Cauchy’s Infinitesimals, His Sum Theorem, and Foundational Paradigms

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

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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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  • Tiziana Bascelli
    • 1
  • Piotr Błaszczyk
    • 2
  • Alexandre Borovik
    • 3
  • Vladimir Kanovei
    • 4
  • Karin U. Katz
    • 5
  • Mikhail G. Katz
    • 5
  • Semen S. Kutateladze
    • 6
  • Thomas McGaffey
    • 7
  • David M. Schaps
    • 8
  • David Sherry
    • 9
  1. 1.Lyceum Gymnasium “F. Corradini”ThieneItaly
  2. 2.Institute of MathematicsPedagogical University of CracowKrakówPoland
  3. 3.School of MathematicsUniversity of ManchesterManchesterUnited Kingdom
  4. 4.IPPI, Moscow, and MIITMoscowRussia
  5. 5.Department of MathematicsBar Ilan UniversityRamat GanIsrael
  6. 6.Sobolev Institute of MathematicsNovosibirsk State UniversityNovosibirskRussia
  7. 7.Rice UniversityHoustonUSA
  8. 8.Department of Classical StudiesBar Ilan UniversityRamat GanIsrael
  9. 9.Department of PhilosophyNorthern Arizona UniversityFlagstaffUS

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