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

Tiziana Bascelli; Piotr Blaszczyk; Alexandre Borovik; Vladimir Kanovei; Karin U. Katz; Mikhail G. Katz; Semen S. Kutateladze; Thomas McGaffey; David M. Schaps; David Sherry

doi:10.1007/s10699-017-9534-y

arXiv · 1704.07723

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

Tiziana Bascelli·Piotr Blaszczyk·Alexandre Borovik·Vladimir Kanovei·Karin U. Katz·Mikhail G. Katz·Semen S. Kutateladze·Thomas McGaffey·David M. Schaps·David Sherry

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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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Tiziana Bascelli, Piotr Blaszczyk, Alexandre Borovik, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, Semen S. Kutateladze, Thomas McGaffey, David M. Schaps, David Sherry. 2017-04-25. Cauchy's infinitesimals, his sum theorem, and foundational paradigms. https://doi.org/10.1007/s10699-017-9534-y

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