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Unificatory Understanding and Explanatory Proofs

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Abstract

One of the central aims of the philosophical analysis of mathematical explanation is to determine how one can distinguish explanatory proofs from non-explanatory proofs. In this paper, I take a closer look at the current status of the debate, and what the challenges for the philosophical analysis of explanatory proofs are. In order to provide an answer to these challenges, I suggest we start from analysing the concept understanding. More precisely, I will defend four claims: (1) understanding is a condition for explanation, (2) unificatory understanding is a type of explanatory understanding, (3) unificatory understanding is valuable in mathematics, and (4) mathematical proofs can contribute to unificatory understanding. As a result, in a context where the epistemic aim is to unify mathematical results, I argue it is fruitful to make a distinction between proofs based on their explanatory value.

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Notes

  1. Until the seventeenth century, as Mancosu (1996) demonstrates, it was not uncommon to think about a sort of causal explanation in mathematics due to the influence of Aristotle’s ideas.

  2. See Hempel (1963), p. 126.

  3. See Kitcher (1989), p. 437.

  4. See, for example, Mancosu’s distinction between local and global cases of mathematical explanation (2008).

  5. Aesthetics, intricacy, precision and utility.

  6. See Frans and Weber (2017).

  7. An earlier version of this account can be found in Delarivière et al. (2017).

  8. Such values might be important for judgements concerning good or bad explanations. We can expect that we prefer a simple explanatory proof above a complicated explanatory proof. The question at hand now is, nevertheless, when we can say a proof is explanatory or not.

  9. See, for example, the collections of papers in De Regt et al. (2009), Grimm et al. (2016).

  10. Note that this does not signify that understanding without explanation is impossible. It might be that we acquire understanding in other ways. One might say that it is possible to gain an understanding of a mathematical method, for example, through familiarising ourselves with this method through experience.

  11. For example, Schurz adds a causality requirement to his model, and the use of terminologies such as data and hypotheses complicate an obvious application to mathematics. Furthermore, the fact that mathematical truths follow deductively from axioms means that they are never genuinely unassimilated.

  12. Note that this is a somewhat intuitive reading of the concept coherence and not, as a referee pointed out, a formal criterion of coherence. The idea, as will be demonstrated in the remainder of the paper, is that there is a relevant epistemic difference in proving related theorems using similar proofs or using dissimilar proofs. In order to make this into a full account of explanation and understanding, the notions of coherence and similarity should be developed more in detail. In this paper, I will not take on this challenge.

  13. As in the discussion provided by Lange (2014).

  14. The following work is not discussed in detail here. Macbeth (2012) discusses the relation between formal proofs and mathematical understanding, where understanding is related to contentful mathematical ideas. Folina (2018) proposes a structuralist perspective to improve our grasp on mathematical understanding. In the philosophy of mathematics education, there is attention to the topic [e.g. Sierpinska (1990)], but I am hesitant to build on such models since the situation of understanding in the classroom is significantly shaped by the presence of an instructor and teacher.

  15. Note that I have identified these cases based on a conceptual analysis in the reflective approach. Before making claims about the activities of actual mathematicians, one needs to confront these findings with mathematical practice. If they recognise these cases as explanatory, we have a valid starting point to assess the notion of mathematical explanation and unificatory understanding in mathematical practice. If they do not recognise theses cases as explanatory, there is still a valid starting point for future work. After all, if the conceptual analysis is reliable, we can start discussing why mathematicians are not interested in providing these type of explanations or why they do not identify certain activities as providing explanations.

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  1. Vrije Universiteit Brussel, Pleinlaan 2, 1050, Brussels, Belgium

    Joachim Frans

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Frans, J. Unificatory Understanding and Explanatory Proofs. Found Sci 26, 1105–1127 (2021). https://doi.org/10.1007/s10699-020-09654-4

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