Skip to main content Accessibility help




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.


Corresponding author



Hide All
Adams, C. C. (1994). The Knot Book. New York: W. H. Freeman.
Alexander, J. W. (1928). Topological invariants of knots and links. Transactions of the American Mathematical Society, 30(2), 275306.
Ammon, S. (2015). Einige Überlegungen zur generativen und instrumentellen Operativität von technischen Bildern. In Depner, H., editor. Visuelle Philosophie. Würzburg: Königshausen & Neumann.
Avigad, J., Dean, E., & Mumma, J. (2009). A formal system for Euclid’s Elements . The Review of Symbolic Logic, 2(04), 700768.
Bredon, G. E. (1993). Topology and Geometry, Vol. 139. New York: Springer.
Brown, J. R. (2008). Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures. New York and London: Routledge.
Carter, J. (2010). Diagrams and proofs in analysis. International Studies in the Philosophy of Science, 24(1), 114.
Colyvan, M. (2012). An Introduction to the Philosophy of Mathematics. Cambridge: Cambridge University Press.
De Toffoli, S. & Giardino, V. (2014). Roles and forms of diagrams in knot theory. Erkenntnis, 79(3), 829842.
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. 315336.
Dutilh-Novaes, C. (2012). Formal Languages in Logic: A Philosophical and Cognitive Analysis. Cambridge: Cambridge University Press.
Feferman, S. (2012). And so on…: reasoning with infinite diagrams. Synthese, 186(1), 371386.
Giaquinto, M. (1994). Epistemology of visual thinking in elementary real analysis. The British Journal for the Philosophy of Science, 45(3), 789813.
Giaquinto, M. (2007). Visual Thinking in Mathematics. Oxford University Press.
Giaquinto, M. (2008). Visualizing in mathematics. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford: Oxford University Press.
Giardino, V. & Greenberg, G. (2015). Introduction: Varieties of iconicity. Review of Philosophy and Psychology, 6(1), 125.
Goodman, N. (1976). Languages of Art (second edition). Indianapolis: Hackett Publishing Company.
Grosholz, E. R. (2007). Representation and Productive Ambiguity in Mathematics and the Sciences. Oxford: Springer.
Halimi, B. (2012). Diagrams as sketches. Synthese, 186(1), 387409.
Ishiguro, H. (1990). Leibniz’s Philosophy of Logic and Language. Cambridge University Press.
Kanizsa, G. (1980). Grammatica del vedere: saggi su percezione e gestalt. Il mulino.
Krämer, S. (1988). Symbolische Maschinen: die Idee der Formalisierung in geschichtlichem Abriß. Wissenschaftliche Buchgesellschaft.
Krämer, S. (2003). Writing, notational iconicity, calculus: On writing as a cultural technique. Modern Language Notes, 118(3), 518537.
Lang, S. (2002). Algebra (revised third edition). Springer.
Larkin, J. H. & Simon, H. A. (1987). Why a diagram is (sometimes) worth ten thousand words. Cognitive Science, 11(1), 65100.
Macbeth, D. (2012a). Diagrammatic reasoning in Frege’s Begriffsschrift . Synthese, 186(1), 289314.
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. 2954.
Macbeth, D. (2012c). Seeing how it goes: Paper-and-pencil reasoning in mathematical practice. Philosophia Mathematica, 20(1), 5885.
Mancosu, P. (2008). The Philosophy of Mathematical Practice. Oxford University Press.
Mancosu, P., Jørgensen, K. F., & Pedersen, S. A. (2005). Visualization, Explanation and Reasoning Styles in Mathematics, Vol. 327. Springer.
Manders, K. (2008). The Euclidean diagram. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford University Press, pp. 112183.
Netz, R. (1998). Greek mathematical diagrams: Their use and their meaning. For the Learning of Mathematics, 18(3), 3339.
Panofsky, E. (1991). Perspective as Symbolic Form (revised edition). New York: Zone Books.
Perini, L. (2005). Visual representations and confirmation. Philosophy of Science, 72(5), 913926.
Rosch, E. (1999). Reclaiming concepts. Journal of Consciousness Studies, 6(11–12), 1112.
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. 20972102). Austin, TX: Cognitive Science Society.
Shin, S.-J. (2004). Heterogeneous reasoning and its logic. The Bulletin of Symbolic Logic, 10(1), 86106.
Shin, S.-J., Lemon, O., & Mumma, J. (Fall 2013). Diagrams. In Zalta, , editor. Stanford Encylcopedia for Philosophy. Available at:
Simmons, H. (2011). An Introduction to Category Theory. Cambridge University Press.
Weber, Z. (2013). Figures, formulae, and functors. In Shin, S.-J. and Moktefi, A., editors. Visual Reasoning with Diagrams. Basel: Birkhäuser, pp. 153170.
Weyl (1995 (original in Gerlman 1932)). Topology and abstract algebra as two roads of mathematical comprehension. The American Mathematical Monthly, 102(5), 453460.
Winther, R. G. (2017). When Maps Become the World. Chicago: The University of Chicago Press.
Wüthrich, A. (2010). The Genesis of Feynman Diagrams, Vol. 26. Netherlands: Springer Science & Business Media.


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed