Journal for General Philosophy of Science

, Volume 48, Issue 2, pp 195–238 | Cite as

Interpreting the Infinitesimal Mathematics of Leibniz and Euler

  • Jacques Bair
  • Piotr Błaszczyk
  • Robert Ely
  • Valérie Henry
  • Vladimir Kanovei
  • Karin U. Katz
  • Mikhail G. Katz
  • Semen S. Kutateladze
  • Thomas McGaffey
  • Patrick Reeder
  • David M. Schaps
  • David Sherry
  • Steven Shnider
Article

Abstract

We apply Benacerraf’s distinction between mathematical ontology and mathematical practice (or the structures mathematicians use in practice) to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves.

Keywords

Archimedean axiom Infinite product Infinitesimal Law of continuity Law of homogeneity Principle of cancellation Procedure Standard part principle Ontology Mathematical practice Euler Leibniz 

Mathematics Subject Classification

Primary 01A50; Secondary 26E35 01A85 03A05 

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Jacques Bair
    • 1
  • Piotr Błaszczyk
    • 2
  • Robert Ely
    • 3
  • Valérie Henry
    • 4
  • Vladimir Kanovei
    • 5
  • Karin U. Katz
    • 6
  • Mikhail G. Katz
    • 6
  • Semen S. Kutateladze
    • 7
  • Thomas McGaffey
    • 8
  • Patrick Reeder
    • 9
  • David M. Schaps
    • 10
  • David Sherry
    • 11
  • Steven Shnider
    • 6
  1. 1.HEC-ULGUniversity of LiegeLiègeBelgium
  2. 2.Institute of MathematicsPedagogical University of CracowKrakówPoland
  3. 3.Department of MathematicsUniversity of IdahoMoscowUSA
  4. 4.Department of MathematicsUniversity of NamurNamurBelgium
  5. 5.IPPI, Moscow, and MIITMoscowRussia
  6. 6.Department of MathematicsBar Ilan UniversityRamat GanIsrael
  7. 7.Sobolev Institute of MathematicsNovosibirsk State UniversityNovosibirskRussia
  8. 8.Rice UniversityHoustonUSA
  9. 9.Kenyon CollegeGambierUSA
  10. 10.Department of Classical StudiesBar Ilan UniversityRamat GanIsrael
  11. 11.Department of PhilosophyNorthern Arizona UniversityFlagstaffUSA

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