从积分概念的发展看数学分析基本概念的发展(续2)

Evolution of the Concept of Integration:Development of Fundamental Conceptions of Mathematical Analysis(Ⅲ)

全文分四部分概述了积分概念的发展历史,此为第三部分,主要介绍黎曼积分.从傅里叶的影响开始,详细讨论关于函数概念大争论的情况,特别是狄利克雷对于傅里叶定理的证明,及其对于黎曼积分的影响.介绍黎曼是怎样区别于柯西的.也详细介绍魏尔斯特拉斯所领导的分析的算术化对于整个数学的发展的影响.

This paper surveys the development of the integration concept.It consists of four parts.This is the third part,which is a survey of Riemann integral.It starts with the impact of Fourier,introduces in detail the controversy over the conception of functions,especially Dirichlet's proof of Fourier's fundamental theorem and its influence on Riemann's theory,and the distinction of Cauchy's and Riemann's integrals.This part ends with a detailed treatment of the arithmetization by Weierstrass and his disciples,and their deep influence on mathematics as a whole.