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Leonhard Euler Mathematical Intelligencer 15b
Leonhard Euler Journal for General Philosophy of Science 17b

Benacerraf emphasized the distinction between mathematical ontology and mathematical practice (or the structures mathematicians use in practice). With emphasis on practice and structures, we 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. Thus, Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit; 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 exploited infinite integers and the associated infinite products. We analyze these in the context of his infinite product decomposition for the sine function. We compare Euler's principle of cancellation 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.

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