A cognitive analysis of Cauchy’s conceptions of function, continuity, limit and infinitesimal, with implications for teaching the calculus

Abstract

In this paper we use theoretical frameworks from mathematics education and cognitive psychology to analyse Cauchy's ideas of function, continuity, limit and infinitesimal expressed in his Cours D'Analyse. Our analysis focuses on the development of mathematical thinking from human perception and action into more sophisticated forms of reasoning and proof, offering different insights from those afforded by historical or mathematical analyses. It highlights the conceptual power of Cauchy's vision and the fundamental change involved in passing from the dynamic variability of the calculus to the modern set-theoretic formulation of mathematical analysis. This offers a re-evaluation of the relationship between the natural geometry and algebra of elementary calculus that continues to be used in applied mathematics, and the formal set theory of mathematical analysis that develops in pure mathematics and evolves into the logical development of non-standard analysis using infinitesimal concepts. It suggests that educational theories developed to evaluate student learning are themselves based on the conceptions of the experts who formulate them. It encourages us to reflect on the principles that we use to analyse the developing mathematical thinking of students, and to make an effort to understand the rationale of differing theoretical viewpoints.

Teaching and Learning of Calculus

“…For example, perceiving a sequence as a long list of numbers that never ends, as well as believing that the meaning of terms like "approach" or "tend to" depends on the context are conceptions that may function as obstacles to thinking faithfully about limit. Similarly, Tall and Katz (2014) argued that some students' conceptions of sequence convergence can be found in historical understandings. Particularly, they showed a subtle parallel between mathematicians' work with the limit process before Cauchy and students' conceptions of limit.…”

Teaching and Learning of Calculus

“…It is widely acknowledged that the development of the history of analysis led at least to two modern versions strongly linked to the structure of the mathematics continuum: standard analysis anticipated by the epsilontic limit definition of Weierstrass (in the context of an Archimedean continuum) and non-standard analysis based on infinitesimal-enriched continuum which includes not only infinitesimals but also their inverses. As Błaszczyk et al (2013) did, Borovik and Katz (2012) and Tall and Katz (2014) argued that it is an oversimplification to interpret Cauchy's work according to an Archimedean context. They exposed definitions stated by Cauchy, including that of continuity, intermediate value theorem and the summation of series, to show that infinitesimals actually have not been eliminated in his contributions.…”

From Pythagoreans and Weierstrassians to True Infinitesimal Calculus

“…Our approach enables the teacher to prepare the students for -δ by explaining the concepts first using a rigorous infinitesimal approach. Studies of methodology involving modern infinitesimals include [16,21,35,40,42,47].…”