Perceiving the Infinite and the Infinitesimal World: Unveiling and Optical Diagrams and the Construction of Mathematical Concepts

Lorenzo Magnani (lmagnani@cc.gatech.edu)
University of Pavia, Pavia, Italy
and Georgia Institute of Technology, Atlanta, GA, USA

Riccardo Dossena (rdossena@pure-reason.unipv.it)
University of Pavia, Pavia, Italy

Abstract:

Many important concepts of the calculus are difficult to grasp, and they may appear epistemologically unjustified. For example, how does a real function appear in ``small'' neighborhoods of its points? How does it appear at infinity? Diagrams allow us to overcome the difficulty in constructing representations of mathematical critical situations and objects. For example they actually reveal the behavior of a real function not ``close to'' a point (as in the standard limit theory) but ``in'' the point. We are interested in our research in the diagrams which play an optical role - microscopes and ``microscopes within microscopes'', telescopes, windows, a mirror role (to externalize rough mental models), and an unveiling role (to help create new and interesting mathematical concepts, theories, and structures). In this paper we describe some examples of optical diagrams as a particular kind of epistemic mediator able to perform the explanatory abductive task of providing a better understanding of the calculus, through a non-standard model of analysis. We also maintain they can be used in many other different epistemological and cognitive situations.