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我们翻译这篇文章的理由 当代文明的科技成就、人们便捷的日常生活，都离不开数学理论的铺垫架构。但我们不是对其敬而远之，就是将其视为机械性的手头工具。实际上，数学是世间铁律、是有力的武器：借助数学，虚伪的政策即刻被揭穿、民众不会被轻易愚弄操纵。数学是文明瑰宝、是无可篡改的谶言：借助数学，思想能够在人类肉体所不能及的高维空间自由翱翔。数学是万物的真理、是直指人心的力量，认识到数学之美、之自由，你也就理解了爱与永恒。 👇 节选：《爱与数学》 作者：EdwardFrenkel 译者：王雅婧 校对：刘小康 推荐：罗玉池 策划：王雅婧 & 泮海伦 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. 那里有个神秘世界，一个存在于平行宇宙中的秘密世界，它典雅美丽，与我们错综复杂地交织在一起。这就是绝大多数人都无法领略的数学世界，而本书将邀你一齐对它进行探索。 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.” 通往数学秘境的道路上，绕不开的是我们的认知与数学本身的矛盾：虽然数学与我们的日常生活息息相关，我们每一次网购、发短信、在互联网上搜索或者使用GPS设备，都少不了数学公式和算法在背后支撑，但大多数人对数学望而却步。用诗人汉斯·马格努斯·恩岑斯贝格尔（Hans Magnus Enzensberger）的话来说，数学已经成为“我们文化中的盲点，只有少数精英在点化下才能在这片异域上扎稳脚跟。”他说，几乎没有人会“咬牙切齿地说他们一想到读小说、欣赏图片或看电影就苦不堪言。”但是，“受过教育的聪明人”却经常会用“夹杂着蔑视与傲慢的口吻”说，数学是“彻头彻尾的折磨”，是“一场噩梦”之类的话，所以他们“讨厌数学”。 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. 为什么会这样呢？我认为主要有两个原因。一是数学比其他学科更抽象，因此入门不易。二是学校学到的不过是数学知识的冰山一角，而绝大部分课本内容都是一千多年前就已经建立的。从那以后，数学已有了突飞猛进的发展，但是现代数学的宝藏在大多数人眼中一直饰以未知的色彩。 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“艺术课”上，如果老师只教你如何粉刷篱笆，会有怎样的结果？如果老师从未在课上带你领略达芬奇和毕加索的作品呢？这能让你学会鉴赏艺术吗？你还会对艺术抱有好奇吗？答案是否定的。你可能会说，“在学校学习艺术是浪费时间。我要是需要粉刷篱笆，肯定会雇人来做啊。”这个类比听起来固然有些荒谬，但学校中的数学教育就是如此，所以在大多数人眼中，学习数学而无意义，和在篱笆旁等着油漆变干没有区别。在今天，尽管欣赏大师们的画作已不是难事，但大师们的数学研究却一直锁于高阁，世人只能略窥一二。 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，数学之所以迷人，绝不仅仅在于其美感。伽利略曾言：“自然法则由数学语言所写。”数学是描述现实、揭示世界运作原理的语言，这种通用语言现已成为检验真理的黄金标准。在科学技术的推动下，数学日益成为权力、财富和进步之源。因此，精通数学这门新语言的人将站在进步的前沿。 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! 人们对数学常抱有误解，认为数学只能充当“工具箱”。就说生物学家吧，他们会做一些实地工作，收集数据、然后试图建立一个拟合这些数据的数学模型（可能是由数学家协助建立的）。虽然这是一个重要的研究模式，但数学让我们洞见了更多：它使我们能够实现突破性的、思维方式上的飞跃。没有数学，我们很可能会陷入僵局。例如，爱因斯坦在理解引力造成的空间弯曲时，并没有试图将任何数据代入到方程中。事实上，他手头也没有可以用于验证的数据。当时没有人能想象，我们的空间竟然是弯曲的，每个人都“觉得”世界是平的！但是爱因斯坦明白，这是将他的狭义相对论（连同他对引力和加速度具有同等效应的敏锐认知）推广到非惯性系统的唯一途径。