What's so sexy about math? | Cédric Villani

譯者: Nan Liu 審譯者: Chloe Fu

What is it that French people do better than all the others?

法國人比其他國家的人 都擅長的是什麼？

If you would take polls,

如果你進行民意調查的話，

the top three answers might be:

排在前三位的答案可能是：

love, wine and whining.

愛情、葡萄酒和抱怨。

(Laughter)

（笑聲）

Maybe.

也許是這樣。

But let me suggest a fourth one:

但是讓我再加一項：

mathematics.

數學。

Did you know that Paris has more mathematicians

你知道巴黎的數學家

than any other city in the world?

比世界上其他任何一座城市的都多嗎？

And more streets with mathematicians' names, too.

而且也有更多 帶有數學家名字的街道。

And if you look at the statistics of the Fields Medal,

如果你看看菲爾茲獎的數據，

often called the Nobel Prize for mathematics,

它通常被稱作數學領域的諾貝爾獎，

and always awarded to mathematicians below the age of 40,

並且總是頒發給 40 歲以下的數學家，

you will find that France has more Fields medalists per inhabitant

你會發現法國人均菲爾茲獎得主數量

than any other country.

多過其他任何一個國家。

What is it that we find so sexy in math?

我們認為數學很迷人的地方是什麼？

After all, it seems to be dull and abstract,

畢竟，它看起來枯燥又抽象，

just numbers and computations and rules to apply.

僅僅是數字、計算和運算規則。

Mathematics may be abstract,

數學可能是抽象的，

but it's not dull

但是它不是枯燥的，

and it's not about computing.

而且它不是關於計算。

It is about reasoning

它是關於推理

and proving our core activity.

以及證明你的核心任務。

It is about imagination,

它是關於想像力，

the talent which we most praise.

我們最讚賞的一種能力。

It is about finding the truth.

它是關於發現真相。

There's nothing like the feeling which invades you

沒有任何事情可以和

when after months of hard thinking,

經過幾個月苦思終於得出 能解決問題的正確推理時

you finally understand the right reasoning to solve your problem.

充盈你的那種感受相比。

The great mathematician André Weil likened this --

偉大的數學家安德雷·韋伊 把這種感覺比作──

no kidding --

不是開玩笑的──

to sexual pleasure.

性愉悅。

But noted that this feeling can last for hours, or even days.

但是請注意這種感覺能持續 幾個小時，甚至幾天，

The reward may be big.

隨之而來的獎賞也可能很豐厚。

Hidden mathematical truths permeate our whole physical world.

我們的現實世界中充滿了 未被發現的數學真相。

They are inaccessible to our senses

我們的感官無法感知到它們，

but can be seen through mathematical lenses.

但我們可以用數學的眼光看見它們。

Close your eyes for moment

請把你們的眼睛閉上一會兒，

and think of what is occurring right now around you.

然後想一想現在 你們身邊正發生著什麼。

Invisible particles from the air around are bumping on you

空氣中每秒都有上億 不可見的微粒撞在你的身上，

by the billions and billions at each second,

all in complete chaos.

一片混亂。

And still,

然而，

their statistics can be accurately predicted by mathematical physics.

它們的數據可以被 數學物理準確地預測。

And open your eyes now

現在請睜開你們的眼睛

to the statistics of the velocities of these particles.

來看看這些顆粒移動速率的數據。

The famous bell-shaped Gauss Curve,

這個著名的鍾型高斯曲線，

or the Law of Errors --

或者是誤差定律──

of deviations with respect to the mean behavior.

相對於平均數的絕對偏差。

This curve tells about the statistics of velocities of particles

這條線描述了顆粒移動速率，

in the same way as a demographic curve

就如同

would tell about the statistics of ages of individuals.

人口曲線描述個體年齡。

It's one of the most important curves ever.

它是迄今為止最重要的曲線之一。

It keeps on occurring again and again,

它一次又一次地

from many theories and many experiments,

出現在眾多理論與實驗中，

as a great example of the universality

作為對我們數學家 十分寶貴、普遍性的

which is so dear to us mathematicians.

一個重要例證。

Of this curve,

關於這條曲線，

the famous scientist Francis Galton said,

著名的科學家 弗朗西斯·高爾頓說過，

"It would have been deified by the Greeks if they had known it.

