温伯格《解释世界》第二章 音乐和数学

即使泰勒斯和他的继任者知道他们需要从他们的物质理论中去得出可验证的结果，他们也几乎无法去实现，其中部分原因在于希腊数学的局限性。巴比伦人那时在算术方面已经取得伟大成就，他们采用60进制而不是10进制。他们也发展了简单的代数，比如解各种二次方程方法（虽然没有用符合表示）。但是早期希腊人的数学很大程度上只是几何。我们前面讲过，柏拉图时代的数学家已经发现了三角形和多面体定律。欧几里德《几何原本》中的许多几何知识在欧几里德时代（约公元前300年）之前已经广为人知。但是在那时希腊只掌握有限的算术，更不用说代数，三角学，或微积分。 最早应用算术方法研究的现象可能是音乐。这是毕达哥拉斯学派所为。毕达哥拉斯是萨摩斯爱奥尼亚岛人，大约在公元前530年移居到意大利南部。他在那里的克罗托内希腊城创立了一个邪教组织，该组织一直延续到公元前300年。 采用“邪教”一词是合适的。早期毕达哥拉斯学派没有留下任何文字记录，但是根据其他作者的故事记述，毕达哥拉斯学派相信灵魂的轮回转世。他们穿着白色长袍，禁止食用豆子，因为豆子与人类胎儿相似。他们组织了一种神权政体，在他们的统治下克罗托内人民在公元前510年摧毁了邻城锡巴里斯。 毕达哥拉斯学派与科学史相关的一面是他们对数学的投入。据亚里士多德《形而上学》记载“他们被称为毕达哥拉斯学派，投身于数学，他们率先引领了数学研究，由于潜心于数学，他们认为数学原则代表万物原则。” 他们重视数学可能源于对音乐的观察。他们注意到在弹奏弦乐器时，如果粗度，成分和弹性一样的两根弦长度之比为小整数，那么奏出的音乐很优美。最简单的情形是一根弦是另一根的一半长。用现代术语，我们说这两根弦相差八度。我们用同样的字母标记他们的声音。如果一根弦是另一根的三分之二长，奏出的两个音符形成“五度音程”，一种非常动听的和弦。如果一根弦是另一根的四分之三长，他们发出的动听的和弦叫“四度音程”。相反，如果两根弦长度之比不是小整数（比如比值为100000/314159），或完全不是整数，那么发出的声音很刺耳，很不动听。我们现在知道这有两个原因，与两根弦一起弹奏时发出的声音周期以及泛音的匹配有关。（见技术说明3）。毕达哥拉斯学派不明白这些道理, 那时也没有其他人明白，这个机理直到17世纪才由法国神父马林·梅森做出解释。相反，据亚里士多德记载，毕达哥拉斯学派则断定“整个世界皆为音阶”。这个观点持续很久。比如西塞罗在《论共和国》里讲述了一个故事，古罗马统帅大西庇阿的幽灵将他的孙子引入到音乐王国。 毕达哥拉斯学派取得的最大进展是在纯数学领域，不是在物理领域。毕达哥拉斯定理众所周知，即直角三角形斜边构成的正方形的面积等于两个直角边构成的正方形的面积之和。没人知道毕达哥拉斯学派中的哪一位证明这个定理，或是如何证明的。基于比例理论可以给出简单的证明，该理论由柏拉图时代毕达哥拉斯学派塔伦通的阿契塔提出。（见技术说明4，欧几里德《几何原本》卷1命题46给出的证明更加复杂）。阿契塔也解决了一个著名的遗留问题：给定一个立方体，应用纯粹的几何方法做另一个立方体，体积刚好是给定立方体的两倍。 毕达哥拉斯定理直接带来了另一项伟大的发现：几何图形中含有不能用整数之比来表示的长度。如果直角三角形两个直角边长为1（不管单位），那么由这两个边长构成的两个正方形面积之和为 12+12=2， 根据毕达哥拉斯定理，斜边长度一定是一个平方为2 的数。很容易证明平方为2 的数无法用整数之比来表示（见技术说明5）。在欧几里德《几何原本》卷X中给出了证明，亚里士多德早先在《前分析篇》中作为反证法的例子提到了这个定理，但没有给出出处。传说这项发现是由生活在意大利南部梅塔蓬图姆的希帕索斯作出，由于泄漏了这个发现，他被毕达哥拉斯学派的人放逐或可能被杀害。 今天我们把像2 的平方根这样的数叫做无理数—他们不能由整数的比值来表示。据柏拉图记载，昔兰尼城的西奥多勒斯证实3，5，6，… 15，17（即除1，4，9，16等本身是其他整数平方以外的所有整数，这点柏拉图没有明说）等等的平方根也同样是无理数。但是早期希腊人不这样表述，而像柏拉图所表述的，面积为2，3，5 等正方形的边长与单位长度“不可通约”。早期希腊人只知道有理数，对他们而言像2的平方根这样的数只具有几何意义，这个局限制约了算术的发展。 柏拉图学院延续了关注纯数学的传统。据说学院入口有个标识：不懂几何者莫入。柏拉图本人不是数学家，但是他对数学很有热情，这其中部分原因可能是由于他在去西西里指导叙拉古城戴奥尼夏二世的旅途中遇到了毕达哥拉斯学派的阿尔库塔斯。学院里一位对柏拉图影响巨大的数学家是雅典的泰阿泰德。他是柏拉图一个对话录中的标题人物，也是另一部的主要对象。人们把五个正多面体的发现归功于泰阿泰德，正如我们前面介绍过的，五实体为柏拉图的元素理论提供了基础。欧几里德《几何原本》给出的这五个实体是仅有的五个凸面实体的证明可能源自泰阿泰德。（注）。他对现今称为无理数的理论也做出了贡献。 （注：事实上（见技术说明2），无论泰阿泰德证明了什么，《几何原本》并没有真正证明它所声称的对仅存五个凸面实体可能性的证明。《几何原本》确实证明了对于正多面体，仅存在五种多面体每个面的边数和在每个顶点交汇的面数的组合。但是它并没有证明对于这些数的每种组合只有一种凸面的可能。） 公元前四世纪最伟大的古希腊数学家可能是尼多斯的欧多克索斯，他是阿尔希塔斯的学生，与柏拉图是同时代人。欧多克索斯一生大部分时间居住于小亚细亚港口城市尼多斯，他曾经在柏拉图学院学习，后来又返回学院任教。欧多克索斯没有著作流传下来，但是人们认为他解决了许多数学领域的难题。比如证明同底同高的圆锥体体积是圆柱体的三分之一。（我不知道欧多克索斯不用微积分如何做出证明）。他对数学最大的贡献是引入了严谨风格，定理可以从清晰表述的公理中推导出来。后来欧几里德著作中采用的就是这种风格。事实上，欧几里德《几何原本》中的许多细节都源于欧多克索斯。 欧多克索斯和毕达哥拉斯学派在数学方面取得了伟大成就，但是他们对自然科学带来的影响好坏参半。其一是欧几里德的《几何原本》采用的推理式写作风格被自然科学工作者无止尽地模仿，有时非常不合适。我们会看到亚里士多德关于自然科学的著作很少涉及数学，但是有时会出现对数学推理拙劣的模仿。比如在他的《物理学》一书中对运动的讨论：“然后A将用C时间穿过B, 用E时间穿过更稀薄的D （若B和D距离相等），时间正比于障碍物的密度。设想B为水，D为空气。”希腊最伟大的物理著作可能是阿基米德的《论浮体》。在第四章我们将详细介绍。该书写的像数学教科书，先有不容置疑的假设，然后是推导出的命题。阿基米德足够聪明，他会选择正确的假设，但是现在科学研究需要的是推理，归纳和猜想。 比这种写作风格更为严重的问题（虽然与此有关）是人们受数学研究而产生的一个错误理解：即通过独立思维可以得出真相。在《理想国》讨论哲学家国王教育时，柏拉图描写苏格拉底争辩说天文学研究应该像研究几何一样。按苏格拉底的意思，仰望天空可能会激发思维，正如观察几何图形可能有助于数学一样，但是这两种情形下获得的知识都来源于纯粹思维。在《理想国》中苏格拉底解释道：“我们应该把天体仅仅作为图形来帮助我们研究其他领域，就像我们面对特殊几何图形一些。” 数学是我们进行物理理论推导的工具。不只如此，它还是一种无可替代的表达物理科学原理的语言。它经常启发出新的认知自然科学的观点。同时，科学的需求也推动了数学的发展。理论物理学家爱德华·威滕的研究工作加深了人们对数学的认识，因此他在1990年被授予数学界最高大奖—菲尔兹奖。但是数学不是自然科学。不结合观察，数学本身不能告诉我们对于世界的任何认知。数学定理也不可能通过观察外部世界得以确定或否定。 这点在古代并不明确，即使在现代早期也是如此。我们前面看到柏拉图和毕达哥拉斯把数学或三角这样的数学概念作为自然界的基本物质组成，我们也会看到一些哲学家认为数理天文学是数学的分枝，而不属于自然科学。 数学与天文学的界限现在已经很明确。我们对数学仍然感到神秘，为什么一些与自然世界毫无关系的数学进展却常常在物理理论中得以实用。物理学家尤金·维格纳在一篇著名的文章中写道：“数学不合情理的有效性”。但是通常我们还是可以很容易区分数学观念与科学原理—科学原理可以通过对现实世界的观察得以最终证实。 有时当代数学家和物理学家会持有不同意见，这通常发生在有关数学严谨性方面。从19世纪早期开始，纯数学研究人员认为严谨至关重要，定义与假设必须准确，推论必须在绝对确定下做出。物理学家更加机会主义，只要求足够的准确，不会导致严重的错误即可。在我的有关量子场论专著前言中，我承认“这本书的部分内容会让热衷数学的读者落泪。” 这样一来会给我们带来沟通问题。一些数学家告诉我他们常常发觉物理学文献令人恼火地含糊。而像我这样需要高级数学工具的物理学家会发觉由于数学家的追求严谨造成他们的文章太过复杂，让人失去兴趣。 一些热衷于数学的物理学家做出了崇高地努力，力图将现代基本粒子物理模型—量子场论—做的更加严谨，而且他们也已经取得了很大进展。但是上半世纪发展起来的基本粒子标准模型完全没有基于对更高数学严谨性的追求。 欧几里德后的希腊数学持续繁荣。在第四章我们会看到希腊化时代数学家阿基米德和阿波罗的伟大成就。

