抽象是有目的的，
04:57
which is that it now suddenly
becomes very widely applicable,
becomes very widely applicable,
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那就是，它的應用範圍
突然間就變得很廣，
突然間就變得很廣，
05:02
because A, B and C could be anything.
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因為 a、b、c 可是以任何東西。
05:06
For example, they could be
three types of privilege:
three types of privilege:
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比如，它們可能是三種特權：
05:10
rich, white and male.
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富裕、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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接著，下一層，有富裕的白人。
05:18
Here we have rich male people.
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這裡是富裕的男性。
05:20
Here we have white male people.
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這裡是白人男性。
05:22
Then we have rich, white and male.
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接著有富裕白人男性。
05:27
And finally, people with none
of those types of privilege.
of those types of privilege.
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最後，是三種特權都沒有的人。
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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現在我要把其他的形容詞
也放回來供強調之用。
也放回來供強調之用。
05:33
So here we have rich, white
non-male people,
non-male people,
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這裡的是富裕白人非男性，
05:36
to remind us that there are
nonbinary people we need to include.
nonbinary people we need to include.
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以提醒我們納入那些無法
用男女二分法區分的人。
用男女二分法區分的人。
05:39
Here we have rich, nonwhite male people.
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這裡的是富裕非白人男性。
05:42
Here we have non-rich, white male people,
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這裡是非富裕白人男性、
05:45
rich, nonwhite, non-male,
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富裕非白人非男性、
05:48
non-rich, white, non-male
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非富裕白人非男性、
05:51
and non-rich, nonwhite, male.
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非富裕非白人男性。
05:53
And at the bottom,
with the least privilege,
with the least privilege,
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在最底下，特權最少的是
05:55
non-rich, nonwhite, non-male people.
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非富裕非白人非男性。
05:59
We have gone from a diagram
of factors of 30
of factors of 30
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我們已經從 30 的因子圖，
06:03
to a diagram of interaction
of different types of privilege.
of different types of privilege.
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來到不同特權的交互作用圖。
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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我想，從這張圖中，
我們可以學到很多。
我們可以學到很多。
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，
每個箭頭代表
直接失去一種特權。
直接失去一種特權。
06:19
Sometimes people mistakenly think
that white privilege means
that white privilege means
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有時，大家會誤以為
白人特權意味著
白人特權意味著
06:23
all white people are better off
than all nonwhite people.
than all nonwhite people.
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所有白人的境況都比非白人更好。
06:28
Some people point at superrich
black sports stars and say,
black sports stars and say,
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有些人會指著超級富裕的
黑人運動明星，說：
黑人運動明星，說：
06:32
"See? They're really rich.
White privilege doesn't exist."
White privilege doesn't exist."
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「看到了嗎？他們很富裕。
白人特權並不存在。」
白人特權並不存在。」
06:36
But that's not what the theory
of white privilege says.
of white privilege says.
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但白人特權的理論
並不是這樣說的。
並不是這樣說的。
06:39
It says that if that superrich sports star
had all the same characteristics
had all the same characteristics
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它說，如果超級富裕的運動明星
在所有的特性都不變的情況下，
在所有的特性都不變的情況下，
06:44
but they were also white,
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只是變成白人，
06:45
we would expect them
to be better off in society.
to be better off in society.
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我們會預期他們
在社會中有更好的境況。
在社會中有更好的境況。
06:51
There is something else
we can understand from this diagram
we can understand from this diagram
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從這張圖，我們還能
了解另外一件事，
了解另外一件事，
06:54
if we look along a row.
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我們可以沿著橫列來看。
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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如果看第二列的最上面一列，
也就是有兩種特權的人，
也就是有兩種特權的人，
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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我們可能可以看到，
他們並非全都很平等。
他們並非全都很平等。
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，
在社會中，富裕白人女性的境況
07:10
than poor white men,
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可能會比貧窮白人男性更好，
07:12
and rich black men are probably
somewhere in between.
somewhere in between.
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而富裕黑人男性的境況
可能會在兩者之間。
可能會在兩者之間。
07:15
So it's really more skewed like this,
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所以它是有點偏斜的，像這樣，
07:18
and the same on the bottom level.
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最下面一層亦是如此。
07:20
But we can actually take it further
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但我們可以再進一步，
07:23
and look at the interactions
between those two middle levels.
between those two middle levels.
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去看中間兩層的交互作用。
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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因為，在社會中，
富裕非白人非男性的境況可能會
富裕非白人非男性的境況可能會
好過貧窮白人男性。
07:33
than poor white men.
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07:35
Think about some extreme
examples, like Michelle Obama,
examples, like Michelle Obama,
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想想看極端的例子，
比如蜜雪兒歐巴馬，
比如蜜雪兒歐巴馬，
07:39
Oprah Winfrey.
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歐普拉溫芙蕾。
07:40
They're definitely better off
than poor, white, unemployed homeless men.
than poor, white, unemployed homeless men.
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她們的境況肯定比貧窮沒工作
且無家可歸的白人男性更好。
且無家可歸的白人男性更好。
07:46
So actually, the diagram
is more skewed like this.
is more skewed like this.
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所以，其實，這張圖的
偏斜應該是像這樣。
偏斜應該是像這樣。
07:49
And that tension exists
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且那種緊繃狀況存在於
07:52
between the layers
of privilege in the diagram
of privilege in the diagram
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圖上各特權分層
07:55
and the absolute privilege
that people experience in society.
that people experience in society.
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和社會中大家體驗到的
絕對特權之間。
絕對特權之間。
07:59
And this has helped me to understand
why some poor white men
why some poor white men
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這協助我了解，
為什麼有些貧窮白人男性
為什麼有些貧窮白人男性
08:02
are so angry in society at the moment.
