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

Research output: Contribution to journalArticle

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.

Original languageEnglish
Pages (from-to)195–238
Number of pages44
JournalJournal for General Philosophy of Science
Volume48
Issue number2
DOIs
Publication statusPublished - Jun 2017

Keywords

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

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    Bair, J., Błaszczyk, P., Ely, R., Henry, V., Kanovei, V., Katz, K. U., Katz, M. G., Kutateladze, S. S., McGaffey, T., Reeder, P., Schaps, D. M., Sherry, D., & Shnider, S. (2017). Interpreting the Infinitesimal Mathematics of Leibniz and Euler. Journal for General Philosophy of Science, 48(2), 195–238. https://doi.org/10.1007/s10838-016-9334-z