Proof in the Wording: Two modalities from Ancient Chinese Algorithms
Résumé
In the mathematical writings that have come down to us from ancient China, proofs did not aim at establishing the truth of theorems but rather the correctness of algorithms. In previous publications, I put forward the idea that the relationship between the text of an algorithm and the proof of its correctness was not as simple as historians of mathematics spontaneously have assumes. Indeed, texts of algorithms very often point out reasons for which algorithms are correct. In this article, I clarify what this statement means, relying on evidence from the Chinese books The Nine Chapters on Mathematical Procedures and Book of Mathematical Procedures. I shed light on two main ways in which the text for an algorithm can indicate reasons why the algorithm is correct. In both cases, commentators seem to read these reasons in the texts and echo them in the shaping of their proofs. First, I discuss a practice that we find also in other scholarly cultures: the transparency of the structure of the list of operations, which allows a sequence of interpretations of the operations and eventually leads to the meaning of the final result. Second, I examine the use of indirect prescriptions: terms for operations those indicate the intentions motivating their use. Regarding texts shaped in this second way and despite a fundamental similarity between the two books, there is a key difference between them, which allows me to suggest a new hypothesis on the history of mathematical proof.
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