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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.

Fermat

Leibniz

Cauchy

More on infinitesimals

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