Abstract
The following is a preliminary and relatively brief, exploratory discussion of the nature of early Chinese mathematics, especially geometry, considered largely in terms of one specific example: the Pythagorean Theorem, known in its Chinese version as the Gou-Gu theorem. In addition to drawing some fundamental comparisons with western traditions, especially with Greek mathematics, some general observations are also made concerning the character and development of early Chinese mathematics. Above all, why did Chinese mathematics develop as it did, as far as it did, but never in the abstract, axiomatic way that it did in Greece? Many scholars have suggested that answers to these kinds of questions are to be found in social and cultural factors in China. Some favor the sociological approach, emphasizing for example that the Chinese were by nature primarily concerned with practical problems and their solutions, and therefore had no interest in developing a highly theoretical mathematics. Others have stressed philosophical factors, taking another widely-held view that Confucianism placed no value on theoretical knowledge, which in turn worked against the development of abstract mathematics of the Greek sort. While both of these views contain elements of truth, and certainly play a role in understanding why the Chinese did not develop a more abstract, deductive sort of mathematics along Greek lines, a different approach is offered here. To the extent that knowledge is transmitted and recorded in language, oral and written, logical and linguistic factors cannot help but have played a part in accounting for the development of Chinese mathematics.
Joseph W. Dauben spent the spring of 1988 in China as one of nine Americans selected by the U.S. National Academy of Sciences to represent the United States under the terms of a Visiting Exchange Scholar Program co-sponsored with the Academia Sinica of the People’s Republic of China. The material discussed in this lecture is based upon research he conducted while in China, primarily at the Institute for History of Natural Science of the Chinese Academy of Sciences in Beijing. Subsequent revisions were made during a semester spent as a member of the Institute of History, National Tsing-Hua University in Taiwan, during the fall term, 1991, with research supported in part by grants from the National Endowment for the Humanities (Grant No. RH-20958-90), the Research Foundation of the City University of New York (PSC-CUNY Award No. 662089), and the National Science Council of the Republic of China (Grant VP-91006).
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References
See the dated but now one of the earliest studies of Chinese mathematics, Alexander Wylie’s Jottings on the Science of Chinese Arithmetic, Shanghae Almanac and Miscellany for 1853. Shanghai, 1853. See also the article based on a translation of Wylie’s Jottings by K. L. Blernatzki, Die Arithmetik der Chinesen, Journal für die reine und angewandte Mathematik 52 (1856), 59–94. For editions of the Jiu Zhang Suan Shu [Nine Chapters on the Mathematical Art], including Liu Hui’s commentary, see Qian Bao-Cong, Suan Jing Shi Shu [Ten Mathematical Classics]. Beijing, 1963, pp. 81–258; and Guo Shu-Chun, Jiu Zhang Suan Shu [Nine Chapters on the Mathematical Art]. Shen-Yang: Liao-Ning Education Press, 1990.
J. Legge, The Chinese Classics. Oxford, 1893. Vol. I, p. 4.
Liu Hui is also known for another important treatise, the so-called Sea Island Manual (because the first problem is devoted to calculating the height and distance of an island at sea). This “manual„ contains only nine problems, and was originally designed as an appendix to Liu Hui’s edition of the Nine Chapters. See Li Yan and Du Shi-Ran, Chinese Mathematics. A Concise History, trans. J.N. Crossley and A. W.-C. Lun. Oxford: Clarendon Press, 1987, p. 75.
The date of Liu Hui’s edition and commentary on the Jiu Zhang Suan Shu is based on a sentence in the manuscript which mentions that it was written in the fourth year of the Jing Yuan reign of King Chen Liu of Wei, which dates it exactly to 263 A.D. See Li and Du 1987, p. 65.
Joseph Needham, Science and Civilization in China. Vol. 3: Mathematics and the Sciences of the Heavens and the Earth. Cambridge: Cambridge University Press, 1959, p. 235.
L. Gauchet, Note sur la généralisation de l’extraction de la racine carrée chez les anciens auteurs chinois, et quelques problèmes du Chiu Chang Suan Shu, Toung Pao 15 (1914), p. 531.
Recent archaeological excavations of tombs at Zhang Jia Shan in Hu Bei have produced a work written on bamboo strips, the Suan Shu Shu, dating from the first half of the second century, B.C. or earlier. It is in the question-answer form like the Nine Chapters. More than 60 different types of calculation are included, and some problems are very close to ones found in the Jiu Zhang Suan Shu. See Li and Du 1987, p. 56.
