Diagrammatic reasoning in mathematics

I worked extensively on what I considered a central instance of diagramming, that is, the use of cognitive artifacts in particular in the practice of mathematics. In my view, actual mathematical practice is a key explanandum of any account of reasoning through cognitive artifacts: mathematics can be considered as the paradigmatic form of valid reasoning relying on external representations of different nature, and the scientific discipline where we kind of “play” with our representational capacities, exploring and pushing further and further the limits of what we are able to conceive. To do it, we need the support of the representational tools that are introduced in newer and newer practices.

Diagrams are an integral part of the practice mathematics: mathematicians draw figures on the blackboard to teach students or to show some result to their colleagues; in some cases – but not as often as compared to their impressive presence in more informal exchanges – the same diagrams or a more polished version of them are finally published in text– books, encyclopedias or research articles. 

Several questions can be asked, at different levels. First, definitional questions: what are diagrams? Second, questions on the way they are used: how do diagrams work in the practice of mathematics? Third, on the cognitive advantages and disadvantages in dealing with them: do they help us exploring our research objects? how? or on the contrary do they constitute a limitation to what we can conceive? Fourth, about their legitimacy in mathematics: is reasoning with them valid? do we get to results on which we can have confidence by using them?

Some papers:

• Giardino, V. (2023), Diagrammatic Proofs in Mathematics (Almost) 20 Years of Research, Section Proofs. In: Sriraman, B. (ed) Handbook of the History and Philosophy of Mathematical Practice, Springer, pp. 2045-2067.

• Giardino, V. (2023), The practice of mathematics: cognitive resources and conceptual content, Topoi, 42, 259–270.

• Giardino (2023), New Perspectives: An Introduction, Section New Perspectives. In: Sriraman, B. (ed) Handbook of the History and Philosophy of Mathematical Practice.

• Giardino,V. – Wöpking,J (2019), Aspect seeing and mathematical representations, AVANT. Trends in Interdisciplinary Studies, 2.

• Eckes, C. – Giardino, V. (2018), The Classificatory Function of Diagrams: Two Examples from Mathematics, Lecture Notes in Computer Science 10871, 120 - 136.

• Giardino,V (2017), Diagrammatic reasoning in mathematics. In: L. Magnani and T.Bertolotti (eds). Spinger Handbook of Model-Based Science, pp. 499 – 522.

• Giardino, V. (2017), The Practical Turn in Philosophy of Mathematics: A Portrait of a Young Discipline, Phenomenology and Mind, 12, 18 – 28.

• De Toffoli, S. – Giardino, V. (2016), Envisioning Trasformations. The Practice of Topology. In: Larvor, B. (ed). Mathematical Cultures, Birkha ̈user Science, Springer, pp. 25 – 50.

• 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, 308, 315 – 336.

• De Toffoli, S. – Giardino, V. (2014), Forms and Roles of Diagrams in Knot Theory, Erkenntnis, 79(4), 829 – 842.

• Giardino, V. (2010), Intuition and visualization in mathematical problem solving, Topoi, 29(1), 29 – 39.