Excursus into the History of Calculus

This is a revised version of the course notes handed to each participant at the limits of mathematics short course, Orono, Maine, June 1994.

The example of the calculus is used to explain how simple, practical math was made enormously complex by imposing on it the Western religiously-colored notion of mathematics as "perfect". We describe a pedagogical experiment to make math easy by teaching "calculus without limits" using the new realistic philosophy of zeroism, different from Platonic idealism or formalist metaphysics. Despite its demonstrated advantages, it is being resisted because of the existing colonial ha...

The goal of this paper consists of developing a new (more physical and numerical in comparison with standard and non-standard analysis approaches) point of view on Calculus with functions assuming infinite and infinitesimal values. It uses recently introduced infinite and infinitesimal numbers being in accordance with the principle 'The part is less than the whole' observed in the physical world around us. These numbers have a strong practical advantage with respect to tradit...

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 ...

We present a characterization of the completeness of the field of real numbers in the form of a \emph{collection of several equivalent statements} borrowed from algebra, real analysis, general topology, and non-standard analysis. We also discuss the completeness of non-Archimedean fields and present several examples of such fields. As an application, we exploit the characterization of the reals to argue that the Leibniz-Euler infinitesimal calculus in the $17^\textrm{th}$-$18...

We present a characterization of the completeness of the field of real numbers in the form of a \emph{collection of ten equivalent statements} borrowed from algebra, real analysis, general topology and non-standard analysis. We also discuss the completeness of non-Archimedean fields and present several examples of such fields. As an application we exploit one of our results to argue that the Leibniz infinitesimal calculus in the $18^\textrm{th}$ century was already a rigorous...

This is a short overview of the influence of mathematicians and their ideas on the creative contribution of Mikhailo Lomonosov on the occasion of the tercentenary of his birth.

This paper introduces DD calculus and describes the basic calculus concepts of derivative and integral in a direct and non-traditional way, without limit definition: Derivative is computed from the point-slope equation of a tangent line and integral is defined as the height increment of a curve. This direct approach to calculus has three distinct features: (i) it defines derivative and (definite) integral without using limits, (ii) it defines derivative and antiderivative sim...

We discuss several aspects of infinity in the history of mathematics.

The present paper presents some reflections of the author on divergent series and their role and place in mathematics over the centuries. The point of view presented here is limited to differential equations and dynamical systems.