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.
The Universe in Zero Words:
The Story of Mathematics as Told Through Equations
By Dana Mackenzie
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!”
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.
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.
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.
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.
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.
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.
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!
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.
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.
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.
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.
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。
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