《古今数学思想》读后感
看完了《古今数学思想》，从奇迹文库网上下载的电子书，是谁写的谁翻译的，是什么时候哪里出版的，这个电子文件里都没有写，从网上书讯中看到的是美国的莫里斯·克莱因著，张理京、张锦炎、江泽涵译，上海科学技术出版社２００２年７月１日第一版第一次印刷。从内容上看，这本书应该在上个世纪八十年在中国已经有过翻译版本，因为它讨论的数学史到１９５０年就为止了。一共四大本，从考古上的数学发现一直到２０世纪中叶，主要讲的是数学在西方的发展，按照时间顺序把数学的各个科目逐个的细说，援引了大量的原始文献，比如说数学家的书信、论文、著作等；此书涉及到的都是纯粹数学方面的东西，对于应用数学在第一本书里说的篇幅较多了，至于还来出现的概率统计方面的数学就根本没提了；此书除了古印度数学外没有涉及到亚洲更多。这些在网络上已经有大量的书评了。
他讲的不完全是数学，书里也说得明白，限于篇幅只能大概说说某些方面的主要进展，所以即使是把这四本书看完了也只是对数学本身的发展有一个很粗浅的了解，关键的所得是知道当时的人们是怎么想的，这也是我最关心的地方。相比那些累牍的数学知识来说，我关心的是他们怎么想的，怎么就想到这些的，知道了这些之后对于理解数学、创造和发展自己的想法是非常有用的。寻找到数学思想发展的脉络，还可以对人们思想发展的一些规律做到很好的总结。在看这些书的同时我也和周围的朋友经常提到数学，他们大多对这个话题望而却步，或者觉得我说的这些没什么意思，总是他们认为这些优秀的思想是晦涩的离人类很远的不易接受的。嗯，我也曾经对数学抱有这样的想法，当我翻开一本儿数学论文集的时候，简直是立刻就被里面的那些天书般的论述搞得昏头胀脑。现在我了解到了他们是怎么想的之后，就感觉亲切多了，并且也会被他们的精彩的思考论述搞得神经很兴奋。嗯，其实都很容易理解，如果你明白那些概念那些性质是什么，而且知道他们使用的方法是怎么来的怎么用的，那五里雾也就从容的看破了。看破了之后，你得到的就是快乐。有些问题可能对智商都要求不高，嗯，当然如果是高智商的话得到的快乐就会更多一些。
从数学的发展过程能够看出勇气对数学家们的用处是多么的大，它是一部人类争取思想自由的历史，有些类似于宗教中自我的修炼，但很多数学家的拼搏也不是来自于对宗教的信仰，可能是信仰真理，我感觉他们信仰的是人类心智本身。他们的工作很少受到政治的左右，但大多受到了当时学术时尚的影响，而且从来不缺少反抗学术时尚的人，他们可能需要更大的勇气，比如最早提出射影几何的蒙热，最早提出群概念的伽罗华，最早说到超限数的康托，他们的想法在一段时间里没有受到人们的重视，或者受到了学术界的攻击，他们可能从自己的理论中得到了快乐，但现实生活并没有给他们很好的待遇，生活得并不好。我觉得数学是个非常脑淫的科目，而且我认为只要是在思考就不会出现胡闹，当然接受我的这个想法可能也是需要勇气的。
数学家可能不是专业的，十七十八世纪的欧洲数学家都是业余爱好者，他们往往也不是专门搞一种数学，而是与数学有关无关的他们都会拿来思考思考。在十八世纪这种业余精神造就了数学史上一个充满勇士的年代，他们使用微积分而不去考虑微积分的逻辑基础，他们勇往直前虽然有很多工作被后来人认为是粗糙的，但也大多被后来人认为是正确的。数学可以不符合哲学、不符合伦理学、不符合逻辑学，这些对于数学来说都不重要，它追求的是思考的有效和有味道，数学就是数学。我现在认识到的数学是这样一种学问，它算不上科学，虽然它具有科学的诸多性质，它可能是客观存在的，人们在发现它的时候无尽的得到美的享受，它也可以是人为的，人们在构造它们的时候同时也在观察它享受它。如果说纯粹数学是有什么目的或者意义的话，它的确是给人们提供了大量的现成的思考的结论和丰富的思维方式，嗯，但这对于搞数学的人来说并不是主要的，只能算是个数学的副产物。我想它更像是语文、艺术，是个人修养的一部分，是在人们自己呆着无聊的时候，即使一无所有也能够得到极大快乐的一种本领。欣赏别人的数学和拥有自己的数学都是快乐的，这和下围棋是同样的道理。
人们被现代数学吓到了，不管现代数学对于现代人们的生活是多么的友善和紧密相关，人们看到一件新东西往往会去探讨它的物理原理它的构造和使用技巧，而不去或者说拒绝触摸它们其中的数学，大多认为自己不是专业搞这个的就不必为此操心了。我得说随着社会分工的细化人们越来越缺少业余精神，数学史提醒我的就是业余精神的伟大。当人们的精神专业到了成为机器的程度，那么人生的乐趣也就没有了，也只能看看那些昂贵或者廉价的娱乐影视实际上这明显的也是隔靴搔痒，做做梦。做梦可能是人类被容许的最后的快乐了吧。从这里想到了文学，虽然现在中国识字的人多了，可是能够通过文字的阅读和书写得到乐趣的人仍旧很少，所以我觉得扫盲不扫盲都是一样的，任何时期中国的非文盲和总人口数比例都是恒定的，这与国家在教育上的投入看上去也无关。应用数学是把数学当作工具来研发来使用的，而纯粹数学和纯粹文学一样，它应当是人生的一部分，而不是谁所专有谁所不能有的。你不理解现代数学，你完全可以自己搞自己的数学么，你的勇气到哪里去了？我们说的个性到哪里去了？还是那句话，个性不是懒惰的理由。不是说只有去打虎才是勇者，独立的详细缜密的思考然后去作为，需要的勇气可能是更大的。锻炼么，谁说锻炼只能发生在体育场和健身房里？
