Interpreting the infinitesimal mathematics of Leibniz and Euler

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 17th and 18th 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 is 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

62 pages, to appear in Journal for General Philosophy of Science

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Cited by in corpus (15)

• Procedures of Leibnizian infinitesimal calculus: An account in three modern frameworks
• Toward a history of mathematics focused on procedures
• Cauchy's infinitesimals, his sum theorem, and foundational paradigms
• Infinitesimal analysis without the Axiom of Choice
• Gregory's sixth operation
• Leibniz on bodies and infinities: rerum natura and mathematical fictions
• Continuity between Cauchy and Bolzano: Issues of antecedents and priority
• Infinite lotteries, spinners, and the applicability of hyperreals
• Controversies in the foundations of analysis: Comments on Schubring's Conflicts
• Fermat's dilemma: Why did he keep mum on infinitesimals? and the European theological context
• Small oscillations of the pendulum, Euler's method, and adequality
• Is Leibnizian calculus embeddable in first order logic?
• Bolzano's Infinite Quantities
• Metric completions, the Heine-Borel property, and approachability
• Episodes from the history of infinitesimals