我們翻譯這篇文章的理由
當代文明的科技成就、人們便捷的日常生活，都離不開數學理論的鋪墊架構。但我們不是對其敬而遠之，就是將其視為機械性的手頭工具。實際上，數學是世間鐵律、是有力的武器：藉助數學，虛偽的政策即刻被揭穿、民眾不會被輕易愚弄操縱。數學是文明瑰寶、是無可篡改的讖言：藉助數學，思想能夠在人類肉體所不能及的高維空間自由翱翔。數學是萬物的真理、是直指人心的力量，認識到數學之美、之自由，你也就理解了愛與永恆。
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節選：《愛與數學》
作者：EdwardFrenkel
譯者：王雅婧
校對：劉小康
推薦：羅玉池
策劃：王雅婧 & 泮海倫
There’s a secret world out there. A hidden parallel universe of beauty and elegance, intricately intertwined with ours. It’s the world of mathematics. And it’s invisible to most of us. This book is an invitation to discover this world.
那裡有個神秘世界，一個存在於平行宇宙中的秘密世界，它典雅美麗，與我們錯綜複雜地交織在一起。這就是絕大多數人都無法領略的數學世界，而本書將邀你一齊對它進行探索。
Consider this paradox: On the one hand, mathematics is woven in the very fabric of our daily lives. Every time we make an online purchase, send a text message, do a search on the Internet, or use a GPS device, mathematical formulas and algorithms are at play. On the other hand, most people are daunted by math. It has become, in the words of poet Hans Magnus Enzensberger, 「a blind spot in our culture – alien territory, in which only the elite, the initiated few have managed to entrench themselves.」 It’s rare, he says, that we 「encounter a person who asserts vehemently that the mere thought of reading a novel, or looking at a picture, or seeing a movie causes him insufferable torment,」 but 「sensible, educated people」 often say 「with a remarkable blend of defiance and pride」 that math is 「pure torture」 or a 「nightmare」 that 「turns them off.」
通往數學秘境的道路上，繞不開的是我們的認知與數學本身的矛盾：雖然數學與我們的日常生活息息相關，我們每一次網購、發簡訊、在網際網路上搜索或者使用GPS設備，都少不了數學公式和算法在背後支撐，但大多數人對數學望而卻步。用詩人漢斯•馬格努斯•恩岑斯貝格爾（Hans Magnus Enzensberger）的話來說，數學已經成為「我們文化中的盲點，只有少數精英在點化下才能在這片異域上扎穩腳跟。」他說，幾乎沒有人會「咬牙切齒地說他們一想到讀小說、欣賞圖片或看電影就苦不堪言。」但是，「受過教育的聰明人」卻經常會用「夾雜著蔑視與傲慢的口吻」說，數學是「徹頭徹尾的折磨」，是「一場噩夢」之類的話，所以他們「討厭數學」。
How is this anomaly possible? I see two main reasons. First, mathematics is more abstract than other subjects, hence not as accessible. Second, what we study in school is only a tiny part of math, much of it established more than a millennium ago. Mathematics has advanced tremendously since then, but the treasures of modern math have been kept hidden from most of us.
為什麼會這樣呢？我認為主要有兩個原因。一是數學比其他學科更抽象，因此入門不易。二是學校學到的不過是數學知識的冰山一角，而絕大部分課本內容都是一千多年前就已經建立的。從那以後，數學已有了突飛猛進的發展，但是現代數學的寶藏在大多數人眼中一直飾以未知的色彩。
What if at school you had to take an 「art class」 in which you were only taught how to paint a fence? What if you were never shown the paintings of Leonardo da Vinci and Picasso? Would that make you appreciate art? Would you want to learn more about it? I doubt it. You would probably say something like this: 「Learning art at school was a waste of my time. If I ever need to have my fence painted, I』ll just hire people to do this for me.」 Of course, this sounds ridiculous, but this is how math is taught, and so in the eyes of most of us it becomes the equivalent of watching paint dry. While the paintings of the great masters are readily available, the math of the great masters is locked away.
