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TED:数学的魅力是什么?

2019-08-29  香光庄


TED英语演讲课

给心灵放个假吧

数学(mathematics或maths,来自希腊语,“máthēma”;经常被缩写为“math”),是研究数量、结构、变化、空间以及信息等概念的一门学科,从某种角度看属于形式科学的一种。数学家和哲学家对数学的确切范围和定义有一系列的看法。

而在人类历史发展和社会生活中,数学也发挥着不可替代的作用,也是学习和研究现代科学技术必不可少的基本工具。

数学的魅力究竟在哪呢?

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

法国人在什么方面做得比别人好呢?

If you would take polls,

如果你去做个投票调查,

the top three answers might be: love,

排名前三的或许是:爱情,

wine and whining.

红酒,和发牢骚。

Maybe.

可能吧。

But let me suggest a fourth style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>但我还想提出第四个答案:数学。

Did you know that Paris has more mathematicians than any other city in the world?

你们知道巴黎的数学家比世界上其它任何一个城市 都要多吗?

And more streets with mathematicians' names, too.

而且以数学家的名字命名的街道数量也更多。

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

如果你查查菲尔兹奖的统计,

often called the Nobel Prize for mathematics,

它也经常被称作“诺贝尔数学奖”,

and always awarded to mathematicians below the age of 40,

只授予40岁以下的数学家,

you will find that France has more Fields medalists per inhabitant than any other country.

你会发现,就人均获奖数量来说, 法国是世界第一。

What is it that we find so sexy in math?

我们在数学中到底发现了什么让人着迷的东西?

After all, it seems to be dull and abstract,

毕竟它看上去那么无聊、抽象,

just numbers and computations and rules to apply.

只是数字、计算、定理而已。

Mathematics may be abstract,

数学可能很抽象,

but it's not dull and it's not about computing.

但是它并不无聊, 而且它并不都是计算。

It is about reasoning and proving our core activity.

它是有关逻辑的推理, 让我们的所作所为都有理有据。

It is about imagination,

它有关丰富的想象,

the talent which we most praise.

我们最常歌颂的人类天赋。

It is about finding the truth.

它还有关真理的追寻。

There's nothing like the feeling which invades you when after months of hard thinking,

当你苦思冥想数月之后, 终于找到问题的正确解法那一刻,

you finally understand the right reasoning to solve your problem.

那种感受真的无与伦比。

The great mathematician André Weil likened this -- no kidding -- to sexual pleasure.

伟大的数学家安德雷·韦依 把这种感受比作(不开玩笑的说) 比作性快感。

But noted that this feeling can last for hours,

但是他还说这种感受可以持续数小时

or even days.

甚至数天。

The reward may be big.

这种回报可能难以估量。

Hidden mathematical truths permeate our whole physical world.

隐藏的数学规律渗透在我们整个物质世界中。

They are inaccessible to our senses but can be seen through mathematical lenses.

我们的感官无法接触到它们,但可以通过数学镜片看到它们。

Close your eyes for moment and think of what is occurring right now around you.

闭上眼睛一小会儿, 想一想你周围此时此刻正在发生的事。

Invisible particles from the air around are bumping style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>周围空气中的看不见的粒子每秒钟都在撞击你,每秒钟数十亿次,

all in complete chaos.

完全是混乱无序的状态。

And still, their statistics can be accurately predicted by mathematical physics.

然而, 它们的行为可以用数学物理学精准地预测。

And open your eyes now to the statistics of the velocities of these particles.

现在打开你的眼睛看看这些分子的速率分布统计。

The famous bell-shaped Gauss Curve,

这是著名的钟形高斯曲线,

or the Law of Errors -- of deviations with respect to the mean behavior.

也可以叫做误差律—— 关于分子平均行为的一些偏差。

This curve tells about the statistics of velocities of particles in the same way as a demographic curve would tell about the statistics of ages of individuals.

这条曲线告诉我们粒子的速率分布情况, 正如一条人口统计曲线,能够告诉我们人口的年龄分布情况。

It's style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>这是有史以来最重要的曲线之一。

It keeps style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>它的规律不断地重复,

from many theories and many experiments,

在诸多理论与实验中呈现,

as a great example of the universality which is so dear to us mathematicians.

它是数学的普适性的体现, 而这种性质对我们数学家至关重要。

Of this curve,

关于这个曲线,

the famous scientist Francis Galton said,

著名科学家弗朗西斯·高尔顿说:

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

“如果古希腊人知道这个规律, 他们一定会把它神化的。

It is the supreme law of unreason.'

这是无理性的最高法则。“

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

高尔顿板就是 把这个“神灵”实体化的最佳体现。

Inside this board are narrow tunnels through which tiny balls will fall down randomly,

在这个板子里有一些狭道, 一些掉落的小球会随机通过这里,

going right or left, or left, etc.

