Foundations of Science

, Volume 23, Issue 2, pp 267–296 | Cite as

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

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


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.


Cauchy’s infinitesimal Sum theorem Quantifier alternation Uniform convergence Foundational paradigms 



V. Kanovei was supported in part by the RFBR Grant Number 17-01-00705. M.  Katz was partially funded by the Israel Science Foundation Grant Number 1517/12. We are grateful to Dave L.  Renfro for helpful suggestions.


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Authors and Affiliations

  • Tiziana Bascelli
    • 1
  • Piotr Błaszczyk
    • 2
  • Alexandre Borovik
    • 3
  • Vladimir Kanovei
    • 4
  • Karin U. Katz
    • 5
  • Mikhail G. Katz
    • 5
    Email author
  • 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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