在必修的「藝術課」上，如果老師只教你如何粉刷籬笆，會有怎樣的結果？如果老師從未在課上帶你領略達文西和畢卡索的作品呢？這能讓你學會鑑賞藝術嗎？你還會對藝術抱有好奇嗎？答案是否定的。你可能會說，「在學校學習藝術是浪費時間。我要是需要粉刷籬笆，肯定會僱人來做啊。」這個類比聽起來固然有些荒謬，但學校中的數學教育就是如此，所以在大多數人眼中，學習數學而無意義，和在籬笆旁等著油漆變乾沒有區別。在今天，儘管欣賞大師們的畫作已不是難事，但大師們的數學研究卻一直鎖於高閣，世人只能略窺一二。
However, it’s not just the aesthetic beauty of math that’s captivating. As Galileo famously said, 「The laws of Nature are written in the language of mathematics.」 Math is a way to describe reality and figure out how the world works, a universal language that has become the gold standard of truth. In our world, increasingly driven by science and technology, mathematics is becoming, ever more, the source of power, wealth, and progress. Hence those who are fluent in this new language will be on the cutting edge of progress.
然而，數學之所以迷人，絕不僅僅在於其美感。伽利略曾言：「自然法則由數學語言所寫。」數學是描述現實、揭示世界運作原理的語言，這種通用語言現已成為檢驗真理的黃金標準。在科學技術的推動下，數學日益成為權力、財富和進步之源。因此，精通數學這門新語言的人將站在進步的前沿。
One of the common misconceptions about mathematics is that it can only be used as a 「toolkit」: a biologist, say, would do some field work, collect data, and then try to build a mathematical model fitting these data (perhaps, with some help from a mathematician). While this is an important mode of operation, math offers us a lot more: it enables us to make groundbreaking, paradigm-shifting leaps that we couldn’t make otherwise. For example, Albert Einstein was not trying to fit any data into equations when he understood that gravity causes our space to curve. In fact, there was no such data. No one could even imagine at the time that our space is curved; everyone 「knew」 that our world was flat! But Einstein understood that this was the only way to generalize his special relativity theory to non-inertial systems, coupled with his insight that gravity and acceleration have the same effect. This was a high-level intellectual exercise within the realm of math, one in which Einstein relied on the work of a mathematician, Bernhard Riemann, completed fifty years earlier. The human brain is wired in such a way that we simply cannot imagine curved spaces of dimension greater than two; we can only access them through mathematics. And guess what, Einstein was right – our universe is curved, and furthermore, it’s expanding. That’s the power of mathematics I am talking about!
人們對數學常抱有誤解，認為數學只能充當「工具箱」。就說生物學家吧，他們會做一些實地工作，收集數據、然後試圖建立一個擬合這些數據的數學模型（可能是由數學家協助建立的）。雖然這是一個重要的研究模式，但數學讓我們洞見了更多：它使我們能夠實現突破性的、思維方式上的飛躍。沒有數學，我們很可能會陷入僵局。例如，愛因斯坦在理解引力造成的空間彎曲時，並沒有試圖將任何數據代入到方程中。事實上，他手頭也沒有可以用於驗證的數據。當時沒有人能想像，我們的空間竟然是彎曲的，每個人都「覺得」世界是平的！但是愛因斯坦明白，這是將他的狹義相對論（連同他對引力和加速度具有同等效應的敏銳認知）推廣到非慣性系統的唯一途徑。這是數學領域的一道高級智力練習，黎曼早在50年前就已完成了準備工作。人腦的結構決定了我們很難想像高於二維的黎曼空間，我們只能借用數學走近它們。你猜最後發生了什麼——愛因斯坦是對的！我們的宇宙是彎曲的，而且還在膨脹中。這就是我所說的數學的力量！
Many examples like this may be found, and not only in physics, but in other areas of science (we will discuss some of them below). History shows that science and technology are transformed by mathematical ideas at an accelerated pace; even mathematical theories that are initially viewed as abstract and esoteric later become indispensable for applications. Charles Darwin, whose work at first did not rely on math, later wrote in his autobiography: 「I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics, for men thus endowed seem to have an extra sense.」 I take it as prescient advice to the next generations to capitalize on mathematics』 immense potential.