「如果希臘人知道這條曲線的話， 它一定會被他們神化的。

It is the supreme law of unreason."

它是無理性的終極法則。」

And there's no better way to materialize that supreme goddess than Galton's Board.

沒有比高爾頓板更好的方式 來展現這位至尊女神。

Inside this board are narrow tunnels

在這個實驗板裡面是狹窄的通道，

through which tiny balls will fall down randomly,

小球會通過這些通道隨機落下，

going right or left, or left, etc.

向右或者向左，或者向左，等等。

All in complete randomness and chaos.

一切都是完全隨機和任意的。

Let's see what happens when we look at all these random trajectories together.

我們來看看把這些隨機的路徑 放在一起觀察將會如何。

(Board shaking)

（搖晃實驗板）

This is a bit of a sport,

這有點像一種運動，

because we need to resolve some traffic jams in there.

因為我們需要解決一些交通擁堵。

Aha.

啊哈。

We think that randomness is going to play me a trick on stage.

我們認為隨機性 將會在這個台上展現一些極妙。

There it is.

就是這樣。

Our supreme goddess of unreason.

我們無理性的至尊女神。

the Gauss Curve,

高斯曲線，

trapped here inside this transparent box as Dream in "The Sandman" comics.

被困在了這個透明的實驗板裡， 就像《睡魔》中的夢一樣。

For you I have shown it,

我已經給你們展示了這條曲線，

but to my students I explain why it could not be any other curve.

但是對我的學生，我要解釋 為什麼不會是其它的曲線。

And this is touching the mystery of that goddess,

這就涉及到這位至尊女神的神祕性，

replacing a beautiful coincidence by a beautiful explanation.

用一個完美的解釋 代替一個美麗的巧合。

All of science is like this.

所有的科學都如此。

And beautiful mathematical explanations are not only for our pleasure.

完美的數學解釋不僅僅讓我們愉快，

They also change our vision of the world.

它們也會改變我們對世界的看法。

For instance,

比如，

Einstein,

愛因斯坦，

Perrin,

佩蘭，

Smoluchowski,

斯莫魯霍夫斯基，

they used the mathematical analysis of random trajectories

他們用隨機路徑的數學分析

and the Gauss Curve

和高斯曲線，

to explain and prove that our world is made of atoms.

來解釋並且證明我們的世界 是由原子組成的。

It was not the first time

這不是第一次

that mathematics was revolutionizing our view of the world.

數學革新了我們對世界的看法。

More than 2,000 years ago,

2000 多年前，

at the time of the ancient Greeks,

在古希臘的時候，

it already occurred.

這就發生了。

In those days,

在那個時候，

only a small fraction of the world had been explored,

只有一小部分的世界已經被探索，

and the Earth might have seemed infinite.

而且地球似乎是無窮大的。

But clever Eratosthenes,

但是聰明的艾拉托瑟尼，

using mathematics,

用數學，

was able to measure the Earth with an amazing accuracy of two percent.

成功地以 2% 的 驚人精確度測量了地球。

Here's another example.

還有另一個例子。

In 1673, Jean Richer noticed

在 1673 年，簡·里歇爾發現

that a pendulum swings slightly slower in Cayenne than in Paris.

鐘擺在卡宴比在巴黎 擺動得略微慢一些。

From this observation alone, and clever mathematics,

僅僅是通過這個發現和巧妙的數學，

Newton rightly deduced

牛頓正確地推斷出

that the Earth is a wee bit flattened at the poles,

地球的兩極有細微的扁平，

like 0.3 percent --

像是 0.3% 那麼多──

so tiny that you wouldn't even notice it on the real view of the Earth.

太小了以至於你甚至無法在 地球的真實圖像上注意到。

These stories show that mathematics

這些故事說明了數學

is able to make us go out of our intuition

可以讓我們走出直覺、

measure the Earth which seems infinite,

測量似乎是無窮大的地球、

see atoms which are invisible

看到肉眼不可見的原子

or detect an imperceptible variation of shape.

或者是發現難以察覺的形狀差異。

And if there is just one thing that you should take home from this talk,

這個演講中你最應當領悟到的

it is this:

應該是：

mathematics allows us to go beyond the intuition

數學讓我們超越直覺

and explore territories which do not fit within our grasp.

