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

摘要

In this paper we use theoretical frameworks from mathematics education andcognitive psychology to analyse Cauchy's ideas of function, continuity, limitand infinitesimal expressed in his Cours D'Analyse. Our analysis focuses on thedevelopment of mathematical thinking from human perception and action into moresophisticated forms of reasoning and proof, offering different insights fromthose afforded by historical or mathematical analyses. It highlights theconceptual power of Cauchy's vision and the fundamental change involved inpassing from the dynamic variability of the calculus to the modernset-theoretic formulation of mathematical analysis. This offers a re-evaluationof the relationship between the natural geometry and algebra of elementarycalculus that continues to be used in applied mathematics, and the formal settheory of mathematical analysis that develops in pure mathematics and evolvesinto the logical development of non-standard analysis using infinitesimalconcepts. It suggests that educational theories developed to evaluate studentlearning are themselves based on the conceptions of the experts who formulatethem. It encourages us to reflect on the principles that we use to analyse thedeveloping mathematical thinking of students, and to make an effort tounderstand the rationale of differing theoretical viewpoints.