What is it that French people do better than all the others? If you would take polls, the top three answers might be: love, wine and whining.
法國人比其他國家的人 都擅長的是什麼? 如果你進行民意調查的話, 排在前三位的答案可能是: 愛情、葡萄酒和抱怨。
(Laughter)
(笑聲)
Maybe. But let me suggest a fourth one: mathematics. Did you know that Paris has more mathematicians than any other city in the world? And more streets with mathematicians' names, too. And if you look at the statistics of the Fields Medal, often called the Nobel Prize for mathematics, and always awarded to mathematicians below the age of 40, you will find that France has more Fields medalists per inhabitant than any other country.
也許是這樣。 但是讓我再加一項: 數學。 你知道巴黎的數學家 比世界上其他任何一座城市的都多嗎? 而且也有更多 帶有數學家名字的街道。 如果你看看菲爾茲獎的數據, 它通常被稱作數學領域的諾貝爾獎, 並且總是頒發給 40 歲以下的數學家, 你會發現法國人均菲爾茲獎得主數量 多過其他任何一個國家。
What is it that we find so sexy in math? After all, it seems to be dull and abstract, just numbers and computations and rules to apply. Mathematics may be abstract, but it's not dull and it's not about computing. It is about reasoning and proving our core activity. It is about imagination, the talent which we most praise. It is about finding the truth. There's nothing like the feeling which invades you when after months of hard thinking, you finally understand the right reasoning to solve your problem. The great mathematician André Weil likened this -- no kidding -- to sexual pleasure. But noted that this feeling can last for hours, or even days.
我們認為數學很迷人的地方是什麼? 畢竟,它看起來枯燥又抽象, 僅僅是數字、計算和運算規則。 數學可能是抽象的, 但是它不是枯燥的, 而且它不是關於計算。 它是關於推理 以及證明你的核心任務。 它是關於想像力, 我們最讚賞的一種能力。 它是關於發現真相。 沒有任何事情可以和 經過幾個月苦思終於得出 能解決問題的正確推理時 充盈你的那種感受相比。 偉大的數學家安德雷·韋伊 把這種感覺比作── 不是開玩笑的── 性愉悅。 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. Hidden mathematical truths permeate our whole physical world. They are inaccessible to our senses but can be seen through mathematical lenses. Close your eyes for moment and think of what is occurring right now around you. Invisible particles from the air around are bumping on you by the billions and billions at each second, all in complete chaos. And still, their statistics can be accurately predicted by mathematical physics. And open your eyes now to the statistics of the velocities of these particles.
隨之而來的獎賞也可能很豐厚。 我們的現實世界中充滿了 未被發現的數學真相。 我們的感官無法感知到它們, 但我們可以用數學的眼光看見它們。 請把你們的眼睛閉上一會兒, 然後想一想現在 你們身邊正發生著什麼。 空氣中每秒都有上億 不可見的微粒撞在你的身上, 一片混亂。 然而, 它們的數據可以被 數學物理準確地預測。 現在請睜開你們的眼睛 來看看這些顆粒移動速率的數據。
The famous bell-shaped Gauss Curve, or the Law of Errors -- of deviations with respect to the mean behavior. 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. It's one of the most important curves ever. It keeps on occurring again and again, from many theories and many experiments, as a great example of the universality which is so dear to us mathematicians.
這個著名的鍾型高斯曲線, 或者是誤差定律── 相對於平均數的絕對偏差。 這條線描述了顆粒移動速率, 就如同 人口曲線描述個體年齡。 它是迄今為止最重要的曲線之一。 它一次又一次地 出現在眾多理論與實驗中, 作為對我們數學家 十分寶貴、普遍性的 一個重要例證。
Of this curve, the famous scientist Francis Galton said, "It would have been deified by the Greeks if they had known it. It is the supreme law of unreason." And there's no better way to materialize that supreme goddess than Galton's Board. Inside this board are narrow tunnels through which tiny balls will fall down randomly, going right or left, or left, etc. All in complete randomness and chaos. Let's see what happens when we look at all these random trajectories together.