这是数学领域的一道高级智力练习，黎曼早在50年前就已完成了准备工作。人脑的结构决定了我们很难想象高于二维的黎曼空间，我们只能借用数学走近它们。你猜最后发生了什么——爱因斯坦是对的！我们的宇宙是弯曲的，而且还在膨胀中。这就是我所说的数学的力量！ 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. 类似例子数不胜数，不仅是在物理学领域，在其他学科领域也是如此（后文将会提及）。历史表明，数学思想加速了科学技术的发展，即使是最初被视为抽象和晦涩的数学理论，后来在实际应用中也发挥出不可或缺的作用。达尔文一开始的工作并不依赖数学，但他后来在自传中写道：“我很遗憾没有理解数学的一些重要原理，至少没有了解得足够深入，因为数学能赋予人们额外的感知力。”此番话语恰如一剂预防针，用以忠告后人，要重视数学的巨大潜力。 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. 小时候，我并未意识到数学秘境的存在，也认为数学只是一门迂腐枯燥的学科。但我也很幸运，高中的最后一年，我遇到了一位专业的数学家，他向我敞开了奇妙的数学世界，让我知道数学和诗歌、艺术和音乐一般，充满了无限的可能性。于是，我深深爱上了数学。 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. 亲爱的读者，当年是我的老师带我走进了数学世界，而我想以这本书为钥匙，打开数学的力量与美丽，让你随着我的脚步进入这个神奇的世界。即使你从未把“数学”和“爱”联系在一起，我们的旅途也定会顺利无虞。数学与你将日渐亲密，甚至交融为一体，进而颠覆你的整个世界观。 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. 数学不同于其他知识。虽然我们对物理世界的认识总被扭曲，但我们对数学真理的认知却不可动摇。数学是客观存在、经久不变的必要真理。不管身处何处，数学公式或定理对于任何人来说都是一样的，不以性别、宗教或肤色而转移，历经千年亦是如此。更令人惊讶的是，数学知识是全人类的共有财产，没有人能为数学知识申请专利，它们注定要被共享。这个世界上，如此深邃精致而触手可及的知识，非数学莫属。有这样的知识宝库存在，几乎难以置信。它如此珍贵，不能只是“一小撮精英”的专属。数学属于我们所有人。 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. 数学的核心功能之一是重组信息，这也是梵高的笔触和普通油漆点的区别。随着3D打印的出现，我们所熟知的现实有了颠覆性的变化：一切事物都在从实物向信息和数据转变。我们现在可以将PDF文件转换为一本书，或是把MP3文件转换成音乐，在不久的将来，我们也可以根据需要，利用3D打印机将信息转换为实物。在这个美丽新世界中，数学的作用将日益凸显，它不仅是组织和整理信息的方式，也是促进信息转化为物质现实的手段。 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. 在本书中，我将描述过去五十年来数学界中最伟大的思想之一——朗兰兹纲领（Langlands Program）。很多人认为朗兰兹纲领是数学的大一统理论。代数、几何、数论、分析和量子物理等领域之间乍一看相去甚远，但迷人的朗兰兹纲领仍精妙地将它们联系在了一起。如果我们把这些领域看成是数学秘境中的一块块大陆，那么朗兰兹纲领就是终极的传送门装置，让我们可以在各个大陆中瞬时往返。 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. 朗兰兹纲领由数学家罗伯特·朗兰兹于20世纪60年代末期提出（朗兰兹目前在普林斯顿的高级研究所工作，他的办公室就是当年爱因斯坦的那间）。朗兰兹纲领源于开创性的数学对称理论，其研究雏形最早可追溯到200年前，一位年仅20岁的法国天才完成了研究工作，不久之后他就死于一场决斗。后来的惊人发现丰富了这位天才少年的研究成果，它不仅为费马大定理的证明提供了思路，并彻底改变了我们对数和方程的认知。另一个深刻的见解是，数学有它自己的罗塞塔石碑，本身就充满了难解的类比和比喻。这些类比好似秘境中蜿蜒的溪水，让朗兰兹纲领的思想得以渗透到几何和量子物理领域，从看似混乱的状态中创造出秩序与和谐。 