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

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)
DOI 10.1007/s10699-005-3003-8
 Save to my reading list
Follow the author(s)
My bibliography
Export citation
Find it on Scholar
Edit this record
Mark as duplicate
Revision history Request removal from index
Download options
PhilPapers Archive

Upload a copy of this paper     Check publisher's policy on self-archival     Papers currently archived: 15,865
External links
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
L. Magnani (2001). Abduction, Reason, and Science. Kluwer Academic/Plenum Publishers.
Charles S. Peirce (1931/1960). Collected Papers. Cambridge, Belknap Press of Harvard University Press.

View all 10 references / Add more references

Citations of this work BETA
Karin U. Katz & Mikhail G. Katz (2011). Cauchy's Continuum. Perspectives on Science 19 (4):426-452.

View all 7 citations / Add more citations

Similar books and articles

Monthly downloads

Added to index


Total downloads

24 ( #122,389 of 1,724,875 )

Recent downloads (6 months)

6 ( #110,368 of 1,724,875 )

How can I increase my downloads?

My notes
Sign in to use this feature

Start a new thread
There  are no threads in this forum
Nothing in this forum yet.