An unexpected tool for understanding inequality: abstract math | Eugenia Cheng

譯者: Lilian Chiu 審譯者: Bruce Sung

（掌聲）

The world is awash with divisive arguments,

這個世界被淹沒了，

充斥著造成不和的爭論、

conflict,

衝突、

fake news,

假新聞、

victimhood,

受害、

exploitation, prejudice, bigotry, blame, shouting

剝削、偏見、不容忍、責怪、叫喊，

and minuscule attention spans.

以及短暫注意力。

It can sometimes seem that we are doomed to take sides,

有時，似乎我們注定得要選邊站，

be stuck in echo chambers

被困在回聲室裡，

and never agree again.

永遠不會再有共識。

It can sometimes seem like a race to the bottom,

有時，感覺就像是逐底競爭，

where everyone is calling out somebody else's privilege

每個人都在大聲指出別人的特權，

and vying to show that they are the most hard-done-by person

在談話中搶著要展現出

他們是最受委屈的人。

in the conversation.

How can we make sense

在這個不合理的世界上， 我們要如何找出意義？

in a world that doesn't?

I have a tool for understanding this confusing world of ours,

我有一個工具，能協助了解 我們這個讓人困惑的世界，

a tool that you might not expect:

這個工具可能是 你們未曾預期過的：

abstract mathematics.

抽象數學。

I am a pure mathematician.

我是個純數學家。

Traditionally, pure maths is like the theory of maths,

傳統上，純數學就像是數學理論，

where applied maths is applied to real problems like building bridges

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，

and flying planes

and controlling traffic flow.

以及控制交通車流。

But I'm going to talk about a way that pure maths applies directly

但我要談的，是將純數學

做為一種思考方式，

to our daily lives

as a way of thinking.

直接應用到我們的日常生活中。

I don't solve quadratic equations to help me with my daily life,

我不用解二次方程來協助日常生活，

but I do use mathematical thinking to help me understand arguments

但我會用數學思维來幫我理解論點，

and to empathize with other people.

及去同理他人。

And so pure maths helps me with the entire human world.

所以，純數學幫我 處理整個人類世界。

But before I talk about the entire human world,

但在我談整個人類世界之前，

I need to talk about something that you might think of

我得要先談談你可能認為

as irrelevant schools maths:

學了沒用的學校數學：

factors of numbers.

數字的因數。

We're going to start by thinking about the factors of 30.

我們從思考 30 的因數開始。

Now, if this makes you shudder with bad memories of school maths lessons,

如果這會引起你過去不好的 學校數學課回憶而讓你發抖，

I sympathize, because I found school maths lessons boring, too.

我感同身受，因為我也覺得 學校數學課很無聊。

But I'm pretty sure we are going to take this in a direction

但我很確定，我們今天 會走不同的方向，

that is very different from what happened at school.

絕對不是學校的那一套。

So what are the factors of 30?

所以，30 的因數是什麼？

Well, they're the numbers that go into 30.

就是能夠整除 30 的數字。

Maybe you can remember them. We'll work them out.

也許你能想起來。 我們來把它們解出來。

It's one, two, three,

答案是 1、2、3、

five, six,

5、6、

10, 15 and 30.

10、15，及 30。

It's not very interesting.

這沒什麼樂趣。

It's a bunch of numbers in a straight line.

就只是一連串的數字。

We can make it more interesting

我們可以讓它更有趣，

by thinking about which of these numbers are also factors of each other

想想看這些數字中， 有哪些也是彼此的因數？

and drawing a picture, a bit like a family tree,

畫一張圖，有點像是家族樹狀圖，

to show those relationships.

把那些關係畫出來。

So 30 is going to be at the top like a kind of great-grandparent.

所以，最頂端是 30， 有點像是曾祖父。

Six, 10 and 15 go into 30.

30 可以分出 6、10，和 15。

Five goes into 10 and 15.

10 和 15 可以分出 5。

Two goes into six and 10.

6 和 10 則可以分出 2。

Three goes into six and 15.

6 和 15 能分出 3。

And one goes into two, three and five.

2、3、5 則能分出 1。

So now we see that 10 is not divisible by three,

現在我們就能看到， 10 無法被 3 整除，

but that this is the corners of a cube,

但，這是立方體的一個角，

which is, I think, a bit more interesting

我認為，這樣會比 一連串的數字更有趣些。

than a bunch of numbers in a straight line.

We can see something more here. There's a hierarchy going on.

這裡還可以看出更多資訊。 有一種階層存在。

At the bottom level is the number one,

在最底層是數字 1，

then there's the numbers two, three and five,

接著是數字 2、3、5。

and nothing goes into those except one and themselves.

除了 1 和它們本身之外， 無法分出其他數字。

You might remember this means they're prime.

你們可能記得， 這就表示它們是質數。

At the next level up, we have six, 10 and 15,

再向上一層，是 6、10、15，

and each of those is a product of two prime factors.

