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Proof, Explanation, and Justification in Mathematical Practice

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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.

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Notes

  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).

  2. On the question, “What are mathematical explanations?” see Inglis and Mejía-Ramos (2019). Again, this question is beyond the scope of this paper. The focus of this paper is “the problem of whether mathematical explanations occur within mathematics itself” (Mancosu 2018). That is, in mathematical practice (specifically, in the published work of practicing mathematicians), are there “proofs that merely prove” or “proofs that are in some way enlightening” as well?

  3. Cf. Pease et al. (2018) who report some empirical support for their conjecture that “there is such a thing as explanation in mathematics.” Their empirical study, however, was not designed to find out how frequently explanations and justifications occur in mathematics.

  4. The empirical methods employed in this paper are the methods of data science and corpus linguistics, such as text mining and corpus analysis, rather than the empirical methods of social science. For examples of the former methods applied to questions in the philosophy of mathematics and logic, see Pease et al. (2018) and Mizrahi (2019). For an example of the latter methods applied to questions in the philosophy of mathematics, see Inglis and Aberdein (2014).

  5. Cf. Pease et al. (2018) who use ‘expla*’ and ‘underst*’ as explanation indicators.

  6. Dutilh Novaes (2019, 73) argues that, “In an explanatory proof, there should be no surprises: each step in the proof must be clear and evident, eliciting immediate understanding in whoever inspects the proof, thus ruling out unexpected ‘turns’”.

  7. For more on the relationship between philosophy of mathematics and argumentation theory, see Pease et al. (2009). Ashton and Mizrahi (2018a) use a similar methodology and the tools of data science to investigate appeals to intuition in philosophy. See also Ashton and Mizrahi (2018b) and Mizrahi (2019).

  8. The jcodes for the other mathematics journals tested in this empirical study are as follows: American Mathematical Monthly (amermathmont), Annals of Mathematics (annamath), Journal of Computational Mathematics (jcompmath), and Journal of the American Mathematical Society (jamermathsoci).

  9. See, e.g., McLarty (2008) for a use of a case study in the philosophy of mathematics.

  10. On methodological issues in philosophy of mathematics, see Cellucci (2013). On the use of case studies in philosophy of science, see Mizrahi (2020).

  11. For more on the application of text mining and corpus analysis methods to philosophy of logic and mathematics, see Pease et al. (2018) and Mizrahi (2019).

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Acknowledgements

I am grateful to two anonymous reviewers of the Journal for General Philosophy of Science for their helpful comments on an earlier draft of this paper.

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  1. School of Arts and Communication, Florida Institute of Technology, 150 W. University Blvd., Melbourne, FL, 32901, USA

    Moti Mizrahi

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Mizrahi, M. Proof, Explanation, and Justification in Mathematical Practice. J Gen Philos Sci 51, 551–568 (2020). https://doi.org/10.1007/s10838-020-09521-7

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