Concepts of infinity in Chinese mathematics.

摘要

This dissertation investigates the concept of infinity in Chinese mathematics by focusing on a series of case studies devoted to specific works or ways in which the concept has appeared and been used throughout the history of mathematics in China. The conventional view holds that ancient Chinese classic texts before 221 BCE introduced mathematical concepts of the infinite. By questioning the validity of employing the Greek concept of pure geometric magnitudes, which has no counterpart in ancient Chinese mathematics, and by analyzing the possible contexts in which these Chinese texts allegedly used concepts of infinity, I conclude that the conventional interpretations are untenable. The second major aspect regarding the appearance of the infinite in ancient Chinese mathematics concerns the phrase "yi mian ming zhi" in the Jiuzhang Suanshu, which has been interpreted as evidence for a Chinese concept of irrational number. My conclusion is that this interpretation of "mian" is also invalid.; The remainder of the thesis is devoted to introductions of Western concepts of infinity to China. I identify the original source for the first Chinese translation (1859) of Books VII through XV of Euclid's Elements, and also discuss how the concept of incommensurability was introduced through an analysis of the translation of certain definitions and propositions from Book X. The introduction of infinitary concepts at the core of the calculus is dealt with primarily through a discussion of the translations of the calculus textbook by Elias Loomis in 1851, and by analyzing the article on "Fluxions" by William Wallace in the Encyclopedia Britannica (1855). The story of the introduction of transfinite set theory and mathematical logic is told mainly through a discussion of Bertrand Russell's lectures on mathematical logic in China and Chinese translations of his two books.; The final analysis of concepts of infinity in modern Chinese mathematics treated in this dissertation focuses on translations of Karl Marx's Mathematical Manuscripts. These were especially significant for debates in China over foundations of the calculus during the Cultural Revolution (1966--1976), and played a central role in influencing the introduction of Abraham Robinson's controversial nonstandard analysis in the 1970s.