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Unit 4
Paul Dirac （保罗·狄拉克）had an eye for beauty. In one essay, from May 1963, the British Nobel laureate（获奖者）referred to beauty nine times. It makes four appearances in four consecutive sentences. In the article, he painted a picture of how physicists saw nature. But the word beauty never defined a sunset, nor a flower, or nature in any traditional sense Dirac（狄拉克） was talking quantum theory and gravity. The beauty lay in the mathematics.
【保罗·狄拉克有鉴赏力。在1963年5月的一篇文章中，这位英国诺贝尔奖得主九次提到“美”。它在四个连续的句子中出现了四次。在这篇文章中，他描绘了一幅物理学家如何看待自然的图景。但“美”这个词从来没有定义过日落、花朵或任何传统意义上的自然，狄拉克说的是量子理论和引力。美在于数学。】
What does it mean for math to be beautiful? It is not about the appearance of the symbols on the page. That, at best, is secondary. Math becomes beautiful through the power and elegance of its arguments and formulae through the bridges it builds between previously unconnected worlds. When it surprises. For those who learn the language, math has the same capacity for beauty as art, music, a full blanket of stars on the darkest night.
【数学美丽意味着什么？ 这与页面上符号的外观无关。 这充其量是次要的。 数学通过其论证和公式的力量和优雅，通过它在以前未连接的世界之间建立的桥梁而变得美丽。 当它令人惊讶时。 对于那些学习语言的人来说，数学与艺术、音乐、最黑暗的夜晚铺满星星一样具有美感。】
“The slow movement of the Mozart clarinet concerto （莫扎特单簧管协奏曲） is a really beautiful piece of music, but I don't print off a page of the score and put that on my wall. It's not about that. It's about the music and the ideas and the emotional response.” say Vicky Neale, a mathematician at Oxford University. “It's the same with a piece of mathematics. It's not how it looks; it's about the underlying thought processes
【“莫扎特单簧管协奏曲的缓慢乐章是一首非常美丽的音乐，但我不会打印一页乐谱并将其挂在墙上。 不是这个问题。 这是关于音乐、想法和情感反应。” 牛津大学数学家维琪·尼尔 (Vicky Neale) 说道。 “数学也是如此。 事实并非如此。 这是关于潜在的思维过程】
Brain scans of mathematicians show that gazing（盯着）at formulae considered beautiful by the beholder elicits activity in the same emotional region as great art and music. The more beautiful the formula, the greater the activity in the medial orbitofrontal cortex. “So far as the brain is concerned, math has beauty just like art, there is common neurophysiological ground, says Sir Michael Atiya, an honorary professor of mathematics at Edinburgh University.
【对数学家的脑部扫描表明，凝视旁观者认为美丽的公式会引发与伟大艺术和音乐相同的情感区域的活动。 公式越漂亮，内侧眶额皮质的活动就越大。 爱丁堡大学数学名誉教授迈克尔·阿蒂亚爵士表示：“就大脑而言，数学就像艺术一样具有美感，有共同的神经生理学基础。”】
Ask mathematicians about the most beautiful equation and one crops up time and again. Written in the 18th century by the Swiss mathematician, Leonhard Euler? The relation is short and simple: e+1=0. It is neat and compact even to the naive eye. But the beauty comes from a deeper understanding: Here the five most important mathematical constants are brought together. Euler's formula marries the world of circles, imaginary numbers and exponentials.
【向数学家询问最美丽的方程式，其中一个会一次又一次地出现。 由瑞士数学家莱昂哈德·欧拉于 18 世纪撰写？ 该关系简短而简单：e+1=0。 即使对于天真的眼睛来说，它也是整洁和紧凑的。 但美妙之处来自于更深入的理解：这里汇集了五个最重要的数学常数。 欧拉公式将圆、虚数和指数的世界结合在一起】
The beauty of other formulae may be more obvious. With E=mc , Albert Einstein built a bridge between energy and mass, two concepts that had previously seemed worlds apart. Maggie Adrien Pocock（玛吉·阿德琳·波科克）, the space scientist, declared it the most beautiful equation and she is in good company. “Why is it so beautiful? Because it comes to life. Now energy will have mass and mass can be put into energy. These four symbols capture a complete world. It's difficult to imagine a shorter formula with more power” says Robert Dijk Graaf（罗伯特·戴克·格拉夫）, director of the Institute for Advanced Study in Princeton, where Einstein was one of the first faculty members.
