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What is at Stake in Mathematical Proofs from Third-Century China?

Published online by Cambridge University Press:  26 September 2008

Karine Chemla
Karine Chemla
Affiliation:
REHSEIS (CNRS, Paris, and Universite Paris VII)
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Abstract

To highlight speculative trends specific to the mathematical tradition that developed in China, the paper analyzes an excerpt of a third-century commentary on a mathematical classic, which arguably contains a proof. The paper shows that the following three tasks cannot be dissociated one from the other: (1) to discuss how the ancient text should be read; (2) to describe the practice of mathematical proof to which this text bears witness; (3) to bring to light connections between philosophy and mathematics that it demonstrates were established in China. To this end the paper defines its use of the word “proof” and outlines a program for an international history of mathematical proof. It describes the sense in which the text conveys a proof and shows how it simultaneously fulfills algorithmic ends, bringing to light a formal pattern that appears to be fundamental both for mathematics and for other domains of reality. The interest in transformations that mathematical writings demonstrate in China at that time seems to have been influenced by philosophical developments based on The Book of Changes (Yi-jing), which the excerpt quotes. This quotation within a mathematical context makes it possible to suggest an interpretation for a rather difficult philosophical statement.

Type
Article
Information
Science in Context , Volume 10 , Issue 2 , Summer 1997 , pp. 227 - 251
Copyright
Copyright © Cambridge University Press 1997