译注1：伽罗瓦（Évariste Galois, 1811-1832）,著名法国数学家，与阿贝尔（Niels Henrik Abel）并称现代“群论”创始者； 译注2：罗塞塔石碑，是一块制作于公元前196年的花岗闪长岩石碑，同时刻有古埃及法老托勒密五世诏书的三种不同语言版本，使得近代的考古学家得以有机会对照各语言版本的内容后，解读出已经失传千余年的埃及象形文之意义与结构，而成为今日研究古埃及历史的重要里程碑。 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. 我告诉你们这些内容，是为了揭示数学中鲜为人知的一面，其中包括了灵感、深思，以及惊人的发现。数学是打破传统壁垒的一种方法，是在寻求真理的过程中自由想象的一种表达。无穷集合论的创始人格奥尔格·康托尔（Georg Cantor）写道：“数学的本质在于它的自由。”数学教会我们严谨地分析现实、研究事实，无论数学指向何方，我们都会义无反顾地追随。数学使我们摆脱教条和偏见，培养我们的创新能力。可以说，数学提供了超越学科本身的工具。 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. 一旦这些数学工具落入坏人之手，也会产生惊人的破坏力，因而迫使我们去审视数学在现实世界中的影响。例如，全球经济危机很大程度上是由于金融市场普遍对数学模型使用不当。许多决策者自身是数学盲，在并没有完全理解这些模型的情况下，仍在贪婪的驱使下轻妄地使用了模型，最终使整个金融系统受到重创。他们狡黠地利用信息的不对称，毫不担心有人揭发，认定其他人也不会去了解这些数学模型的原理。也许，如果能有更多的人了解数学模型和金融系统的运作规律，我们就不会被愚弄这么久了。 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. 再举一个例子：1996年，美国政府任命的一个委员会秘密召开会议，修改了消费物价指数（CPI）的计算公式，而CPI是衡量通货膨胀的指数，它间接决定了税收登记、社会保障、医疗保险和其他指数化的支付。数千万的美国民众受此影响，但很少有人公开讨论这一新公式及其后果。最近，又有人试图利用这个神秘公式作为操纵美国经济的后门。 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. 如果每个人都能掌握一定的数学知识，诸如此类的幕后交易将要少得多。数学等于严谨加上完备的知识再乘以忠于事实。在一个日益以数学为动力的世界中，我们都应该以必要的数学知识和工具傍身，以保护我们的自身利益不受少数当权者武断决定的影响。没有数学，就没有自由。 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. 数学同艺术、文学和音乐一样，是文化遗产的一部分。作为人类，我们渴望新的发现，获取新知，更好地理解宇宙和我们在其中的位置。哎，虽然我们不能像哥伦布再发现新大陆，也不能成为第一个登上月球的人，但如果我告诉你，你无需横渡海洋或翱翔太空即可将世界奇观一窥究竟呢？奇观就在这里，交织在当下的现实之中。从某种意义上来说，它更是存在于我们之中。数学指引着宇宙的流向，潜藏在各类形状和曲线之后，掌控着小到原子大到恒星的世间万物。 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. 本书即是步入富饶而惊艳的数学秘境的邀请函。我将它献给没有数学背景的各位读者。如果你觉得数学很难、无法理解，或是你曾对数学心存惶恐、却又好奇有什么值得探究，那么这本书很适合你。 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. 人们常常认为，只有钻研数学多年才能领会数学的美妙，这显然不切实际。甚至还有人认为，大多数人在数学方面有先天的学习障碍。对此我不敢苟同。很多人没有上过物理课或生物课，但他们都或多或少听过太阳系、原子和基本粒子、DNA的双螺旋等概念，甚至对这些概念略知一二。要是说这些深奥的概念是我们文化和集体意识的一部分，因而得以被人熟知，没有人会感到惊讶。同样，如果对数学概念和思想加以恰当地解释，无须多年辛苦的数学学习，每个人都可以掌握重要的数学概念和思想。