1. What’s going on mathematically?
Trying to understand a theorem, and the proof techniques used.
2. What is the emotional and logistical context?
This is relatively straightforward. It takes concentration, but the concentration sort of builds on itself. It’s like reading a really engaging story – you get roped in. So it has to be pretty quiet and peaceful, but my attention is not that frail and I’ll get into “the zone” pretty quickly. It’s a good one for when I feel like feeling like a mathematician, but have some procrastination/anxiety/staleness. So I may start out slow, but pretty quickly feel happy. (I tried to do this yesterday, after rock climbing all day, and while eating dinner, getting devoured by mozzies, and sitting in an uncomfortable chair, but I failed. After a good night’s sleep and breakfast and coffee, it was effortless.) I must do it with a pad of paper, to scribble out drawings, diagrams, and recapitulations (i.e. I rewrite the sentence on my own, to somehow concretize every single word).
3. What thoughts are there?
It’s in no sense easy, but is easier than some other flavors; because I get so roped in, it’s easy to proceed. But it really demands that you harness all your knowledge about the concepts on hand – every sentence usually has two or three clauses, each of which requires an intense “Why is this true?” To figure this out, I need to draw on things I learned a long time ago, or recently, or vague connections between ideas, and often look up definitions or other results. When I can’t figure out steps, I keep track of those failures, or come up with various provisional explanations, whose correctness I can usually establish by the end of the proof, using the test of consistency. Some leaps are simply too great, and I quickly give up hope of understanding them; I used to ask my advisor about all of these technicalities, and sometimes still do. In the end, I (hopefully) can find a simple conceptual organization (“what’s really going on?”, “what’s the essence?”, “what’s the shape of the proof?”) within the proof, quite different from the original narrative.
4. What quality of awareness?
Quite single-pointed and dialogical, narrative. It’s very intimate and microscopic, tracing the smallest steps of logic and understanding, at a nice slow pace; it’s not about the destination, but the process; any understanding is good understanding, and fortunately the ideas often resonate beautifully. The provisional explanations take a wider focus, but usually are born and die without too much trouble and time.
5. What emotions?
Happy and content; peaceful. Especially when I can take my time with the ideas, and enjoy them. I get frustrated when I can’t figure out parts, and annoyed when there are flippant comments that I have no choice but to take on faith (…sometimes these are even wrong!).
6. What does it resolve to, after how much time?
It usually spurs on more non-linear work, pretty quickly.
7. How frequent is this flavor?
It seems healthy to work through a proof every once in a while, since there are so many helpful clues hidden in the proof techniques of theorems. It’s probably the most helpful part of figuring out my research – giving me the highest density of new research ideas – and so I should do it as often as I can.
8. What are good/bad ways to change or follow it up?
Good: dive right into applying the proof techniques. Bad: get really frustrated at not understanding or forgetting things.
This blog post by Terence Tao has links to several comments on how to read mathematics.