有些往右,有些往左。

All in complete randomness and chaos.

完全是随机的、混乱的。

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

让我们看看这些随机路线会呈现怎样的规律。

This is a bit of a sport,

这其实算是锻炼身体,

because we need to resolve some traffic jams in there.

因为我们得疏通一些拥堵的状况。

Aha.

啊哈。

We think that randomness is going to play me a trick style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>看来随机性要在这个舞台上跟我开个小玩笑了。

There it is.

好了!

Our supreme goddess of unreason.

这个无理性的至高无上的神,

the Gauss Curve,

高斯曲线,

trapped here inside this transparent box as Dream in 'The Sandman'

被困在这个透明的盒子里,就像《睡魔》漫画里的梦魇

comics.

一样。

For you I have shown it,

我向各位展示了这个规律,

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

但向我的学生,我要解释为什么它不可能是任何其它的曲线。

And this is touching the mystery of that goddess,

这就近乎揭开了这个神灵的面纱,

replacing a beautiful coincidence by a beautiful explanation.

把一个美丽的巧合变成一个赏心悦目的数学解释。

All of science is like this.

一切的科学都是这样的。

And beautiful mathematical explanations are not style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>美丽的数学解法并不只是为了我们自己开心。

They also change our vision of the world.

它们也改变了我们对世界的看法。

For instance, Einstein, Perrin, Smoluchowski,

举个例子, 爱因斯坦、 佩兰、 斯莫鲁霍夫斯基,

they used the mathematical analysis of random trajectories and the Gauss Curve to explain and prove that our world is made of atoms.

他们对粒子的随机轨迹进行了数学分析,再加上高斯曲线, 他们解释并证明了我们的世界由原子组成。

It was not the first time that mathematics was revolutionizing our view of the world.

这并不是第一次, 数学已经多次颠覆了我们的世界观。

More than 2,000 years ago,

两千多年前,

at the time of the ancient Greeks,

在古希腊的时代,

it already occurred.

就已经颠覆了。

In those days,

在那个时代,

only a small fraction of the world had been explored,

人们只探索了世界的很小一部分,

and the Earth might have seemed infinite.

而地球看上去无边无际。

But clever Eratosthenes, using mathematics,

但聪明的埃拉托斯提尼,运用数学,

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

成功的测量了地球的大小,误差只有惊人的2%。

Here's another example.

还有另一个例子。

In 1673, Jean Richer noticed that a pendulum swings slightly slower in Cayenne than in Paris.

1673年,让·里奇注意到卡宴的钟摆摆动速度比巴黎略慢。

From this observation alone,

只用这一个现象,

and clever mathematics,

以及一些巧妙的数学推导,

Newton rightly deduced that the Earth is a wee bit flattened at the poles,

牛顿正确地推断出地球在两极地区稍稍扁一些,

like 0.3 percent -- so tiny that you wouldn't even notice it style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>大概只有0.3%, 这种细微的差别在 观察地球全貌时根本无法发现。

These stories show that mathematics is able to make us go out of our intuition measure the Earth which seems infinite,

这些故事说明了, 数学能够让我们超越自己的直觉, 测量看似不可测的地球尺寸,

see atoms which are invisible or detect an imperceptible variation of shape.

观察看不见的原子, 或是检测肉眼不可识别的微小形变。

And if there is just style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>如果你们只能从我的演讲中了解到一样东西,

it is this: mathematics allows us to go beyond the intuition and explore territories which do not fit within our grasp.

那应该就是:数学让我们超越人类直觉, 并且探索我们所无法触及的领域。

Here's a modern example you will all relate to: searching the Internet.

这有个例子各位都非常熟悉:上网。

The World Wide Web,

万维网,

more than style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>有着超过十亿个网页, 难道你想全部搜索一遍吗?

Computing power helps,

计算机可能有帮助,

but it would be useless without the mathematical modeling to find the information hidden in the data.

但是如果没有了数学模型, 它就是一堆废铁, 无法搜寻数据中隐藏的信息。

Let's work out a baby problem.

让我们做一道很简单的题。

Imagine that you're a detective working style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>想象你是一个侦探, 正在调查一个犯罪案件,

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

很多人参与其中,并且各执一词。

Who do you want to interview first?

你想先询问谁呢?

Sensible answer: prime witnesses.

合理的答案是:主要的目击者。

You see, suppose that there is person number seven,

想想看, 假设有一位7号证人,

tells you a story,

告诉了你一件事情,

but when you ask where he got if from,

但当你问他从哪里听说的,

he points to person number three as a source.

他说3号证人是消息来源。

And maybe person number three, in turn,

有可能3号证人

points at person number style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>也相应地指向1号证人作为主要消息来源。

Now number style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>现在1号证人是主要目击者了,

so I definitely want to interview him -- priority.

