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Experimenting with Diagrams in Mathematics

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Diagrammatic Representation and Inference (Diagrams 2020)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12169))

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Abstract

In this paper a way of understanding differences between diagrams is proposed, based on how their spatial properties relate to conceptual content they refer to and on how this content is assigned to diagrams. It is also argued that some uses of one of the distinguished classes of diagrams can be construed as a specific kind of experiment.

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Notes

  1. 1.

    In [5] it is proposed to take Cartesian diagrams as the third subclass of mathematical diagrams.

  2. 2.

    It should be noted that Stanisław Ulam came up with this way of representing the set of natural numbers while playing around with a pencil and sheet of paper. This fits well the described practice of “experimenting with representation”.

  3. 3.

    Cartesian diagrams are various visualizations that make use of the Cartesian coordinate system to represent objects that can be characterized by two numeric values such as a function with single real argument and value, two numeric properties of an object or a set of pairs of real numbers which satisfy a certain property P(x,y).

  4. 4.

    It may also happen that the translation is made into a geometric language, e.g. when properties of numbers are investigated after representing them as line segments or areas. In this case the geometric diagram should also be treated as “conventional” diagram.

References

  1. Barceló Aspeitia, A.A.: Mathematical pictures. In: Chapman, P., Stapleton, G., Moktefi, A., Perez-Kriz, S., Bellucci, F. (eds.) Diagrams 2018. LNCS (LNAI), vol. 10871, pp. 137–147. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-91376-6_15

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  6. Macbeth, D.: Seeing how it goes: paper-and-pencil reasoning in mathematical practice. Philosophia Mathematica (III) 20, 58–85 (2012)

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Correspondence to Michał Sochański .

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Sochański, M. (2020). Experimenting with Diagrams in Mathematics. In: Pietarinen, AV., Chapman, P., Bosveld-de Smet, L., Giardino, V., Corter, J., Linker, S. (eds) Diagrammatic Representation and Inference. Diagrams 2020. Lecture Notes in Computer Science(), vol 12169. Springer, Cham. https://doi.org/10.1007/978-3-030-54249-8_45

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  • DOI: https://doi.org/10.1007/978-3-030-54249-8_45

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  • Print ISBN: 978-3-030-54248-1

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