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Carter, Jessica. Diagrams and Proofs in Analysis
2010, International Studies in the Philosophy of Science, 24(1): 1-14.
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Added by: Fenner Stanley Tanswell
Abstract: This article discusses the role of diagrams in mathematical reasoning in the light of a case study in analysis. In the example presented certain combinatorial expressions were first found by using diagrams. In the published proofs the pictures were replaced by reasoning about permutation groups. This article argues that, even though the diagrams are not present in the published papers, they still play a role in the formulation of the proofs. It is shown that they play a role in concept formation as well as representations of proofs. In addition we note that 'visualization' is used in two different ways. In the first sense 'visualization' denotes our inner mental pictures, which enable us to see that a certain fact holds, whereas in the other sense 'visualization' denotes a diagram or representation of something.

Comment (from this Blueprint): In this paper, Carter discusses a case study from free probability theory in which diagrams were used to inspire definitions and proof strategies. Interestingly, the diagrams were not present in the published results making them dispensable in one sense, but Carter argues that they are essential in the sense that their discovery relied on the visualisation supplied by the diagrams.

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McCallum, Kate. Untangling Knots: Embodied Diagramming Practices in Knot Theory
2019, Journal of Humanistic Mathematics, 9(1): 178-199.
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Added by: Fenner Stanley Tanswell
Abstract: The low visibility and specialised languages of mathematical work pose challenges for the ethnographic study of communication in mathematics, but observation-based study can offer a real-world grounding to questions about the nature of its methods. This paper uses theoretical ideas from linguistic pragmatics to examine how mutual understandings of diagrams are achieved in the course of conference presentations. Presenters use shared knowledge to train others to interpret diagrams in the ways favoured by the community of experts, directing an audience’s attention so as to develop a shared understanding of a diagram’s features and possible manipulations. In this way, expectations about the intentions of others and appeals to knowledge about the manipulation of objects play a part in the development and communication of concepts in mathematical discourse.

Comment (from this Blueprint): McCallum is an ethnographer and artist, who in this piece explores the way in which mathematicians use diagrams in conference presentations, especially in knot theory. She emphasises that there are a large number of ways that diagrams can facilitate communication and understanding. The diagrams are dynamic in many way, and she shows how the way in which a speaker interacts with the diagram (through drawing, erasing, labelling, positioning, emphasising etc.) is part of explaining the mathematics it represents.

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