【其他公式的美妙可能更明显。 通过 E=mc ，阿尔伯特·爱因斯坦在能量和质量之间架起了一座桥梁，这两个概念以前似乎是天壤之别。 太空科学家玛吉·阿德琳·波科克 (Maggie Aderin Pocock) 宣称这是最美丽的方程式，而且她身边也有很多人。 “为什么这么漂亮？ 因为它栩栩如生。 现在能量将具有质量，并且质量可以转化为能量。 这四个符号捕捉了一个完整的世界。 很难想象一个更短、更强大的公式。”普林斯顿高等研究院院长罗伯特·戴克·格拉夫 (Robert Dijk Graaf) 说道，爱因斯坦是该研究所的首批教员之一】
“One of the reasons there's almost an objective beauty in mathematics is that we use the word beautiful also to indicate the raw power in an idea. The equations or results in mathematics that are seen to be beautiful are almost like poems. The power per variable is something that is part of the experience. Just seeing a huge part of mathematics or nature being described with just a few symbols gives a great sense of elegance or beauty Dijkgraaf adds. “A second element is you feel its beauty is reflecting reality. It's reflecting a sense of order that's out there as part of the laws of nature”
【“数学中几乎有一种客观的美，其中一个原因是我们用‘美’这个词来表示一个想法的原始力量。数学中的方程式或结果被认为是美的，几乎就像诗一样。每个变量的能量是游戏体验的一部分。Dijkgraaf补充说，仅仅看到数学或自然的很大一部分被仅仅几个符号所描述，就会给人一种非常优雅或美丽的感觉。“第二个因素是你觉得它的美反映了现实。它反映了一种秩序感，这是自然法则的一部分。”】
The power of an equation to connect what seems like completely unrelated realms of mathematics comes up often. Marcus du Sautoy, a math professor at Oxford, has more than a soft spot for Riemann’s formula. Published by Bernhard Riemann in 1859 (the same year Charles Species stunned the world with On the Origin of Species), the formula reveals how many primes exist below a given number, where primes are whole numbers divisible only by themselves and 1, such as 2, 3, 5, 7 and 11. While one side of the equation describes the primes, the other is controlled zeros.
【一个方程的力量，连接看似完全不相关的数学领域经常出现。牛津大学的数学教授马库斯·杜·索托伊对黎曼公式的偏爱不止于此。这个公式由Bernhard Riemann于1859年发表(同年Charles Species以《物种起源》震惊了世界)，它揭示了一个给定数字下面有多少个素数，其中素数是只能被自己和1整除的整数，比如2,3,5,7和11。方程的一边是质数，另一边是可控的零。】
“This formula turns these indivisible prime numbers, into something completely different,” says du Sautoy “On the one side, you've got these indivisible prime numbers and then Riemann takes you on this journey to somewhere completely unexpected, to these things which we now call the Riemann zeros. Each of these zeros gives rise to a note - and it's the combination of these notes together which tell us how the primes on the other side are distributed across all numbers.”
【“这个公式把这些不可分的质数变成了完全不同的东西，”杜索托伊说，“一方面，你有这些不可分的质数，然后黎曼带你踏上旅程，到达一个完全意想不到的地方，到达我们现在称之为黎曼零的东西。每一个零都会产生一个音符，这些音符的组合告诉我们另一边的质数是如何分布在所有数字上的。”】
More than 2,000 years ago, the ancient Greek mathematician, Euclid, solved numerical puzzle so beautifully that it still makes Neale smile every time it comes to mind. “When I think about beauty in mathematics, my first thoughts are not about equations. For me it's much more about an argument, a line of thinking, or a particular proof” she says.