很多人也被“数学”这个名字迷惑了，好像必须是和数有关的才叫数学，所以对于现代数学中那些脱离了数字的学问就不能理解了，我现在感觉不管是大自然里存在的还是我们自己设想的，所有对象，都可以放在数学中去研究一番，思考的时候一定会遇到问题的，去解决这许多问题这个动作本身就是数学了。它可能是希尔伯特说的一大堆符号的推演，也有可能是欧氏几何中的图形，我更愿意把数学所处理的对象看成是抽象的东西。比如说自然数，１、２、３……，它实际上是一个有头没尾的序列，你可以想象在地上摆小棍子，一串的摆下去一直到无穷大，这一堆棍子就是数学的基础了，它们有顺序，可能还会有方位，有数量，人们给这些棍子遍用上了符号，于是自然数产生了，然后就是零，没有棍子的时候，然后是负数、整数、分数、有理数、无理数、复数，如果涉及到了方程求解未知数，那就有了代数数和超越数，还有了矩阵和微积分，然后从一串棍子变成无穷串棍子，就有了集合，然后就是群、域、环等等概念，然后是度量、角度，出现了几何，当这些棍子不被放在平面上时，又有了球面几何和立体几何，当你通过棍子的摆放发现时间空间和数量之间的关系或者没什么关系的时候，就会出现射影几何和非欧几何，然后是更高的唯度和次数，然后抛弃棍子，去思考或者说观察这些头脑里的新构造是什么样子的时候，你就脱离开了古典数学，进入到了现代数学了。当然还可以通过其他途径，排列组合、绘图、光学、天文学等等，方法太多了，从小说里也可以进入到很有趣的思考中来。当人们脱离开了棍子的束缚，它的思想就不再被压抑了，无拘无束了，那是大快乐。鸟儿通过翅膀在空中飞翔，人类通过数学、流体力学、材料学等等也同样飞到了空中。这可能是对自我对本身的突破，我认为这种突破是追求快乐的人们在有生之年必要做的事情。
追求生之快乐，才会体会到死亡的悲哀。生也无趣，面对死亡也就不会去再多的求生了。当我看到那些数学英雄们的成就时，欧拉、高斯、柯西、庞加莱、牛顿、阿基米德……，我也好像被解放了，或许我不会有他们那样的成就，但这并不妨碍我去体会美妙和快乐。定是要有一定的精力的付出的，坐享其成终归没有真正参与创造那么痛快淋漓。嗯，还有好些想法我还没有琢磨透彻，呵，就先写到这儿吧。
- Re: 銆婂彜浠婃暟瀛︽�濇兂銆嬭鍚庢劅posted on 11/11/2007
拜读。请问，有英原版的e-book吗？ - Re: 銆婂彜浠婃暟瀛︽�濇兂銆嬭鍚庢劅posted on 11/11/2007
啊，没有:(
- Re: 銆婂彜浠婃暟瀛︽�濇兂銆嬭鍚庢劅posted on 11/12/2007
朱老剑客 wrote:
《古今数学思想》读后感
当人们脱离开了棍子的束缚，它的思想就不再被压抑了，无拘无束了，那是大快乐。鸟儿通过翅膀在空中飞翔，人类通过数学、流体力学、材料学等等也同样飞到了空中。这可能是对自我对本身的突破，我认为这种突破是追求快乐的人们在有生之年必要做的事情。
Man starts to live only when he stops inundating himself with eating, working, sleeping and etc, in other words, only when he strives to live outside himself :)) - Re: 銆婂彜浠婃暟瀛︽�濇兂銆嬭鍚庢劅posted on 11/12/2007
朱老，还记得我吗
有趣的是你现在开始关心数学了。：） 有机会细细讨论
对了，橄榄树我再也连不上了。你呢？ - Re: 銆婂彜浠婃暟瀛︽�濇兂銆嬭鍚庢劅posted on 11/12/2007
"Mathematical Thought from Ancient to Modern Times", Morris Kline. - Re: 銆婂彜浠婃暟瀛︽�濇兂銆嬭鍚庢劅posted on 11/12/2007
- posted on 11/12/2007
COPYRIGHT NOTICE:
Edited by Victor J. Katz:
The Mathematics of Egypt, Mesopotamia, China, India, and Islam is published by Princeton University Press and copyrighted, © 2007, by Princeton University Press. All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher, except for reading and browsing via the World Wide Web. Users are not permitted to mount this file on any network servers.
Follow links Class Use and other Permissions. For more information, send email to: permissions@pupress.princeton.edu
Introduction
