When reading a proof, we not only want to be able to check its correctness, but desire to understand it. Philosophers of mathematical practice have started to analyze what it means to understand a proof, for example see (Avigad 2008). In this talk, I will consider the different senses in which a reader may understand a particular step in a proof, the associated benefits of such understanding, and how it may be promoted or stifled. To do this, I will examine case studies from number theory, comparing different proofs of the same theorem. Using these case studies, I identify two types of understanding associated with proof steps: (i) understanding where the proof step comes from; (ii) understanding how the proof step contributes to the overall argument. I suggest that these two kinds of understanding work together to allow the reader to reuse the proof step effectively in other contexts, and thus have a practical benefit. I then argue that the choice of notation, as well as the structure of the proof, can foster, or suppress, such understanding.
Jeremy Avigad, “Understanding Proofs,” The Philosophy of Mathematical Practice (Paolo Mancosu, editor), Oxford University Press, Great Clarendon Street, Oxford OX2 6DP, 2008 pp. 317-353.