内容概述
柏拉图深受毕达哥拉斯学派影响，其著作中存在许多关于“数”的神秘表述。《理想国》第八卷的一段神谕式文字困扰了注释者两千多年，直到 1923 年英国数学家格蕾丝·奇泽姆·扬在《伦敦数学会会刊》上发表文章，提出一个惊人结论：柏拉图猜测（甚至可能证明了）唯一同时满足 x²+y²=z² 和 x³+y³+z³=w³ 的互素整数解是 x=3, y=4, z=5, w=6。
中文正文
60° 柏拉图的数命论
柏拉图深受毕达哥拉斯学派的影响，从他本人的数学信仰来看，他的内心似乎是相信数字命运的。在他所有的著作中，再没有什么玄之又玄的词句像《理想国》第八卷那段话更令人疑惑：
神圣的产生物有一个完善的数的周期；而有灭亡的产生物的周期只是一个最小的数——一定的乘法（控制的和被控制的，包括三级四项的），用它通过使有相同单位的有理数相似或不相似，或通过加法或减法，得出一个最后的得数。
其4对3的基本比例，和5结合，再乘3次，产生两个和谐；其中之一是相等因子相乘和100乘同次方结合的产物，另一个是有的相等有的不相等的因子相乘的产物，即，其一或为有理数（各减1）的对角线平方乘100，或为无理数（各减2）平方乘100，另一为3的立方乘100。这全部的几何数乃是优生或劣生的决定性因素。如果你们的护卫者弄错了，在不是生育的好时节里让新郎新娘结了婚，生育的孩子就不会是优秀的或幸运的。
读者也许想试着解读这段奇异的神谕似的句子。不过，就连柏拉图最亲近的追随者们也没能解开它，过了2300年，我们才有一个满意的解释。1923年，20世纪初的英国著名数学家扬（Grace Chisholm Young）在《伦敦数学会纪事》上发表了一篇迷人的文章。我们要读扬女士的原文，才能欣赏她那透彻的分析。她的一个非凡结论说，柏拉图猜想（也可能证明了），同时满足
x² + y² = z² 和 x³ + y³ + z³ = w³
的互素的整数，只有x=3，y=4，z=5，w=6。
29 这段文字有许多译法（“不译”也是一种，例如 John L. Davies 和 David J. Vaughan 的英译本就选择“不译”，“因为我们发现那是不可能的”；他们提供的解释似乎也不令人满意）。即使译出来了，我们也看不明白。这里借用了商务印书馆“汉译世界学术名著丛书”《理想国》的文本（郭斌和、张竹明译），因为它的文字似乎最接近本书引用的英译本。相对说来，牛津大学 Jowett 的英译本似乎更“浅显”一点，没有那么多奇怪的数学文字。
英文正文
60° Some particularly elusive Platonic numerology. Plato was much influenced by the early Pythagoreans and it would seem that, at heart, he was a numerologist so far as his own mathematical beliefs went. There is no profundity in all of his works that has given his commentators more trouble than the following passage from Book VIII of his Republic:
For that which, though created, is divine, a recurring period exists, which is embraced by a perfect number. For that which is human, however, by that one for which it first occurs that the increasings of the dominant and the dominated, when they take three spaces and four boundaries making similar and dissimilar and increasing and decreasing, cause all to appear familiar and expressible.
Whose base, modified, as four to three, then married to five, three times increased, yields two harmonies: one equal multiplied by equal, a hundred times the same: the other equal in length to the former, but oblong, a hundred of the numbers upon expressible diameters of five, each diminished by one, or by two if inexpressible, and a hundred cubes of three. This sum now, a geometrical number, is lord over all these affairs, over better and worse births; and when in ignorance of them, the guardians unite the brides and bridegrooms wrongly, the children will not be well-endowed, either in their constitutions or in their fates.
The reader may care to try to unravel the mathematical basis of this strange and oracular passage. The passage defied analysis even by Plato’s immediate successors, and it was not until some 2300 years later that a satisfying explanation was given. The elucidation appeared in a fascinating article in the Proceedings of the London Mathematical Society of 1925, and was given by Grace Chisholm Young, an eminent British mathematician of the early twentieth century. One must read Mrs. Young’s article to appreciate her penetrating analysis, but one remarkable conclusion reached by Mrs. Young is that Plato guessed, and possibly proved, that the only relatively prime whole numbers x, y, z, w simultaneously satisfying
x² + y² = z² and x³ + y³ + z³ = w³,
are x = 3, y = 4, z = 5, w = 6.
