In response to Grasshoppers and linear proofs, Avery Andrews said:
Maybe a related question is how much time people do/ought spend on really mastering the proofs of theorems in textbooks, ‘mastering’ being, say, able to explain it in any desired amount of detail at least 2 weeks after last looking at it.
There are two different goals:
- Mastering the proof of a theorem in a textbook so that you can explain it in any desired amount of detail…
- Mastering a proof of the theorem so that you can explain it in any desired amount of detail…
My observation is that most research mathematicians don’t attempt (1); they are satisfied with (2). Trying to understand a written proof in detail can be quite difficult:
- The author may use misleading language.
- The author may jump over a piece of reasoning that to them is obvious but not to you.
- The author may mention a previous step or a theorem that justifies the current step, but get the reference wrong.
And so on.
In my observation the typical mathematician will look at the proof, perhaps getting some idea of the overall strategy of the whole proof or a particular part, and then think about it independently until they come up with a proof or part of it. This may or may not be what the author had in mind. But by thinking through it the reader will solidify their understanding of the proof in a way that reading and rereading step by step is unlikely to do.
When you construct your knowledge like that you are likely to have it in a permanent, well semi-permanent, way.
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