關於這條曲線, 著名的科學家 弗朗西斯·高爾頓說過, 「如果希臘人知道這條曲線的話, 它一定會被他們神化的。 它是無理性的終極法則。」 沒有比高爾頓板更好的方式 來展現這位至尊女神。 在這個實驗板裡面是狹窄的通道, 小球會通過這些通道隨機落下, 向右或者向左,或者向左,等等。 一切都是完全隨機和任意的。 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, because we need to resolve some traffic jams in there. Aha. We think that randomness is going to play me a trick on stage.
這有點像一種運動, 因為我們需要解決一些交通擁堵。 啊哈。 我們認為隨機性 將會在這個台上展現一些極妙。
There it is. Our supreme goddess of unreason. the Gauss Curve, trapped here inside this transparent box as Dream in "The Sandman" comics. For you I have shown it, but to my students I explain why it could not be any other curve. And this is touching the mystery of that goddess, replacing a beautiful coincidence by a beautiful explanation.
就是這樣。 我們無理性的至尊女神。 高斯曲線, 被困在了這個透明的實驗板裡, 就像《睡魔》中的夢一樣。 我已經給你們展示了這條曲線, 但是對我的學生,我要解釋 為什麼不會是其它的曲線。 這就涉及到這位至尊女神的神祕性, 用一個完美的解釋 代替一個美麗的巧合。
All of science is like this. And beautiful mathematical explanations are not only for our pleasure. They also change our vision of the world. For instance, Einstein, Perrin, Smoluchowski, they used the mathematical analysis of random trajectories and the Gauss Curve to explain and prove that our world is made of atoms.
所有的科學都如此。 完美的數學解釋不僅僅讓我們愉快, 它們也會改變我們對世界的看法。 比如, 愛因斯坦, 佩蘭, 斯莫魯霍夫斯基, 他們用隨機路徑的數學分析 和高斯曲線, 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 that mathematics was revolutionizing our view of the world. More than 2,000 years ago, at the time of the ancient Greeks, it already occurred. In those days, only a small fraction of the world had been explored, and the Earth might have seemed infinite. But clever Eratosthenes, using mathematics, was able to measure the Earth with an amazing accuracy of two percent.
這不是第一次 數學革新了我們對世界的看法。 2000 多年前, 在古希臘的時候, 這就發生了。 在那個時候, 只有一小部分的世界已經被探索, 而且地球似乎是無窮大的。 但是聰明的艾拉托瑟尼, 用數學, 成功地以 2% 的 驚人精確度測量了地球。
Here's another example. In 1673, Jean Richer noticed that a pendulum swings slightly slower in Cayenne than in Paris. From this observation alone, and clever mathematics, Newton rightly deduced that the Earth is a wee bit flattened at the poles, like 0.3 percent -- so tiny that you wouldn't even notice it on the real view of the Earth.
還有另一個例子。 在 1673 年,簡·里歇爾發現 鐘擺在卡宴比在巴黎 擺動得略微慢一些。 僅僅是通過這個發現和巧妙的數學, 牛頓正確地推斷出 地球的兩極有細微的扁平, 像是 0.3% 那麼多── 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 is able to make us go out of our intuition measure the Earth which seems infinite, see atoms which are invisible or detect an imperceptible variation of shape. And if there is just one thing that you should take home from this talk, it is this: mathematics allows us to go beyond the intuition and explore territories which do not fit within our grasp.
這些故事說明了數學 可以讓我們走出直覺、 測量似乎是無窮大的地球、 看到肉眼不可見的原子 或者是發現難以察覺的形狀差異。 這個演講中你最應當領悟到的 應該是: 數學讓我們超越直覺 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: searching the Internet. The World Wide Web, more than one billion web pages -- do you want to go through them all? Computing power helps, but it would be useless without the mathematical modeling to find the information hidden in the data.
