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.