‘Chasing’ the diagram—the use of visualizations in algebraic reasoning

Review of Symbolic Logic 10 (1):158-186 (2017)
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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.

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Silvia De Toffoli
University School of Advanced Studies IUSS Pavia

Citations of this work

Reconciling Rigor and Intuition.Silvia De Toffoli - 2020 - Erkenntnis 86 (6):1783-1802.
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Reliability of mathematical inference.Jeremy Avigad - 2020 - Synthese 198 (8):7377-7399.
Who's Afraid of Mathematical Diagrams?Silvia De Toffoli - 2023 - Philosophers' Imprint 23 (1).
The philosophy of logical practice.Ben Martin - 2022 - Metaphilosophy 53 (2-3):267-283.

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References found in this work

Languages of Art.Nelson Goodman - 1970 - Philosophy and Rhetoric 3 (1):62-63.
The Philosophy of Mathematical Practice.Paolo Mancosu (ed.) - 2008 - Oxford, England: Oxford University Press.
An introduction to the philosophy of mathematics.Mark Colyvan - 2012 - Cambridge: Cambridge University Press.

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