Even if Thales and his successors had understood that from their theories of matter they needed to derive consequences that could be compared with observation, they would have found the task prohibitively difficult, in part because of the limitations of Greek mathematics. The Babylonians had achieved great competence in arithmetic, using a number system based on 60 rather than 10. They had also developed some simple techniques of algebra, such as rules (though these were not expressed in symbols) for solving various quadratic equations. But for the early Greeks, mathematics was largely geometric. As we have seen, mathematicians by Plato’s time had already discovered theorems about triangles and polyhedrons. Much of the geometry found in Euclid’s Elements was already well known before the time of Euclid, around 300 BC. But even by then the Greeks had only a limited understanding of arithmetic, let alone algebra, trigonometry, or calculus. The phenomenon that was studied earliest using methods of arithmetic may have been music. This was the work of the followers of Pythagoras. A native of the Ionian island of Samos, Pythagoras emigrated to southern Italy around 530 BC. There, in the Greek city of Croton, he founded a cult that lasted until the 300s BC. The word “cult” seems appropriate. The early Pythagoreans left no writings of their own, but according to the stories told by other writers1 the Pythagoreans believed in the transmigration of souls. They are supposed to have worn white robes and forbidden the eating of beans, because that vegetable resembled the human fetus. They organized a kind of theocracy, and under their rule the people of Croton destroyed the neighboring city of Sybaris in 510 BC. What is relevant to the history of science is that the Pythagoreans also developed a passion for mathematics. According to Aristotle’s Metaphysics,2 “the Pythagoreans, as they are called, devoted themselves to mathematics: they were the first to advance this study, and having been brought up in it, they thought its principles were the principles of all things.” Their emphasis on mathematics may have stemmed from an observation about music. They noted that in playing a stringed instrument, if two strings of equal thickness, composition, and tension are plucked at the same time, the sound is pleasant if the lengths of the strings are in a ratio of small whole numbers. In the simplest case, one string is just half the length of the other. In modern terms, we say that the sounds of these two strings are an octave apart, and we label the sounds they produce with the same letter of the alphabet. If one string is two-thirds the length of the other, the two notes produced are said to form a “fifth,” a particularly pleasing chord. If one string is three-fourths the length of the other, they produce a pleasant chord called a “fourth.” By contrast, if the lengths of the two strings are not in a ratio of small whole numbers (for instance if the length of one string is, say, 100,000/314,159 