所以我一定想要先去采访他。

And from the graph we also see that person number four is a prime witness.

从这幅图中,我们同样看到4号证人,是一位主要目击者。

And maybe I even want to interview him first,

我可能更想先去采访他,

because there are more people who refer to him.

因为他被提及的次数比1号还要多。

OK, that was easy,

好吧,这还算简单的,

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

但是如果你有一大群人要作证呢?

And this graph,

这张图

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

我可以把它当作 一件复杂案件的所有证人,

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

但也可以把它看作是互相链接的网页,

referring to each other for contents.

互相引用其中的内容。

Which style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>哪些网页最有权威性呢?

Not so clear.

还不太清楚。

Enter PageRank, style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>于是PageRank问世了, 它是谷歌最早的基石之一。

This algorithm uses the laws of mathematical randomness to determine automatically the most relevant web pages,

这种算法运用了数学随机性的定律, 来自动判断哪些网页关联最多,

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

与我们在高尔顿板实验中运用随机性的方法一样。

So let's send into this graph a bunch of tiny,

那就把一堆小小的数码玻璃珠放到这个图表中,

digital marbles and let them go randomly through the graph.

让它们随机的在图中穿行。

Each time they arrive at some site,

每当它们到达某个网页,

they will go out through some link chosen at random to the next style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>它们就会随机选择一个链接,

And again, and again, and again.

然后跳转到另一页,一遍又一遍重复。

And with small, growing piles,

用这些小小的光点,

we'll keep the record of how many times each site has been visited by these digital marbles.

我们记录下每个网页被访问的次数, 就用这些数码珠子。

Here we go.

开始吧。

Randomness, randomness.

一切随机。

And from time to time,

有时候呢,

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

我们就完全随机跳跃,以增加乐趣。

And look at this: from the chaos will emerge the solution.

看看这个:在一片混乱中产生了一个答案。

The highest piles correspond to those sites which somehow are better connected than the others,

这里最高的几堆对应着那些相对来说链接更多的网页,

more pointed at than the others.

被引用更多次的网页。

And here we see clearly which are the web pages we want to first try.

在这里我们清晰地看到, 哪一些是我们最想先看的网页。

Once again, the solution emerges from the randomness.

再一次, 问题的解答来源于随机性。

Of course, since that time,

当然,从那以后,

Google has come up with much more sophisticated algorithms,

谷歌已经发明出 数不胜数的复杂算法,

but already this was beautiful.

但是这个算法已经很好了。

And still, just style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>然而, 这只是沧海一粟。

With the advent of digital area,

随着数字领域的飞速发展,

more and more problems lend themselves to mathematical analysis,

越来越多的问题需要用数学分析来解决,

making the job of mathematician a more and more useful style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>让数学家这个工作变得越来越实用,

to the extent that a few years ago,

以至于大约几年前,

it was ranked number style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>它在数百个职业中排名第一, 这份排名是有关最好和最差的职业, 由华尔街日报在2009年发表。

Mathematician -- best job in the world.

数学家是世界上最好的工作。

That's because of the applications: communication theory, information theory, game theory, compressed sensing, machine learning, graph analysis, harmonic analysis.

这是因为它应用广泛:通讯理论、 信息理论、 博弈论、 压缩传感、 机器学习、 图表分析、 谐波分析。

And why not stochastic processes, linear programming,

为什么不是随机过程,线性规划,

or fluid simulation?

或者流体模拟。

Each of these fields have monster industrial applications.

以上每一个领域都有规模巨大的工业应用。

And through them,

透过它们可以看出,

there is big money in mathematics.

数学的商机是无限的。

And let me concede that when it comes to making money from the math,

我必须承认, 谈到用数学赚钱,

the Americans are by a long shot the world champions, with clever,

美国人可是遥遥领先全世界,

emblematic billionaires and amazing, giant companies, all resting, ultimately,

有一群标志性绝顶聪明的领导者, 还有让人大开眼界的商业巨头,

on good algorithm.

归根结底都不约而同地依赖好的算法。

Now with all this beauty,

数学兼具着美、

usefulness and wealth,

实用性, 以及无限商机,

mathematics does look more sexy.

它似乎的确更有魅力了。

But don't you think that the life a mathematical researcher is an easy style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>但是你千万别以为数学家的生活很轻松。

It is filled with perplexity, frustration,

它充满了困惑,沮丧,

a desperate fight for understanding.

是追求真知的绝望之战。

Let me evoke for you style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>我给大家说一说我的数学生涯中最特别的一天。

Or should I say,

或者我该说,

one of the most striking nights.

最特别的一晚。

At that time,

那个时候,

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

我待在普林斯顿大学的高等研究所里,

the home of Albert Einstein and arguably the most holy place for mathematical research in the world.

