‘Chasing’ The Diagram - The Use of Visualizations in Algebraic Reasoning

Review of Symbolic Logic 10 (1):158-186 (2017)
Authors
Silvia De Toffoli
Princeton University
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.
Keywords Diagrams  Visualization  Mathematical Notations  Homological Algebra
Categories (categorize this paper)
DOI 10.1017/s1755020316000277
Options
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

Our Archive
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

No references found.

Add more references

Citations of this work BETA

Add more citations

Similar books and articles

Diagrams as Sketches.Brice Halimi - 2012 - Synthese 186 (1):387-409.
The Tinctures and Implicit Quantification Over Worlds.Jay Zeman - 1997 - In Paul Forster & Jacqueline Brunning (eds.), The Rule of Reason: The Philosophy of C.S. Peirce. University of Toronto Press. pp. 96-119.
Diagrams and Proofs in Analysis.Jessica Carter - 2010 - International Studies in the Philosophy of Science 24 (1):1 – 14.
The Role of Diagrams in Mathematical Arguments.David Sherry - 2009 - Foundations of Science 14 (1-2):59-74.
On Automating Diagrammatic Proofs of Arithmetic Arguments.Mateja Jamnik, Alan Bundy & Ian Green - 1999 - Journal of Logic, Language and Information 8 (3):297-321.
Diagrams in Biology.Laura Perini - 2013 - The Knowledge Engineering Review 28 (3):273-286.
What Perception is Doing, and What It is Not Doing, in Mathematical Reasoning.Dennis Lomas - 2002 - British Journal for the Philosophy of Science 53 (2):205-223.
Diagram-Based Geometric Practice.Kenneth Manders - 2008 - In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford University Press. pp. 65--79.

Analytics

Added to PP index
2017-08-23

Total downloads
182 ( #33,164 of 2,308,512 )

Recent downloads (6 months)
66 ( #5,552 of 2,308,512 )

How can I increase my downloads?

Monthly downloads

My notes

Sign in to use this feature