times the length of the other), or not in a ratio of whole numbers at all, then the sound is jarring and unpleasant. We now know that there are two reasons for this, having to do with the periodicity of the sound produced by the two strings played together, and the matching of the overtones produced by each string (see Technical Note 3). None of this was understood by the Pythagoreans, or indeed by anyone else until the work of the French priest Marin Mersenne in the seventeenth century. Instead, the Pythagoreans according to Aristotle judged “the whole heaven to be a musical scale.”3 This idea had a long afterlife. For instance, Cicero, in On the Republic, tells a story in which the ghost of the great Roman general Scipio Africanus introduces his grandson to the music of the spheres. It was in pure mathematics rather than in physics that the Pythagoreans made the greatest progress. Everyone has heard of the Pythagorean theorem, that the area of a square whose edge is the hypotenuse of a right triangle equals the sum of the areas of the two squares whose edges are the other two sides of the triangle. No one knows which if any of the Pythagoreans proved this theorem, or how. It is possible to give a simple proof based on a theory of proportions, a theory due to the Pythagorean Archytas of Tarentum, a contemporary of Plato. (See Technical Note 4. The proof given as Proposition 46 of Book I of Euclid’s Elements is more complicated.) Archytas also solved a famous outstanding problem: given a cube, use purely geometric methods to construct another cube of precisely twice the volume. The Pythagorean theorem led directly to another great discovery: geometric constructions can involve lengths that cannot be expressed as ratios of whole numbers. If the two sides of a right triangle adjacent to the right angle each have a length (in some units of measurement) equal to 1, then the total area of the two squares with these edges is 12 + 12 = 2, so according to the Pythagorean theorem the length of the hypotenuse must be a number whose square is 2. But it is easy to show that a number whose square is 2 cannot be expressed as a ratio of whole numbers. (See Technical Note 5.) The proof is given in Book X of Euclid’s Elements, and mentioned earlier by Aristotle in his Prior Analytics4 as an example of a reductio ad impossibile, but without giving the original source. There is a legend that this discovery is due to the Pythagorean Hippasus, possibly of Metapontum in southern Italy, and that he was exiled or murdered by the Pythagoreans for revealing it. We might today describe this as the discovery that numbers like the square root of 2 are irrational— they cannot be expressed as ratios of whole numbers. According to Plato,5 it was shown by Theodorus of Cyrene that the square roots of 3, 5, 6, . . . , 15, 17, etc. (that is, though Plato does not say so, the square roots of all the whole numbers other than the numbers 1, 4, 9, 16, etc., that are the squares of whole numbers) are irrational in the same sense. But the early Greeks would not have expressed it this way. Rather, as the translation of Plato has it, the sides of squares whose areas are 2, 3, 5, etc., square feet are “incommensurate” with a single foot. The early Greeks had no conception of any but rational numbers, so for them quantities like the square root of 2 could be given only a geometric significance, and this constraint further