A century ago, mathematics history began with the Greeks, then skipped a thousand years and continued with developments in the European Renaissance. There was sometimes a brief mention that the “Arabs” preserved Greek knowledge during the dark ages so that it was available
for translation into Latin beginning in the twelfth century, and perhaps even a note that algebra was initially developed in the lands of Islam before being transmitted to Europe. Indian and Chinese mathematics barely rated a footnote.
Since that time, however, we have learned much. First of all, it turned out that the Greeks had predecessors. There was mathematics both in ancient Egypt and in ancient Mesopotamia. Archaeologists discovered original material from these civilizations and deciphered
the ancient texts. In addition, the mathematical ideas stemming from China and India gradually came to the attention of historians. In the nineteenth century, there had been occasional mention of these ideas in fairly obscure sources in the West, and there had even been translations into English or other western languages of certain mathematical texts. But it was only in the late twentieth century that major attempts began to be made to understand the mathematical ideas of these two great civilizations and to try to integrate them into a worldwide history of mathematics. Similarly, the nineteenth century saw numerous translations
of Islamic mathematical sources from the Arabic, primarily into French and German. But it was only in the last half of the twentieth century that historians began to put together these mathematical ideas and attempted to develop an accurate history of the mathematics of Islam, a history beyond the long-known preservation of Greek texts and the algebra of al-Khwarizmi. Yet, even as late as 1972, Morris Kline’s monumental work Mathematical Thought from Ancient to Modern Times contained but 12 pages on Mesopotamia, 9 pages on Egypt, and 17 pages combined on India and the Islamic world (with nothing at all on China) in its total of 1211 pages.
It will be useful, then, to give a brief review of the study of the mathematics of Egypt, Mesopotamia, China, India, and Islam to help put this Sourcebook in context.