類似例子數不勝數，不僅是在物理學領域，在其他學科領域也是如此（後文將會提及）。歷史表明，數學思想加速了科學技術的發展，即使是最初被視為抽象和晦澀的數學理論，後來在實際應用中也發揮出不可或缺的作用。達爾文一開始的工作並不依賴數學，但他後來在自傳中寫道：「我很遺憾沒有理解數學的一些重要原理，至少沒有了解得足夠深入，因為數學能賦予人們額外的感知力。」此番話語恰如一劑預防針，用以忠告後人，要重視數學的巨大潛力。
When I was growing up, I wasn’t aware of the hidden world of mathematics. Like most people, I thought math was a stale, boring subject. But I was lucky:
in my last year of high school I met a professional mathematician who opened the magical world of math to me. I learned that mathematics is full of infinite possibilities as well as elegance and beauty, just like poetry, art, and music. I fell in love with math.
小時候，我並未意識到數學秘境的存在，也認為數學只是一門迂腐枯燥的學科。但我也很幸運，高中的最後一年，我遇到了一位專業的數學家，他向我敞開了奇妙的數學世界，讓我知道數學和詩歌、藝術和音樂一般，充滿了無限的可能性。於是，我深深愛上了數學。
Dear reader, with this book I want to do for you what my teachers and men- tors did for me: unlock the power and beauty of mathematics, and enable you to enter this magical world the way I did, even if you are the sort of person who has never used the words 「math」 and 「love」 in the same sentence. Mathematics will get under your skin just like it did under mine, and your worldview will never be the same.
親愛的讀者，當年是我的老師帶我走進了數學世界，而我想以這本書為鑰匙，打開數學的力量與美麗，讓你隨著我的腳步進入這個神奇的世界。即使你從未把「數學」和「愛」聯繫在一起，我們的旅途也定會順利無虞。數學與你將日漸親密，甚至交融為一體，進而顛覆你的整個世界觀。
Mathematical knowledge is unlike any other knowledge. While our perception of the physical world can always be distorted, our perception of mathematical truths can’t be. They are objective, persistent, necessary truths. A mathematical formula or theorem means the same thing to anyone anywhere – no matter what gender, religion, or skin color; it will mean the same thing to anyone a thousand years from now. And what’s also amazing is that we own all of them. No one can patent a mathematical formula, it’s ours to share. There is nothing in this world that is so deep and exquisite and yet so readily available to all. That such a reservoir of knowledge really exists is nearly unbelievable. It’s too precious to be given away to the 「initiated few.」 It belongs to all of us.
數學不同於其他知識。雖然我們對物理世界的認識總被扭曲，但我們對數學真理的認知卻不可動搖。數學是客觀存在、經久不變的必要真理。不管身處何處，數學公式或定理對於任何人來說都是一樣的，不以性別、宗教或膚色而轉移，歷經千年亦是如此。更令人驚訝的是，數學知識是全人類的共有財產，沒有人能為數學知識申請專利，它們註定要被共享。這個世界上，如此深邃精緻而觸手可及的知識，非數學莫屬。有這樣的知識寶庫存在，幾乎難以置信。它如此珍貴，不能只是「一小撮精英」的專屬。數學屬於我們所有人。
One of the key functions of mathematics is the ordering of information. This is what distinguishes the brush strokes of Van Gogh from a mere blob of paint. With the advent of 3D printing, the reality we are used to is undergoing a radical transformation: everything is migrating from the sphere of physical objects to the sphere of information and data. We will soon be able to convert information into matter on demand by using 3D printers just as easily as we now convert a PDF file into a book or an MP3 file into a piece of music. In this brave new world, the role of mathematics will become even more central: as the way to organize and order information, and as the means to facilitate the conversion of information into physical reality.
