Perceiving the infinite and the infinitesimal world: Unveiling and optical diagrams in mathematics [Book Review]

Foundations of Science 10 (1):7-23 (2005)
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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.

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Lorenzo Magnani
Universita' degli Studi di Pavia

References found in this work

Non-standard Analysis.Gert Heinz Müller - 2016 - Princeton University Press.
Conceptual Revolutions.Paul Thagard - 1992 - Princeton: Princeton University Press.
Collected papers.Charles S. Peirce - 1931 - Cambridge,: Belknap Press of Harvard University Press.
Abduction, Reason, and Science.L. Magnani - 2001 - Kluwer Academic/Plenum Publishers.

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