英语学习拓展
常用词汇
elusive音标/ɪˈluːsɪv/
难以捉摸的；难以理解的。
原文用法：Some particularly elusive Platonic numerology。
numerology音标/ˌnjuːməˈrɒlədʒi/
数字命理学；数字神秘主义。
原文用法：同上。
profundity音标/prəˈfʌndəti/
深奥；深刻。
原文用法：There is no profundity in all of his works。
commentator音标/ˈkɒmənteɪtə(r)/
注释者；评论家。
原文用法：has given his commentators more trouble。
oracular音标/ɒˈrækjələ(r)/
神谕的；预言的。
原文用法：this strange and oracular passage。
divine音标/dɪˈvaɪn/
神圣的。
原文用法：though created, is divine。
recurring音标/rɪˈkɜːrɪŋ/
循环的；重复的。
原文用法：a recurring period exists。
embrace音标/ɪmˈbreɪs/
包含；拥抱。
原文用法：which is embraced by a perfect number。
dominant音标/ˈdɒmɪnənt/
支配的；主导的。
原文用法：the increasings of the dominant and the dominated。
boundary音标/ˈbaʊndri/
边界。
原文用法：three spaces and four boundaries。
expressible音标/ɪkˈspresəbl/
可表达的。
原文用法：cause all to appear familiar and expressible。
harmony音标/ˈhɑːməni/
和谐。
原文用法：yields two harmonies。
oblong音标/ˈɒblɒŋ/
长方形的；拉长的。
原文用法：but oblong。
diameter音标/daɪˈæmɪtə(r)/
直径。
原文用法：expressible diameters of five。
inexpressible音标/ˌɪnɪkˈspresəbl/
不可表达的。
原文用法：or by two if inexpressible。
cube音标/kjuːb/
立方。
原文用法：a hundred cubes of three。
well-endowed音标/wel ɪnˈdaʊd/
天赋好的；禀赋优良的。
原文用法：the children will not be well-endowed。
constitution音标/ˌkɒnstɪˈtjuːʃn/
体质；构成。
原文用法：either in their constitutions or in their fates。
unravel音标/ʌnˈrævl/
解开；阐明。
原文用法：try to unravel the mathematical basis。
elucidation音标/ɪˌluːsɪˈdeɪʃn/
阐明；解释。
原文用法：The elucidation appeared in a fascinating article。
eminent音标/ˈemɪnənt/
杰出的。
原文用法：an eminent British mathematician。
penetrating音标/ˈpenɪtreɪtɪŋ/
深刻的；透彻的。
原文用法：her penetrating analysis。
remarkable音标/rɪˈmɑːkəbl/
非凡的；显著的。
原文用法：one remarkable conclusion。
relatively prime音标/ˌrelətɪvli ˈpraɪm/
互素的。
原文用法：the only relatively prime whole numbers。
simultaneously音标/ˌsɪmlˈteɪniəsli/
同时地。
原文用法：simultaneously satisfying。
同义词与近义词
对应核心词汇：elusive
对应核心词汇：elusive。
辨析：elusive 指难以把握；obscure 指晦涩难懂；perplexing 指令人困惑。
对应核心词汇：numerology
对应核心词汇：numerology。
辨析：numerology 指数字命理学；number mysticism 更强调神秘主义。
对应核心词汇：profundity
对应核心词汇：profundity。
辨析：profundity 指深奥的特质；depth 泛指深度。
对应核心词汇：commentator
对应核心词汇：commentator。
辨析：commentator 指注释者；interpreter 指解释者；exegete 指经文注释家。
对应核心词汇：divine
对应核心词汇：divine。
辨析：divine 指神圣的；holy 指宗教上的圣洁；sacred 指庄严神圣的。
对应核心词汇：recurring
对应核心词汇：recurring。
辨析：recurring 指重复出现；cyclical 指循环的；periodic 指周期性的。
对应核心词汇：embrace
对应核心词汇：embrace。
辨析：embrace 本义拥抱，引申为包含；encompass 指全面包含；contain 指内部含有。
对应核心词汇：expressible
对应核心词汇：expressible。
辨析：expressible 指可用语言表达；representable 指可表示；describable 指可描述。
对应核心词汇：unravel
对应核心词汇：unravel。
辨析：unravel 指解开复杂问题；decipher 指破译；explain 泛指解释。
对应核心词汇：elucidation
对应核心词汇：elucidation。
辨析：elucidation 指深入阐明；explanation 泛指解释；clarification 指澄清。
对应核心词汇：eminent
对应核心词汇：eminent。
辨析：eminent 指在某一领域杰出的；distinguished 指卓越的；prominent 指突出的。
对应核心词汇：relatively prime
对应核心词汇：relatively prime。
辨析：二者同义，指最大公约数为 1。
常用短语与固定搭配
- Platonic numerology：柏拉图式数命论。原文用法：Some particularly elusive Platonic numerology。
- early Pythagoreans：早期毕达哥拉斯学派。原文用法：much influenced by the early Pythagoreans。
- at heart：在本质上。原文用法：at heart, he was a numerologist。
- so far as … is concerned：就……而言。原文用法：so far as his own mathematical beliefs went。
- give … more trouble than …：给……带来的麻烦超过……。原文用法：has given his commentators more trouble than the following passage。
- recurring period：循环周期。原文用法：a recurring period exists。
- perfect number：完全数。原文用法：embraced by a perfect number。
- dominant and dominated：主导者与被主导者。原文用法：the increasings of the dominant and the dominated。
- similar and dissimilar：相似的与不相似的。原文用法：making similar and dissimilar。
- increasing and decreasing：增加的与减少的。原文用法：and increasing and decreasing。
- four to three：四比三。原文用法：as four to three。
- married to five：与五结合。原文用法：then married to five。
