1 Ways Mathematicians Read Proofs Annie Selden Originally written May 2004 Updated December 17, 2017 Just as with the reading of any text, one's reading of mathematical proofs depends upon one's purposes. Does one just want to get the gist of an argument so one is well-informed about current mathematical discoveries generally, or in one's subfield, such as number theory or differential equations? Does one want to find out whether a result, and its proof, are relevant to one's own research? Does one think one can actually use a result in one's own research? Has one been asked to blind referee a manuscript for publication in a journal? Is one looking over a dissertation as a member of a PhD student's committee? Is one grading a graduate course examination, consisting entirely of proofs and counterexamples, in a course such as real analysis? Is one grading an examination, consisting of proofs, in undergraduate transition-to proof course? Some empirical work has been done on these topics. We have suggested a hypothetical mathematician’s validation of a proof of a calculus theorem (Appendix, Selden & Selden, 1995), not so much because we think this particular proof would necessarily be validated by mathematicians, but rather because it was rather short and illustrated the points we were trying to make. We did an exploratory study of beginning transition-to-proof course students validation of purported proofs (Selden & Selden, 2003). Subsequently, Weber (2008) asked mathematicians to validate purported number theory proofs, assuming certain contexts, using some of the purported proofs of our validation paper (Selden & Selden, 2003) and other more difficult ones. Further studies have been done on proof comprehension to see how mathematicians read proofs, including a large survey study (Mejia-Ramos & Weber, 2014; Weber, Inglis, Mejia-Ramos, 2014). Recently, Moore (2016) has asked mathematicians to evaluate transition-to-proof course students’ proof attempts and comment on them. This short paper is meant to provoke further thought, as well as research on the topic of validation and on reading proofs more generally. Validation is both a separate activity from proving, when one is reading someone else's proof, and an integral part of it, when one is reading and checking over one's own proofs in the hope of eventual publication or for submission in a course. Validation and reading can occur at various levels of depth, or carefulness, when reading/checking a proof. One can: 1) Read the entire proof relatively quickly to get the structure or "gist" of the argument. This is a form of reading or skimming to gain information. 2) Read a part of a proof to understand a new technique that one might want to apply in one's own research. This can have aspects of reading both for understanding and for correctness.