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此時對於社會感到如此憤怒。
08:06
Because they are considered to be high up
in this cuboid of privilege,
in this cuboid of privilege,
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因為在這個特權立方體中，
他們被視為處在高位，
他們被視為處在高位，
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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但就絕對特權來說，
他們其實沒有感覺到它的效應。
他們其實沒有感覺到它的效應。
08:15
And I believe that understanding
the root of that anger
the root of that anger
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我相信，了解那憤怒的根源，
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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比起只是反過來對他們感到憤怒，
是更有建設性的做法，
是更有建設性的做法，
08:25
Seeing these abstract structures
can also help us switch contexts
can also help us switch contexts
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看這些抽象的結構
也能協助我們調換情境，
也能協助我們調換情境，
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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了解不同的人在頂端的
不同情境中會如何。
不同情境中會如何。
08:33
In our original diagram,
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在我們原始的圖中，
08:35
rich white men were at the top,
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富裕白人男性是在頂端的人，
08:37
but if we restricted
our attention to non-men,
our attention to non-men,
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但如果我們把注意力侷限在非男性，
08:41
we would see that they are here,
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他們在這些地方，
08:42
and now the rich, white
non-men are at the top.
non-men are at the top.
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現在，富裕白人非男性
變成是在頂端。
變成是在頂端。
08:45
So we could move to
a whole context of women,
a whole context of women,
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我們可以改成完全
只看女性的情境，
只看女性的情境，
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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現在我們的三種特權就變成是
富裕、白人、順性別。
富裕、白人、順性別。
08:53
Remember that "cisgendered" means
that your gender identity does match
that your gender identity does match
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「順性別」的意思就是
你的性別認同確實符合
你的性別認同確實符合
08:57
the gender you were assigned at birth.
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你出生時具有的性別。
09:00
So now we see that rich, white cis women
occupy the analogous situation
occupy the analogous situation
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現在，我們看到富裕白人
順性別女性佔據的地位，
順性別女性佔據的地位，
09:06
that rich white men did
in broader society.
in broader society.
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和在更廣的社會中
富裕白人男性的地位類似。
富裕白人男性的地位類似。
09:09
And this has helped me understand
why there is so much anger
why there is so much anger
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這協助我了解，
為什麼會有這麼多的怒火
為什麼會有這麼多的怒火
09:12
towards rich white women,
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是針對富裕白人女性，
09:14
especially in some parts
of the feminist movement at the moment,
of the feminist movement at the moment,
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特別是在此時女權主義
運動的一些地方，
運動的一些地方，
09:17
because perhaps they're prone
to seeing themselves as underprivileged
to seeing themselves as underprivileged
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因為也許她們傾向會認為，
相對於白人男性，她們沒有特權，
09:21
relative to white men,
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09:23
and they forget how overprivileged
they are relative to nonwhite women.
they are relative to nonwhite women.
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而她們忘記了相對於非白人女性，
她們其實有太多的特權。
她們其實有太多的特權。
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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我們都可以用這些
抽象結構來協助我們，
抽象結構來協助我們，
在我們比較有特權和比較
沒特權的不同情況中轉換。
沒特權的不同情況中轉換。
09:36
in which we are more privileged
and less privileged.
and less privileged.
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09:38
We are all more privileged than somebody
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我們都比某些人更有特權，
09:41
and less privileged than somebody else.
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也都有某些人比我們更有特權。
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，身為亞洲人，
我知道也感覺得到
我知道也感覺得到
09:49
I am less privileged than white people
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我的特權不如白人，
09:52
because of white privilege.
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因為白人特權。
09:53
But I also understand
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但我也了解，
09:55
that I am probably among
the most privileged of nonwhite people,
the most privileged of nonwhite people,
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在非白人當中，
我可能是最有特權的，
我可能是最有特權的，
09:59
and this helps me pivot
between those two contexts.
between those two contexts.
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這就協助我在兩種情境當中轉換。
10:03
And in terms of wealth,
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就財富來說，
10:05
I don't think I'm super rich.
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我不認為我超有錢。
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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我不像有些人有錢到不用工作。
10:10
But I am doing fine,
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但我過得很好，
10:11
and that's a much better
situation to be in
situation to be in
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我的情況比很多人好很多了，
10:13
than people who are really struggling,
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比如真的在掙扎的人，
10:15
maybe are unemployed
or working at minimum wage.
or working at minimum wage.
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沒有工作或是工作領最低薪資的人。
10:20
I perform these pivots in my head
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我在我的腦中進行這些轉換，
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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來協助我了解從他人的觀點
看到的是什麼，
看到的是什麼，
10:30
which brings me to this
possibly surprising conclusion:
possibly surprising conclusion:
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這讓我得到了這個
或許非常驚人的結論：
或許非常驚人的結論：
10:35
that abstract mathematics
is highly relevant to our daily lives
is highly relevant to our daily lives
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抽象數學
和我們的日常生活息息相關，
10:42
and can even help us to understand
and empathize with other people.
and empathize with other people.
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甚至能夠協助我們
了解並同情他人。
10:50
My wish is that everybody would try
to understand other people more
to understand other people more
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我的希望是每個人都能
嘗試對別人多一點了解，
嘗試對別人多一點了解，
10:56
and work with them together,
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和他們一起合作，
10:58
rather than competing with them
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而不是和他們競爭，
11:00
and trying to show that they're wrong.
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試圖證明他們是錯的。
11:04
And I believe that abstract
mathematical thinking
mathematical thinking
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我相信抽象數學思考
能協助我們做到這個目標。
11:08
can help us achieve that.
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11:12
Thank you.
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謝謝。
11:13
(Applause)
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（掌聲）
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