- three times increased：增加三次。原文用法：three times increased。
- equal multiplied by equal：相等乘以相等。原文用法：one equal multiplied by equal。
- a hundred times the same：相同的一百倍。原文用法：a hundred times the same。
- equal in length to the former：长度与前一个相等。原文用法：the other equal in length to the former。
- in ignorance of：无知于；不了解。原文用法：when in ignorance of them。
- well-endowed：天赋好的。原文用法：the children will not be well-endowed。
- unravel the mathematical basis：解开数学基础。原文用法：unravel the mathematical basis of this strange and oracular passage。
- oracular passage：神谕般的段落。原文用法：同上。
- defied analysis：使分析束手无策。原文用法：The passage defied analysis。
- immediate successors：直接继承人；直接追随者。原文用法：Plato’s immediate successors。
- not until … that：直到……才。原文用法：it was not until some 2300 years later that a satisfying explanation was given。
- satisfying explanation：令人满意的解释。原文用法：同上。
- Proceedings of the London Mathematical Society：《伦敦数学会会刊》。原文用法：in the Proceedings of the London Mathematical Society。
- penetrating analysis：透彻的分析。原文用法：her penetrating analysis。
- relatively prime whole numbers：互素整数。原文用法：the only relatively prime whole numbers。
- simultaneously satisfying：同时满足。原文用法：simultaneously satisfying x² + y² = z² and x³ + y³ + z³ = w³。
常用句式
- Pattern: There is no profundity in all of his works that has given his commentators more trouble than …：他所有著作中的深奥之处，没有比……更让注释者头疼的了。原文例句：There is no profundity in all of his works that has given his commentators more trouble than the following passage from Book VIII of his Republic.
- Pattern: The reader may care to try to …：读者也许想试着……。原文例句：The reader may care to try to unravel the mathematical basis of this strange and oracular passage.
- Pattern: It was not until … that …：直到……才……。原文例句：it was not until some 2300 years later that a satisfying explanation was given.
- Pattern: One must read … to appreciate …：必须阅读……才能欣赏……。原文例句：One must read Mrs. Young’s article to appreciate her penetrating analysis.
- Pattern: One remarkable conclusion reached by … is that …：……得出的一个非凡结论是……。原文例句：one remarkable conclusion reached by Mrs. Young is that Plato guessed, and possibly proved, that …
长难句解析
Whose base, modified, as four to three, then married to five, three times increased, yields two harmonies: one equal multiplied by equal, a hundred times the same: the other equal in length to the former, but oblong, a hundred of the numbers upon expressible diameters of five, each diminished by one, or by two if inexpressible, and a hundred cubes of three.
- 主干：Whose base yields two harmonies.
- 修饰与从句：
- modified, as four to three 为过去分词短语，修饰 base； - then married to five, three times increased 为并列的过去分词短语，继续描述 base 的运算； - 冒号后为 two harmonies 的同位语，分别解释“第一种和谐”和“第二种和谐”； - equal in length to the former, but oblong 为形容词短语，修饰 the other； - upon expressible diameters of five 为介词短语，修饰 numbers； - each diminished by one, or by two if inexpressible 为独立分词结构，补充说明 numbers。
- 逻辑关系：句子描述一个被称为“几何数”的神秘数如何通过比例 4:3、与 5 结合、三次放大等运算产生“两个和谐”。语言充满象征与比喻，逐字翻译困难，体现了柏拉图文本的神秘色彩。
语言使用特点
- 大量使用古典哲学与数学术语，语言高度象征化；
- 排比结构（similar and dissimilar, increasing and decreasing）增强修辞效果；
- 使用 whose base … yields … 这种物主代词 + 名词主语结构，使抽象概念拟人化；
- 句子结构复杂，省略与插入频繁，要求读者耐心拆解。
迁移练习
- 解释 relatively prime 的含义，并判断 8 和 15 是否互素。
- 验证 (3, 4, 5, 6) 是否同时满足 x² + y² = z² 和 x³ + y³ + z³ = w³。
- 用 It was not until … that … 描述一个你或他人经过长期努力才理解某件事的经历。
点击展开参考答案
- Relatively prime means having no common factor other than 1. Since gcd(8, 15) = 1, they are relatively prime.
- 3² + 4² = 9 + 16 = 25 = 5²; 3³ + 4³ + 5³ = 27 + 64 + 125 = 216 = 6³. So (3, 4, 5, 6) satisfies both equations.
- （示例）It was not until I read the original article that I fully understood Plato’s mysterious passage.