这里曾是爱因斯坦多年的家, 也很可能是世界上数学研究的神圣之颠。

And that night I was working and working style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>那天晚上我苦思冥想, 寻找一个非常隐晦的证明,

which was incomplete.

非常不完整。

It was all about understanding the paradoxical stability property of plasmas,

它是有关于等离子体的矛盾稳定特性的了解,

which are a crowd of electrons.

这里指的是一团电子云。

In the perfect world of plasma,

在等离子体的理想世界,

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

是没有任何碰撞的, 而且没有任何摩擦力, 使其像我们习惯的那么稳定。

But still, if you slightly perturb a plasma equilibrium,

然而, 如果你轻微打破等离子体平衡,

you will find that the resulting electric field spontaneously vanishes,

你会发现相应产生的电场会自发的消失,

or damps out,

或者是减弱,

as if by some mysterious friction force.

好像受到了某种神秘摩擦力的影响。

This paradoxical effect,

这种矛盾的特性,

called the Landau damping,

叫做朗道阻尼,

is style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>是等离子物理中最重要的现象之一,

and it was discovered through mathematical ideas.

而且它是由数学思想推导出来的。

But still, a full mathematical understanding of this phenomenon was missing.

然而, 对此现象的完整数学理解还不完善。

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

和我以前的学生和主要合作者克莱门特·穆特一起,

in Paris at the time,

我们那时在巴黎,

we had been working for months and months style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>我们为了寻找这个证法已经花了好几个月。

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

实际上, 我还以为我们可以解决这个问题。

But the truth is,

然而事实上,

the proof was just not working.

那种证法完全无效。

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

即使是一百多页的复杂数学推导,

and a bunch discoveries,

还有一大堆的新发现,

and huge calculation,

巨大的计算量,

it was not working.

依然得不出什么结论。

And that night in Princeton,

在普林斯顿的那个晚上,

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

证明中的一个小缺口让我近乎疯狂。

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

我对它使出浑身解数,

and still nothing was working.

但是依旧没有进展。

1 a.m., 2 a.m., 3 a.m., not working.

凌晨一点、两点、三点, 毫无进展。

Around 4 a.m.,

大概凌晨四点的时候,

I go to bed in low spirits.

我无精打采的上床。

Then a few hours later,

几个小时后,

waking up and go, 'Ah,

我从床上爬起来, “啊,

it's time to get the kids to school --'

该送孩子们上学了。”

What is this?

这是什么?

There was this voice in my head, I swear.

我确定,我的脑袋里有个声音。

'Take the second term to the other side,

“把第二个任期带到另一边,

Fourier transform and invert in L2.'

傅里叶展开然后在L2域反变换。”

Damn it, that was the start of the solution!

可恶!这才要开始解了啊!

You see, I thought I had taken some rest,

我以为我自己在休息,

but really my brain had continued to work style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>但实际上,我的大脑一直在思考这个问题。

In those moments,

在那些时刻,

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

你不会想到你的职业生涯或是你的同事,

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

这只是你自己与问题之间的斗争。

That being said,

但说到这里,

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

如果你因努力工作而得到升职,当然是很好的事情。

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

在我们完成了朗道阻尼方面的重大研究后,

I was lucky enough to get the most coveted Fields Medal from the hands of the President of India,

我很幸运地获得了我梦寐以求的菲尔兹奖,我从印度总统手中接过此奖,

in Hyderabad style='margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-family: Arial, Helvetica, sans-serif; font-size: 14px;       text-align: left;       background-color: rgb(244, 250, 244);  '>那是在2010年8月19日, 在海德拉巴城。

2010 -- an honor that mathematicians never dare to dream,

2010年-这是数学家们从来不敢梦想的荣誉,

a day that I will remember until I live.

我也会将这天永远铭记在心。

What do you think,

对于这样的情况,

on such an occasion?

你们觉得怎样呢?

Pride, yes?

很自豪,对吧?

And gratitude to the many collaborators who made this possible.

还有对主要合作者的感激之情。

And because it was a collective adventure,

而且因为这是一个集体研究,

you need to share it,

你需要把成果公开,

not just with your collaborators.

而非只是与合作者共享。

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

我相信每个人都可以欣赏数学研究的刺激感,

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

并且分享精彩研究过程中的人和事。

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

我在昂利·庞加莱研究所与我的团队工作,

together with partners and artists of mathematical communication worldwide,

还有一些其他的合伙人、世界各地的数学交流艺术家,

so that we can found our own,

于是我们就可以创立我们自己的,

very special museum of mathematics there.

非常特别的数学博物馆。

So in a few years,

再过几年,

when you come to Paris,

当你来到巴黎,

after tasting the great,

在你们品尝过美味酥脆的

crispy baguette and macaroon,

法国长面包和马卡龙(蛋白杏仁饼甜点)之后,

please come and visit us at Institut Henri Poincaré,

请各位也来我们的 昂利·庞加莱研究所转一转,

and share the mathematical dream with us.