數學的核心功能之一是重組信息，這也是梵谷的筆觸和普通油漆點的區別。隨著3D列印的出現，我們所熟知的現實有了顛覆性的變化：一切事物都在從實物向信息和數據轉變。我們現在可以將PDF文件轉換為一本書，或是把MP3文件轉換成音樂，在不久的將來，我們也可以根據需要，利用3D印表機將信息轉換為實物。在這個美麗新世界中，數學的作用將日益凸顯，它不僅是組織和整理信息的方式，也是促進信息轉化為物質現實的手段。
In this book, I will describe one of the biggest ideas to come out of mathematics in the last fifty years: the Langlands Program, considered by many as the Grand Unified Theory of mathematics. It’s a fascinating theory that weaves a web of tantalizing connections between mathematical fields that at first glance seem to be light years apart: algebra, geometry, number theory, analysis, and quantum physics. If we think of those fields as continents in the hidden world of mathematics, then the Langlands Program is the ultimate teleportation device, capable of getting us instantly from one of them to another, and back.
在本書中，我將描述過去五十年來數學界中最偉大的思想之一——朗蘭茲綱領（Langlands Program）。很多人認為朗蘭茲綱領是數學的大一統理論。代數、幾何、數論、分析和量子物理等領域之間乍一看相去甚遠，但迷人的朗蘭茲綱領仍精妙地將它們聯繫在了一起。如果我們把這些領域看成是數學秘境中的一塊塊大陸，那麼朗蘭茲綱領就是終極的傳送門裝置，讓我們可以在各個大陸中瞬時往返。
Launched in the late 1960s by Robert Langlands, the mathematician who currently occupies Albert Einstein’s office at the Institute for Advanced Study in Princeton, the Langlands Program had its roots in a groundbreaking mathematical theory of symmetry. Its foundations were laid two centuries ago by a French prodigy, just before he was killed in a duel, at age twenty. It was subsequently enriched by another stunning discovery, which not only led to the proof of Fermat’s Last Theorem, but revolutionized the way we think about numbers and equations. Yet another penetrating insight was that mathematics has its own Rosetta stone and is full of mysterious analogies and metaphors. Following these analogies as creeks in the enchanted land of math, the ideas of the Langlands Program spilled into the realms of geometry and quantum physics, creating order and harmony out of seeming chaos.
朗蘭茲綱領由數學家羅伯特•朗蘭茲於20世紀60年代末期提出（朗蘭茲目前在普林斯頓的高級研究所工作，他的辦公室就是當年愛因斯坦的那間）。朗蘭茲綱領源於開創性的數學對稱理論，其研究雛形最早可追溯到200年前，一位年僅20歲的法國天才完成了研究工作，不久之後他就死於一場決鬥。後來的驚人發現豐富了這位天才少年的研究成果，它不僅為費馬大定理的證明提供了思路，並徹底改變了我們對數和方程的認知。另一個深刻的見解是，數學有它自己的羅塞塔石碑，本身就充滿了難解的類比和比喻。這些類比好似秘境中蜿蜒的溪水，讓朗蘭茲綱領的思想得以滲透到幾何和量子物理領域，從看似混亂的狀態中創造出秩序與和諧。
譯註1：伽羅瓦（Évariste Galois, 1811-1832）,著名法國數學家，與阿貝爾（Niels Henrik Abel）並稱現代「群論」創始者；
譯註2：羅塞塔石碑，是一塊製作於公元前196年的花崗閃長巖石碑，同時刻有古埃及法老託勒密五世詔書的三種不同語言版本，使得近代的考古學家得以有機會對照各語言版本的內容後，解讀出已經失傳千餘年的埃及象形文之意義與結構，而成為今日研究古埃及歷史的重要裡程碑。
I want to tell you about all this to expose the sides of mathematics we rarely get to see: inspiration, profound ideas, startling revelations. Mathematics is a way to break the barriers of the conventional, an expression of unbounded imagination in the search for truth. Georg Cantor, creator of the theory of infinity, wrote: 「The essence of mathematics lies in its freedom.」 Mathematics teaches us to rigorously analyze reality, study the facts, follow them wherever they lead. It liberates us from dogmas and prejudice, nurtures the capacity for innovation. It thus provides tools that transcend the subject itself.