impeded the development of arithmetic. The tradition of concern with pure mathematics was continued in Plato’s Academy. Supposedly there was a sign over its entrance, saying that no one should enter who was ignorant of geometry. Plato himself was no mathematician, but he was enthusiastic about mathematics, perhaps in part because, during the journey to Sicily to tutor Dionysius the Younger of Syracuse, he had met the Pythagorean Archytas. One of the mathematicians at the Academy who had a great influence on Plato was Theaetetus of Athens, who was the title character of one of Plato’s dialogues and the subject of another. Theaetetus is credited with the discovery of the five regular solids that, as we have seen, provided a basis for Plato’s theory of the elements. The proof* offered in Euclid’s Elements that these are the only possible convex regular solids may be due to Theaetetus, and Theaetetus also contributed to the theory of what are today called irrational numbers. The greatest Hellenic mathematician of the fourth century BC was probably Eudoxus of Cnidus, a pupil of Archytas and a contemporary of Plato. Though resident much of his life in the city of Cnidus on the coast of Asia Minor, Eudoxus was a student at Plato’s Academy, and returned later to teach there. No writings of Eudoxus survive, but he is credited with solving a great number of difficult mathematical problems, such as showing that the volume of a cone is one-third the volume of the cylinder with the same base and height. (I have no idea how Eudoxus could have done this without calculus.) But his greatest contribution to mathematics was the introduction of a rigorous style, in which theorems are deduced from clearly stated axioms. It is this style that we find later in the writings of Euclid. Indeed, many of the details in Euclid’s Elements have been attributed to Eudoxus. Though a great intellectual achievement in itself, the development of mathematics by Eudoxus and the Pythagoreans was a mixed blessing for natural science. For one thing, the deductive style of mathematical writing, enshrined in Euclid’s Elements, was endlessly imitated by workers in natural science, where it is not so appropriate. As we will see, Aristotle’s writing on natural science involves little mathematics, but at times it sounds like a parody of mathematical reasoning, as in his discussion of motion in Physics: “A, then, will move through B in a time C, and through D, which is thinner, in time E (if the length of B is equal to D), in proportion to the density of the hindering body. For let B be water and D be air.”6 Perhaps the greatest work of Greek physics is On Floating Bodies by Archimedes, to be discussed in Chapter 4. This book is written like a mathematics text, with unquestioned postulates followed by deduced propositions. Archimedes was smart enough to choose the right postulates, but scientific research is more honestly reported as a tangle of deduction, induction, and guesswork. More important than the question of style, though related to it, is a false goal inspired by mathematics: to reach certain truth by the unaided intellect. In his discussion of the education of philosopher kings in the Republic, Plato has Socrates argue that astronomy should be done in the same way as geometry. According to Socrates, looking at the sky may be helpful as a spur to the intellect, in the same way that looking at a geometric diagram may