与我们共享一个数学的梦。

Thank you.

谢谢。

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

法国人在什么方面做得比别人好呢?

If you would take polls,

如果你去做个投票调查,

the top three answers might be: love,

排名前三的或许是:爱情,

wine and whining.

红酒,和发牢骚。

Maybe.

可能吧。

But let me suggest a fourth one: mathematics.

但我还想提出第四个答案:数学。

Did you know that Paris has more mathematicians than any other city in the world?

你们知道巴黎的数学家比世界上其它任何一个城市 都要多吗?

And more streets with mathematicians' names, too.

而且以数学家的名字命名的街道数量也更多。

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

如果你查查菲尔兹奖的统计,

often called the Nobel Prize for mathematics,

它也经常被称作“诺贝尔数学奖”,

and always awarded to mathematicians below the age of 40,

只授予40岁以下的数学家,

you will find that France has more Fields medalists per inhabitant than any other country.

你会发现,就人均获奖数量来说, 法国是世界第一。

What is it that we find so sexy in math?

我们在数学中到底发现了什么让人着迷的东西?

After all, it seems to be dull and abstract,

毕竟它看上去那么无聊、抽象,

just numbers and computations and rules to apply.

只是数字、计算、定理而已。

Mathematics may be abstract,

数学可能很抽象,

but it's not dull and it's not about computing.

但是它并不无聊, 而且它并不都是计算。

It is about reasoning and proving our core activity.

它是有关逻辑的推理, 让我们的所作所为都有理有据。

It is about imagination,

它有关丰富的想象,

the talent which we most praise.

我们最常歌颂的人类天赋。

It is about finding the truth.

它还有关真理的追寻。

There's nothing like the feeling which invades you when after months of hard thinking,

当你苦思冥想数月之后, 终于找到问题的正确解法那一刻,

you finally understand the right reasoning to solve your problem.

那种感受真的无与伦比。

The great mathematician André Weil likened this -- no kidding -- to sexual pleasure.

伟大的数学家安德雷·韦依 把这种感受比作(不开玩笑的说) 比作性快感。

But noted that this feeling can last for hours,

但是他还说这种感受可以持续数小时

or even days.

甚至数天。

The reward may be big.

这种回报可能难以估量。

Hidden mathematical truths permeate our whole physical world.

隐藏的数学规律渗透在我们整个物质世界中。

They are inaccessible to our senses but can be seen through mathematical lenses.

我们的感官无法接触到它们,但可以通过数学镜片看到它们。

Close your eyes for moment and think of what is occurring right now around you.

闭上眼睛一小会儿, 想一想你周围此时此刻正在发生的事。

Invisible particles from the air around are bumping on you by the billions and billions at each second,

周围空气中的看不见的粒子每秒钟都在撞击你,每秒钟数十亿次,

all in complete chaos.

完全是混乱无序的状态。

And still, their statistics can be accurately predicted by mathematical physics.

然而, 它们的行为可以用数学物理学精准地预测。

And open your eyes now to the statistics of the velocities of these particles.

现在打开你的眼睛看看这些分子的速率分布统计。

The famous bell-shaped Gauss Curve,

这是著名的钟形高斯曲线,

or the Law of Errors -- of deviations with respect to the mean behavior.

也可以叫做误差律—— 关于分子平均行为的一些偏差。

This curve tells about the statistics of velocities of particles in the same way as a demographic curve would tell about the statistics of ages of individuals.

这条曲线告诉我们粒子的速率分布情况, 正如一条人口统计曲线,能够告诉我们人口的年龄分布情况。

It's one of the most important curves ever.

这是有史以来最重要的曲线之一。

It keeps on occurring again and again,

它的规律不断地重复,

from many theories and many experiments,

在诸多理论与实验中呈现,

as a great example of the universality which is so dear to us mathematicians.

它是数学的普适性的体现, 而这种性质对我们数学家至关重要。

Of this curve,

关于这个曲线,

the famous scientist Francis Galton said,

著名科学家弗朗西斯·高尔顿说:

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

“如果古希腊人知道这个规律, 他们一定会把它神化的。

It is the supreme law of unreason.'

这是无理性的最高法则。“

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

高尔顿板就是 把这个“神灵”实体化的最佳体现。

Inside this board are narrow tunnels through which tiny balls will fall down randomly,

在这个板子里有一些狭道, 一些掉落的小球会随机通过这里,

going right or left, or left, etc.

有些往右,有些往左。

All in complete randomness and chaos.

完全是随机的、混乱的。

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

让我们看看这些随机路线会呈现怎样的规律。

This is a bit of a sport,

这其实算是锻炼身体,

because we need to resolve some traffic jams in there.