我告訴你們這些內容，是為了揭示數學中鮮為人知的一面，其中包括了靈感、深思，以及驚人的發現。數學是打破傳統壁壘的一種方法，是在尋求真理的過程中自由想像的一種表達。無窮集合論的創始人格奧爾格•康託爾（Georg Cantor）寫道：「數學的本質在於它的自由。」數學教會我們嚴謹地分析現實、研究事實，無論數學指向何方，我們都會義無反顧地追隨。數學使我們擺脫教條和偏見，培養我們的創新能力。可以說，數學提供了超越學科本身的工具。
Love & Math
The Heart of Hidden Reality. By Edward Frenkel. Basic Books.
These tools can be used for good and for ill, forcing us to reckon with math’s real-world effects. For example, the global economic crisis was caused to a large extent by the widespread use of inadequate mathematical models in the financial markets. Many of the decision makers didn’t fully understand these models due to their mathematical illiteracy, but were arrogantly using them anyway – driven by greed – until this practice almost wrecked the entire system. They were taking unfair advantage of the asymmetric access to information and hoping that no one would call their bluff because others weren’t inclined to ask how these mathematical models worked either. Perhaps, if more people understood how these models functioned, how the system really worked, we wouldn’t have been fooled for so long.
一旦這些數學工具落入壞人之手，也會產生驚人的破壞力，因而迫使我們去審視數學在現實世界中的影響。例如，全球經濟危機很大程度上是由於金融市場普遍對數學模型使用不當。許多決策者自身是數學盲，在並沒有完全理解這些模型的情況下，仍在貪婪的驅使下輕妄地使用了模型，最終使整個金融系統受到重創。他們狡黠地利用信息的不對稱，毫不擔心有人揭發，認定其他人也不會去了解這些數學模型的原理。也許，如果能有更多的人了解數學模型和金融系統的運作規律，我們就不會被愚弄這麼久了。
As another example, consider this: in 1996, a commission appointed by the U.S. government gathered in secret and altered a formula for the Consumer Price Index, the measure of inflation that determines the tax brackets, Social Security, Medicare, and other indexed payments. Tens of millions of Americans were affected, but there was little public discussion of the new formula and its consequences. And recently there was another attempt to exploit this arcane formula as a backdoor on the U.S. economy.
再舉一個例子：1996年，美國政府任命的一個委員會秘密召開會議，修改了消費物價指數（CPI）的計算公式，而CPI是衡量通貨膨脹的指數，它間接決定了稅收登記、社會保障、醫療保險和其他指數化的支付。數千萬的美國民眾受此影響，但很少有人公開討論這一新公式及其後果。最近，又有人試圖利用這個神秘公式作為操縱美國經濟的後門。
Far fewer of these sorts of backroom deals could be made in a mathematically literate society. Mathematics equals rigor plus intellectual integrity times reliance on facts. We should all have access to the mathematical knowledge and tools needed to protect us from arbitrary decisions made by the powerful few in an increasingly math-driven world. Where there is no mathematics, there is no freedom.
如果每個人都能掌握一定的數學知識，諸如此類的幕後交易將要少得多。數學等於嚴謹加上完備的知識再乘以忠於事實。在一個日益以數學為動力的世界中，我們都應該以必要的數學知識和工具傍身，以保護我們的自身利益不受少數當權者武斷決定的影響。沒有數學，就沒有自由。
Mathematics is as much part of our cultural heritage as art, literature, and music. As humans, we have a hunger to discover something new, reach new meaning, understand better the universe and our place in it. Alas, we can’t discover a new continent like Columbus or be the first to set foot on the Moon. But what if I told you that you don’t have to sail across an ocean or fly into space to discover the wonders of the world? They are right here, intertwined with our present reality. In a sense, within us. Mathematics directs the flow of the universe, lurks behind its shapes and curves, holds the reins of everything from tiny atoms to the biggest stars.
