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十九世纪的实分析:它的前景和回顾

已有 368 次阅读 2019-11-19 20:12 |个人分类:异类微积分|系统分类:观点评述| 蝴蝶模型, 连续性, 无穷小量, “极限”, 标准部分

19TH CENTURY REAL ANALYSIS, FORWARD ANDBACKWARD

JACQUES BAIR, PIOTR B LASZCZYK, PETER HEINIG,VLADIMIR KANOVEI, AND MIKHAIL G. KATZ

【摘要】柯西的工作在很大程度上推动了19世纪的实分析。柯西提出了变量、极限和无穷小量的概念,但是这些术语的意义与它们现在的意义不一致。

实分析的发展具有目的论的性质,柯西的后继者在以这个假说为主体的概念框架下朝着既定的结果工作。所以,G和S认为,柯西作品中对limite的引用必然意味着柯西在研究阿基米德连续体,而无穷小只是一种方便的修辞手法,柯西在脑海中对阿基米德极限有一个完整的解释。然而,根据现代无穷小分析的标准原理,柯西利用极限的过程还存在另一种形式化的方式,更符合柯西对于无穷小的大量使用。

我们对一个错误的概念提出了质疑, 据说柯西曾经在巴黎理工不得不以这个概念为依据教授无穷小量。我们将说明,争议主要是围绕严密性问题,这是与无穷小量的问题无关的。与柯西同时代的de Prony对柯西的方法进行了批判,对柯西和当时的人们阐明了严密性的意义。仔细阅读柯西的作品,就会动摇人们对柯西在分析史上的地位的既定看法,并指出他不仅是个epsion的先驱,同样也是一个无穷小技术的先驱者。

关键词:蝴蝶模型,连续性,无穷小量,“极限”,标准部分,变量,科技,德普罗尼


Abstract 19th century real analysis received a major impetusfrom Cauchy’s work. Cauchy mentions variable quantities, limits,and infinitesimals, but the meaning he attached to these terms isnot identical to their modern meaning.

Some Cauchy historians work in a conceptual scheme domi-nated by an assumption of a teleological nature of the evolutionof real analysis toward a preordained outcome. Thus, Gilain andSiegmund-Schultze assume that references to limite in Cauchy’swork necessarily imply that Cauchy was working with an Archi-medean continuum, whereas infinitesimals were merely a conve-nient figure of speech, for which Cauchy had in mind a completejustification in terms of Archimedean limits. However, there is an-other formalisation of Cauchy’s procedures exploiting his limite,more consistent with Cauchy’s ubiquitous use of infinitesimals, interms of the standard part principle of modern infinitesimal anal-ysis.

We challenge a misconception according to which Cauchy wasallegedly forced to teach infinitesimals at the Ecole Polytechnique.We show that the debate there concerned mainly the issue of rigor,a separate one from infinitesimals. A critique of Cauchy’s approachby his contemporaryde Prony sheds light on the meaning of rigor toCauchy and his contemporaries. An attentive reading of Cauchy’swork challenges received views on Cauchy’s role in the history ofanalysis, and indicates that he was a pioneer of infinitesimal tech-niques as much as a harbinger of the Epsilontik.

Keywords: butterfly model; continuity; infinitesimals; limite;standard part; variable quantity; Cauchy; de Prony

论文链接:

19th century real analysis, forward and backwardarxiv.orghttp://dx.doi.org/10.14708/am.v13i1.6440dx.doi.org





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