Valérie Henry

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 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.

Une modélisation d'un zoom au moyen de microscopes virtuels

In this paper, we explain how a computer works when “zooms” are made around a point on a planar curve. This modelisation leads to an easy and algorithmic method to find the (vertical or not vertical) tangents for the studied curve. Mots-cle: Zoom, microscope virtuel, tangente, point de non derivabilite, limite de courbes, convergence uniforme, didactique des mathematiques. ZDM Subject Classification: I10, P7, R20.

From Newton's Fluxions to Virtual Microscopes

The method of fluxions was originally given by Newton among others in order to determine the tangent to a curve. In this note, we will formulate this method by the light of some modern mathematical tools: using the concept of limit, but also with hyperreal numbers and their standard parts and with dual numbers; another way is the use of virtual microscopes both in the contexts of classical and non standard analysis.

Cauchy, Infinitesimals and ghosts of departed quantifiers

Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinsons frameworks. The latter provides closer proxies for the procedures of the classical masters. Thus, Leibnizs distinction between assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robinsons framework, while Leibnizs law of homogeneity with the implied notion of equality up to negligible terms finds a mathematical formalisation in terms of standard part. It is hard to provide parallel formalisations in a Weierstrassian framework but scholars since Ishiguro have engaged in a quest for ghosts of departed quantifiers to provide a Weierstrassian account for Leibnizs infinitesimals. Euler similarly had notions of equality up to negligible terms, of which he distinguished two types: geometric and arithmetic. Euler routinely used product decompositions into a specific infinite number of factors, and used the binomial formula with an infinite exponent. Such procedures have immediate hyperfinite analogues in Robinsons framework, while in a Weierstrassian framework they can only be reinterpreted by means of paraphrases departing significantly from Eulers own presentation. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinsons framework but scholars working in a Weierstrassian framework bend over backwards either to claim that Cauchy was vague or to engage in a quest for ghosts of departed quantifiers in his work. Cauchys procedures in the context of his 1853 sum theorem (for series of continuous functions) are more readily understood from the viewpoint of Robinsons framework, where one can exploit tools such as the pointwise definition of the concept of uniform convergence. Keywords: historiography; infinitesimal; Latin model; butterfly model

An Introduction to Differentials Based on Hyperreal Numbers and Infinite Microscopes.

Abstract In this article, we propose to introduce the differential of a function through a non-classical way, lying on hyperreals and infinite microscopes. This approach is based on the developments of nonstandard analysis, wants to be more intuitive than the classical one and tries to emphasize the functional and geometric aspects of the differential. In the second part of the work, we analyze the results of an experiment made with undergraduate students who had been taught calculus by a non standard way for nearly two years.