數學同藝術、文學和音樂一樣，是文化遺產的一部分。作為人類，我們渴望新的發現，獲取新知，更好地理解宇宙和我們在其中的位置。哎，雖然我們不能像哥倫布再發現新大陸，也不能成為第一個登上月球的人，但如果我告訴你，你無需橫渡海洋或翱翔太空即可將世界奇觀一窺究竟呢？奇觀就在這裡，交織在當下的現實之中。從某種意義上來說，它更是存在於我們之中。數學指引著宇宙的流向，潛藏在各類形狀和曲線之後，掌控著小到原子大到恆星的世間萬物。
This book is an invitation to this rich and dazzling world. I wrote it for readers without any background in mathematics. If you think that math is hard, that you won’t get it, if you are terrified by math, but at the same time curious whether there is something there worth knowing – then this book is for you.
本書即是步入富饒而驚豔的數學秘境的邀請函。我將它獻給沒有數學背景的各位讀者。如果你覺得數學很難、無法理解，或是你曾對數學心存惶恐、卻又好奇有什麼值得探究，那麼這本書很適合你。
There is a common fallacy that one has to study mathematics for years to appreciate it. Some even think that most people have an innate learning disability when it comes to math. I disagree: most of us have heard of and have at least a rudimentary understanding of such concepts as the solar system, atoms and elementary particles, the double helix of DNA, and much more, without taking courses in physics and biology. And nobody is surprised that these sophisticated ideas are part of our culture, our collective consciousness. Likewise, everybody can grasp key mathematical concepts and ideas, if they are explained in the right way. To do this, it is not necessary to study math for years; in many cases, we can cut right to the point and jump over tedious steps.
人們常常認為，只有鑽研數學多年才能領會數學的美妙，這顯然不切實際。甚至還有人認為，大多數人在數學方面有先天的學習障礙。對此我不敢苟同。很多人沒有上過物理課或生物課，但他們都或多或少聽過太陽系、原子和基本粒子、DNA的雙螺旋等概念，甚至對這些概念略知一二。要是說這些深奧的概念是我們文化和集體意識的一部分，因而得以被人熟知，沒有人會感到驚訝。同樣，如果對數學概念和思想加以恰當地解釋，無須多年辛苦的數學學習，每個人都可以掌握重要的數學概念和思想。很多時候，我們可以跳過冗餘的步驟、直擊主題。
The problem is: while the world at large is always talking about planets, atoms, and DNA, chances are no one has ever talked to you about the fascinating ideas of modern math, such as symmetry groups, novel numerical systems in which 2 and 2 isn’t always 4, and beautiful geometric shapes like Riemann surfaces. It’s like they keep showing you a little cat and telling you that this is what a tiger looks like. But actually the tiger is an entirely different animal. I』ll show it to you in all of its splendor, and you』ll be able to appreciate its 「fearful symmetry,」 as William Blake eloquently said.
問題是，當整個世界都在討論行星、原子和DNA的時候，很少有人會跟你討論現代數學的迷人思想，比如對稱群、2加2不總是等於4的新記數系統，以及諸如黎曼曲面的美麗幾何圖形。現在談到數學有如照貓畫虎，可兩者是全然不同的動物。而我將向你展示數學的輝煌，用威廉•布萊克（William Blake）的話來說，就是讓你欣賞到它「令人敬畏的對稱美」。
Don’t get me wrong: reading this book won’t by itself make you a mathematician. Nor am I advocating that everyone should become a mathematician. Think about it this way: learning a small number of chords will enable you to play quite a few songs on a guitar. It won’t make you the world’s best guitar player, but it will enrich your life. In this book I will show you the chords of modern math, which have been hidden from you. And I promise that this will enrich your life.