be helpful in mathematics, but in both cases real knowledge comes solely through thought. Socrates explains in the Republic that “we should use the heavenly bodies merely as illustrations to help us study the other realm, as we would if we were faced with exceptional geometric figures.”7 Mathematics is the means by which we deduce the consequences of physical principles. More than that, it is the indispensable language in which the principles of physical science are expressed. It often inspires new ideas about the natural sciences, and in turn the needs of science often drive developments in mathematics. The work of a theoretical physicist, Edward Witten, has provided so much insight into mathematics that in 1990 he was awarded one of the highest awards in mathematics, the Fields Medal. But mathematics is not a natural science. Mathematics in itself, without observation, cannot tell us anything about the world. And mathematical theorems can be neither verified nor refuted by observation of the world. This was not clear in the ancient world, nor indeed even in early modern times. We have seen that Plato and the Pythagoreans considered mathematical objects such as numbers or triangles to be the fundamental constituents of nature, and we shall see that some philosophers regarded mathematical astronomy as a branch of mathematics, not of natural science. The distinction between mathematics and science is pretty well settled. It remains mysterious to us why mathematics that is invented for reasons having nothing to do with nature often turns out to be useful in physical theories. In a famous article,8 the physicist Eugene Wigner has written of “the unreasonable effectiveness of mathematics.” But we generally have no trouble in distinguishing the ideas of mathematics from principles of science, principles that are ultimately justified by observation of the world. Where conflicts now sometimes arise between mathematicians and scientists, it is generally over the issue of mathematical rigor. Since the early nineteenth century, researchers in pure mathematics have regarded rigor as essential; definitions and assumptions must be precise, and deductions must follow with absolute certainty. Physicists are more opportunistic, demanding only enough precision and certainty to give them a good chance of avoiding serious mistakes. In the preface of my own treatise on the quantum theory of fields, I admit that “there are parts of this book that will bring tears to the eyes of the mathematically inclined reader.” This leads to problems in communication. Mathematicians have told me that they often find the literature of physics infuriatingly vague. Physicists like myself who need advanced mathematical tools often find that the mathematicians’ search for rigor makes their writings complicated in ways that are of little physical interest. There has been a noble effort by mathematically inclined physicists to put the formalism of modern elementary particle physics—the quantum theory of fields—on a mathematically rigorous basis, and some interesting progress has been made. But nothing in the development over the past half century of the Standard Model of elementary particles has depended on reaching a higher level of mathematical rigor. Greek mathematics continued to thrive after Euclid. In Chapter 4 we will come to the great achievements of the later Hellenistic mathematicians Archimedes and Apollonius.