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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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References

BIBLIOGRAPHY

Adams, C. C. (1994). The Knot Book. New York: W. H. Freeman.Google Scholar
Alexander, J. W. (1928). Topological invariants of knots and links. Transactions of the American Mathematical Society, 30(2), 275–306.Google Scholar
Ammon, S. (2015). Einige Überlegungen zur generativen und instrumentellen Operativität von technischen Bildern. In Depner, H., editor. Visuelle Philosophie. Würzburg: Königshausen & Neumann.Google Scholar
Avigad, J., Dean, E., & Mumma, J. (2009). A formal system for Euclid’s Elements . The Review of Symbolic Logic, 2(04), 700–768.Google Scholar
Bredon, G. E. (1993). Topology and Geometry, Vol. 139. New York: Springer.Google Scholar
Brown, J. R. (2008). Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures. New York and London: Routledge.Google Scholar
Carter, J. (2010). Diagrams and proofs in analysis. International Studies in the Philosophy of Science, 24(1), 1–14.Google Scholar
Colyvan, M. (2012). An Introduction to the Philosophy of Mathematics. Cambridge: Cambridge University Press.Google Scholar
De Toffoli, S. & Giardino, V. (2014). Roles and forms of diagrams in knot theory. Erkenntnis, 79(3), 829–842.Google Scholar
De Toffoli, S. & Giardino, V. (2015). An Inquiry into the Practice of Proving in Low-dimensional Topology. Boston Studies in the Philosophy and History of Science, Vol. 308, Chapter 15. Boston: Springer, pp. 315–336.Google Scholar
Dutilh-Novaes, C. (2012). Formal Languages in Logic: A Philosophical and Cognitive Analysis. Cambridge: Cambridge University Press.Google Scholar
Feferman, S. (2012). And so on…: reasoning with infinite diagrams. Synthese, 186(1), 371–386.Google Scholar
Giaquinto, M. (1994). Epistemology of visual thinking in elementary real analysis. The British Journal for the Philosophy of Science, 45(3), 789–813.Google Scholar
Giaquinto, M. (2007). Visual Thinking in Mathematics. Oxford University Press.Google Scholar
Giaquinto, M. (2008). Visualizing in mathematics. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford: Oxford University Press.Google Scholar
Giardino, V. & Greenberg, G. (2015). Introduction: Varieties of iconicity. Review of Philosophy and Psychology, 6(1), 1–25.Google Scholar
Goodman, N. (1976). Languages of Art (second edition). Indianapolis: Hackett Publishing Company.Google Scholar
Grosholz, E. R. (2007). Representation and Productive Ambiguity in Mathematics and the Sciences. Oxford: Springer.Google Scholar
Halimi, B. (2012). Diagrams as sketches. Synthese, 186(1), 387–409.Google Scholar
Ishiguro, H. (1990). Leibniz’s Philosophy of Logic and Language. Cambridge University Press.Google Scholar
Kanizsa, G. (1980). Grammatica del vedere: saggi su percezione e gestalt. Il mulino.Google Scholar
Krämer, S. (1988). Symbolische Maschinen: die Idee der Formalisierung in geschichtlichem Abriß. Wissenschaftliche Buchgesellschaft.Google Scholar
Krämer, S. (2003). Writing, notational iconicity, calculus: On writing as a cultural technique. Modern Language Notes, 118(3), 518–537.Google Scholar
Lang, S. (2002). Algebra (revised third edition). Springer.Google Scholar
Larkin, J. H. & Simon, H. A. (1987). Why a diagram is (sometimes) worth ten thousand words. Cognitive Science, 11(1), 65–100.Google Scholar
Macbeth, D. (2012a). Diagrammatic reasoning in Frege’s Begriffsschrift . Synthese, 186(1), 289–314.CrossRefGoogle Scholar
Macbeth, D. (2012b). Proof and understanding in mathematical practice. In Giardino, V., Moktefi, A., Mols, S., & Bendegem, J. P. V., editors. From Practice to Results in Logic and Mathematics. Philosophia Scientiae, Vol. 16. Kimé, pp. 29–54.Google Scholar
Macbeth, D. (2012c). Seeing how it goes: Paper-and-pencil reasoning in mathematical practice. Philosophia Mathematica, 20(1), 58–85.Google Scholar
Mancosu, P. (2008). The Philosophy of Mathematical Practice. Oxford University Press.Google Scholar
Mancosu, P., Jørgensen, K. F., & Pedersen, S. A. (2005). Visualization, Explanation and Reasoning Styles in Mathematics, Vol. 327. Springer.Google Scholar
Manders, K. (2008). The Euclidean diagram. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford University Press, pp. 112–183.Google Scholar
Netz, R. (1998). Greek mathematical diagrams: Their use and their meaning. For the Learning of Mathematics, 18(3), 33–39.Google Scholar
Panofsky, E. (1991). Perspective as Symbolic Form (revised edition). New York: Zone Books.Google Scholar
Perini, L. (2005). Visual representations and confirmation. Philosophy of Science, 72(5), 913–926.CrossRefGoogle Scholar
Rosch, E. (1999). Reclaiming concepts. Journal of Consciousness Studies, 6(11–12), 11–12.Google Scholar
Schlimm, D. & Neth, H. (2008). Modeling ancient and modern arithmetic practices: Addition and multiplication with arabic and roman numerals In Sloutsky, V. Love, B. & McRae, K. (Editors), 30th Annual Conference of the Cognitive Science Society (pp. 2097–2102). Austin, TX: Cognitive Science Society.Google Scholar
Shin, S.-J. (2004). Heterogeneous reasoning and its logic. The Bulletin of Symbolic Logic, 10(1), 86–106.Google Scholar
Shin, S.-J., Lemon, O., & Mumma, J. (Fall 2013). Diagrams. In Zalta, , editor. Stanford Encylcopedia for Philosophy. Available at: http://plato.stanford.edu/archives/fall2013/entries/diagrams/.Google Scholar
Simmons, H. (2011). An Introduction to Category Theory. Cambridge University Press.Google Scholar
Weber, Z. (2013). Figures, formulae, and functors. In Shin, S.-J. and Moktefi, A., editors. Visual Reasoning with Diagrams. Basel: Birkhäuser, pp. 153–170.Google Scholar
Weyl (1995 (original in Gerlman 1932)). Topology and abstract algebra as two roads of mathematical comprehension. The American Mathematical Monthly, 102(5), 453–460.Google Scholar
Winther, R. G. (2017). When Maps Become the World. Chicago: The University of Chicago Press.Google Scholar
Wüthrich, A. (2010). The Genesis of Feynman Diagrams, Vol. 26. Netherlands: Springer Science & Business Media.Google Scholar