並且探索不在我們掌控範圍內的領域。

Here's a modern example you will all relate to:

這是一個和你們都相關的現代案例：

searching the Internet.

上網。

The World Wide Web,

網際網路，

more than one billion web pages --

超過十億的網頁──

do you want to go through them all?

你們想要瀏覽完所有的網頁嗎？

Computing power helps,

計算機的能力可以提供幫助，

but it would be useless without the mathematical modeling

但是它是無用的，如果沒有數學模型

to find the information hidden in the data.

來幫助找到隱藏在數據中的信息。

Let's work out a baby problem.

讓我們解決一個和嬰兒有關的問題。

Imagine that you're a detective working on a crime case,

想像一下你是一名偵探， 在偵查一個犯罪案，

and there are many people who have their version of the facts.

然後很多人都有各自版本的事實。

Who do you want to interview first?

你想要先詢問誰？

Sensible answer:

合理的答案：

prime witnesses.

首要證人。

You see,

要知道，

suppose that there is person number seven,

假設七號人物

tells you a story,

告訴了你一件事，

but when you ask where he got if from,

但是你問他，他是怎麼知道的，

he points to person number three as a source.

他說消息來在三號人物。

And maybe person number three, in turn,

然後可能三號人物

points at person number one as the primary source.

又說消息來自一號人物。

Now number one is a prime witness,

現在一號人物就是首要證人。

so I definitely want to interview him -- priority.

所以我肯定想要詢問他──首要任務。

And from the graph

從圖表中

we also see that person number four is a prime witness.

我們也可以看出四號人物 也是首要證人。

And maybe I even want to interview him first,

也許我想要先詢問他，

because there are more people who refer to him.

因為有更多人的提到了他。

OK, that was easy,

好的，那很容易，

but now what about if you have a big bunch of people who will testify?

但是現在如果你有一大群人 需要詢問該怎麼辦呢？

And this graph,

這幅圖表，

I may think of it as all people who testify in a complicated crime case,

我可以把它看作是要在一個複雜的 犯罪案中作證詞的所有人，

but it may just as well be web pages pointing to each other,

但是它也可以是彼此聯繫的網頁，

referring to each other for contents.

通過彼此參考來獲取信息。

Which ones are the most authoritative?

最具權威的是哪些呢？

Not so clear.

不是很清楚。

Enter PageRank,

輸入網頁排序，

one of the early cornerstones of Google.

一個 Google 早期的奠基石。

This algorithm uses the laws of mathematical randomness

這個算法使用數學隨機法則

to determine automatically the most relevant web pages,

來自動確定最相關的網頁，

in the same way as we used randomness in the Galton Board experiment.

和我們在高爾頓實驗板實驗中 所使用的隨機性方法一樣。

So let's send into this graph

讓我們把

a bunch of tiny, digital marbles

一束極小的電子彈珠 輸送到這個圖表中，

and let them go randomly through the graph.

然後讓它們在圖表中隨機移動。

Each time they arrive at some site,

每一次它們都會到達某個地方，

they will go out through some link chosen at random to the next one.

它們會通過隨機選擇的、 通往下一個地方的通道出去。

And again, and again, and again.

一次，一次， 又一次。

And with small, growing piles,

根據這些小的，不斷變大的堆積，

we'll keep the record of how many times each site has been visited

我們會記錄每一個地方被

by these digital marbles.

這些電子彈珠到訪的次數。

Here we go.

我們開始吧。

Randomness, randomness.

隨機，隨機。

And from time to time,

不時地，

also let's make jumps completely randomly to increase the fun.

讓我們也完全隨機地跳躍 來增加趣味性。

And look at this:

看看這個：

from the chaos will emerge the solution.

答案會從混亂中顯現出來。

The highest piles correspond to those sites

最高的堆積對應那些地點

which somehow are better connected than the others,

那些出於某種原因 比其它地點有更多連接的地點，

more pointed at than the others.

比其它地點受到更多指示的地點。

And here we see clearly

在這裡我們可以清楚地看見

which are the web pages we want to first try.

哪些是我們想要先看的網頁。

Once again,

再一次，

the solution emerges from the randomness.

答案從隨機性中顯現。

Of course, since that time,

當然，從那時以後，

Google has come up with much more sophisticated algorithms,

Google 設計出了更複雜的算法，

but already this was beautiful.

