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.

Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others

“…Sections 6 through 8 Y. Sergeyev [152], and D. Spalt [156]. These were dealt with respectively in KatzSherry [102], Bascelli et al [15], Kanovei-Katz-Sherry [87], Bair et al [7], BorovikKatz [28], B laszczyk et al [25], B laszczyk et al [23], Bascelli et al [16], B laszczyk et al [26], Gutman et al [58], Katz-Katz [94]. 6 See e.g., a quotation in Delfini-Lobry [35] from Berkeley Physics Course, Crawford [33] and Section 6.2. explore additional aspects of Robinson's framework.…”

“…Let H be an infinite hypernatural number (more formally, H ∈ * N \ N) and z ∈ C. We retrieve formulas of the sort that already appeared in Euler; see Bair et al [7]. In the following we will exploit hypercomplex numbers…”

“…For a broad outline of such a program see the studies [8] (2013), [27] (2013), [15] (2014), [25] [87] (2015), [7] (2017);…”

From Pythagoreans and Weierstrassians to True Infinitesimal Calculus

“…2 Thus, in discussing Kepler's Second Law, Toeplitz does not hesitate to exploit both (infinitesimal) differentials and the notion of utter smallness as a pedagogical device:…”

“…Readers can find other relatively unfriendly treatments of Robinson's framework, often in the context of other historical discussions of mathematics, in works by J. Earman [13], K. Easwaran [14], H. M. Edwards [15], G. Ferraro [17], J. Gray [18], P. Halmos [20], H. Ishiguro [23], G. Schubring [38], and Y. Sergeyev [39], and rebuttals and counterarguments against some of these in [2,3,4,7,8,9,10,19,24,27,29].…”