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Metaphor and Model-Based Reasoning in Maxwell’s Mathematical Physics

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Model-Based Reasoning in Science and Technology

Part of the book series: Studies in Applied Philosophy, Epistemology and Rational Ethics ((SAPERE,volume 8))

Abstract

The role of model-based reasoning in experimental and theoretical scientific thinking has been extensively studied. However, little work has been done on the role of mathematical representations in such thinking. I will describe how the nature of mathematical expressions in physics can be analyzed using an extension of the metaphoric analysis of mathematics. Lakoff and Núñez [29] argued that embodied metaphors underlie basic mathematical ideas (e.g., the concept of “number” is based on the embodied operations of “collecting objects”), with more complex expressions developed via conceptual blends from simpler expressions (e.g., “addition” as “combining collections”). In physics, however, the need to represent physical processes and observed entities (including measurements) places different demands on the blending processes. In model-based reasoning, conceptual blends must often be based on immediately available embodiments as well as highly developed mathematical expressions that draw upon long term working memory. Thus, Faraday’s representations of magnetic fields as “lines of force” were modeled by Maxwell as vectors. In the paper, I compare Faraday’s experimental investigation of the magnetic field within a magnet to Maxwell’s mathematical treatment of the same problem. Both can be understood by unpacking the metaphoric underpinnings as physical representations. The implications for analogical and model-based reasoning accounts of scientific thinking are discussed.

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Notes

  1. 1.

    The embodiment of metaphor will also be assumed here, and is important to the notion of model based reasoning as a species of abduction (e.g., [33]). For further discussion of these issues, see Cat [3], Gooding [21], Nersessian [39], and Tweney [51]. Simpson [42, 41], while emphasizing the rhetoric of Maxwell's Treatise, is advancing a similar argument.

  2. 2.

    Note also that my approach differs from accounts that regard metaphor as a somewhat loose use of similarity, while analogy has been regarded as founded on more severe constraints. See Gentner and Jerzioski [17], which adopts such a view. I am using the two terms in unconventional fashion, with metaphor referring to implicit comparisons and analogy to those drawn explicitly.

  3. 3.

    A good brief introduction to Faraday's work is James [25]. For Maxwell, a good beginning is Everitt [8]. Both works acknowledge the nuances!

  4. 4.

    Maxwell's famous derivation, suggesting that light was an electromagnetic manifestation, appeared initially in the second paper, was re-expressed in the third paper, and finalized at the end of the Treatise.

  5. 5.

    Thomson [47] had considered a problem similar to Maxwell's, in determining the force on a unit pole placed within a small cavity in a magnet. However he did not resolve the directional paradox between the directions of what were later called B and H by Maxwell. Smith and Wise ([44], pp. 279–281) describe Thomson's approach and indicate that he did not fully publish his results.

  6. 6.

    It is interesting to note the similarity of this maneuver to that used by Lakatos in Proofs and refutations, which used a similar ploy to discuss the nature of discovery in mathematical proof [29].

  7. 7.

    Indeed, the terms, analogy and metaphor, have had a flexible boundary in much of the writing about their use in science. Thus, for example, much of what Bradie [1] has written about metaphor applies equally to analogy.

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Acknowledgments

Thanks are due especially to Howard Fisher, who has saved me from many errors and is not responsible for remaining ones! I have benefitted greatly from discussions of Maxwell with John Clement, Howard Fisher, Frank James, Nancy Nersessian, and Thomas Simpson. The paper’s ultimate origin stems from discussions with the late David Gooding and with Elke Kurz-Milcke. The proximate origin is a paper given at MBR012 in Sestri Levante, Italy, in June, 2012; I am grateful for the questions and comments of the other participants and to Lorenzo Magnani for his support. Matt Lira and Frank James provided helpful comments on an early draft, for which I am grateful.

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  1. Bowling Green State University, Bowling Green, OH, 43403, USA

    Ryan D. Tweney

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Correspondence to Ryan D. Tweney .

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  1. Department of Humanities Philosophy Section, University of Pavia, Pavia, Italy

    Lorenzo Magnani

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Tweney, R.D. (2014). Metaphor and Model-Based Reasoning in Maxwell’s Mathematical Physics. In: Magnani, L. (eds) Model-Based Reasoning in Science and Technology. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37428-9_21

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