但是這個已經足夠好了。

And still,

但是，

just one problem in a million.

有一個萬裡挑一的問題。

With the advent of digital area,

隨著數碼時代的到來，

more and more problems lend themselves to mathematical analysis,

越來越多的問題需要數學分析來解釋，

making the job of mathematician a more and more useful one,

讓數學家的工作越來越有用，

to the extent that a few years ago,

以至於達到了這樣的程度，幾年前，

it was ranked number one among hundreds of jobs

它排在幾百個工作之首，

in a study about the best and worst jobs

資料來自一份關於最佳工作 和最差工作的研究，

published by the Wall Street Journal in 2009.

2009 年由華爾街日報出版。

Mathematician --

數學家──

best job in the world.

世界上最好的工作。

That's because of the applications:

那是因為各種應用：

communication theory,

通信原理、

information theory,

信息原理、

game theory,

博弈論、

compressed sensing,

壓縮傳感、

machine learning,

機器學習、

graph analysis,

圖表分析、

harmonic analysis.

和聲分析。

And why not stochastic processes,

應該還有隨機過程、

linear programming,

線性編程，

or fluid simulation?

或者是流體仿真吧？

Each of these fields have monster industrial applications.

這些領域中的每一個 都有龐大的工業應用。

And through them,

通過它們，

there is big money in mathematics.

數學可以帶來豐厚的收入。

And let me concede

請允許我承認

that when it comes to making money from the math,

當涉及到通過數學賺錢，

the Americans are by a long shot the world champions,

美國人是遙遙領先的世界冠軍，

with clever, emblematic billionaires and amazing, giant companies,

有著機敏的、具代表性的億萬富翁、

令人驚異的大公司，

all resting, ultimately, on good algorithm.

所有的這些公司最終都依靠好的算法。

Now with all this beauty, usefulness and wealth,

現在，有了這種美麗、實用性和財富，

mathematics does look more sexy.

數學的確看起來更迷人了。

But don't you think

但是不要認為

that the life a mathematical researcher is an easy one.

數學研究員的生活很容易。

It is filled with perplexity,

它充滿了複雜性、

frustration,

沮喪，

a desperate fight for understanding.

為了解釋而進行的絕望的鬥爭。

Let me evoke for you

請讓我為你回憶一下

one of the most striking days in my mathematician's life.

我數學生涯中頗具震撼的一天。

Or should I say,

或許我應該說，

one of the most striking nights.

最震撼的夜晚之一。

At that time,

那個時候，

I was staying at the Institute for Advanced Studies in Princeton --

我在普林斯頓高等研究院工作──

for many years, the home of Albert Einstein

很多年來都是 阿爾伯特·愛因斯坦的住所，

and arguably the most holy place for mathematical research in the world.

而且可以說是世界上 最神聖的數學研究的地方。

And that night I was working and working on an elusive proof,

那個晚上，我在推理一個很難的證明，

which was incomplete.

一個那時尚未完成的證明。

It was all about understanding

它完全是關於理解

the paradoxical stability property of plasmas,

等離子體的矛盾的穩定特性，

which are a crowd of electrons.

等離子體是一團電子。

In the perfect world of plasma,

在理想的等離子體世界中，

there are no collisions

是沒有碰撞

and no friction to provide the stability like we are used to.

也沒有摩擦力來提供 我們所習慣的穩定性。

But still,

但是，

if you slightly perturb a plasma equilibrium,

如果你稍微破壞了 等離子體的平衡狀態，

you will find that the resulting electric field

你會發現隨之而來的電場

spontaneously vanishes,

自然消失了，

or damps out,

或者減弱了，

as if by some mysterious friction force.

就像是由一些神秘的 摩擦力造成的一樣。

This paradoxical effect,

這種矛盾的效應，

called the Landau damping,

被稱作朗道阻尼，

is one of the most important in plasma physics,

是等離子體物理學中 最重要的效應之一，

and it was discovered through mathematical ideas.

它是通過數學概念發現的。

But still,

但是，

a full mathematical understanding of this phenomenon was missing.

一套對於這個現象完整的數學理解

那時並不存在。

And together with my former student and main collaborator Clément Mouhot,

和我以前的學生， 也是我主要的合作者克萊蒙·穆奧，

in Paris at the time,

那時在巴黎，

we had been working for months and months on such a proof.