For a detailed discussion of this “Xiang Bu„ principle, see Wu Wenchun, The Out-In Complementary Principle, in: Ancient China’s Technology and Science. Beijing: Foreign Languages Press, 1983, pp. 66–89.
Lam Lay-Yong, A Critical Study of the Yang Hui Suan Fa. A Thirteenth-century Chinese Mathematical Treatise. Singapore: Singapore University Press, 1977, p. 83.
Lam 1977, p. 47.
Readers not familiar with these methods but wishing to know more about the application of similarity of triangles and the double difference method should consult the article by Wu Wenchun, The Out-In Complementary Principle, especially the section on “Gnomon, Shadow and Double Differences„, in: Ancient China’s Technology and Science, Beijing: Foreign Languages Press, 1983, pp. 66–89, esp. pp. 68–70.
He Shao-Geng, Method for Determining Segment Areas and Evaluation of π, in: Ancient China’s Technology and Science. Beijing: Foreign Languages Press, 1983, pp. 90–98, esp. 92.
Donald B. Wagner, An Early Chinese Derivation of the Volume of a Pyramid: Liu Hui, Third Century A.D., Historia Mathematica 6 (1979), 164–188, esp. p. 183.
Alfred Bloom, The Linguistic Shaping of Thought. A Study in the Impact of Language on Thinking in China and the West. Hillsadle, New Jersey: Lawrence Erlbaum Associates, 1981, p. 33. Bloom’s findings have been critically reviewed by M. Garrett, Theoretical Buffalo, Conceptual Kangaroos, and Counterfactual Fish, Early China 9–10 (1983–1985); and by Kuang-Ming Wu, Counterfactuals, Universals and Chinese Thinking, Philosophy East and West 37 (1987).
Bloom 1981, p. 36.
Chan Wing-Tsit, A Source Book in Chinese Philosophy. Princeton: Princeton University Press, 1963. See especially Chapter 10, “Debates on Metaphysical Concepts: the Logicians„, p. 232.
Bloom 1981, p. 48.
Bloom 1981, p. 53. Bloom’s views have been questioned, in particular by Terry Kit-Fong Au, Counter/actuals: In reply to Alfred Bloom, Cognition 17 (1984), 289–302; and by Lisa Garbern Liu, Reasoning Counter/actually in Chinese: Are there any Obstacles, Cognition 21 (1985), 239–270; see also the article by Halvor Eifring, The Chinese Counterfactual, Journal of Chinese Linguistics 16 (1988), 193–217.
For a detailed discussion of the circumstances of the context of Pythagorean mathematics and the unexpected discovery of incommensurable magnitudes, see Joseph W. Dauben, Conceptual Revolutions and the History of Mathematics. Two studies in the growth of knowledge, in: Transformation and Tradition in the Sciences, ed. E. Mendelsohn. Cambridge, England: Cambridge University Press, 1984, pp. 81–103, especially the section devoted to “The Pythagorean Discovery of Incommensurable Magnitudes„, pp. 84–89.
Scholium to Euclid, Elementa, X, 1 in: Opera Omnia, ed. J. L. Heiberg. Leipzig: Teubner, 1888, p. 417. For other accounts of the drowning episode, see Iamblichus, De Vita Pythagorica Liber, XXXIV, 247, and XVIII, 88, ed. Ludwig Teubner. Leipzig: Teubner, 1937, pp. 132 and 52, respectively, and Iamblichus, De Communi Mathe-matica Scientia Liber, XXV, ed. Nicola Festa. Leipzig: Teubner, 1891, pp. 76–78. Pappus, however, viewed the story of the drowning as a “parable„. See The Commentary of Pappus on Book X of Euclid’s Elements, Book I, sec. 2, eds. G. Junge and W. Thomson. Cambridge, Mass.: Harvard University Press, 1930; reprinted, New York: Johnson Reprint Corp., 1968, p. 64.
Morris Kline, Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1972, p. 50.
Joseph Needham, Science and Civilization in China. Volume 2: History of Scientific Thought. Cambridge: Cambridge University Press, 1956, p. 202. See also Bloom 1981, pp. 57–58.
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Dauben, J.W. (1992). The “Pythagorean theorem„ and Chinese Mathematics Liu Hui’s Commentary on the 勾股 (Gou-Gu) Theorem in Chapter Nine of the Jiu Zhang Suan Shu . In: Demidov, S.S., Rowe, D., Folkerts, M., Scriba, C.J. (eds) Amphora. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8599-7_7
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