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

• Karine Chemla
Chapter

## 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!

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