CAUCHY’S INFINITESIMALS, HIS SUM THEOREM, AND FOUNDATIONAL PARADIGMS

TIZIANA BASCELLI, PIOTR BL ASZCZYK, ALEXANDRE BOROVIK, VLADIMIR KANOVEI, KARIN U. KATZ, MIKHAIL G. KATZ, SEMEN S. KUTATELADZE, THOMAS MCGAFFEY, DAVID M. SCHAPS, AND DAVID SHERRY

摘要：柯西和定理是今天本科数学分析中关于函数列收敛的一个基本的结果的原型。我们试图解释柯西的证明，并讨论涉及比较不同的解释范式的相关问题。柯西的证明常用现代框架下的魏尔斯特拉斯范式来阐释。我们仔细分析了柯西的证明，在一个不同的现代框架中找到了更接近的阐释。

关键词:柯西的无穷小;和定理;变量替换;一致收敛;基本范式。

Abstract. Cauchy’s sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy’s proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy’s proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy’s proof closely and show that it finds closer proxies in a different modern framework.

Keywords: Cauchy’s infinitesimal; sum theorem; quantifier alternation; uniform convergence; foundational paradigms.

http://dx.doi.org/10.1007/s10699-017-9534-y​dx.doi.orghttps://arxiv.org/abs/1704.07723​arxiv.orgMatches for: MR=3803893​mathscinet.ams.org