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Interpreting the Infinitesimal Mathematics of Leibniz and Euler. / Bair, Jacques; Błaszczyk, Piotr; Ely, Robert и др.
в: Journal for General Philosophy of Science, Том 48, № 2, 01.06.2017, стр. 195-238.Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование
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TY - JOUR
T1 - Interpreting the Infinitesimal Mathematics of Leibniz and Euler
AU - Bair, Jacques
AU - Błaszczyk, Piotr
AU - Ely, Robert
AU - Henry, Valérie
AU - Kanovei, Vladimir
AU - Katz, Karin U.
AU - Katz, Mikhail G.
AU - Kutateladze, Semen S.
AU - McGaffey, Thomas
AU - Reeder, Patrick
AU - Schaps, David M.
AU - Sherry, David
AU - Shnider, Steven
PY - 2017/6/1
Y1 - 2017/6/1
N2 - 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.
AB - 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.
KW - Archimedean axiom
KW - Euler
KW - Infinite product
KW - Infinitesimal
KW - Law of continuity
KW - Law of homogeneity
KW - Leibniz
KW - Mathematical practice
KW - Ontology
KW - Principle of cancellation
KW - Procedure
KW - Standard part principle
UR - http://www.scopus.com/inward/record.url?scp=84978654577&partnerID=8YFLogxK
U2 - 10.1007/s10838-016-9334-z
DO - 10.1007/s10838-016-9334-z
M3 - Article
AN - SCOPUS:84978654577
VL - 48
SP - 195
EP - 238
JO - Journal for General Philosophy of Science
JF - Journal for General Philosophy of Science
SN - 0925-4560
IS - 2
ER -
ID: 9049127