這是一個和你們都相關的現代案例: 上網。 網際網路, 超過十億的網頁── 你們想要瀏覽完所有的網頁嗎? 計算機的能力可以提供幫助, 但是它是無用的,如果沒有數學模型 來幫助找到隱藏在數據中的信息。
Let's work out a baby problem. Imagine that you're a detective working on a crime case, and there are many people who have their version of the facts. Who do you want to interview first? Sensible answer: prime witnesses. You see, suppose that there is person number seven, tells you a story, but when you ask where he got if from, he points to person number three as a source. And maybe person number three, in turn, points at person number one as the primary source. Now number one is a prime witness, so I definitely want to interview him -- priority. And from the graph we also see that person number four is a prime witness. And maybe I even want to interview him first, because there are more people who refer to him.
讓我們解決一個和嬰兒有關的問題。 想像一下你是一名偵探, 在偵查一個犯罪案, 然後很多人都有各自版本的事實。 你想要先詢問誰? 合理的答案: 首要證人。 要知道, 假設七號人物 告訴了你一件事, 但是你問他,他是怎麼知道的, 他說消息來在三號人物。 然後可能三號人物 又說消息來自一號人物。 現在一號人物就是首要證人。 所以我肯定想要詢問他──首要任務。 從圖表中 我們也可以看出四號人物 也是首要證人。 也許我想要先詢問他, 因為有更多人的提到了他。
OK, that was easy, but now what about if you have a big bunch of people who will testify? And this graph, I may think of it as all people who testify in a complicated crime case, but it may just as well be web pages pointing to each other, referring to each other for contents. Which ones are the most authoritative? Not so clear.
好的,那很容易, 但是現在如果你有一大群人 需要詢問該怎麼辦呢? 這幅圖表, 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, 但是它也可以是彼此聯繫的網頁, 通過彼此參考來獲取信息。 最具權威的是哪些呢? 不是很清楚。
Enter PageRank, one of the early cornerstones of Google. This algorithm uses the laws of mathematical randomness to determine automatically the most relevant web pages, in the same way as we used randomness in the Galton Board experiment. So let's send into this graph a bunch of tiny, digital marbles and let them go randomly through the graph. Each time they arrive at some site, they will go out through some link chosen at random to the next one. And again, and again, and again. And with small, growing piles, we'll keep the record of how many times each site has been visited by these digital marbles.
輸入網頁排序, 一個 Google 早期的奠基石。 這個算法使用數學隨機法則 來自動確定最相關的網頁, 和我們在高爾頓實驗板實驗中 所使用的隨機性方法一樣。 讓我們把 一束極小的電子彈珠 輸送到這個圖表中, 然後讓它們在圖表中隨機移動。 每一次它們都會到達某個地方, 它們會通過隨機選擇的、 通往下一個地方的通道出去。 一次,一次, 又一次。 根據這些小的,不斷變大的堆積, 我們會記錄每一個地方被 這些電子彈珠到訪的次數。
Here we go. Randomness, randomness. And from time to time, also let's make jumps completely randomly to increase the fun.
我們開始吧。 隨機,隨機。 不時地, 讓我們也完全隨機地跳躍 來增加趣味性。
And look at this: from the chaos will emerge the solution. The highest piles correspond to those sites which somehow are better connected than the others, more pointed at than the others. And here we see clearly which are the web pages we want to first try. Once again, the solution emerges from the randomness. Of course, since that time, Google has come up with much more sophisticated algorithms, but already this was beautiful.
看看這個: 答案會從混亂中顯現出來。 最高的堆積對應那些地點 那些出於某種原因 比其它地點有更多連接的地點, 比其它地點受到更多指示的地點。 在這裡我們可以清楚地看見 哪些是我們想要先看的網頁。 再一次, 答案從隨機性中顯現。 當然,從那時以後, Google 設計出了更複雜的算法, 但是這個已經足夠好了。
And still, just one problem in a million. With the advent of digital area, more and more problems lend themselves to mathematical analysis, making the job of mathematician a more and more useful one, to the extent that a few years ago, 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.
