Argument Abstraction is commonly valued as being essential to mathematics or even consubstantial with it. In relation to this belief, mathematical texts from Antiquity-be they from Babylon, Egypt, or China-, which are composed of seemingly concrete problems and algorithms solving them, have been considered to be practice-oriented and deprived of theory. This paper offers an alternative view on both issues. Relying on evidence given by third-century commentaries on The Nine Chapters on Mathematical Procedures, a Chinese treatise composed around the beginning of the Common Era, this paper argues that, in the scholarly mathematical tradition of ancient China, generality was more valued than abstraction. In this respect, problems must be interpreted as paradigms, in the sense grammarians use the term. One of the goals of theoretical endeavor was to exhibit the most general operations possible, and this purpose can be read in quite a few specific features of mathematical practice and concepts. Moreover, it is shown that abstraction is not absent from The Nine Chapters, but that it entertains with generality a relationship that requires analysis and can by no means be taken for granted. These ancient sources hence constitute an invitation to develop a critical approach toward the relation of mathematics to abstraction and generality taken separately, as well as the relation of these two values with each other.

First century Chinese, fifth century Indian, and Arabic documents from the 9th century onwards, contain similar tabular procedures to extract square and cube roots on place-value numeration systems. Moreover, an 11th century Chinese astronomer, Jia Xian, as well as al-Samaw'al, a 12th century Arab mathematician, extracted roots of higher order with the so-called Ruffini-Horner procedure. This article attempts to define a textual method to organize this corpus, by distinguishing relevant criteria for identifying similarities and differences from a historical as well as conceptual point of view. The first part analyses three different states of the descriptions of algorithms in China between the 1st and the 11th centuries, all of which exhibit a definite historical stability. The rewriting which allows one to proceed progressively from one state to the next shows a uniformity in the components of the algorithm, which culminates in procedures of the type Ruffini-Horner. Textual criteria demonstrate a greater affinity of certain algorithms, such as those described by Kūshyār ibn Labbān (ca 1000) with Chinese rather than with Indian texts, which are in turn closer to algorithms described by al-Khwārizmī. Criteria of the same kind link the algorithms of Jia Xian and al-Samaw'al on the one hand, and those of Kūshyār and al-Samaw'al on the other.

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