[2013.06.15]纤云无语探苍穹——用方程式描述的数学故事

2013-6-16 09:34| 发布者: migmig| 查看: 78988| 评论: 21|原作者: 悠悠万事97

摘要: 用方程式描述的数学故事

The Universe in Zero Words:
The Story of Mathematics as Told Through Equations
By Dana Mackenzie
没有文字的世界
——用方程式描述的数学故事

达纳•麦肯齐著

One afternoon in Rio de Janeiro, the Nobel Prize-winning physist Richard Feynman was eating dinner in his favorite restaurant. It wasn’t actually dinnertime yet, so the dining room was quiet … until the abacus salesman walked in. The waiters, who were presumably not interested in buying an abacus, challenged the salesman to prove that he could do arithmetic faster than their customer. Feynman agreed to the challenge.

里约热内卢的一个下午，诺贝尔奖金得主、物理学家理查德•费曼[sup]1[/sup] 正在他喜欢的一家餐馆里用餐。其实这还不到吃晚饭的时间，所以餐厅里静悄悄的……但当一位算盘推销员走进来之后，一切就都不同了。侍应生们应该对买算盘没啥兴趣，但他们向推销员起哄，要他证明，他做算术题能比他们的一位顾客更快。费曼同意进行这一挑战。

At first, the contest wasn’t even close. On the addition problems, Feynman wrote, the abacus salesman “beat me hollow.” He would have the answer before Feynman even finished writing down the numbers. But then the salesman started getting cocky. He challenged Feynman to multiplication problems. Feynman still lost to the abacus, but not by as much. The salesman, not satisfied with his narrow margin of victory, challenged Feynman to harder and harder problems, and got more and more flustered. Finally he played his trump card. “Raios cubicos!” the salesman said. “Cube roots!”

开始时比赛完全一边倒。做加法时，费曼用笔算，算盘推销员把他打得“落花流水”。还不等费曼把数字写完，推销员就已经报出了答案。但接着推销员就趾高气扬起来了。他提出要跟费曼比赛乘法。这一盘费曼依旧败北，但输得没有第一次惨。推销员对自己没有大获全胜不满意，又不断地在越来越难的问题上向费曼挑战，但他的优势却越来越小，人也变得越来越慌张了。最后他祭出了杀手锏：“立方根！”推销员说。

Obviously, by this point the competition was more about pride than about selling an abacus. It’s difficult to imagine why a restaurant manager would ever need to compute a cube root. But Feynman agreed, provided that the waiters, who were watching the competition and enjoying it immensely, would choose the number. The number they picked was 1729.03.

显然，到了这一步，竞赛跟出卖算盘已经没多大关系了，更重要的是荣誉之争。很难想象一家餐馆的经理为什么会有一天需要计算立方根。但费曼同意了，条件是让兴致盎然地在周围观战的侍应生出题。他们选定了1729.03这个数字。

The abacist set to work with a passion, hunching over the abacus, his fingers flying too fast for the eye to follow. Meanwhile, Feynman writes, he was just sitting there. The waiters asked him what he was doing, and he tapped his head: “Thinking!” Within a few seconds, Feynman had written down five digits of the answer (12.002). After a while, the abacus salesman triumphantly announced “12!” and then a few minutes later, “12.0!” By this time Feynman had added several more digits to his answer. The waiters laughed at the salesman, who left in humiliation, beaten by the power of pure thought.

算盘高手热情洋溢地投入了工作。他伏在算盘上运指如飞，让观战者目不暇给。费曼写道: 与此同时，他却坐在那里一动也不动。侍应生们问他在干什么，他点了点自己的脑袋说：“思考！”几秒钟之内费曼就写下了五位数的答案（12.002）。过了一会儿，算盘推销员得意洋洋地喊出了“12”！几分钟后他又报出了“12.0”！但到这时，费曼的答案上已经又多出了几位数字。侍应生们嘲笑那位推销员。他在纯粹的思考面前惨遭败绩，铩羽而去。