To begin with, our most important source on Egyptian mathematics, the Rhind Mathematical Papyrus, was discovered, probably in the ruins of a building in Thebes, in the middle of the nineteenth century and bought in Luxor by Alexander Henry Rhind in 1856. Rhind died in 1863 and his executor sold the papyrus, in two pieces, to the British Museum in 1865. Meanwhile, some fragments from the break turned up in New York, having been acquired also in Luxor by the American dealer Edwin Smith in 1862. These are now in the
2 | Victor Katz
Brooklyn Museum. The first translation of the Rhind Papyrus was into German in 1877. The first English translation, with commentary, was made in 1923 by Thomas Peet of the University of Liverpool. Similarly, the Moscow Mathematical Papyrus was purchased around 1893 by V. S. Golenishchev and acquired about twenty years later by the Moscow Museum of Fine Arts. The first notice of its contents appeared in a brief discussion by B. A. Turaev, conservator
of the Egyptian section of the museum, in 1917. He wrote chiefly about problem 14, the determination of the volume of a frustum of a square pyramid, noting that this showed “the presence in Egyptian mathematics of a problem that is not to be found in Euclid.” The first complete edition of the papyrus was published in 1930 in German by W. W. Struve. The first complete English translation was published by Marshall Clagett in 1999.
Thus, by early in the twentieth century, the basic outlines of Egyptian mathematics were well understood—at least the outlines as they could be inferred from these two papyri. And gradually the knowledge of Egyptian mathematics embedded in these papyri and other sources became part of the global story of mathematics, with one of the earliest discussions being in Otto Neugebauer’s Vorlesungen über Geschichte der antiken Mathematischen Wissenschaften (more usually known as Vorgriechische Mathematik) of 1934, and further discussions and analysis by B. L. Van der Waerden in his Science Awakening of 1954. A more recent survey is by James Ritter in Mathematics Across Cultures.
A similar story can be told about Mesopotamian mathematics. Archaeologists had begun to unearth the clay tablets of Mesopotamia beginning in the middle of the nineteenth century, and it was soon realized that some of the tablets contained mathematical tables or problems. But it was not until 1906 that Hermann Hilprecht, director of the University of Pennsylvania’s excavations in what is now Iraq, published a book discussing tablets containing multiplication and reciprocal tables and reviewed the additional sources that had been published earlier, if without much understanding. In 1907, David Eugene Smith brought some of Hilprecht’s work to the attention of the mathematical world in an article in the Bulletin of the American Mathematical Society, and then incorporated some of these ideas into his 1923 History of Mathematics.
Meanwhile, other archaeologists were adding to Hilprecht’s work and began publishing some of the Mesopotamian mathematical problems. The study of cuneiform mathematics changed dramatically, however, in the late 1920s, when François Thureau-Dangin and Otto Neugebauer independently began systematic programs of deciphering and publishing these tablets. In particular, Neugebauer published two large collections: Mathematische Keilschrift-Texte in 1935–37 and (with Abraham Sachs) Mathematical Cuneiform Texts in 1945. He then summarized his work for the more general mathematical public in his 1951 classic, The Exact Sciences in Antiquity. Van der Waerden’s Science Awakening was also influential in publicizing Mesopotamian mathematics. Jens Høyrup’s survey of the historiography of Mesopotamian mathematics provides further details.
Virtually the first mention of Chinese mathematics in a European language was in several articles in 1852 by Alexander Wylie entitled “Jottings on the Science of the Chinese: Arithmetic,” appearing in the North China Herald, a rather obscure Shanghai journal. However, they were translated in part into German by Karl L. Biernatzki and published in Crelle’s Journal in 1856. Six years later they also appeared in French. It was through these articles
that Westerners learned of what is now called the Chinese Remainder problem and how it was initially solved in fourth-century China, as well as about the ten Chinese classics and the Chinese algebra of the thirteenth century. Thus, by the end of the nineteenth century,
Introduction | 3
European historians of mathematics could write about Chinese mathematics, although, since they did not have access to the original material, their works often contained errors.
