相关论文: A cognitive analysis of Cauchy's conceptions of fu…
19th century real analysis received a major impetus from Cauchy's work. Cauchy mentions variable quantities, limits, and infinitesimals, but the meaning he attached to these terms is not identical to their modern meaning. Some Cauchy…
Cauchy published his Cours d'Analyse 200 years ago. We analyze Cauchy's take on the concepts of rigor, continuity, and limit, and explore a pair of approaches in the literature to the meaning of his infinitesimal analysis and his sum…
Cauchy's contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the epsilon-delta doctrine in the context of an…
Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy's proof, and discuss the related epistemological questions involved in…
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…
We discuss the repercussions of the development of infinitesimal calculus into modern analysis, beginning with viewpoints expressed in the nineteenth and twentieth centuries and relating them to the natural cognitive development of…
Like his colleagues de Prony, Petit, and Poisson at the Ecole Polytechnique, Cauchy used infinitesimals in the Leibniz-Euler tradition both in his research and teaching. Cauchy applied infinitesimals in an 1826 work in differential geometry…
Mathematical conception of infinite quantities forms a cornerstone of many disciplines of modern mathematics --- from differential calculus to set theory. In fact, it could be argued that the most significant revolutions in mathematics in…
This paper frames calculus as a global, centuries-long development rather than a subject that began only with Newton and Leibniz. Drawing on ideas from Greek, Indian, Islamic, and later European mathematics, it highlights how concepts like…
We propose a novel foundation for calculus that focuses on the notion of approximations while avoiding the use of limits altogether. Continuity is defined as approximation at a point, while differentiability is defined as approximation with…
Classical computations can not capture the essence of infinite computations very well. This paper will focus on a class of infinite computations called convergent infinite computations}. A logic for convergent infinite computations is…
Foundations of Science recently published a rebuttal to a portion of our essay it published two years ago. The author, G. Schubring, argues that our 2013 text treated unfairly his 2005 book, Conflicts between generalization, rigor, and…
The paper discusses relationships between aesthetics theory and mathematical models of mind. Mathematical theory describes abilities for concepts, emotions, instincts, imagination, adaptation, learning, cognition, language, approximate…
We show how to extract a monotonic learning algorithm from a classical proof of a geometric statement by interpreting the proof by means of interactive realizability, a realizability sematics for classical logic. The statement is about the…
In teaching infinitesimal calculus we sought to present basic concepts like continuity and convergence by comparing and contrasting various definitions, rather than presenting "the definition" to the students as a monolithic absolute. We…
This article exemplifies a novel approach to the teaching of introductory differential calculus using the modern notion of ``infinitesimal'' as opposed to the traditional approach using the notion of ``limit''. I illustrate the power of the…
Technology is currently ubiquitous and is also part of the educational system at all levels. It started with communication technology systems, and later continued with digital competence. Nowadays, although these previous concepts are still…
Questions concerning origin of mathematical knowledge and roles of language and intuition (imagery) in mathematical thoughts are long standing and widely debated. By introspection, mathematicians usually have some beliefs regarding these…
A close look at students' written work on examinations offers a wealth of information about their performance, their knowledge of the subject, their strengths, weaknesses and misconceptions, and their overall level of mathematical skills…
We offer a view of mathematics as an experimental science where axioms play the role of foundational theories like general relativity and quantum mechanics in physics. Under this view, axioms are provisional and inferred from experience…