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The Parallel Structure of Mathematical Reasoning

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The Argument of Mathematics

Part of the book series: Logic, Epistemology, and the Unity of Science ((LEUS,volume 30))

Abstract

This chapter defends an account of mathematical reasoning as comprised of two parallel structures. The argumentational structure is composed of arguments by means of which mathematicians seek to persuade each other of their results or, more generally, to achieve goals appropriate for whatever dialogue they are having. The inferential structure is composed of derivations which offer a formal counterpart to these arguments. The precise relationship between the two structures may be understood in terms of the range of argumentation schemes which may be instantiated by steps of the argumentational structure. Just as different views about the foundations of mathematics may be characterized in terms of the admissibility of steps in the inferential structure, different views about mathematical practice may be characterized in terms of the admissibility of steps in the argumentational structure.

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Notes

  1. 1.

    Such ‘pointing’ may bring to mind Jody Azzouni’s ‘derivation indicator view’ of mathematical practice (Azzouni, 2004). However, Azzouni’s ‘indicating’ describes a looser correspondence, closer to that holding in general between the two structures. Moreover, at least on some construals, such as that in (Dove, 2013, 304), Azzouni’s conception of derivation is broader than mine.

  2. 2.

    This would seem to be an attempt to characterize what would later be described as abductive reasoning, or inference to the best explanation.

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Acknowledgements

Previous versions of parts of this chapter were delivered in Vienna, St Andrews, Gainesville, FL, Windsor, ON, and Birmingham. I am grateful to the audiences for helpful discussion. I am also grateful to Ian Dove for insightful comments and to Alison Pease for ideas developed during our collaboration on (Pease and Aberdein, 2011).

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  1. Department of Humanities and Communication, Florida Institute of Technology, 150 West University Boulevard, Melbourne, FL, 32901-6975, USA

    Andrew Aberdein

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  1. , Dept. Humanities & Communication, Florida Institute of Technology, W. University Blvd. 150, Melbourne, 32901-6975, Florida, USA

    Andrew Aberdein

  2. , Department of Philosophy, University of Nevada, Maryland Parkway 4505, Las Vegas, 89154, Nevada, USA

    Ian J Dove

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Aberdein, A. (2013). The Parallel Structure of Mathematical Reasoning. In: Aberdein, A., Dove, I. (eds) The Argument of Mathematics. Logic, Epistemology, and the Unity of Science, vol 30. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6534-4_18

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