The first detailed study of Chinese mathematics written in English by a scholar who could read Chinese was Mathematics in China and Japan, published in 1913 by the Japanese scholar Yoshio Mikami. Thus David Eugene Smith, who co-authored a work solely on Japanese mathematics
with Mikami, could include substantial sections on Chinese mathematics in his own History of 1923. Although other historians contributed some material on China during the first half of the twentieth century, it was not until 1959 that a significant new historical study appeared, volume 3 of Joseph Needham’s Science and Civilization in China, entitled Mathematics and the Sciences of the Heavens and the Earth. One of Needham’s chief collaborators
on this work was Wang Ling, a Chinese researcher who had written a dissertation on the Nine Chapters at Cambridge University. Needham’s work was followed by the section on China in A. P. Yushkevich’s history of medieval mathematics (1961) in Russian, a book that was in turn translated into German in 1964. Since that time, there has been a concerted effort by both Chinese and Western historian of mathematics to make available translations of the major Chinese texts into Western languages.
The knowledge in the West of Indian mathematics occurred much earlier than that of Chinese mathematics, in part because the British ruled much of India from the eighteenth century
on. For example, Henry Thomas Colebrooke collected Sanskrit mathematical and astronomical
texts in the early nineteenth century and published, in 1817, his Algebra with Arithmetic and Mensuration from the Sanscrit of Brahmegupta and Bhascara. Thus parts of the major texts of two of the most important medieval Indian mathematicians were available in English, along with excerpts from Sanskrit commentaries on these works. Then in 1835, Charles Whish published a paper dealing with the fifteenth-century work in Kerala on infinite series, and Ebenezer Burgess in 1860 published a translation of the Sürya-siddhänta, a major early Indian work on mathematical astronomy. Hendrik Kern in 1874 produced an edition of
-
the Aryabhaôïya of Aryabhata, while George Thibaut wrote a detailed essay on the Üulbasütras, which was published, along with his translation of the Baudhäyana Üulbasütra, in the late 1870s. The research on medieval Indian mathematics by Indian researchers around the same time, including Bäpu Deva Sästrï, Sudhäkara Dvivedï, and S. B. Dikshit, although originally published in Sanskrit or Hindi, paved the way for additional translations into English.
Despite the availability of some Sanskrit mathematical texts in English, it still took many years before Indian contributions to the world of mathematics were recognized in major European historical works. Of course, European scholars knew about the Indian origins of the decimal place-value system. But in part because of a tendency in Europe to attribute Indian mathematical ideas to the Greeks and also because of the sometimes exaggerated claims by Indian historians about Indian accomplishments, a balanced treatment of the history of mathematics in India was difficult to achieve. Probably the best of such works was the History of Indian Mathematics: A Source Book, published in two volumes by the Indian mathematicians
Bibhutibhusan Datta and Avadhesh Narayan Singh in 1935 and 1938. In recent years, numerous Indian scholars have produced new Sanskrit editions of ancient texts, some of which have never before been published. And new translations, generally into English, are also being produced regularly, both in India and elsewhere.
As to the mathematics of Islam, from the time of the Renaissance Europeans were aware that algebra was not only an Arabic word, but also essentially an Islamic creation. Most early algebra works in Europe in fact recognized that the first algebra works in that continent were
4 | Victor Katz
translations of the work of al-Khwärizmï and other Islamic authors. There was also some awareness that much of plane and spherical trigonometry could be attributed to Islamic authors. Thus, although the first pure trigonometrical work in Europe, On Triangles by Regiomontanus, written around 1463, did not cite Islamic sources, Gerolamo Cardano noted a century later that much of the material there on spherical trigonometry was taken from the twelfth-century work of the Spanish Islamic scholar Jäbir ibn AflaH.
By the seventeenth century, European mathematics had in many areas reached, and in some areas surpassed, the level of its Greek and Arabic sources. Nevertheless, given the continuous
contact of Europe with Islamic countries, a steady stream of Arabic manuscripts, including mathematical ones, began to arrive in Europe. Leading universities appointed professors
of Arabic, and among the sources they read were mathematical works. For example, the work of Êadr al-æüsï (the son of NaËïr al-Dïn al-æüsï) on the parallel postulate, written originally
in 1298, was published in Rome in 1594 with a Latin title page. This work was studied by John Wallis in England, who then wrote about its ideas as he developed his own thoughts on the postulate. Still later, Newton’s friend, Edmond Halley, translated into Latin Apollonius’s Cutting-off of a Ratio, a work that had been lost in Greek but had been preserved via an Arabic translation.