它們每一個都是 兩個質數因數的乘積。

So six is two times three,

6 就是 2 乘以 3，

10 is two times five,

10 就是 2 乘以 5，

15 is three times five.

15 就是 3 乘以 5。

And then at the top, we have 30,

接著，最頂層是 30，

which is a product of three prime numbers --

它是三個質數的乘積——

two times three times five.

2 乘以 3 乘以 5。

So I could redraw this diagram using those numbers instead.

所以，我用那些數字 把這張圖重新畫過。

We see that we've got two, three and five at the top,

我們可以看見， 最頂層是 2、3、5，

we have pairs of numbers at the next level,

下面一層則是成對的數字，

and we have single elements at the next level

再下面一層是單一個元素，

and then the empty set at the bottom.

最底下則是一個空集合。

And each of those arrows shows losing one of your numbers in the set.

每一個箭頭表示 從集合中拿掉一個數字。

Now maybe it can be clear

也許這樣更清楚顯示，

that it doesn't really matter what those numbers are.

那些數字是什麼其實無所謂。

In fact, it doesn't matter what they are.

事實上，它們是什麼都無所謂。

So we could replace them with something like A, B and C instead,

我們可以用 a、b、c 來取代它們，

and we get the same picture.

我們還是會有同樣的圖。

So now this has become very abstract.

現在，它就變得非常抽象了。

The numbers have turned into letters.

數字被轉成英文字。

But there is a point to this abstraction,

但，轉成這種抽象是有目的的，

which is that it now suddenly becomes very widely applicable,

那就是，它的應用範圍 突然間就變得很廣，

because A, B and C could be anything.

因為 a、b、c 可是以任何東西。

For example, they could be three types of privilege:

比如，它們可能是三種特權：

rich, white and male.

富裕、白人、男性。

So then at the next level, we have rich white people.

接著，下一層，有富裕的白人。

Here we have rich male people.

這裡是富裕的男性。

Here we have white male people.

這裡是白人男性。

Then we have rich, white and male.

接著有富裕白人男性。

And finally, people with none of those types of privilege.

最後，是三種特權都沒有的人。

And I'm going to put back in the rest of the adjectives for emphasis.

現在我要把其他的形容詞 也放回來供強調之用。

So here we have rich, white non-male people,

這裡的是富裕白人非男性，

to remind us that there are nonbinary people we need to include.

以提醒我們納入那些無法 用男女二分法區分的人。

Here we have rich, nonwhite male people.

這裡的是富裕非白人男性。

Here we have non-rich, white male people,

這裡是非富裕白人男性、

rich, nonwhite, non-male,

富裕非白人非男性、

non-rich, white, non-male

非富裕白人非男性、

and non-rich, nonwhite, male.

非富裕非白人男性。

And at the bottom, with the least privilege,

在最底下，特權最少的是

non-rich, nonwhite, non-male people.

非富裕非白人非男性。

We have gone from a diagram of factors of 30

我們已經從 30 的因子圖，

to a diagram of interaction of different types of privilege.

來到不同特權的交互作用圖。

And there are many things we can learn from this diagram, I think.

我想，從這張圖中， 我們可以學到很多。

The first is that each arrow represents a direct loss of one type of privilege.

首先，

每個箭頭代表 直接失去一種特權。

Sometimes people mistakenly think that white privilege means

有時，大家會誤以為 白人特權意味著

all white people are better off than all nonwhite people.

所有白人的境況都比非白人更好。

Some people point at superrich black sports stars and say,

有些人會指著超級富裕的 黑人運動明星，說：

"See? They're really rich. White privilege doesn't exist."

「看到了嗎？他們很富裕。 白人特權並不存在。」

But that's not what the theory of white privilege says.

但白人特權的理論 並不是這樣說的。

It says that if that superrich sports star had all the same characteristics

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，

but they were also white,

只是變成白人，

we would expect them to be better off in society.

我們會預期他們 在社會中有更好的境況。

There is something else we can understand from this diagram

從這張圖，我們還能 了解另外一件事，

if we look along a row.

我們可以沿著橫列來看。

If we look along the second-to-top row, where people have two types of privilege,

如果看第二列的最上面一列， 也就是有兩種特權的人，

we might be able to see that they're not all particularly equal.

我們可能可以看到， 他們並非全都很平等。

For example, rich white women are probably much better off in society

比如，

在社會中，富裕白人女性的境況

than poor white men,

可能會比貧窮白人男性更好，

and rich black men are probably somewhere in between.

而富裕黑人男性的境況 可能會在兩者之間。

So it's really more skewed like this,

所以它是有點偏斜的，像這樣，

and the same on the bottom level.

最下面一層亦是如此。

But we can actually take it further

但我們可以再進一步，

and look at the interactions between those two middle levels.

去看中間兩層的交互作用。

Because rich, nonwhite non-men might well be better off in society

因為，在社會中， 富裕非白人非男性的境況可能會

好過貧窮白人男性。

than poor white men.

