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‘CHASING’ THE DIAGRAM—THE USE OF VISUALIZATIONS IN ALGEBRAIC REASONING

Published online by Cambridge University Press:  28 October 2016

SILVIA DE TOFFOLI
SILVIA DE TOFFOLI*
Affiliation:
Department of Philosophy, Stanford University
*
*DEPARTMENT OF PHILOSOPHY STANFORD UNIVERSITY 450 SERRA MALL, STANFORD CA 94305, USA E-mail: silviadt@stanford.edu
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Abstract

The aim of this article is to investigate the roles of commutative diagrams (CDs) in a specific mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation that goes beyond the traditional bipartition of mathematical representations into diagrammatic and linguistic. It will be argued that one of the reasons why CDs form a good notation is that they are highly mathematically tractable: experts can obtain valid results by ‘calculating’ with CDs. These calculations, take the form of ‘diagram chases’. In order to draw inferences, experts move algebraic elements around the diagrams. It will be argued that these diagrams are dynamic. It is thanks to their dynamicity that CDs can externalize the relevant reasoning and allow experts to draw conclusions directly by manipulating them. Lastly, it will be shown that CDs play essential roles in the context of proof as well as in other phases of the mathematical enterprise, such as discovery and conjecture formation.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 10 , Issue 1 , March 2017 , pp. 158 - 186
Copyright
Copyright © Association for Symbolic Logic 2016 

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