請別誤會，閱讀本書並不會讓你成為數學家，我也不主張每個人都應該精通數學。你可以這樣想，學習簡單的和弦就可以彈奏不少的吉他歌曲了，雖然這不會讓你成為世界上最好的吉他手，但足以豐富你的生活。在本書中，我將向你們展示現代數學的和弦，保證在你學會隱匿於視線之外的和弦後，生活會更加精彩。
One of my teachers, the great Israel Gelfand, used to say: 「People think they don’t understand math, but it’s all about how you explain it to them. If you ask a drunkard what number is larger, 2/3 or 3/5, he won’t be able to tell you. But if you rephrase the question: what is better, 2 bottles of vodka for 3 people or 3 bottles of vodka for 5 people, he will tell you right away: 2 bottles for 3 people, of course.」
我的一位老師、偉大的伊斯拉埃爾•蓋爾範德（Israel Gelfand）曾經說過：「人們認為自己不懂數學，但關鍵是如何向他們解釋數學。如果你問一個醉漢2/3和3/5哪一個更大，他什麼都說不出來。但如果你稍改措辭，問他三個人喝兩瓶伏特加還是五個人喝三瓶伏特加更好，他會馬上告訴你：當然是三個人喝兩瓶伏特加更好。」
My goal is to explain this stuff to you in terms that you will understand.
我的目標正是用你能理解的語言去解釋數學。
I will also talk about my experience of growing up in the former Soviet Union, where mathematics became an outpost of freedom in the face of an oppressive regime. I was denied entrance to Moscow State University because of the discriminatory policies of the Soviet Union. The doors were slammed shut in front of me. I was an outcast. But I didn’t give up. I would sneak into the University to attend lectures and seminars. I would read math books on my own, sometimes late at night. And in the end, I was able to hack the system. They didn’t let me in through the front door; I flew in through a window. When you are in love, who can stop you?
書中還會談一談我在前蘇聯的成長經歷。高壓政權之下，數學成了自由的前哨。由於蘇聯的歧視政策，我被莫斯科國立大學拒之門外。數學研究的大門在我面前砰地關上了，我成為了局外人。但我沒有放棄，我會偷偷溜進大學去聽講座和研討會。我還閱讀數學書自學，有的時候一直讀到深夜。最終，我「黑」進了系統。他們不讓我從前門進去，那我就從窗戶飛進去。當你陷入愛情時，誰能阻止你追愛的腳步？
Two brilliant mathematicians took me under their wings and became my mentors. With their guidance, I started doing mathematical research. I was still a college student, but I was already pushing the boundaries of the unknown. This was the most exciting time of my life, and I did it even though I was sure that the discriminatory policies would never allow me to have a job as a mathematician in the Soviet Union.
兩位傑出的數學家偷偷接納了我，成為我的導師。在他們的指導下，我開始了數學研究。雖然還是個大學生，但我已在開拓未知的領域，這是我一生中最激動人心的時光。儘管我確信，歧視政策永遠不會允許我在蘇聯成為一名數學家，但我還是做到了。
But there was a surprise in store: my first mathematical papers were smuggled abroad and became known, and I got invited to Harvard University as a Visiting Professor at age twenty-one. Miraculously, at exactly the same time perestroika in the Soviet Union lifted the iron curtain, and citizens were allowed to travel abroad. So there I was, a Harvard professor without a Ph.D., hacking the system once again. I continued on my academic path, which led me to research on the frontiers of the Langlands Program and enabled me to participate in some of the major advances in this area during the last twenty years. In what follows, I will describe spectacular results obtained by brilliant scientists as well as what happened behind the scenes.