Like all good tales, Feynman’s duel with the abacist has many layers of meaning. On the most superficial level, it is a story about genius; the Nobel Prize winner beating the machine. However, Feynman’s intention when he told this story about himself was quite different. He was not a boastful man. In the context of his book, the point of the story was that ordinary people—not Nobel Prize winners, not geniuses—could do just the same thing as he did, with a little bit of number sense and mathematical knowledge. There were two secrets behind his seemingly magical feat. First, he needed to know that 1728 was a perfect cube: 123 = 1728 (not common knowledge, perhaps, but it’s something most physicists would be aware of, because a cubic foot is 123 or 1728 cubic inches.) And he needed to know a famous equation from calculus, called Taylor’s formula—a very general approximation method that allows you to go from the exact equation:
1728[sup]1/3[/sup] = 12
to the approximate equation: 1729.03[sup]1/3[/sup] ≈ 12.002

这是一个很好的故事。一切好的故事都含有多层意义，费曼与算盘高手对决的这一故事也不例外。从最表面的意义上说，这是一个关于天才的故事；诺贝尔奖金得主击败了机器。然而，费曼在讲述这个有关自己的故事时有着与此大不相同的目的。他不是一个喜欢自夸的人。从他书中讲述的前因后果中可以看出，这个故事要说明的是：对数字有一定感觉、有一定数学知识的普通人也能跟他做得一样好。这些人用不着染指诺贝尔奖金，用不着是天才。他的技巧看上去如同魔法，但后面隐藏着两个秘密。首先，他需要知道1728是一个完全立方数：12[sup]3[/sup] = 1728（或许这并不是人人都知道的常识，但大部分学物理的人都会知道，因为1立方英尺是12[sup]3[/sup] 或者说1728立方英寸[sup]2[/sup]。）。而且他需要知道微积分中一个叫做泰勒公式的著名等式；这是一个非常普适的近似方法，可以让人通过已有的准确等式得到近似式，即从

1728[sup]1/3[/sup] = 12
得到 1729.03[sup]1/3[/sup] ≈ 12.002

Equations are the lifeblood of mathematics and science. They are the brush strokes that mathematicians use to create their art, or the secret code that they use to express their ideas about the universe. That is not to say that equations are the only tool that mathematicians use; words and diagrams are important, too. Nevertheless, when push comes to shove—for instance, when they have to compute the cube root of 1729.03—equations convey information with an economy and precision that words or abaci can never match.

公式是数学与科学的命脉。它们是数学家用来建造自己的艺术殿堂的一砖一石，或者说是他们用来表达他们有关宇宙的想法的密码。这并不是说，公式是数学家使用的唯一工具；语言与图表也很重要。但无论如何，在他们必须应付紧急情况时，例如在必须计算1729.03的立方根时，公式就能向他们传达简捷而又准确的信息，这是语言或者算盘永远无法比拟的。

The rest of the world, outside of science, does not speak the language of equations, and thus a vast cultural gap has emerged between those who understand them and those who do not. This book is an attempt to build a bridge across that chasm. It is intended for the reader who would like to understand mathematics on its own terms, and who would like to appreciate mathematics as an art. Surely we would not attempt to discuss the works of Rembrandt or Van Gogh without actually looking at their paintings. Why, then, should we talk about Isaac Newton or Albert Einstein without exhibiting their “paintings”? The following chapters will try to explain in words—even if words are feeble and inaccurate—what these equations mean and why they are justly treasured by those who know them.

在科学以外的世界中，人们不使用公式这种语言，因此在理解公式的人和不理解公式的人之间横亘着一条宏大的文化鸿沟。本书是在这一鸿沟上架设桥梁的一次尝试。本书的对象是那些愿意理解数学本身的意义、也愿意把数学作为一种艺术来欣赏的读者。毫无疑问，如果我们试图讨论伦勃朗[sup]3[/sup] 或者凡高[sup]4 [/sup]的作品，我们就必须观看他们的油画。既然如此，在说到艾萨克•牛顿或者阿尔伯特•爱因斯坦时，我们难道能够不去展示他们的“画作”吗？尽管语言贫乏而又不那么准确，但在以下各章中，我还是试图用语言来解释这些公式的意义，以及那些理解它们的人恰如其分地视它们如珍宝的原因。

Let's go back now to Richard Feynman and that abacus salesman, because there is more to say about them. In all likelihood, neither of them knew that they were playing out a scene that had already been enacted centuries before, when Arabic numerals first arrived in Europe.

