1. What mathematical activities? What level of rigor?
After a period of creative research (of days, weeks, months,…), it’s necessary to consolidate and pull together results. This involves very carefully retracing steps, chronologically, and lining up ideas and proofs. Initially, results are scattered throughout my notes; or proofs haven’t been written down; or some statements are wrong, or outdated, or improved upon.
Because the original path to the result is almost always not the most direct, everything must be restructured. The goal here is to present ideas and proofs in the conventional form, an explanation to a particular audience. So there is a pure logic component, of lining up arguments correctly, but also a conversational component, as I decide how much detail to include, how much exposition, how much rigor.
This task is relatively easy and straightforward. It can feel administrative at times, for example when compiling a list of references. Virtually all math is type-set in Tex, so writing up involves hours and hours of typing Tex code, which is not very intellectually gripping.
I try to keep a running list of random ideas or questions that pop into my head as I’m writing something up. But I won’t pursue these until I’ve finished, since switching back and forth seems to make the writing up process less efficient.
2. What relevant interactions with other mathematicians?
This is maybe the most independent extended math experience I know. I might need to check some work with someone else, but presumably at this stage I’ve already solidified the results. It’s helpful to ask for tips on Tex syntax. I might have someone check that I’ve included the right amount of justification and exposition for my target audience. When submitting a paper, there is a well-established process of refereeing, which involves recursive feedback and reworking, and this can drag out past any self-contained “writing up” experience.
3. How does it feel, what is the mood?
Pulling things together can be affirming, and satisfying. It feels good to solidify knowledge. Writing up can be relaxing, or mildly frustrating. It’s unnerving when I find a mistake I made a long time ago, and have to fix it.
4. What state of mind? stable vs. chaotic? focused vs. dispersed?
Pulling together feels easy, methodical, and uniquely compartmentalized. I only need to worry about one proof or handful of ideas at a time, and can safely ignore the periphery. Of course, I try to stay open to the occasional random new idea or question, but I intentionally stop my mind from wandering too much from the concrete task at hand. Writing up results is conversational and performative – my mind traces through the ideas as though I were explaining them out loud, in real time.
This kind of math is also relatively easy to turn on and off. Sometimes while doing it my mind wanders from math, and I get lost in a daydream. It’s maybe the closest that math comes to being a “day job.”
5. What type of self-reflection during the experience, and did it help?
As mentioned above, I try to keep a balance between capturing any possibly-valuable peripheral thoughts, and not getting too distracted from finishing the write-up. So I allow myself to use the restructuring and reviewing as an opportunity to gain perspective, but this perspective only comes if I keep some distance and don’t get wrapped up in following new leads. Maintaining this balance requires some self-reflection. In fact, it seems that the better I’m attuned to this balance, the closer I can get to simultaneously maximizing perspective and efficiency.
6. An everyday metaphor for the experience?
Pulling together and writing up is just like washing dishes. The goal is to sanitize all the mess of discovery, and to dry off any trace of the restructuring. We present a stack of dry, clean, glistening ideas, full of order and necessity, untouched by humans. These spotless ideas are complete in themselves, but sit ready to be used and rearranged as vessels and tools for someone else’s new mess.
The dish-washing process is narrative (to me), relaxing, and mechanical. I can let my mind wander, to some extent. There are definitely more efficient and less efficient ways to do it.
7. An example of a good day and a bad day?
A good day ends with a few new pages of nice, clean Texed math. On a bad day, I find a gap or hole, and can’t fix it.
8. What did you do when you were stuck?
Getting stuck might mean finding a gap in some argument; this needs to be fixed. Or it might mean that I lost or can’t find some proof, so I have to reprove it. Or it might be that I don’t know the Tex syntax for the symbol I want, which means I have to hunt through Tex documentation.
9. When and why did it end?
It ends when the results typed up and pretty.
Terry Tao has advice on writing mathematics.