但後來發生的事著實在我意料之外：我的第一篇數學論文流傳到國外並引起了關注。21歲的時候，哈佛大學邀我成為客座教授。更巧的是，彼時蘇聯正進行戈巴契夫改革。鐵幕被拉開了，民眾被允許出國旅遊。於是，我一個沒有博士學位的哈佛教授，再一次「黑」進了系統。我得以繼續我的學術道路，開展朗蘭茲綱領的前沿研究，並涉足過去20年中數學領域的一些重大進展。在接下來的文章中，我將描述傑出科學家取得的驚人成果及其背後的故事。
This book is also about love. Once, I had a vision of a mathematician discovering the 「formula of love,」 and this became the premise of a film Rites of Love and Math, which I will talk about later in the book. Whenever I show the film, someone always asks: 「Does a formula of love really exist?」
「愛」是本書的另一主題。我曾想像數學家發現了「愛的公式」，這也是我拍攝電影《愛與數學的儀式》（Rites of Love and Math）的前提（後文將有提及）。每當我放映這部電影時，總會有人問我：「真的有『愛的公式』嗎？」
My response: 「Every formula we create is a formula of love.」 Mathematics is the source of timeless profound knowledge, which goes to the heart of all matter and unites us across cultures, continents, and centuries. My dream is that all of us will be able to see, appreciate, and marvel at the magic beauty and exquisite harmony of these ideas, formulas, and equations, for this will give so much more meaning to our love for this world and for each other.
我的回答是：「我們創造的每一個公式都是『愛的公式』。」數學不斷給予我們永恆深奧的知識，它深入到所有物質的核心，跨越文化、空間和時間將我們凝聚在一起。我夢想有一天，所有人都能欣賞驚嘆這些想法、公式和方程的美妙與精緻，因為這將賦予我們對世界和彼此的愛以更多意義。
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參考閱讀：
純粹的數學世界，對於絕大多數人來說，都充滿了距離感。不過已經有許多電影作品，可以幫助我們從更具人情味的角度，窺探數學或是數學家的世界。
《美麗心靈》自然是一部不容錯過的關於數學家的電影，講述了著名數學家納什如何在精神困境中實現自己在數學領域的突破。
https://movie.douban.com/subject/1306029/comments?status=P
而另一部值得一提的經典則是《心靈捕手》，故事圍繞馬特達蒙所飾演的年輕清潔工，如何在教授的幫助下，發掘自己的數學天分，走出了童年陰影而展開。
https://movie.douban.com/subject/1292656/
關於數學天才，紀錄片《危險的知識》也廣受好評，或許提供了另一個視角走進這些天才式的人物。在這部紀錄片裡，導演聚焦了四位數學家：喬治·康託爾、路德維格・玻爾茲曼、哥德爾和阿蘭·圖靈。他們中的大多數最終因為精神錯亂而自殺，但他們在數學世界裡留下的智慧仍舊深刻的影響著後世。
此外，還有許多優秀的紀錄片，從宏觀層面精彩地探討了「數學」，這個看似深奧且遙遠的話題。
BBC拍攝的《解碼數學》就「分別從「『數字、形狀、預測』三個方向來探尋隱藏在宗教、建築、藝術、生物等大自然和人類生活中的終極密碼——數學」，在這個紀錄片中，我們或許可以發現，不論是中世紀教堂，還是神秘的巨石陣，不論是飛機航線的規劃，還是音樂的演奏，甚至是動畫設計與分形等等，我們生活的方方面面都滲透著數學的奧妙。
https://movie.douban.com/subject/7051249/
BBC的另一個紀錄片《數學的故事》中還講述了有關古代中國的數學故事。https://movie.douban.com/subject/4139917/
另一部法國的高分紀錄片，則似乎相對更加深奧，據說其中所探討的四維空間中的諸多理論很是燒腦，勇者可以嘗試一下。
當然，如果你更喜歡讀書，還有一本關於數學簡史的新書可以推薦給你，《數學簡史：確定性的消失》。本書作者莫裡斯·克萊因是數學史大家，在書中，他探討了「數千年來數學在直覺、邏輯、應用之間穿梭往復的炫目旅程，再現真實數學的發展過程，闡述數學的起源、數學的繁榮和科學的數學化，直到當代數學的現狀：數學與確定性（邏輯，嚴密性，完備性）漸行漸遠。」
https://book.douban.com/subject/30296440/
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