让我们重新谈起理查德•费曼和算盘推销员吧，因为关于他们还有别的事情要说。非常可能的是，他们都不知道，他们这场竞赛的擂台其实在许多个世纪之前就已经搭起，那正是阿拉伯数字刚刚来到欧洲的时刻。

When the new number system appeared around the beginning of the thirteenth century, many people were deeply suspicious of it. They had to learn nine new and unfamiliar symbols: 1, 2, 3, 4, 5, 6, 7, 8, and 9—or, to be more precise, they had to learn the somewhat distorted thirteenth-century versions thereof. The new symbols looked to some people like occult runes, instead of the nice solid Roman letters (I, V, X, etc.) they were accustomed to. To make things worse, they were Arabic—not even Christian—which made them appear even more suspicious to a deeply religious society. And finally, they included an innovation that was especially hard to grasp: the number zero, a something that meant nothing.

当这一新的数字系统在大约十三世纪初出现的时候，许多人对它颇有疑虑。他们必须学习九个自己不熟悉的新符号：1，2，3，4，5，6，7，8，9；嗯，其实更准确地说，是与我们熟知的那些符号略有不同的十三世纪版本。对于某些人来说，这些新符号看上去不像他们习惯的罗马字母（I、V、X等等）那么好看，那么硬朗，而像是神秘的如尼[sup]5[/sup] 符号。而让事情雪上加霜的是，它们甚至不是基督教世界的产物，而是阿拉伯的泊来品，这就更让一个笃信宗教的社会感到怀疑了。而且，最后，这些符号中还包括了一个更令人难以把握的新玩意，数字零，一个意味着什么都没有的东西。

Nevertheless, Arabic numbers had an undeniable power. Unlike Roman numerals, which were useful for writing numbers but impractical for calculating with them, the decimal place-value system made it possible to do both. In a sense, Arabic numbers democratized mathematics. In many ancient societies, only a specially trained class of scribes could do arithmetic. With decimal notation, you did not need special training or special tools, only your brain and a pen.

尽管如此，阿拉伯数字的力量是无可抗拒的。罗马数字在书写数字时很有用，但用它们计算则实在不敢恭维；而十进位制数字无论写或算都没有问题。从某种意义上说，阿拉伯数字让数学民主化了。在许多古代社会中，只有经过特殊训练的书吏阶层才能演算算术。但有了十进位制之后，人们再也不需要特殊训练或者特殊工具了，只要动脑子，再加上一支笔就成。

The struggle between the old and new number systems went on for a very long time—well over two centuries. And, in fact, open competitions were held between abacists (people who used mechanical tools to do arithmetic) and algorists (people who used the new algorithmic methods). So Feynman and the abacus salesman were re-fighting a very old duel!

新老数字系统之间的对决经历了漫长的岁月——远远超过两个世纪。而且事实上，在算盘高手（使用机械工具做算术的人）和算学大师（使用新算法的人）之间也曾有过多次公开较量。所以，费曼和算盘推销员之间的对撼是一场非常古老的决斗的重演！

WE KNOW HOW the battle ended. Nowadays, everyone in Western society uses decimal numbers. Grade school students learn the algorithms for adding, subtracting, multiplying, and dividing. So clearly, the algorists won. But Feynman’s story shows that the reasons may not be as simple as you think. On some problems, the abacists were undoubtedly faster. Remember that the abacus salesman “beat him hollow” at addition. But the decimal system provides a deeper insight into numbers than a mechanical device does. So the harder the problem, the better the algorist will perform. As science progressed during the Renaissance, mathematicians would need to perform even more sophisticated calculations than cube roots. Thus, the algorists won for two reasons: at the high end, the decimal system was more compatible with advanced mathematics; while at the low end, the decimal system empowered everyone to do arithmetic.