但是, 有一個萬裡挑一的問題。 隨著數碼時代的到來, 越來越多的問題需要數學分析來解釋, 讓數學家的工作越來越有用, 以至於達到了這樣的程度,幾年前, 它排在幾百個工作之首, 資料來自一份關於最佳工作 和最差工作的研究, 2009 年由華爾街日報出版。
Mathematician -- best job in the world. That's because of the applications: communication theory, information theory, game theory, compressed sensing, machine learning, graph analysis, harmonic analysis. And why not stochastic processes, linear programming, or fluid simulation? Each of these fields have monster industrial applications. And through them, there is big money in mathematics. And let me concede that when it comes to making money from the math, the Americans are by a long shot the world champions, with clever, emblematic billionaires and amazing, giant companies, all resting, ultimately, on good algorithm.
數學家── 世界上最好的工作。 那是因為各種應用: 通信原理、 信息原理、 博弈論、 壓縮傳感、 機器學習、 圖表分析、 和聲分析。 應該還有隨機過程、 線性編程, 或者是流體仿真吧? 這些領域中的每一個 都有龐大的工業應用。 通過它們, 數學可以帶來豐厚的收入。 請允許我承認 當涉及到通過數學賺錢, 美國人是遙遙領先的世界冠軍, 有著機敏的、具代表性的億萬富翁、 令人驚異的大公司, 所有的這些公司最終都依靠好的算法。
Now with all this beauty, usefulness and wealth, mathematics does look more sexy. But don't you think that the life a mathematical researcher is an easy one. It is filled with perplexity, frustration, a desperate fight for understanding.
現在,有了這種美麗、實用性和財富, 數學的確看起來更迷人了。 但是不要認為 數學研究員的生活很容易。 它充滿了複雜性、 沮喪, 為了解釋而進行的絕望的鬥爭。
Let me evoke for you one of the most striking days in my mathematician's life. Or should I say, one of the most striking nights. At that time, I was staying at the Institute for Advanced Studies in Princeton -- for many years, the home of Albert Einstein and arguably the most holy place for mathematical research in the world. And that night I was working and working on an elusive proof, which was incomplete. It was all about understanding the paradoxical stability property of plasmas, which are a crowd of electrons. In the perfect world of plasma, there are no collisions and no friction to provide the stability like we are used to. But still, if you slightly perturb a plasma equilibrium, you will find that the resulting electric field spontaneously vanishes, or damps out, as if by some mysterious friction force.
請讓我為你回憶一下 我數學生涯中頗具震撼的一天。 或許我應該說, 最震撼的夜晚之一。 那個時候, 我在普林斯頓高等研究院工作── 很多年來都是 阿爾伯特·愛因斯坦的住所, 而且可以說是世界上 最神聖的數學研究的地方。 那個晚上,我在推理一個很難的證明, 一個那時尚未完成的證明。 它完全是關於理解 等離子體的矛盾的穩定特性, 等離子體是一團電子。 在理想的等離子體世界中, 是沒有碰撞 也沒有摩擦力來提供 我們所習慣的穩定性。 但是, 如果你稍微破壞了 等離子體的平衡狀態, 你會發現隨之而來的電場 自然消失了, 或者減弱了, 就像是由一些神秘的 摩擦力造成的一樣。
This paradoxical effect, called the Landau damping, is one of the most important in plasma physics, and it was discovered through mathematical ideas. But still, a full mathematical understanding of this phenomenon was missing. And together with my former student and main collaborator Clément Mouhot, in Paris at the time, we had been working for months and months on such a proof. Actually, I had already announced by mistake that we could solve it. But the truth is, the proof was just not working. In spite of more than 100 pages of complicated, mathematical arguments, and a bunch discoveries, and huge calculation, it was not working. And that night in Princeton, a certain gap in the chain of arguments was driving me crazy. I was putting in there all my energy and experience and tricks, and still nothing was working. 1 a.m., 2 a.m., 3 a.m., not working. Around 4 a.m., I go to bed in low spirits. Then a few hours later, waking up and go, "Ah, it's time to get the kids to school --" What is this? There was this voice in my head, I swear. "Take the second term to the other side, Fourier transform and invert in L2."
