Forms and Roles of Diagrams in Knot Theory

Erkenntnis 79 (4):829-842 (2014)
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The aim of this article is to explain why knot diagrams are an effective notation in topology. Their cognitive features and epistemic roles will be assessed. First, it will be argued that different interpretations of a figure give rise to different diagrams and as a consequence various levels of representation for knots will be identified. Second, it will be shown that knot diagrams are dynamic by pointing at the moves which are commonly applied to them. For this reason, experts must develop a specific form of enhanced manipulative imagination, in order to draw inferences from knot diagrams by performing epistemic actions. Moreover, it will be argued that knot diagrams not only can promote discovery, but also provide evidence. This case study is an experimentation ground to evaluate the role of space and action in making inferences by reasoning diagrammatically.

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Author Profiles

Silvia De Toffoli
University School of Advanced Studies IUSS Pavia
Valeria Giardino
Centre National de la Recherche Scientifique

Citations of this work

What are mathematical diagrams?Silvia De Toffoli - 2022 - Synthese 200 (2):1-29.
Reconciling Rigor and Intuition.Silvia De Toffoli - 2020 - Erkenntnis 86 (6):1783-1802.
Reliability of mathematical inference.Jeremy Avigad - 2020 - Synthese 198 (8):7377-7399.
An Inquiry into the Practice of Proving in Low-Dimensional Topology.Silvia De Toffoli & Valeria Giardino - 2015 - In Gabriele Lolli, Giorgio Venturi & Marco Panza (eds.), From Logic to Practice. Zurich, Switzerland: Springer International Publishing. pp. 315-336.

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References found in this work

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.
Visual Thinking in Mathematics: An Epistemological Study.Marcus Giaquinto - 2007 - Oxford, England: Oxford University Press.
The Philosophy of Mathematical Practice.Paolo Mancosu - 2009 - Studia Logica 92 (1):137-141.

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