我们知道这场斗争的结局。如今，西方社会中的每一人都在使用十进位制数字。小学生用这种方法学习算术的加减乘除。所以，很明显的是，十进位算法取得了胜利。但费曼的故事告诉我们，背后的原因可能并不像人们想象的那么简单。对于某些问题，使用机械无疑要快些。记得吧，算盘推销员在加法问题上把费曼打得“落花流水”。但与机械装置相比，十进位制启迪人们，让他们对于数字有了更为深邃的洞察力。所以，问题越难，算学大师的表现就越好。当科学在文艺复兴时期发展、进步的时候，数学家就需要进行比求取立方根更为深奥的计算。因此，算学大师获胜的原因有二：其一，从高端来说，十进位制数字与高等数学更为匹配；其二，从低端来说，十进位制数字让人人都能做算术。

But before we start feeling too smug about our “superior” number system, the tale offers several cautionary lessons. First is a message that is far from obvious to most people: There are many different ways to do mathematics. The way you learned in school is only one of numerous possibilities. Especially when we study the history of mathematics, we find that other civilizations used different notations and had different styles of reasoning, and those styles often made very good sense for that society. We should not assume they are “inferior.” An abacus salesman can still beat a Nobel Prize winner at addition and multiplication.

且慢。在开始对自己“优越”的数字系统过分自鸣得意之前，我们还应该注意到，这个故事还给我们上了几堂有关谨慎的课程。首先，有一条对大多数人来说远非明显的信息，就是人们可以用许多不同的方法做数学。特别是在研究数学史时，我们会发现，其他文明的人类使用不同的计数法且有不同的推理方式，而那些方式经常合乎他们的社会的情理。我们不应该认为这些方式“低人一等”。一位算盘推销员照样可以在加法和乘法上击败一位诺贝尔奖金得主。

Feynman’s tale exemplifies also how mathematical cultures have collided many times in the past. Often this collision of cultures has benefited both sides. For instance, the Arabs didn’t invent Arabic numbers or the idea of zero—they borrowed them from India.

费曼的故事也是一个例子，它说明了不同的数学文化在历史上是如何多次发生冲突的。这种文化冲突时常让双方获利。例如，阿拉伯人并没有发明阿拉伯数字或者零这个理念，他们是从印度人那里学来的。

Finally, we should recognize that the victory of the algorists may be only temporary. In the present era, we have a new calculating device; it’s called the computer. Any mathematics educator can see signs that our students’ number sense, the inheritance bequeathed to us by the algorists, is eroding. Students today do not understand numbers as well as they once did. They rely on the computer’s perfection, and they are unable to check its answers in case they type the numbers in wrong. We again find ourselves in a contest between two paradigms, and it is by no means certain how the battle will end. Perhaps our society will decide, as in ancient times, that the average person does not need to understand numbers and that we can entrust this knowledge to an elite caste. If so, the bridge to science and higher mathematics will become closed to many more people than it is today.

最后，我们应该认识到，算学大师的胜利可能只是暂时的。当今之世，我们有了一种叫做计算机的新型计算机器。任何数学教育工作者都能够看到以下迹象：当代的学生正在逐步丢失算学大师为我们留下的遗产——对于数字的感觉。今天的学生们对数字的了解不如过去了。他们依赖于计算机的尽善尽美；万一他们打错了键盘，他们没有能力检查计算机的结果是否正确。现在，我们又一次发现，我们正处于两种观念对抗的年代，而人们还完全不清楚这一战役会以何种结局收场。或许，我们的社会会像古时候一样认为，一般人没有必要了解数字，这种知识可以交由特别的精英人士处理。如果情况果真如此，将有比今天多得多的人发现：对于他们来说，通往科学与高等数学的桥梁将无异于一架不可企及的天梯。

  1. 理查德•费曼（Richard Phillips Feynman，1918 －1988），美国物理学家。1965年诺贝尔物理奖得主。他提出的费曼图、费曼规则和重整化的计算方法是研究量子电动力学和粒子物理学的重要工具。
  2. 英制1英尺 = 12 英寸。1英寸 ≈ 2.54厘米。
  3. 伦勃朗（Rembrandt，1606 – 1669），荷兰画家。
  4. 凡高（Van Gogh，1853 – 1890），荷兰画家。
  5. 如尼符号（runes），古代北欧人使用的字母和文字，西欧人过去有时认为它们带有神秘色彩。