這種矛盾的效應, 被稱作朗道阻尼, 是等離子體物理學中 最重要的效應之一, 它是通過數學概念發現的。 但是, 一套對於這個現象完整的數學理解 那時並不存在。 和我以前的學生, 也是我主要的合作者克萊蒙·穆奧, 那時在巴黎, 我們為了這樣的一個證明 一起努力了很多個月。 實際上, 我那時錯誤地宣布了我們可以解決它。 但事實是, 那個證明是無效的。 儘管用了超過 100 頁的複雜數學論證, 有了一些發現, 和完成了龐大的計算量, 那個證明還是無效。 在普林斯頓的那晚, 一系列論證中的某個欠缺 幾乎讓我發瘋。 我把我所有的精力、 經驗和技巧都用在那裡了, 但是還是沒有任何結果。 凌晨一點、兩點、三點, 什麼結果都沒有。 大概凌晨四點的時候, 我情緒低落地去睡覺了。 然後幾個小時以後, 我醒來,繼續工作, 「啊,孩子們上學的時間到了──」 這是什麼? 我發誓,在我腦中有這樣一個聲音。 「把第二項移到另一側, 傅立葉變換,然後把 L2 倒置。」
(Laughter)
(笑聲)
Damn it, that was the start of the solution!
見鬼了, 那就是答案的開始!
You see, I thought I had taken some rest, but really my brain had continued to work on it. In those moments, you don't think of your career or your colleagues, it's just a complete battle between the problem and you.
要知道, 我認為我休息了一下, 但是事實上我的大腦繼續 思考著那個問題。 在那些時刻, 你不會想到你的職業 或者是你的同事, 那只是那個問題 和你之間的一場鬥爭。
That being said, it does not harm when you do get a promotion in reward for your hard work. And after we completed our huge analysis of the Landau damping, I was lucky enough to get the most coveted Fields Medal from the hands of the President of India, in Hyderabad on 19 August, 2010 -- an honor that mathematicians never dare to dream, a day that I will remember until I live.
儘管如此, 當你因為你的努力 而得到升職,也沒什麼不好。 我們完成了龐大的朗道阻尼分析之後, 我很幸運 得到了夢寐以求的菲爾茲獎 由印度主席授予, 2010 年 8 月 19 日在海得拉巴── 一個數學家們從不敢奢求的榮譽, 我永遠都會記得的一天。
What do you think, on such an occasion? Pride, yes? And gratitude to the many collaborators who made this possible. And because it was a collective adventure, you need to share it, not just with your collaborators. I believe that everybody can appreciate the thrill of mathematical research, and share the passionate stories of humans and ideas behind it. And I've been working with my staff at Institut Henri Poincaré, together with partners and artists of mathematical communication worldwide, so that we can found our own, very special museum of mathematics there.
你會有什麼想法, 在這樣的一個場合? 自豪,對嗎? 還有感激,對那些 讓這個成為可能的人。 因為這是一個集體的經歷, 你需要分享它, 不僅僅是和你的合作者。 我相信每一個人都可以感受 數學研究的刺激, 並且分享在這背後 充滿激情的人和事。 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, 還有世界各地的 數學通訊夥伴和藝術家, 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, when you come to Paris, after tasting the great, crispy baguette and macaroon, please come and visit us at Institut Henri Poincaré, and share the mathematical dream with us.
所以幾年後, 當你來到巴黎的時候, 品嚐完美味酥脆的 法國長棍麵包和馬卡龍之後, 請來龐加萊研究院拜訪我們, 並且和我們分享數學夢想。
Thank you.
謝謝。
(Applause)
(掌聲)