Proof, Explanation, and Justification in Mathematical Practice
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Proof, Explanation, and Justification in Mathematical Practice Moti Mizrahi1
© Springer Nature B.V. 2020
Abstract In this paper, I propose that applying the methods of data science to “the problem of whether mathematical explanations occur within mathematics itself” (Mancosu 2018) might be a fruitful way to shed new light on the problem. By carefully selecting indicator words for explanation and justification, and then systematically searching for these indicators in databases of scholarly works in mathematics, we can get an idea of how mathematicians use these terms in mathematical practice and with what frequency. The results of this empirical study suggest that mathematical explanations do occur in research articles published in mathematics journals, as indicated by the occurrence of explanation indicators. When compared with the use of justification indicators, however, the data suggest that justifications occur much more frequently than explanations in scholarly mathematical practice. The results also suggest that justificatory proofs occur much more frequently than explanatory proofs, thus suggesting that proof may be playing a larger justificatory role than an explanatory role in scholarly mathematical practice. Keywords Corpus linguistics · Data science · Explanation · Justification · Mathematical practice · Proof · Text mining
1 Introduction One of the central questions in the philosophy of mathematics is whether mathematical explanations occur in mathematical practice (Mancosu 2018).1 According to Hanna et al. (2010, 2), “philosophers of mathematics have turned their attention more and more from the justificatory to the explanatory role of proof” (emphasis in original). This suggests that proof plays an equal, dual role in mathematics: a justificatory role and an explanatory role. As Zelcer (2013, 173) puts it, “the distinction between proofs that merely prove and proofs 1 Cf. Mancuso (2008b, 135) on the problem of “giving an account of mathematical explanation of empirical phenomena,” which is a different problem from the one about the explanatory and justificatory role of proof in mathematics. In this paper, I am concerned with the latter, not the former. See also Mancuso (2018). For recent work on mathematical explanations in science, see Andersen (2018) and Pincock (2015).
* Moti Mizrahi [email protected] 1
School of Arts and Communication, Florida Institute of Technology, 150 W. University Blvd., Melbourne, FL 32901, USA
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that are in some way enlightening […] has attracted the attention of philosophers” lately (emphasis added).2 Now, instances of the word ‘explain’ and its cognates are not difficult to find in mathematics. Here are a few examples (emphasis added): Our proof explains this bump in graph theoretic terms (Cherlin 2016, 342). The above proof explains why we consider weak solutions of FBSDEs associated with the problem (Rozkosz 2013, 1079). There are also instances of mathematicians reflecting on their practice and saying that proofs are ex
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