因为我们得疏通一些拥堵的状况。

Aha.

啊哈。

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

看来随机性要在这个舞台上跟我开个小玩笑了。

There it is.

好了!

Our supreme goddess of unreason.

这个无理性的至高无上的神,

the Gauss Curve,

高斯曲线,

trapped here inside this transparent box as Dream in 'The Sandman'

被困在这个透明的盒子里,就像《睡魔》漫画里的梦魇

comics.

一样。

For you I have shown it,

我向各位展示了这个规律,

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

但向我的学生,我要解释为什么它不可能是任何其它的曲线。

And this is touching the mystery of that goddess,

这就近乎揭开了这个神灵的面纱,

replacing a beautiful coincidence by a beautiful explanation.

把一个美丽的巧合变成一个赏心悦目的数学解释。

All of science is like this.

一切的科学都是这样的。

And beautiful mathematical explanations are not only for our pleasure.

美丽的数学解法并不只是为了我们自己开心。

They also change our vision of the world.

它们也改变了我们对世界的看法。

For instance, Einstein, Perrin, Smoluchowski,

举个例子, 爱因斯坦、 佩兰、 斯莫鲁霍夫斯基,

they used the mathematical analysis of random trajectories and the Gauss Curve to explain and prove that our world is made of atoms.

他们对粒子的随机轨迹进行了数学分析,再加上高斯曲线, 他们解释并证明了我们的世界由原子组成。

It was not the first time that mathematics was revolutionizing our view of the world.

这并不是第一次, 数学已经多次颠覆了我们的世界观。

More than 2,000 years ago,

两千多年前,

at the time of the ancient Greeks,

在古希腊的时代,

it already occurred.

就已经颠覆了。

In those days,

在那个时代,

only a small fraction of the world had been explored,

人们只探索了世界的很小一部分,

and the Earth might have seemed infinite.

而地球看上去无边无际。

But clever Eratosthenes, using mathematics,

但聪明的埃拉托斯提尼,运用数学,

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

成功的测量了地球的大小,误差只有惊人的2%。

Here's another example.

还有另一个例子。

In 1673, Jean Richer noticed that a pendulum swings slightly slower in Cayenne than in Paris.

1673年,让·里奇注意到卡宴的钟摆摆动速度比巴黎略慢。

From this observation alone,

只用这一个现象,

and clever mathematics,

以及一些巧妙的数学推导,

Newton rightly deduced that the Earth is a wee bit flattened at the poles,

牛顿正确地推断出地球在两极地区稍稍扁一些,

like 0.3 percent -- so tiny that you wouldn't even notice it on the real view of the Earth.

大概只有0.3%, 这种细微的差别在 观察地球全貌时根本无法发现。

These stories show that mathematics is able to make us go out of our intuition measure the Earth which seems infinite,

这些故事说明了, 数学能够让我们超越自己的直觉, 测量看似不可测的地球尺寸,

see atoms which are invisible or detect an imperceptible variation of shape.

观察看不见的原子, 或是检测肉眼不可识别的微小形变。

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

如果你们只能从我的演讲中了解到一样东西,

it is this: mathematics allows us to go beyond the intuition and explore territories which do not fit within our grasp.

那应该就是:数学让我们超越人类直觉, 并且探索我们所无法触及的领域。

Here's a modern example you will all relate to: searching the Internet.

这有个例子各位都非常熟悉:上网。

The World Wide Web,

万维网,

more than one billion web pages -- do you want to go through them all?

有着超过十亿个网页, 难道你想全部搜索一遍吗?

Computing power helps,

计算机可能有帮助,

but it would be useless without the mathematical modeling to find the information hidden in the data.

但是如果没有了数学模型, 它就是一堆废铁, 无法搜寻数据中隐藏的信息。

Let's work out a baby problem.

让我们做一道很简单的题。

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

想象你是一个侦探, 正在调查一个犯罪案件,

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

很多人参与其中,并且各执一词。

Who do you want to interview first?

你想先询问谁呢?

Sensible answer: prime witnesses.

合理的答案是:主要的目击者。

You see, suppose that there is person number seven,

想想看, 假设有一位7号证人,

tells you a story,

告诉了你一件事情,

but when you ask where he got if from,

但当你问他从哪里听说的,

he points to person number three as a source.

他说3号证人是消息来源。

And maybe person number three, in turn,

有可能3号证人

points at person number one as the primary source.

也相应地指向1号证人作为主要消息来源。

Now number one is a prime witness,

现在1号证人是主要目击者了,

so I definitely want to interview him -- priority.

所以我一定想要先去采访他。

And from the graph we also see that person number four is a prime witness.

从这幅图中,我们同样看到4号证人,是一位主要目击者。

And maybe I even want to interview him first,

我可能更想先去采访他,

because there are more people who refer to him.