Think about some extreme examples, like Michelle Obama,

想想看極端的例子， 比如蜜雪兒歐巴馬，

Oprah Winfrey.

歐普拉溫芙蕾。

They're definitely better off than poor, white, unemployed homeless men.

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。

So actually, the diagram is more skewed like this.

所以，其實，這張圖的 偏斜應該是像這樣。

And that tension exists

且那種緊繃狀況存在於

between the layers of privilege in the diagram

圖上各特權分層

and the absolute privilege that people experience in society.

和社會中大家體驗到的 絕對特權之間。

And this has helped me to understand why some poor white men

這協助我了解， 為什麼有些貧窮白人男性

are so angry in society at the moment.

此時對於社會感到如此憤怒。

Because they are considered to be high up in this cuboid of privilege,

因為在這個特權立方體中， 他們被視為處在高位，

but in terms of absolute privilege, they don't actually feel the effect of it.

但就絕對特權來說， 他們其實沒有感覺到它的效應。

And I believe that understanding the root of that anger

我相信，了解那憤怒的根源，

is much more productive than just being angry at them in return.

比起只是反過來對他們感到憤怒， 是更有建設性的做法，

Seeing these abstract structures can also help us switch contexts

看這些抽象的結構 也能協助我們調換情境，

and see that different people are at the top in different contexts.

了解不同的人在頂端的 不同情境中會如何。

In our original diagram,

在我們原始的圖中，

rich white men were at the top,

富裕白人男性是在頂端的人，

but if we restricted our attention to non-men,

但如果我們把注意力侷限在非男性，

we would see that they are here,

他們在這些地方，

and now the rich, white non-men are at the top.

現在，富裕白人非男性 變成是在頂端。

So we could move to a whole context of women,

我們可以改成完全 只看女性的情境，

and our three types of privilege could now be rich, white and cisgendered.

現在我們的三種特權就變成是 富裕、白人、順性別。

Remember that "cisgendered" means that your gender identity does match

「順性別」的意思就是 你的性別認同確實符合

the gender you were assigned at birth.

你出生時具有的性別。

So now we see that rich, white cis women occupy the analogous situation

現在，我們看到富裕白人 順性別女性佔據的地位，

that rich white men did in broader society.

和在更廣的社會中 富裕白人男性的地位類似。

And this has helped me understand why there is so much anger

這協助我了解， 為什麼會有這麼多的怒火

towards rich white women,

是針對富裕白人女性，

especially in some parts of the feminist movement at the moment,

特別是在此時女權主義 運動的一些地方，

because perhaps they're prone to seeing themselves as underprivileged

因為也許她們傾向會認為，

相對於白人男性，她們沒有特權，

relative to white men,

and they forget how overprivileged they are relative to nonwhite women.

而她們忘記了相對於非白人女性， 她們其實有太多的特權。

We can all use these abstract structures to help us pivot between situations

我們都可以用這些 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。

in which we are more privileged and less privileged.

We are all more privileged than somebody

我們都比某些人更有特權，

and less privileged than somebody else.

也都有某些人比我們更有特權。

For example, I know and I feel that as an Asian person,

比如，身為亞洲人， 我知道也感覺得到

I am less privileged than white people

我的特權不如白人，

because of white privilege.

因為白人特權。

But I also understand

但我也了解，

that I am probably among the most privileged of nonwhite people,

在非白人當中， 我可能是最有特權的，

and this helps me pivot between those two contexts.

這就協助我在兩種情境當中轉換。

And in terms of wealth,

就財富來說，

I don't think I'm super rich.

我不認為我超有錢。

I'm not as rich as the kind of people who don't have to work.

我不像有些人有錢到不用工作。

But I am doing fine,

但我過得很好，

and that's a much better situation to be in

我的情況比很多人好很多了，

than people who are really struggling,

比如真的在掙扎的人，

maybe are unemployed or working at minimum wage.

沒有工作或是工作領最低薪資的人。

I perform these pivots in my head

我在我的腦中進行這些轉換，

to help me understand experiences from other people's points of view,

來協助我了解從他人的觀點 看到的是什麼，

which brings me to this possibly surprising conclusion:

這讓我得到了這個 或許非常驚人的結論：

that abstract mathematics is highly relevant to our daily lives

抽象數學

和我們的日常生活息息相關，

and can even help us to understand and empathize with other people.

甚至能夠協助我們

了解並同情他人。

My wish is that everybody would try to understand other people more

我的希望是每個人都能 嘗試對別人多一點了解，

and work with them together,

和他們一起合作，

rather than competing with them

而不是和他們競爭，

and trying to show that they're wrong.

試圖證明他們是錯的。

And I believe that abstract mathematical thinking

我相信抽象數學思考

can help us achieve that.

能協助我們做到這個目標。

Thank you.

謝謝。

(Applause)

（掌聲）