On mathematicians’ disagreements on what constitutes a proof
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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.