Generality above Abstraction: The General Expressed in Terms of the Paradigmatic in Mathematics in Ancient China

Abstract

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.

Texts, Textual Acts and the History of Science

“…The question on which we chose to focus, i.e., describing the relationship between the text of the algorithms and the corresponding flows of computation, allows us to bypass this range of consideration in a first phase. However, the conclusions reached could cast light on the issue of the function and use of the writings.15 We can prove that this is how the commentator Liu Hui, the earliest reader whose reading of a problem can be observed, interpreted a similar text of algorithm in The Nine Chapters, seeChemla (2003). In the case of this passage from the Mathematical Classic by Zhang Qiujian, this interpretation is confirmed by Li Chunfeng's commentary to be examined below.…”

“…Note also that this type of generality is to be distinguished from the fact that, in contrast to the "detailed procedures," Li Chunfeng's "procedure" is described in abstract terms. The interest for uniformity in the treatment of the various cases in ancient China can also be captured in the way in which the commentator Liu Hui deals with problem 6.18 in The Nine Chapters, see Chemla (2003).…”

Mathematical symbols as epistemic actions

“…However, these rods were less useful to express general abstract rules other than actual calculations which preserved the concreteness of Chinese mathematics. Consequently, Chinese algebra textbooks never attempted to give an abstract formulation of a general rule, but presented examples that served as paradigms to solve similar problems (Chemla, 2003). In contrast, once European mathematicians had invented symbols to represent variables and unknowns from the 16th century onwards, they could provide general solutions to problems.…”

False position in Leonardo of Pisa's Liber Abbaci

“…This preference has been observed, for example, in Babylonian and Chinese mathematics [Høyrup, 2002;Chemla, 2003]. In later European mathematics it became more common, even for writers strongly influenced by Leonardo, to give general descriptions first and then specific examples as applications [Spiesser, 2000, 376].…”

“…But generally speaking, Leonardo does not offer multiple similar examples on the same theme, and each example seems to be intended as a general description of how to solve problems of that type Chemla [2003]. observes a similar approach in ancient Chinese mathematics.…”