Explanation and Proof in Mathematics pp 253-285 | Cite as

# Proof in the Wording: Two Modalities from Ancient Chinese Algorithms

## Abstract

This paper aims at analyzing the ways in which the description of an algorithm can refer to the reasons for its correctness. I rely on ancient Chinese mathematical sources: the *Book of Mathematical Procedures* (*ca*. 186 B.C.E.), *The Nine Chapters on Mathematical Procedures*, (probably first century C.E.), commentaries on the latter by Liu Hui (completed in 263 C.E.) and by a group of scholars working under Li Chunfeng’s supervision (presented to the throne in 656 C.E.). These sources show two fundamental ways in which an algorithm can indicate the reasons for its correctness. First, the algorithm can be decomposed into steps and sequences of steps, the meaning of which can be formulated with respect to the problem by reference to which the algorithm was formulated; second, the algorithm can use indirect speech acts to prescribe the operations to be executed: Instead of directly prescribing the operation(s), it refers to them either by the material effect they will have in the situation (first case) or by a term indicating both their material effect and their formal intention (second case). The latter description goes along with prescribing not one operation but several at a time, since it is their combination that achieves the aim intended and indicated by the term used. Such modes of indirect prescription occur in both ancient books. However, the second case occurs only in *The Nine Chapters* and is abundantly discussed by the commentators. This may indicate an evolution in the modes of approaching the correctness of algorithms between the dates when the two books were composed.

## Keywords

Mathematical Procedure Critical Edition Emphasis Mine Algebraic Proof Material Level## Notes

### Acknowledgments

It is my pleasure to express my deepest gratitude to John Holt, who had the difficult task of taming my English, and to Sarah-Jane Patterson who helped me in a crucial way to implement these changes. Without them, the paper would not be as readable as it has become. Nevertheless, I remain responsible for all remaining shortcomings. My most sincere thanks to Gila Hanna and Niels Jahnke, for their support and their patience in all circumstances!

## References

- Chemla, K. (1991). Theoretical aspects of the chinese algorithmic tradition (First to Third Century).
*Historia Scientiarum, 42,*75-98; Errata in the following issue.Google Scholar - Chemla, K. (1992). Les fractions comme modèle formel en Chine ancienne. In: K. C. Paul Benoit, & J. Ritter (Eds.)
*Histoire de fractions fractions d’histoire*(pp. 189-207, 405, 410). Basel: Birkhäuser.Google Scholar - Chemla, K. (1996). Relations between procedure and demonstration: Measuring the circle in the “Nine chapters on mathematical procedures” and their commentary by Liu Hiu (3rd century). In: H. N. Jahnke, N. Knoche, & M. Otte (Eds.)
*History of Mathematics and Education: Ideas and Experiences*(pp. 69-112). Goettingen: Vandenhoeck & Ruprecht.Google Scholar - Chemla, K. (1997). What is at stake in mathematical proofs from third-century China?
*Science in Context*,*10,*227-251.Google Scholar - Chemla, K. (1997/1998). Fractions and irrationals between algorithm and proof in ancient China.
*Studies in History of Medicine and Science. New Series*,*15,*31-54.Google Scholar - Chemla, K. (2003). Generality above abstraction: the general expressed in terms of the paradigmatic in mathematics in ancient China.
*Science in context, 16*, 413-458.Google Scholar - Chemla, K. (2006). Documenting a process of abstraction in the mathematics of ancient China. In: C. Anderl, & H. Eifring (Eds.)
*Studies in Chinese Language and Culture - Festschrift in Honor of Christoph Harbsmeier on the Occasion of his 60th Birthday*(pp. 169-194). Oslo: Hermes Academic Publishing and Bookshop A/S.Google Scholar - Chemla, K., & Shuchun, G. (2004).
*Les neuf chapitres. Le Classique mathématique de la Chine ancienne et ses commentaires*. Paris: Dunod, pp. 27-39.Google Scholar - Cullen, C. (2004). The Suan shu shu
*筭數書*‘Writings on Reckoning’: A translation of a Chinese mathematical collection of the second century BC, with explanatory commentary. Needham Research Institute Working Papers Vol. 1. Cambridge: Needham Research Institute.Google Scholar - Guo Shuchun郭書春 (1984). 《九章算術》和劉徽注中之率概念及其應用試析 (Analysis of the concept of
*lü*and its uses in*The Nine Chapters on Mathematical Procedures*and Liu Hui’s commentary) (in Chinese).*Kejishi Jikan 科技史集刊 (Journal for the History of Science and Technology)*,*11*, 21-36.Google Scholar - Guo Shuchun郭書春 (1992).
*Gudai shijie shuxue taidou Liu Hui 古代世界數學泰斗劉徽 (Liu Hui, a leading figure of ancient world mathematics)*, 1st edn. Jinan: Shandong kexue jishu chubanshe.Google Scholar - Høyrup, J. (1990). Algebra and naive geometry: An investigation of some basic aspects of Old Babylonian mathematical thought.
*Altorientalische Forschungen*,*17,*27-69, 262-324.Google Scholar - Li Jimin李繼閔 (1982). Zhongguo gudai de fenshu lilun中國古代的分數理論’. In: W. Wenjun (Ed.)
*‘Jiuzhang suanshu’ yu Liu Hui [The Nine Chapters on Mathematical Procedures and Liu Hui]*(pp. 190-209). Beijing: Beijing Shifan Daxue Chubanshe.Google Scholar - Li Jimin李繼閔 (1990).
*Dongfang shuxue dianji Jiuzhang suanshu ji qi Liu Hui zhu yanjiu 東方數學典籍——《九章算術》及其劉徽注研究 (Research on the Oriental mathematical Classic The Nine Chapters on Mathematical Procedures and on its Commentary by Liu Hui)*, 1 Vol. Xi’an: Shaanxi renmin jiaoyu chubanshe.Google Scholar - Li Yan 李儼 (1958).
*Zhongguo shuxue dagang. Xiuding ben 中國數學大綱 (Outline of the history of mathematics in China. Revised edition)*, 2 Vols. Beijing: Kexue chubanshe.Google Scholar - Peng Hao 彭浩 (2001).
*Zhangjiashan hanjian «Suanshushu» zhushi 張家山漢簡《算術書》注釋(Commentary on the Book of Mathematical Procedures, a writing on bamboo slips dating from the Han and discovered at Zhangjiashan)*. Beijing: Science Press (Kexue chubanshe).Google Scholar - Qian Baocong 錢寶琮 (1964).
*Zhongguo shuxue shi 中國數學史 (History of mathematics in China)*. Beijing: Kexue chubanshe.Google Scholar - Rashed, R. (2007).
*Al-Khwarizmi. Le commencement de l’algèbre. Texte établi, traduit et commenté par R. Rashed*. Sciences dans l’histoire. Paris: Librairie scientifique et technique Albert Blanchard.Google Scholar - Wu Wenjun 吳文俊 (Ed.) (1982).
*‘Jiuzhang suanshu’ yu Liu Hui 九章算術與劉徽 [The Nine Chapters on Mathematical Procedures and Liu Hui]*. Beijing: Beijing Shifan Daxue Chubanshe.Google Scholar - Wu Wenjun 吳文俊, Bai Shangshu 白尚恕, Shen Kangshen 沈康身, LI Di 李迪 (Eds.) (1993).
*Liu Hui yanjiu 劉徽研究 (Research on Liu Hui)*. Xi’an: Shaanxi renmin jiaoyu chubanshe, Jiuzhang chubanshe.Google Scholar