很多时候，我们可以跳过冗余的步骤、直击主题。 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. 问题是，当整个世界都在讨论行星、原子和DNA的时候，很少有人会跟你讨论现代数学的迷人思想，比如对称群、2加2不总是等于4的新记数系统，以及诸如黎曼曲面的美丽几何图形。现在谈到数学有如照猫画虎，可两者是全然不同的动物。而我将向你展示数学的辉煌，用威廉·布莱克（William Blake）的话来说，就是让你欣赏到它“令人敬畏的对称美”。 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. 请别误会，阅读本书并不会让你成为数学家，我也不主张每个人都应该精通数学。你可以这样想，学习简单的和弦就可以弹奏不少的吉他歌曲了，虽然这不会让你成为世界上最好的吉他手，但足以丰富你的生活。在本书中，我将向你们展示现代数学的和弦，保证在你学会隐匿于视线之外的和弦后，生活会更加精彩。 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.” 我的一位老师、伟大的伊斯拉埃尔·盖尔范德（Israel Gelfand）曾经说过：“人们认为自己不懂数学，但关键是如何向他们解释数学。如果你问一个醉汉2/3和3/5哪一个更大，他什么都说不出来。但如果你稍改措辞，问他三个人喝两瓶伏特加还是五个人喝三瓶伏特加更好，他会马上告诉你：当然是三个人喝两瓶伏特加更好。” My goal is to explain this stuff to you in terms that you will understand. 我的目标正是用你能理解的语言去解释数学。 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? 书中还会谈一谈我在前苏联的成长经历。高压政权之下，数学成了自由的前哨。由于苏联的歧视政策，我被莫斯科国立大学拒之门外。数学研究的大门在我面前砰地关上了，我成为了局外人。但我没有放弃，我会偷偷溜进大学去听讲座和研讨会。我还阅读数学书自学，有的时候一直读到深夜。最终，我“黑”进了系统。他们不让我从前门进去，那我就从窗户飞进去。当你陷入爱情时，谁能阻止你追爱的脚步？ 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. 两位杰出的数学家偷偷接纳了我，成为我的导师。在他们的指导下，我开始了数学研究。虽然还是个大学生，但我已在开拓未知的领域，这是我一生中最激动人心的时光。尽管我确信，歧视政策永远不会允许我在苏联成为一名数学家，但我还是做到了。 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. 但后来发生的事着实在我意料之外：我的第一篇数学论文流传到国外并引起了关注。21岁的时候，哈佛大学邀我成为客座教授。更巧的是，彼时苏联正进行戈尔巴乔夫改革。铁幕被拉开了，民众被允许出国旅游。于是，我一个没有博士学位的哈佛教授，再一次“黑”进了系统。我得以继续我的学术道路，开展朗兰兹纲领的前沿研究，并涉足过去20年中数学领域的一些重大进展。在接下来的文章中，我将描述杰出科学家取得的惊人成果及其背后的故事。 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?” “爱”是本书的另一主题。我曾想象数学家发现了“爱的公式”，这也是我拍摄电影《爱与数学的仪式》（Rites of Love and Math）的前提（后文将有提及）。每当我放映这部电影时，总会有人问我：“真的有‘爱的公式’吗？” 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：“我们创造的每一个公式都是‘爱的公式’。”数学不断给予我们永恒深奥的知识，它深入到所有物质的核心，跨越文化、空间和时间将我们凝聚在一起。我梦想有一天，所有人都能欣赏惊叹这些想法、公式和方程的美妙与精致，因为这将赋予我们对世界和彼此的爱以更多意义。 👇 参考阅读：
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