因为他被提及的次数比1号还要多。

OK, that was easy,

好吧,这还算简单的,

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

但是如果你有一大群人要作证呢?

And this graph,

这张图

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

我可以把它当作 一件复杂案件的所有证人,

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

但也可以把它看作是互相链接的网页,

referring to each other for contents.

互相引用其中的内容。

Which ones are the most authoritative?

哪些网页最有权威性呢?

Not so clear.

还不太清楚。

Enter PageRank, one of the early cornerstones of Google.

于是PageRank问世了, 它是谷歌最早的基石之一。

This algorithm uses the laws of mathematical randomness to determine automatically the most relevant web pages,

这种算法运用了数学随机性的定律, 来自动判断哪些网页关联最多,

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

与我们在高尔顿板实验中运用随机性的方法一样。

So let's send into this graph a bunch of tiny,

那就把一堆小小的数码玻璃珠放到这个图表中,

digital marbles and let them go randomly through the graph.

让它们随机的在图中穿行。

Each time they arrive at some site,

每当它们到达某个网页,

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

它们就会随机选择一个链接,

And again, and again, and again.

然后跳转到另一页,一遍又一遍重复。

And with small, growing piles,

用这些小小的光点,

we'll keep the record of how many times each site has been visited by these digital marbles.

我们记录下每个网页被访问的次数, 就用这些数码珠子。

Here we go.

开始吧。

Randomness, randomness.

一切随机。

And from time to time,

有时候呢,

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

我们就完全随机跳跃,以增加乐趣。

And look at this: from the chaos will emerge the solution.

看看这个:在一片混乱中产生了一个答案。

The highest piles correspond to those sites which somehow are better connected than the others,

这里最高的几堆对应着那些相对来说链接更多的网页,

more pointed at than the others.

被引用更多次的网页。

And here we see clearly which are the web pages we want to first try.

在这里我们清晰地看到, 哪一些是我们最想先看的网页。

Once again, the solution emerges from the randomness.

再一次, 问题的解答来源于随机性。

Of course, since that time,

当然,从那以后,

Google has come up with much more sophisticated algorithms,

谷歌已经发明出 数不胜数的复杂算法,

but already this was beautiful.

但是这个算法已经很好了。

And still, just one problem in a million.

然而, 这只是沧海一粟。

With the advent of digital area,

随着数字领域的飞速发展,

more and more problems lend themselves to mathematical analysis,

越来越多的问题需要用数学分析来解决,

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

让数学家这个工作变得越来越实用,

to the extent that a few years ago,

以至于大约几年前,

it was ranked number one among hundreds of jobs in a study about the best and worst jobs published by the Wall Street Journal in 2009.

它在数百个职业中排名第一, 这份排名是有关最好和最差的职业, 由华尔街日报在2009年发表。

Mathematician -- best job in the world.

数学家是世界上最好的工作。

That's because of the applications: communication theory, information theory, game theory, compressed sensing, machine learning, graph analysis, harmonic analysis.

这是因为它应用广泛:通讯理论、 信息理论、 博弈论、 压缩传感、 机器学习、 图表分析、 谐波分析。

And why not stochastic processes, linear programming,

为什么不是随机过程,线性规划,

or fluid simulation?

或者流体模拟。

Each of these fields have monster industrial applications.

以上每一个领域都有规模巨大的工业应用。

And through them,

透过它们可以看出,

there is big money in mathematics.

数学的商机是无限的。

And let me concede that when it comes to making money from the math,

我必须承认, 谈到用数学赚钱,

the Americans are by a long shot the world champions, with clever,

美国人可是遥遥领先全世界,

emblematic billionaires and amazing, giant companies, all resting, ultimately,

有一群标志性绝顶聪明的领导者, 还有让人大开眼界的商业巨头,

on good algorithm.

归根结底都不约而同地依赖好的算法。

Now with all this beauty,

数学兼具着美、

usefulness and wealth,

实用性, 以及无限商机,

mathematics does look more sexy.

它似乎的确更有魅力了。

But don't you think that the life a mathematical researcher is an easy one.

但是你千万别以为数学家的生活很轻松。

It is filled with perplexity, frustration,

它充满了困惑,沮丧,

a desperate fight for understanding.

是追求真知的绝望之战。

Let me evoke for you one of the most striking days in my mathematician's life.

我给大家说一说我的数学生涯中最特别的一天。

Or should I say,

或者我该说,

one of the most striking nights.

最特别的一晚。

At that time,

那个时候,

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

我待在普林斯顿大学的高等研究所里,

the home of Albert Einstein and arguably the most holy place for mathematical research in the world.

这里曾是爱因斯坦多年的家, 也很可能是世界上数学研究的神圣之颠。

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

那天晚上我苦思冥想, 寻找一个非常隐晦的证明,

which was incomplete.

