Cauchy's infinitesimals, his sum theorem, and foundational paradigms

2017/04/30 by Tiziana Bascelli, Piotr Blaszczyk, Alexandre Borovik +7 · 1 citation
Mathematics · #math.HO #math.CA #math.LO #msc:01A55 #msc:01A85 #msc:26E35

paper · pdf · doi:10.1007/s10699-017-9534-y

42 pages; to appear in Foundations of Science

arxiv created 2017/05/09 · arxiv updated 2017/06/30

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.

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