Yet in the seventeenth and eighteenth centuries, when Islamic contributions to mathematics
may well have helped Europeans develop their own mathematics, most Arabic manuscripts lay unread in libraries around the world. It was not until the mid-nineteenth century that European scholars began an extensive program of translating these mathematical manuscripts.
Among those who produced a large number of translations, the names of Heinrich Suter in Switzerland and Franz Woepcke in France stand out. (Their works have recently been collected and republished by the Institut für Geschichte der arabisch-islamischen Wissenschaften.) In the twentieth century, Soviet historians of mathematics began a major program of translations from the Arabic as well. Until the middle of the twentieth century, however, no one in the West had pulled together these translations to try to give a fuller picture
of Islamic mathematics. Probably the first serious history of Islamic mathematics was a section of the general history of medieval mathematics written in 1961 by A. P. Yushkevich, already mentioned earlier. This section was translated into French in 1976 and published as a separate work, Les mathématiques arabes (VIIIe–XVe siècles). Meanwhile, the translation program
continues, and many new works are translated each year from the Arabic, mostly into English or French.
By the end of the twentieth century, all of these scholarly studies and translations of the mathematics of these various civilizations had an impact on the general history of mathematics.
Virtually all recent general history textbooks contain significant sections on the mathematics
of these five civilizations. As this sourcebook demonstrates, there are many ideas that were developed in these five civilizations that later reappeared elsewhere. The question that then arises is how much effect the mathematics of these civilizations had on what is now world mathematics of the twenty first-century. The answer to this question is very much under debate. We know of many confirmed instances of transmission of mathematical ideas from one of these cultures to Europe or from one of these cultures to another, but there are numerous
instances where, although there is circumstantial evidence of transmission, there is no definitive documentary evidence. Whether such will be found as more translations are made and more documents are uncovered in libraries and other institutions around the world is a question for the future to answer.
Introduction | 5
References
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Burgess, Ebenezer. 1860. “The Süryasiddhänta.” Journal of the American Oriental Society,6.
Clagett, Marshall. 1999. Ancient Egyptian Science: A Source Book. Vol. 3, Ancient Egyptian
Mathematics. Philadelphia: American Philosophical Society.
Colebrooke, Henry Thomas. 1817. Algebra with Arithmetic and Mensuration from the Sanscrit
of Brahmegupta and Bhascara. London: John Murray.
Datta, B., and A. N. Singh., 1935/38. History of Hindu Mathematics: A Source Book. 2 vols.
Bombay: Asia Publishing House.
Hilprecht, Hermann V. 1906. Mathematical, Metrological and Chronological Tablets from the
Temple Library of Nippur Philadelphia: University Museum of Pennsylvania.
Høyrup, Jens. 1996. “Changing Trends in the Historiography of Mesopotamian Mathematics:
An Insider’s View.” History of Science 34: 1–32.
Kern, Hendrik. 1874. The Aryabhatiya, with the Commentary Bhatadipika of Paramadicyara.
Leiden: Brill.
Kline, Morris. 1972. Mathematical Thought from Ancient to Modern Times.New York: Oxford
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Thanks, RZP!
It is time to give the Arabs credit long overdue.
rzp wrote:
COPYRIGHT NOTICE:
Edited by Victor J. Katz:
The Mathematics of Egypt, Mesopotamia, China, India, and Islam - Re: 銆婂彜浠婃暟瀛︽�濇兂銆嬭鍚庢劅posted on 11/14/2007
天！我受不了那么长段儿的英文！
vieplivee wrote:
朱老，还记得我吗
有趣的是你现在开始关心数学了。：） 有机会细细讨论
对了，橄榄树我再也连不上了。你呢？
当然记得了，哈哈，我一直关心数学，我是职业搞核武器的。:P
橄榄树我也上不去，马兰说要重新整一个服务器，可是三焦给她联系好人了，她却没有联系。这段时间马兰也很着急，又要找服务器，嗯，可是我一直没有再和她取得联系。
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