On mathematicians’ disagreements on what constitutes a proof

ABSTRACT

We report the results of a study in which we asked 94 mathematicians to evaluate whether five arguments qualified as proofs. We found that mathematicians disagreed as to whether a visual argument and a computer-assisted argument qualified as proofs, but they viewed these proofs as atypical. The mathematicians were also aware that many other mathematicians might not share their judgment and viewed their own judgment as contextual. For typical proofs using standard inferential methods, there was a strong consensus amongst the mathematicians that these proofs were valid. An instructional consequence is that for the standard inferential methods covered in introductory proof courses, we should have the instructional goal that students appreciate why these inferential methods are valid. However, for controversial inferential methods such as visual inferences, students should understand why mathematicians have not reached a consensus on their validity.

KEYWORDS:

• Agreement
• mathematicians
• proof

Acknowledgements

We are grateful to Matthew Inglis, Kristen Lew, Kate Melhuish, and the anonymous reviewers for helpful comments on earlier drafts of this manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Throughout the paper, the term “mathematics educators” refers to researchers in mathematics education.

2 It is not necessarily the case that the mathematical community should be defined in this way (Stillman, Brown, & Czocher, in preparation). We only observe that this is how the mathematical community is usually operationalized in mathematics education studies, in which participants are recruited from mathematics staff and the opinions of research-active mathematicians are cited. We also do not claim that the mathematical community that we described is homogeneous. Indeed, one purpose of this paper is to highlight their heterogeneity.

3 For a notable example, see the well-known debates about the relationship between argumentation and proof (Balacheff, 1999; Boero, 1999; Duval, 1999). These debates hinged on the extent that a proof needs to be a structured argument highlighting logical dependency within an axiomatic system (c.f., Balacheff, 1999; Mariotti, 2006), a matter that is still being disputed today.

4 Stylianides tailored his definition to be appropriate to classrooms and included that the accepted statements should be known by the classroom community and the inferential schemes should be accepted by, or within the conceptual grasp, of the community. In this paper, we only discuss statements and inferential methods that are within the conceptual grasp of the professional mathematical community, so these important nuances for classroom proofs will not be relevant here.

5 These journals include Educational Studies in Mathematics (Mejia-Ramos & Weber, 2014), Cognition and Instruction (Inglis & Mejia-Ramos, 2009a; Lai, Weber, & Mejia-Ramos, 2012), and Research in Mathematics Education (Weber, 2013).

6 Qualtrics is a software company that allows researchers to conduct on-line surveys.