非常不完整。

It was all about understanding the paradoxical stability property of plasmas,

它是有关于等离子体的矛盾稳定特性的了解,

which are a crowd of electrons.

这里指的是一团电子云。

In the perfect world of plasma,

在等离子体的理想世界,

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

是没有任何碰撞的, 而且没有任何摩擦力, 使其像我们习惯的那么稳定。

But still, if you slightly perturb a plasma equilibrium,

然而, 如果你轻微打破等离子体平衡,

you will find that the resulting electric field spontaneously vanishes,

你会发现相应产生的电场会自发的消失,

or damps out,

或者是减弱,

as if by some mysterious friction force.

好像受到了某种神秘摩擦力的影响。

This paradoxical effect,

这种矛盾的特性,

called the Landau damping,

叫做朗道阻尼,

is one of the most important in plasma physics,

是等离子物理中最重要的现象之一,

and it was discovered through mathematical ideas.

而且它是由数学思想推导出来的。

But still, a full mathematical understanding of this phenomenon was missing.

然而, 对此现象的完整数学理解还不完善。

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

和我以前的学生和主要合作者克莱门特·穆特一起,

in Paris at the time,

我们那时在巴黎,

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

我们为了寻找这个证法已经花了好几个月。

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

实际上, 我还以为我们可以解决这个问题。

But the truth is,

然而事实上,

the proof was just not working.

那种证法完全无效。

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

即使是一百多页的复杂数学推导,

and a bunch discoveries,

还有一大堆的新发现,

and huge calculation,

巨大的计算量,

it was not working.

依然得不出什么结论。

And that night in Princeton,

在普林斯顿的那个晚上,

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

证明中的一个小缺口让我近乎疯狂。

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

我对它使出浑身解数,

and still nothing was working.

但是依旧没有进展。

1 a.m., 2 a.m., 3 a.m., not working.

凌晨一点、两点、三点, 毫无进展。

Around 4 a.m.,

大概凌晨四点的时候,

I go to bed in low spirits.

我无精打采的上床。

Then a few hours later,

几个小时后,

waking up and go, 'Ah,

我从床上爬起来, “啊,

it's time to get the kids to school --'

该送孩子们上学了。”

What is this?

这是什么?

There was this voice in my head, I swear.

我确定,我的脑袋里有个声音。

'Take the second term to the other side,

“把第二个任期带到另一边,

Fourier transform and invert in L2.'

傅里叶展开然后在L2域反变换。”

Damn it, that was the start of the solution!

可恶!这才要开始解了啊!

You see, I thought I had taken some rest,

我以为我自己在休息,

but really my brain had continued to work on it.

但实际上,我的大脑一直在思考这个问题。

In those moments,

在那些时刻,

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

你不会想到你的职业生涯或是你的同事,

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

这只是你自己与问题之间的斗争。

That being said,

但说到这里,

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

如果你因努力工作而得到升职,当然是很好的事情。

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

在我们完成了朗道阻尼方面的重大研究后,

I was lucky enough to get the most coveted Fields Medal from the hands of the President of India,

我很幸运地获得了我梦寐以求的菲尔兹奖,我从印度总统手中接过此奖,

in Hyderabad on 19 August,

那是在2010年8月19日, 在海德拉巴城。

2010 -- an honor that mathematicians never dare to dream,

2010年-这是数学家们从来不敢梦想的荣誉,

a day that I will remember until I live.

我也会将这天永远铭记在心。

What do you think,

对于这样的情况,

on such an occasion?

你们觉得怎样呢?

Pride, yes?

很自豪,对吧?

And gratitude to the many collaborators who made this possible.

还有对主要合作者的感激之情。

And because it was a collective adventure,

而且因为这是一个集体研究,

you need to share it,

你需要把成果公开,

not just with your collaborators.

而非只是与合作者共享。

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

我相信每个人都可以欣赏数学研究的刺激感,

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

并且分享精彩研究过程中的人和事。

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

我在昂利·庞加莱研究所与我的团队工作,

together with partners and artists of mathematical communication worldwide,

还有一些其他的合伙人、世界各地的数学交流艺术家,

so that we can found our own,

于是我们就可以创立我们自己的,

very special museum of mathematics there.

非常特别的数学博物馆。

So in a few years,

再过几年,

when you come to Paris,

当你来到巴黎,

after tasting the great,

在你们品尝过美味酥脆的

crispy baguette and macaroon,

法国长面包和马卡龙(蛋白杏仁饼甜点)之后,

please come and visit us at Institut Henri Poincaré,

请各位也来我们的 昂利·庞加莱研究所转一转,

and share the mathematical dream with us.

与我们共享一个数学的梦。

Thank you.

谢谢。

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