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
Learn more about PhilPapers
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|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
No references found.
Citations of this work BETA
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.
Karin Katz & Mikhail Katz (2012). A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography. Foundations of Science 17 (1):51-89.
Alexandre Borovik & Mikhail G. Katz (2012). Who Gave You the Cauchy–Weierstrass Tale? The Dual History of Rigorous Calculus. Foundations of Science 17 (3):245-276.
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.
Karin Katz & Mikhail Katz (2012). Stevin Numbers and Reality. Foundations of Science 17 (2):109-123.
Similar books and articles
Brice Halimi (2012). Diagrams as Sketches. Synthese 186 (1):387-409.
Jessica Carter (2010). Diagrams and Proofs in Analysis. International Studies in the Philosophy of Science 24 (1):1 – 14.
Sun-Joo Shin (1994). Peirce and the Logical Status of Diagrams. History and Philosophy of Logic 15 (1):45-68.
Letitia Meynell (2008). Why Feynman Diagrams Represent. International Studies in the Philosophy of Science 22 (1):39 – 59.
Yaroslav Sergeyev (2009). Numerical Computations and Mathematical Modelling with Infinite and Infinitesimal Numbers. Journal of Applied Mathematics and Computing 29:177-195.
Solomon Feferman (2012). And so On...: Reasoning with Infinite Diagrams. Synthese 186 (1):371 - 386.
Yaroslav Sergeyev (2009). Numerical Point of View on Calculus for Functions Assuming Finite, Infinite, and Infinitesimal Values Over Finite, Infinite, and Infinitesimal Domains. Nonlinear Analysis Series A 71 (12):e1688-e1707.
Added to index2009-01-28
Total downloads16 ( #101,250 of 1,099,017 )
Recent downloads (6 months)5 ( #58,097 of 1,099,017 )
How can I increase my downloads?