Usage of the terms "likewise" and "like" in texts for algorithms. Algorithmic analogies in ancient China (original) (raw)
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Mathematical writings in China relied entirely on the algorithmic mode to express sequences of operations, to justify the correctness of these, and to bring mathematical objects in relation one to another. In this paper, I shall use one example to show how the structural elements in an algorithm convey a mathematical meaning and can be interpreted in the light of the ancient Chinese geometrical tradition. The example stems from an 11th century text by Shen Gua 沈括 and calculates the number of kegs of wine piled up in the form of a truncated pyramid with a rectangular base.
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With starting point in Donald Knuth’s paper “Ancient Babylonian Algorithms”, and using the algebraic reading of pre-Modern mathematical texts as a parallel, the paper discusses the relevance of the algorithm concept, on one hand as an analytical tool for the understanding and comparison of mathematical procedures, on the other as a possible key to how pre-Modern reckoners thought their mathematics and to how they thought about it.
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Many pieces of evidence converge towards the conclusion that generality was the main theoretical value prized by the practitioners of mathematics in ancient China and that it was valued more than abstraction (Chemla 2003). More precisely, these scholars regularly aimed at the greatest generality possible, but did not always achieve or express it through abstract terms. This does not mean, however, that abstraction played no role for them and that they did not find it necessary in some cases to carry out operations of abstraction. In fact, the earliest Chinese mathematical sources handed down through the written tradition, The Gnomon of the Zho1 u (Zho1 ubı4 , probably 1 st century C. E., Cullen 1996), The Nine Chapters on Mathematical Procedures (Jiu3 zha1 ng sua4 nshu4 , below: The Nine Chapters; probably 1 st century C. E.), 1 as well as their commentaries, bear witness to several uses of abstraction. On the other hand, the relationship between generality and abstraction that they evince differs from what can be found in, say, Euclid's Elements. 2 In correlation with this, both the generality
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This essay approaches the knowledge required to write up and use instructions with a specific method. It relies on specific procedures taken from the Chinese canon The Nine Chapters on Mathematical Procedures (九章算術), which, in the author's view, was completed in the first century CE. These procedures enabled readers to do things. To analyse the type of knowledge required to produce these texts of procedures and to use them, the essay puts into play two layers of commentary. The ancient layer was written between the third and the seventh centuries, whereas the later layer was composed between the eleventh and the thirteenth centuries. The author shows that these two layers of commentary read the same text of procedure differently, using different approaches and understanding it differently. The author also shows how the two layers of commentary use mathematical problems to approach a procedure, even though problems are used differently in the two contexts. This illustrates how, i...