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Mathematical Explanation in Practice

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Abstract

The connection between understanding and explanation has recently been of interest to philosophers. Inglis and Mejía-Ramos (Synthese, 2019. https://doi-org.proxy.library.nd.edu/10.1007/s11229-019-02234-5) propose that within mathematics, we should accept a functional account of explanation that characterizes explanations as those things that produce understanding. In this paper, I start with the assumption that this view of mathematical explanation is correct and consider what we can consequently learn about mathematical explanation. I argue that this view of explanation suggests that we should shift the question of explanation away from why-questions and towards a “what’s going on here” question. Additionally, I argue that when we recognize the connection between understanding and explanation we naturally see how more than just proof can be explanatory. I expand this point by detailing how definitions and diagrams can be explanatory. In all, we see that when we take seriously the connection between understanding and explanation, we get a better sense of how explanation arises within mathematics.

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Notes

  1. Wilkenfeld’s account of functional explanation can be found in Wilkenfeld (2014).

  2. See Steiner (1978), Kitcher (1981), Kitcher (1989), and Lange (2017) for the details of these accounts.

  3. I point an interested reader to the following articles that comment on these accounts: D’Alessandro (2019), Hafner and Mancosu (2005), Hamami and Morris (2020), Inglis and Mejía-Ramos (2019).

  4. It’s worth noting that some authors who have written about explanatory proof have acknowledged that explanatory proof is not all there is to mathematical explanation. For instance, Steiner (1978), Lange (2017), and Lange (2018) acknowledge that there is more than just explanatory proof.

  5. Skow (2015) has argued that explanations (i.e., answers to why-questions) do not generally produce understanding. I do not mean to endorse this more general claim, but am instead restricting my claim to the domain of mathematics. As a result, I am suggesting that there may be something about mathematical explanation that makes it significantly different from explanation in other domains.

  6. See Crow (1998) and Kline (1972) for discussions of the introduction of imaginary numbers and the initial skeptical response from the mathematical community.

  7. See Baumberger (2014) for a clear discussion of some of the different kinds of understanding.

  8. For a related discussion, see de Regt (2017) (ch 3) which argues that a pluralist view of scientific understanding is needed.

  9. Note that if a function is differentiable then it is also continuous, and so the belief that if a function is continuous then it is differentiable amounts to a conflation of the two notions.

  10. See Manheim (1964), chapter 4, for a detailed discussion of the discovery of these functions and the functions themselves. There is also a detailed discussion of the work being done in the nineteenth century to improve our understanding of functions and their properties in Kline (1972), chapter 40.

  11. These are the clearest ways that a function can be continuous but non-differentiable at a point. But there is another way that a function can be continuous and non-differentiable at a point. Specifically, this will happen when a function has a vertical tangent line at a point.

  12. See Giaquinto (2011) for a discussion of the visualizability of nowhere differentiable continuous functions and their failure to be “pencil continuous”.

  13. Figure 5 was created by Sandy Ganzell and is used here with his permission.

  14. It is worth noting that the diagram in Fig. 6 corresponds to the virtual trefoil knot. But this is not the standard representation of the virtual trefoil knot, though it is equivalent and so can be manipulated into the standard presentation by means of a series of virtual and classical Reidemeister moves.

  15. An example of such an invariant is the forbidden number, see Crans et al. (2015).

  16. Some other authors have also considered mathematical explanations that appear in forms other than proof. See D’Alessandro (2020), Lange (2018), Lehet (2021).

  17. This account of explanatory definitions was first introduced in Lehet (2021).

  18. This is a fact that we used earlier when proving that for prime natural number, p, \(\sqrt{p}\) is irrational.

  19. For the sake of simplicity, I am limiting the discussion to circles with centers at the origin.

  20. To clarify, these definitions are logically equivalent when the first definition is restricted to taking the center of the circle to be the origin or when the second definition is stated in full generality.

  21. Or at least, we come across approximations of circles that are indistinguishable from the real thing to the human eye.

  22. Again, for the sake of simplicity, I am assuming that the center of the ellipse is the origin.

  23. See Brown (2008), De Toffoli (2017), De Toffoli and Giardino (2014), De Toffoli and Giardino (2016), Giaquinto (2008), Mancosu (2008) for discussions of the epistemic significance of diagrams. Carter (2019) also provides an overview of what philosophical work has been done regarding the use of diagrams in mathematics.

  24. It is worth noting that this diagram is a variation of the one given in Steiner (1978), which uses two isosceles triangles to form a square rather than a rectangle. A square can be formed when we let the hypotenuses of the two triangles overlap.

  25. Here I have only mentioned two examples of texts that have focused on a diagrammatic presentation of material. As Mancosu (2008) points out, the late twentieth century produced many other examples of texts that take a visual approach to mathematics.

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Acknowledgments

I would like to thank several audiences for their feedback on the ideas presented in this paper including those at the Explanation and Understanding within Mathematics Workshop, hosted by Vrije Universiteit Brussel, and at the University of Notre Dame Colloquium. Additionally, I would like to thank Curtis Franks, Colin McLarty, Timothy Bays, and Patricia Blanchette for their helpful feedback and comments on several drafts of this work. I was first introduced virtual knot theory by Sandy Ganzell and thank him for this introduction as well as for Fig. 5 which he created. Lastly, I would like to thank several anonymous referees for their detailed comments.

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    Ellen Lehet

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Lehet, E. Mathematical Explanation in Practice. Axiomathes 31, 553–574 (2021). https://doi.org/10.1007/s10516-021-09557-4

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