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Added by: Fenner Stanley TanswellAbstract: 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.

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Added by: Fenner Stanley TanswellAbstract: The aim of this article is to investigate speciﬁc aspects connected with visualization in the practice of a mathematical subﬁeld: lowdimensional topology. Through a case study, it will be established that visualization can play an epistemic role. The background assumption is that the consideration of the actual practice of mathematics is relevant to address epistemological issues. It will be shown that in lowdimensional topology, justiﬁcations can be based on sequences of pictures. Three theses will be defended. First, the representations used in the practice are an integral part of the mathematical reasoning. As a matter of fact, they convey in a material form the relevant transitions and thus allow experts to draw inferential connections. Second, in lowdimensional topology experts exploit a particular type of manipulative imagination which is connected to intuition of two and threedimensional space and motor agency. This imagination allows recognizing the transformations which connect diﬀerent pictures in an argument. Third, the epistemic—and inferential—actions performed are permissible only within a speciﬁc practice: this form of reasoning is subjectmatter dependent. Local criteria of validity are established to assure the soundness of representationally heterogeneous arguments in lowdimensional topology.
Comment (from this Blueprint): De Toffoli and Giardino look at proof practices in lowdimensional topology, and especially a proof by Rolfsen that relies on epistemic actions on a diagrammatic representation. They make the case that the many diagrams are used to trigger our manipulative imagination to make inferential moves which cannot be reduced to formal statements without loss of intuition.

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Added by: Fenner Stanley TanswellAbstract: In this paper we first explore how Wittgenstein’s philosophy provides a conceptual tools to discuss the possibility of the simultaneous existence of culturally different mathematical practices. We will argue that Wittgenstein’s later work will be a fruitful framework to serve as a philosophical background to investigate ethnomathematics (Wittgenstein 1973). We will give an overview of Wittgenstein’s later work which is referred to by many researchers in the field of ethnomathematics. The central philosophical investigation concerns Wittgenstein’s shift to abandoning the essentialist concept of language and therefore denying the existence of a universal language. Languages—or ‘language games’ as Wittgenstein calls them—are immersed in a form of life, in a cultural or social formation and are embedded in the totality of communal activities. This gives rise to the idea of rationality as an invention or as a construct that emerges in specific local contexts. In the second part of the paper we introduce, analyse and compare the mathematical aspects of two activities known as string figuremaking and sand drawing, to illustrate Wittgenstein’s ideas. Based on an ethnomathematical comparative analysis, we will argue that there is evidence of invariant and distinguishing features of a mathematical rationality, as expressed in both string figuremaking and sand drawing practices, from one society to another. Finally, we suggest that a philosophicalanthropological approach to mathematical practices may allow us to better understand the interrelations between mathematics and cultures. Philosophical investigations may help the reflection on the possibility of culturally determined ethnomathematics, while an anthropological approach, using ethnographical methods, may afford new materials for the analysis of ethnomathematics and its links to the cultural context. This combined approach will help us to better characterize mathematical practices in both sociological and epistemological terms.
Comment (from this Blueprint): Francois and Vandendriessche here present a later Wittgensteinian approach to “ethnomathematics”: mathematics practiced outside of mainstream Western contexts, often focused on indigenous or tribal groups. They focus on two case studies, stringfigure making and sanddrawing, in different geographic and cultural contexts, looking at how these practices are mathematical.

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Added by: Fenner Stanley TanswellAbstract: In recent years, philosophical work directly concerned with the practice of mathematics has intensified, giving rise to a movement known as the philosophy of mathematical practice. In this paper we offer a survey of this movement aimed at mathematics educators. We first describe the core questions philosophers of mathematical practice investigate as well as the philosophical methods they use to tackle them. We then provide a selective overview of work in the philosophy of mathematical practice covering topics including the distinction between formal and informal proofs, visualization and artefacts, mathematical explanation and understanding, value judgments, and mathematical design. We conclude with some remarks on the potential connections between the philosophy of mathematical practice and mathematics education.
Comment (from this Blueprint): While this paper by Hamami & Morris is not a necessary reading, it provides a fairly broad overview of the practical turn in mathematics. Since it was aimed at mathematics educators, it is a very accessible piece, and provides useful directions to further reading beyond what is included in this blueprint.

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Added by: Fenner Stanley TanswellAbstract: The low visibility and specialised languages of mathematical work pose challenges for the ethnographic study of communication in mathematics, but observationbased study can offer a realworld 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.
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.