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References

Chemla, Karine. 1991. “Theoretical Aspects of the Chinese Algorithmic Tradition (First to Third Century).” Historia Scientiarum 42:75–98 [+ errata in the next issue].Google Scholar
Chemla, Karine. 1992a. “Les fractions comme modéle formel en Chine ancienne.” In Histoire defractions, fractions d'histoire, edited by Benoit, P., Chemla, K., and Ritter, J., 188–207. Birkhäuser.Google Scholar
Chemla, Karine. 1992b. “Résonances entre démonstration et procédure: Remarques sur le commentaire de Liu Hui (3e siècle) aux Neuf chapitres sur les procédures mathématiques,” in Regards obliques sur l'argumentation en Chine, Extrême-Orient, Extrême-Occident 14: 91–129, edited by Chemla, Karine, vol. 14, 91–129.Google Scholar
Chemla, Karine. 1994a. “Nombre et opération, chaine et trame du réel mathématique. Re-marques sur le commentaire de Liu Hui aux Neuf chapitres sur les procédures mathématiques” in Sous les nombres, le monde, Extrême-Orient, Extrême-Occident, edited by Volkov, Alexei, vol. 16, 43–70.Google Scholar
Chemla, Karine. 1994b. “Nombres, opérations et équations en divers fonctionnements: Quelques méthodes de comparaison entre des procédures élaborées dans trois mondes différents.” In Nombres, astres, plantes et viscéres: Sept essais sur l'histoire des sciences et des techniques en Asie orientale, edited by Anget, I. Will, P. E., 1–36. Mémoires de l' Institut des Hautes Etudes Chinoises, 35.Google Scholar
Chemla, Karine. 1996a. “Le jeu d'opérations opposées mais complémentaires dans les textes mathématiques chinois anciens. Premières remarques.” In Ceci n'est pas un Festschrift pour Imre Toth, edited by Probst, Siegmund, Chemla, K., Erdély, Agnés, and Moretto, Antonio.Google Scholar
Chemla, Karine. 1996b. “Relations between Procedure and Demonstration: Measuring the Circle in The Nine Chapters on Mathematical Procedures and Their Commen–tary by Liu Hui (3rd Century).” In History of Mathematics and Education: Ideas and Experiences: Proceedings of Conference on the History of Mathematics, Essen, 1992, edited by Jahnke, H. N., Knoche, N., and Otte, M., 69–112. Göttingen: Vandenhoeck & Ruprecht.Google Scholar
Chemla, Karine. 1997. “Croisements entre pensée du changement dans le Yijing et pratiques mathématiques en Chine ancienne.” In En suivant la voie royale, Mélanges en l'honneur de L. Vandermeersch, edited by Gernet, J. and Kalinowski, M., 191–205. Presses de l' Ecole Françhise d'Extrême-Orient.Google Scholar
Chemla, Karine. — Forthcoming. “Philosophical Reflections in Chinese Ancient Mathematical Texts; Liu Hui's reference to the Yijing.” In Proceedings of the Eighth International Conference on the History of Science in East Asia, 1997, edited by Yungsik, Kim.Google Scholar
Chemla, K., and Shuchun, Guo. In preparation. Les neuf chapitres sur les procédures mathématiques ainsi que leurs commentaires. Edition critique et traduction. Paris: Diderot Multimédia Sciences.Google Scholar
Chemla, K., Martin, F., and Pigeot, J., eds. 1995. La valeur de la citation en Chine et au Japan, Extrême-Orient, Extrême-Occident.Google Scholar
Cheng, Anne. 1985. Etude sur le confucianisme Han. L'élaboration d'une tradition exégétique sur les classiques, Mémoires de l' Institut des Hautes Etudes Chi-noises, vol. 21. Collège de France, IHEC.Google Scholar
Fu, Daiwie. 1993. “Problem Domain, Taxonomy, and Comparativity in Histories of Science — with a Case Study in the Comparative History of “Optics”.” In Philosophy and Conceptual History of Science in Taiwan, edited by Lin, Cheng-hun and Fu, Daiwie, 123–48. Boston Studies in the Philosophy of Science, vol. 141. Boston: Kluwer Academic Publishers.CrossRefGoogle Scholar
Fu, Daiwie. 1994. “Imposition of Taxonomy, or the Modern Translation of'Scientific Fragments' from Ancient Chinese Texts into European Languages.” Paper presented at the Colloquium on “Translations from Chinese into European Languages” in E.H.E.S.S., Paris.Google Scholar
Heng, Gao. 1979. Zhouyi dazhuan jinzhu. Qilu shushe.Google Scholar
Graham, Angus. 1993. “The Way and the One in Ho-kuan-tzu.” in Epistemological Issues in Classical Chinese Philosophy, edited by Lenk, Hans and Paul, Gregor, 31–43. Albany: State University of New York Press.Google Scholar
Shuchun, Guo. 1984. “Exploration of the Sources of Liu Hui's Thought” (“Liu Hui sixiang tanyuan”). In Zhongguo zhexueshi yanjiu (Research on the history of philosophy in China), 2:57–62. (In Chinese).Google Scholar
Shuchun, Guo. — Lui Hui, an Authority of Ancient World Mathematics (Gudai shijieshuxue taidou Liu Hui), Shandong kexue jishu chubanshe. (In Chinese).Google Scholar
Juschkewitsch, Adolf Pavlovitch. [1961] 1964. Geschichte der Mathematik im Mittelalter. Leipzig: Teubner. A slightly updated version of the Russian original.Google Scholar
Knoblock, John. 1988. Xunzi: A Translation and Study of the Complete Works. Volume 1, books 1–6. Stanford, Calif.: Stanford University Press.CrossRefGoogle Scholar
Lackner, Michael. 1995. “Citation et éveil: Quelques remarques à propos de l'emploi de la citation chez Zhang Zai.” In Le travail de la citation en Chine et au Japon: Extrême-Orient, Extrême-Occident, vol. 17, edited by Chemla, K., Martin, F., and Pigeot, J., 111–30.Google Scholar
Lackner, Michael. 1996. “La position d'une expression dans un texte: Explorations diagrammatiques de la signification.” In Disposer pour dire, placer pour penser, situer pour agir: Pratiques de la position en Chine, Extrême-Orient, Extrême-Occident, vol. 18, edited by Chemla, K. and Lackner, M., 35–49.Google Scholar
Leblanc, Charles. 1985. Huainantzu: Philosophical Synthesis in Early Han Thought. Hong Kong: Hong Kong University Press.Google Scholar
Jimin, Li. 1990. Research on the Oriental Mathematical Classic The Nine Chapters on Mathematical Procedures and on Its Commentary by Liu Hui (Dongfang shuxue dianji Jiuzhang suanshu ji qi Liu Hui zhu yanjiu), Shaanxi renmin jiaoyu chubanshe.Google Scholar
Yan, Li and Shiran, Du. 1987. Chinese Mathematics: A Concise History. Translated by Crossley, John N. and Lun, Anthony W. C.. Oxford: Oxford Science Publications.Google Scholar
jishi, Lieze. 1979. Collection of Explanations on Liezi, edited by Bojun, Yang. Zhonghua shuju.Google Scholar
Martzloff, Jean-Claude. 1987. Histoire des mathématiques chinoises. Paris: Masson.Google Scholar
Mikami, Yoshio. [1913] 1974. “The Development of Mathematics in China and Japan.” In Abhandlungen zur Geschichte der mathematischen Wissenschaften mil Einschluss ihrer Anwendungen, vol. 30. New York: Chelsea Publishing.Google Scholar
Peterson, Willard J. 1982. “Making Connections: “Commentary on the Attached Verbalizations” of the Book of Change.” Harvard Journal of Asiatic Studies 42:67–116.CrossRefGoogle Scholar
Baocong, Qian. 1963. Suanjing shi shu, (“Ten Classics of Mathematics”). Beijing, Zhonghua shuju, 2 volumes. (In Chinese).Google Scholar
Sivin, Nathan. 1991. “Change and Continuity in Early Cosmology: The Great Commentary to the Book of Changes” Chûgoku kodai kagakushi ron, Kyooto daigaku jinbun kagaku kenkyû jo kenkyû hookoku, 3–43.Google Scholar
Hangsheng, Xu et al. 1989. Weijin xuanxueshi, (History of the Xuanxue during the dynasties Wei and Jin”). Shaanxi shifan daxue chubanshe.Google Scholar