Abstract
This chapter proposes an analysis of analogical arguments in mathematics. Such arguments are used to show that mathematical conjectures are plausible. The core idea of the chapter is that there is a strong link between good analogies and fruitful generalization. Specifically, a good analogical argument in mathematics is one that articulates a clear proof that is ‘fit for imitation’. This idea is first stated as a simple test, and then refined and deepened through a set of models for relationships of similarity that are important in mathematical analogies. The chapter concludes by addressing the philosophical basis for analogical reasoning and the use of analogies in extended mathematical research programs.
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Notes
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Of course, Herschel is speaking of empirical hypotheses. The vera causa requirement cannot be applied literally to mathematical conjectures.
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Indeed, the probabilistic conception of plausibility is important, and can be useful if we wish to model the strength of an analogical argument (in mathematics or elsewhere). My contention here is simply that the modal conception of plausibility is both ubiquitous and fundamental in mathematics.
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For a very different set of tensions between the structure-mapping approach and mathematical analogies, see (Schlimm, 2008).
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The test makes use of the terminology introduced in Sect. 12.2.
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Lacking an account of similarity at this point, I simply assert that there is a natural correspondence here. Shortly, we shall describe this as a case of geometric similarity.
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See Lakatos (1976) for the history and limitations of this formula.
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For a polygon, F (the number of faces) is 1; for a polyhedron, S (the number of solid elements) is 1.
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The full generalization involves concepts of algebraic topology; see Munkres (1984).
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The fraktur variables—representing relations, functions and constants—are free in Ψ. The other relation, function and constant symbols denote entities in the source and target domains.
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A modified result is true: if \({(ab)}^{3} = {a}^{3}{b}^{3}\) for all a, b in T, and the number of elements in T is not divisible by 3, then T is abelian.
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In Section 4.10.
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The proposal can be generalized to cases where geometric similarity is defined using more than one parameter, although this requires the introduction of a metric.
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Again, see Bartha (2010) , section 4.10, for a full discussion.
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If this sum of positive terms converges to a finite limit, then it is absolutely convergent and so the order of summation is unimportant.
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Versions of these principles are identified by van Fraassen (1989).
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Acknowledgements
This chapter is based largely on material in my book (Bartha, 2010), with some additions and clarifications. Sections of that book are reproduced here by kind permission of Oxford University Press. Many issues that are neglected or only briefly mentioned here are discussed in the book. I also wish to acknowledge the helpful contributions of the editors.
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Bartha, P. (2013). Analogical Arguments in Mathematics. 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_12
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