The Nine Chapters on the Mathematical Procedures and Liu Hui’s Mathematical Theory

Abstract

When discussing ancient mathematical theories, scholars often limit themselves to Greek mathematics and, especially to its axiomatic system, which they use as the standard to evaluate traditional mathematics in other cultures: whichever failed to form an axiomatic system is considered to be without theory. Therefore, even those scholars who highly praise the achievements in ancient Chinese mathematics consider that “the greatest deficiency in old Chinese mathematical thought was the absence of the idea of rigorous proofs” and that there is no formal logic in ancient Chinese mathematics; in particular it did not have deductive logic. They further contend that, “in the flight from practice into the realm of pure intellect, Chinese mathematics did not participate,” [5, p. 151] and conclude that Chinese mathematics has no theory.

I think that Liu Hui’s commentary (263 A.D.) to the Nine Chapters on the Mathematical Procedures, hereafter Nine Chapters, completely proved the formulas and solutions in Nine Chapters. It, mainly based on deductive logics, elucidated deep mathematical theories. Even though Nine Chapters itself does not contain mathematical reasoning and proofs, which is a major flaw in the pursuit of mathematical theory in the history of Chinese mathematics, there are certain correct abstract procedures that possess a general applicability which should be considered as mathematical theories in Chinese mathematics. Sir Geoffrey Lloyd, after explaining the difference between Liu Hui’s and Euclid’s mathematics, said “Mais cela ne signifie pas une absence d’intérêt pour la validation des résultats ou pour la recherche d’une systématisation” [3, préface, p. xi]. Based on the Nine Chapters, this article will discuss Liu Hui’s contribution to the mathematical theory in order to stimulate more fruitful discussions.

Needham cited the claim that Chinese mathematics lacks “rigorous proofs,” from his correspondence with the Japanese historian of Chinese mathematics, Yoshio Mikami [三上義天].