Abstract
Examining different cases, I show that, and how, analogies play a variety of roles in the application of mathematics. Different purposes may involve different cognitive and methodological tasks, some more descriptive – clustering –, others more heuristic – problem-solving and guiding metaphors – and others more evidentiary – robustness analysis and statistical inference in single studies and meta-analyses. Classification, generalization and justification cannot be easily reduced to basic standard patterns of analogical reasoning. I discuss two, however, main kinds of cases: the roles of analogy in the application of mathematics and, connectedly, the role of mathematics in the representation and implementation of analogical cognition.
Moreover, while the relevant similarities get a mathematical expression, the case of machine learning, including the case of similarity metrics and cluster analysis, shows that analogical cognition is not ultimately mathematical. And what I call the analogical circle is virtuous. But I also point to mathematical metaphors and, in relation to mathematical representations of similarity in clustering analysis, raise the issue of the possibility of mathematical representation of metaphoric function itself.
The different cases show in a new way also that the dependence on subjective judgment, research purpose, data characteristics and background information render the application of mathematics and analogical cognition, when connected, unavoidably contextual and plural.
The cases I have presented show also how analogy-driven or aided applications of mathematics connect analogical and synthetic tasks, and that they do so in different ways. They show, more specifically, also that each kind of task is a two-sided coin, so to speak, in scientific cognition and methodology. Synthesis involves also some form of differentiation or analysis, and connectedly, similarity is inseparable from dissimilarity.
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Notes
- 1.
I am grateful to Dedre Gentner for providing the source.
- 2.
Maxwell’s methodological thinking about analogies borrowed from his interests in language and reasoning. He was educated and self-educated in the Aristotelian tradition of logic and rhetoric and in the new formulations of inductive scientific methodology of Herschel, Mill and Whewell. He was also an avid reader of romantic literature and poem writer and a prolific illustrator and draftsman of geometrical and ornamental designs (Cat, 2013). He viewed the application of mathematics accordingly, as a practice of exact but vivid language in the general representation of physical phenomena.
- 3.
Maxwell was explicit about the role of formal analogies. Thus, Maxwell claimed (1) that the correctness of physical analogies depends on the mathematical analogy, namely, that related quantities belong in the same mathematical classes (Maxwell, 1856, 157, 1870, 219), and (2) that mathematical classes are in turn based on mathematical analogies (Maxwell, 1870, 227). Those mathematical analogies are entertained in the mind, however, as far from purely symbolic and formal.
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Maxwell’s analogies between mechanics and electricity or magnetism differed in this sense from William Thomson’s. As Nersessian puts it: ‘Maxwell did not make direct analogies between the domains. Rather, he constructed intermediary, hybrid models that embodied constraints from Newtonian sources and the electromagnetic target domain. Abstractive processes, especially generic modeling, enabled integrating selective constraints from the different domains because they were considered at a level of generality that eliminated the domain-specific differences.’ (Nersessian, 2018, 60)
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Cat, J. (2022). Synthesis and Similarity in Science: Analogy in the Application of Mathematics and Application of Mathematics to Analogy. In: Wuppuluri, S., Grayling, A.C. (eds) Metaphors and Analogies in Sciences and Humanities. Synthese Library, vol 453. Springer, Cham. https://doi.org/10.1007/978-3-030-90688-7_6
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