Perceiving the infinite and the infinitesimal world: Unveiling and optical diagrams in mathematics [Book Review]
David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Foundations of Science 10 (1):7-23 (2005)
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.
|Keywords||abduction action-based reasoning diagrams mathematical reasoning|
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References found in this work BETA
Paul Thagard (1992). Conceptual Revolutions. Princeton University Press.
L. Magnani (2001). Abduction, Reason, and Science. Kluwer Academic/Plenum Publishers.
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Citations of this work BETA
Karin Katz & Mikhail Katz (2012). A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography. Foundations of Science 17 (1):51-89.
Karin Katz & Mikhail Katz (2012). Stevin Numbers and Reality. Foundations of Science 17 (2):109-123.
Karin U. Katz & Mikhail G. Katz (2011). Cauchy's Continuum. Perspectives on Science 19 (4):426-452.
Mikhail G. Katz & David Sherry (2013). Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes From Berkeley to Russell and Beyond. [REVIEW] Erkenntnis 78 (3):571-625.
Piotr Błaszczyk, Mikhail G. Katz & David Sherry (2013). Ten Misconceptions From the History of Analysis and Their Debunking. Foundations of Science 18 (1):43-74.
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