我們為了這樣的一個證明 一起努力了很多個月。

Actually,

實際上，

I had already announced by mistake that we could solve it.

我那時錯誤地宣布了我們可以解決它。

But the truth is,

但事實是，

the proof was just not working.

那個證明是無效的。

In spite of more than 100 pages of complicated, mathematical arguments,

儘管用了超過 100 頁的複雜數學論證，

and a bunch discoveries,

有了一些發現，

and huge calculation,

和完成了龐大的計算量，

it was not working.

那個證明還是無效。

And that night in Princeton,

在普林斯頓的那晚，

a certain gap in the chain of arguments was driving me crazy.

一系列論證中的某個欠缺 幾乎讓我發瘋。

I was putting in there all my energy and experience and tricks,

我把我所有的精力、 經驗和技巧都用在那裡了，

and still nothing was working.

但是還是沒有任何結果。

1 a.m., 2 a.m., 3 a.m.,

凌晨一點、兩點、三點，

not working.

什麼結果都沒有。

Around 4 a.m., I go to bed in low spirits.

大概凌晨四點的時候， 我情緒低落地去睡覺了。

Then a few hours later,

然後幾個小時以後，

waking up and go,

我醒來，繼續工作，

"Ah, it's time to get the kids to school --"

「啊，孩子們上學的時間到了──」

What is this?

這是什麼？

There was this voice in my head, I swear.

我發誓，在我腦中有這樣一個聲音。

"Take the second term to the other side,

「把第二項移到另一側，

Fourier transform and invert in L2."

傅立葉變換，然後把 L2 倒置。」

(Laughter)

（笑聲）

Damn it,

見鬼了，

that was the start of the solution!

那就是答案的開始！

You see,

要知道，

I thought I had taken some rest,

我認為我休息了一下，

but really my brain had continued to work on it.

但是事實上我的大腦繼續 思考著那個問題。

In those moments,

在那些時刻，

you don't think of your career or your colleagues,

你不會想到你的職業 或者是你的同事，

it's just a complete battle between the problem and you.

那只是那個問題 和你之間的一場鬥爭。

That being said,

儘管如此，

it does not harm when you do get a promotion in reward for your hard work.

當你因為你的努力 而得到升職，也沒什麼不好。

And after we completed our huge analysis of the Landau damping,

我們完成了龐大的朗道阻尼分析之後，

I was lucky enough

我很幸運

to get the most coveted Fields Medal

得到了夢寐以求的菲爾茲獎

from the hands of the President of India,

由印度主席授予，

in Hyderabad on 19 August, 2010 --

2010 年 8 月 19 日在海得拉巴──

an honor that mathematicians never dare to dream,

一個數學家們從不敢奢求的榮譽，

a day that I will remember until I live.

我永遠都會記得的一天。

What do you think,

你會有什麼想法，

on such an occasion?

在這樣的一個場合？

Pride, yes?

自豪，對嗎？

And gratitude to the many collaborators who made this possible.

還有感激，對那些 讓這個成為可能的人。

And because it was a collective adventure,

因為這是一個集體的經歷，

you need to share it, not just with your collaborators.

你需要分享它， 不僅僅是和你的合作者。

I believe that everybody can appreciate the thrill of mathematical research,

我相信每一個人都可以感受 數學研究的刺激，

and share the passionate stories of humans and ideas behind it.

並且分享在這背後 充滿激情的人和事。

And I've been working with my staff at Institut Henri Poincaré,

我和我的職員在索邦大學 龐加萊研究院已經工作一段時間了，

together with partners and artists of mathematical communication worldwide,

還有世界各地的 數學通訊夥伴和藝術家，

so that we can found our own, very special museum of mathematics there.

以至於我們可以在那裡建立 我們自己的非常特別的數學博物館。

So in a few years,

所以幾年後，

when you come to Paris,

當你來到巴黎的時候，

after tasting the great, crispy baguette and macaroon,

品嚐完美味酥脆的 法國長棍麵包和馬卡龍之後，

please come and visit us at Institut Henri Poincaré,

請來龐加萊研究院拜訪我們，

and share the mathematical dream with us.

並且和我們分享數學夢想。

Thank you.

謝謝。

(Applause)

（掌聲）