Proof, Explanation, and Justification in Mathematical Practice

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.

Epistemic Markers in the Scientific Discourse

“…Computational approaches have turned this intuition into effective algorithms and methods that make it possible to mine word patterns so as to shed light on the semantic content of specific corpora (Aggarwal 2015). Such approaches are starting to be used in history and philosophy of science (HPS) studies (Overton 2013; Pence and Ramsey 2018; Malaterre et al 2019; Noichl 2021; Mizrahi 2020; Khelfaoui et al 2021). Here, we focus on terminological frequency analyses and the identification of correlation patterns between groups of terms (notably, between what we call “epistemic fields”).…”

Modeling and corpus methods in experimental philosophy

“…Among the first group, we find four papers (Mejía-Ramos et al, 2019;Mizrahi, 2020b;Overton, 2013;Pease et al, 2019) that studied how scientists and mathematicians use terms linked to the concept of explanation.…”

“…Among the first group, we find four papers (Mejía‐Ramos et al., 2019; Mizrahi, 2020b; Overton, 2013; Pease et al., 2019) that studied how scientists and mathematicians use terms linked to the concept of explanation. Tallant (2013) and Andow (2015) did similar work with the notion of intuition, and Mizrahi (2020a) worked on the concept of progress.…”

“…However, they seem to implicitly assume that using explanation words and describing oneself as explaining are the same—one might think that this assumption should be motivated. Finally, the approach adopted by Mizrahi (2020b) is the one that raises the most concerns, as it interprets the use of words like “explain” not even as a reference to the concept of explanation, but as indicating that the text excerpt that contains this segment is an explanation. But obviously people do not label their discourse as they are expressing it.…”