【2000多年前，古希腊数学家欧几里得(Euclid)解出了一道非常漂亮的数字谜题，以至于每当想起这个谜题，尼尔都会会心一笑。“当我想到数学之美时，我首先想到的不是方程。对我来说，它更多的是关于一个论点，一条思路，或者一个特定的证明。】
Euclid proved there are infinitely many prime numbers. How did he do it? He began by imagining a universe where the number of primes was not infinite. Given a big enough blackboard, one could chalk them all up.
【欧几里得证明了素数有无穷多个。他是怎么做到的?他开始想象一个质数不是无限的宇宙。只要有一块足够大的黑板，人们就可以把它们都记下来。】
He then asked what happened if all these primes were multiplied together: 2 x 3 x 5 and so on, all the way to the end of the list, and the result added to the number 1. This huge new number provides the answer. Either it is a prime number itself, and so the original list was incomplete, or it is divisible by a smaller prime. But divide Euclid's number by any prime on the list and always there is a 1 left over. The number is not divisible by any prime on the list. "It turns out you reach an absurdity, a contradiction”, says Neale. The original assumption that the number primes is finite must be wrong.
【然后他问，如果把所有这些质数相乘会发生什么:2 × 3 × 5，以此类推，一直到列表的末尾，结果加到数字1上。这个庞大的新数字提供了答案。要么它本身就是一个素数，所以原来的列表是不完整的，要么它能被一个更小的素数整除。但是用欧几里得的数除以数列上的任何质数总是剩下一个1。这个数不能被数列上的任何素数整除。尼尔说:“事实证明，这是荒谬的，是矛盾的。”最初质数是有限的假设肯定是错误的。】
“The proof for me is really beautiful. It takes some thinking to get your head around it, but it doesn't involve learning lots of difficult concepts. It's surprising that you can prove something so difficult in such an elegant way” Neale adds.
【“对我来说，证据真的很漂亮。你需要一些思考才能理解它，但它不需要学习很多困难的概念。你能以如此优雅的方式证明如此困难的事情，真是令人惊讶。”】
Behind beautiful processes lies beautiful mathematics. Well, some of the time. Hannah Fry, a lecturer in the mathematics of cities at UCL spent years staring at the Navier-Stokes equations. “They're a single mathematical sentence that is capable of describing the miraculously beautiful and diverse behavior of almost all of the earth's fluids,” she says. With a grasp of the formulae, we can understand blood flow in the body, make boats glide through the water, and build awesome chocolate enrobers.
【美丽的过程背后是美丽的数学。嗯，有时候是这样。伦敦大学学院(UCL)城市数学讲师汉娜?弗莱(Hannah Fry)花了数年时间研究纳维-斯托克斯方程。她说:“它们是一个简单的数学句子，能够描述地球上几乎所有流体的神奇的美丽和多样的行为。”掌握了这些公式，我们就能理解体内的血液流动，使船只在水中滑行，并制造出令人惊叹的巧克力连衣裙。】
In his 1963 essay, Dirac elevated beauty from an aesthetic response to something far more profound: a route to the truth. “It is more important to have beauty in one equation than to have them fit experiment,” he wrote, continuing: “It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress. Shocking at first pass, Dirac captured what is now a common sentiment: When a beautiful equation seems at odds with nature, the fault may lie not with the math, but in applying it to the wrong aspect of nature.
【在他1963年的文章中，狄拉克将美从一种审美反应提升到更深刻的东西:通往真理的道路。他写道:“在一个方程中拥有美比让它们符合实验更重要。”他继续写道:“似乎如果一个人从一个方程中获得美的角度出发，如果一个人真的有一个正确的见解，他就一定会取得进步。乍一看令人震惊的是，狄拉克抓住了现在的普遍观点:当一个美丽的方程似乎与自然不符时，错误可能不在于数学，而在于将它应用于自然的错误方面。】
“Truth and beauty are closely related but not the same says Atiyah. “You are never sure that you have the truth. All you are doing is striving towards better and better truths and the light that guides you is beauty. Beauty is the torch you hold up and follow in the belief that it will lead you to truth in the end.”
