内容概述
故事讲述19世纪末的数学成就：从关于正整数的公理化假设出发，不增加其他假设，便可依次得到所有整数、所有有理数和所有实数。因为无理数（如 √2）也属于实数，所以在实数系统的意义上，古老的毕达哥拉斯信念——“万物都依赖于整数”——在今天得到了证明。
中文正文
56° 证明毕达哥拉斯
每个学大学数学的同学，都熟悉19世纪末以前的惊人的数学成就，它们从关于正整数的假设出发，不需要更多的假定，就得到了所有整数的集合，然后得到所有有理数的集合、所有实数的集合。因为无理数（如 √2）属于实数，于是我们看到，至少在实数系统下，古老的毕达哥拉斯信仰在今天得到了证明：万物都依赖于所有的数。
英文正文
56° Pythagoras justified. Every student of college mathematics learns of the remarkable mathematical accomplishment made toward the end of the nineteenth century wherein, by starting from a postulational development of the whole numbers (that is, positive integers) and making no further assumptions, one first obtains the set of all integers, then the set of all rational numbers, and then the set of all real numbers. Since the irrational numbers, like √2, are among the real numbers, we see that, at least so far as the real number system is concerned, the ancient Pythagorean belief that everything depends upon the whole numbers is today justified.
英语学习拓展
常用词汇
postulational音标/pəˈstʌʃənl/
公理化的；基于假设的。
原文用法：by starting from a postulational development of the whole numbers。
development音标/dɪˈveləpmənt/
发展；构建。
原文用法：同上。
assumption音标/əˈsʌmpʃn/
假设。
原文用法：making no further assumptions。
obtain音标/əbˈteɪn/
得到；获得。
原文用法：one first obtains the set of all integers。
rational音标/ˈræʃnəl/
有理的。
原文用法：the set of all rational numbers。
irrational音标/ɪˈræʃnəl/
无理的。
原文用法：the irrational numbers。
concern音标/kənˈsɜːn/
涉及；关于。
原文用法：so far as the real number system is concerned。
justify音标/ˈdʒʌstɪfaɪ/
证明……正当；为……辩护。
原文用法：is today justified。
同义词与近义词
对应核心词汇：development
对应核心词汇：development。
辨析：development 侧重逐步发展；construction 强调主动构建；derivation 强调推导。
对应核心词汇：assumption
对应核心词汇：assumption。
辨析：assumption 指论证中默认的假设；hypothesis 指有待验证的假说；premise 指推理的前提。
对应核心词汇：obtain
对应核心词汇：obtain。
辨析：obtain 是正式用语，指通过推理或努力得到；derive 强调由……推导而来；acquire 强调逐步获得。
对应核心词汇：justify
对应核心词汇：justify。
辨析：justify 指证明合理或正确；vindicate 指洗清嫌疑、证明无罪；validate 指验证有效性。
常用短语与固定搭配
- made toward the end of the nineteenth century：在19世纪末取得。原文用法：the remarkable mathematical accomplishment made toward the end of the nineteenth century。
- starting from：从……出发。原文用法：by starting from a postulational development。
- make no further assumptions：不再做进一步假设。原文用法：making no further assumptions。
- first … then … and then：首先……然后……再然后……。原文用法：one first obtains … then … and then …。
- so far as … is concerned：就……而言。原文用法：so far as the real number system is concerned。
- depend upon：依赖于。原文用法：everything depends upon the whole numbers。
常用句式
- Pattern: Every student of … learns of …：每一个学习……的学生都知道……。原文例句：Every student of college mathematics learns of the remarkable mathematical accomplishment made toward the end of the nineteenth century …
- Pattern: Since …, we see that …：由于……，我们可以看出……。原文例句：Since the irrational numbers, like √2, are among the real numbers, we see that … the ancient Pythagorean belief … is today justified.
- Pattern: By doing A and doing B, one first … then … and then …：通过做 A 和做 B，首先……然后……再然后……。原文例句：by starting from a postulational development … and making no further assumptions, one first obtains … then … and then …
长难句解析
Every student of college mathematics learns of the remarkable mathematical accomplishment made toward the end of the nineteenth century wherein, by starting from a postulational development of the whole numbers (that is, positive integers) and making no further assumptions, one first obtains the set of all integers, then the set of all rational numbers, and then the set of all real numbers.
- 主干：Every student learns of the accomplishment.
- 修饰与从句：
- of college mathematics 为介词短语，修饰 student； - made toward the end of the nineteenth century 为过去分词短语，修饰 accomplishment； - wherein 引导定语从句，修饰 century/accomplishment，表示“在这项成就中”； - by starting from … and making no further assumptions 为方式状语； - that is, positive integers 为插入语，解释 whole numbers。
- 逻辑关系：句子先提出“学生都知道这项成就”，再用 where/in which 说明成就的具体内容——从正整数公理出发，不增加假设，依次得到整数、有理数、实数。
语言使用特点
- 使用过去分词短语和介词短语压缩信息，是典型的学术书面语；
- 用 wherein 连接大背景与具体过程，语气正式；
- 通过 first … then … and then 清晰呈现递进关系；
- 用 so far as … is concerned 限定结论的适用范围，体现学术严谨性。
迁移练习
- 用 first … then … and then 描述你学习某个技能或解决某个问题的步骤。
- 用 so far as … is concerned 造一个关于学科兴趣或个人观点的句子。
- 翻译：从正整数的公理化假设出发，不增加更多假设，便可依次得到整数、有理数和实数。
点击展开参考答案
- （示例）First I reviewed the textbook, then I worked through the examples, and then I tried the practice problems on my own.
- （示例）So far as classical music is concerned, I prefer Mozart to Beethoven.
- By starting from a postulational development of the whole numbers and making no further assumptions, one first obtains the set of all integers, then the set of all rational numbers, and then the set of all real numbers.