【阿提亚说:“真与美密切相关，但又不相同。“你永远无法确定你是否掌握了真相。你所做的一切都是为了更好的真理而奋斗，而引导你的光就是美。美是你高举并追随的火炬，你相信它最终会把你引向真理。”】
Something approaching faith in mathematical beauty has led physicists to draw up two of the most compelling descriptions of reality: super symmetry and string theory”. In a super symmetric universe, every known type of particle has a heavier invisible twin. In string theory, reality has 10 dimensions, but six are curled up so tight they are hidden from us. The mathematics behind both theories are often described as beautiful, but it is not at all clear if either is true.
【某种对数学之美的近乎信仰的东西使物理学家们对现实做出了两种最令人信服的描述:超对称理论和弦理论。”在超对称的宇宙中，每一种已知的粒子都有一个更重的看不见的孪生兄弟。在弦理论中，现实有10维，但其中6维卷曲得太紧，我们看不见。这两种理论背后的数学常常被描述为美丽的，但我们根本不清楚哪一个是正确的。】
There is a danger here for mathematicians. Beauty is a fallible guide. "You can literally be seduced by something that is not correct. This is a risk”. says Dijk-Graaf. whose institute motto, “Truth and Beauty”, features one naked and one dressed woman. “Sometimes I feel that physicists, like Odysseus, must tie themselves to the mast of the ship so they are not seduced by the Sirens of mathematics.
【这对数学家来说是危险的。美是不可靠的向导。“你真的会被不正确的东西所诱惑。这是一种风险。”戴格拉说。该学院的校训是“真与美”，画的是一名裸体女子和一名穿着衣服的女子。“有时我觉得物理学家，就像奥德修斯一样，必须把自己绑在船的桅杆上，这样他们才不会被数学的塞壬诱惑。】
It may be that mathematicians and scientists are the only groups that still use the word “beautiful” without hesitation. It is rarely employed by critics of literature, art or music, who perhaps fear it sounds superficial or kitschy.
【也许数学家和科学家是唯一仍然毫不犹豫地使用“美”这个词的群体。文学、艺术或音乐评论家很少使用它，他们可能担心它听起来肤浅或俗气。】
“I'm very proud that in mathematics and science the concept of beauty is still there. ?I think it's an incredibly important concept in our lives," says Dijkgraaf. “These sense of beauty we experience in math and science is a multidimensional sense of beauty. We don't feel it's in any conflict with being deep, or interesting, or powerful, or meaningful. For the mathematician, it's all captured by that one word.
【“我很自豪，在数学和科学中，美的概念仍然存在。我认为这是我们生活中一个非常重要的概念，”迪克格拉夫说。“我们在数学和科学中感受到的美感是一种多维度的美感。我们不觉得它与深度、有趣、强大或有意义有任何冲突。对于数学家来说，这一切都包含在一个词里。】
思考题：Mathematics is known to be a magical tool in solving many of the mysteries in the world, but the author argues that there is beauty in mathematics in addition to its utility. Paul Dirac even advised people to go for the beauty in their equations over experimental verification. Do you think the beauty of mathematics is important? Please summarize the beauty described by Paul Dirac and make explanation on your opinion.
【众所周知，数学是解决世界上许多谜团的神奇工具，但作者认为，数学除了实用性之外，还有它的美。保罗·狄拉克(Paul Dirac)甚至建议人们追求方程式的美，而不是实验验证。你认为数学之美很重要吗?请总结保罗·狄拉克所描述的美，并解释你的观点。】
In my opinion, the beauty of mathematics is indeed important. It goes beyond mere utility and serves as a guidepost for discovering profound truths about the universe. Beautiful mathematical concepts often lead to elegant solutions, offering insights that may transcend immediate applications. The pursuit of beauty in mathematics fosters creativity and a deeper understanding of the interconnectedness of various mathematical ideas.
【在我看来，数学的美丽确实很重要。它超越了纯粹的实用性，是发现关于宇宙的深刻真理的指南。美丽的数学概念常常导致优雅的解决方案，提供超越即时应用的见解。在数学之美的追求中，培养创造力，深化对各种数学思想相互联系的理解。】