1. What’s going on mathematically?
This is a mathematical flavor that happens during a “Pulling together and writing up” season. I have some collection of results that I’ve pulled together, and I’m trying to turn it into a coherent paper or talk. To turn it into a whole, instead of a bunch of pieces. In a good paper, the results are “complete” in some sense. But most of the time, math sprawls continuously off to the horizon. Choosing a paper-size chunk of terrain, and trying to develop it into a coherent whole, is what I call filling out the results.
2. What is the emotional and logistical context?
There’s usually some excitement that things are coming together. For me there’s always some confusion about what’s interesting, or rather what will be considered interesting to others. There’s usually time pressure from a deadline.
3. What thoughts are there?
There’s an element of logic, maybe even necessity. On the one hand, I need to know where to draw the line. To make up an example, imagine that I proved something is true for every positive value of some parameter n. I could stop there, or I could work for a few months longer to try to prove it for every negative value as well, or every real-valued n, or complex-valued n, etc. At a certain point, I draw the line somewhere. This decision takes into consideration the background required, and the proof methods used, and is usually a straightforward decision. On the other hand, maybe I know n can only take values 0, 1, or 2. If I’ve only addressed the n=0 and n=1 case, there’s a feeling of necessity that I should consider the n=2 case. If it is significantly harder, or different, or uninteresting, I should at least mention that this is the case. I think every mathematician would agree that omitting any mention of the n=2 case would be a shortsight.
But mostly the thoughts are centered on aesthetic considerations. The paper needs to “flow”, to be “complete”, to go “far enough” but not “too far” (not to mention the proofs must go “deep enough”, but not “too deep”). These are all culturally-defined aesthetic qualities, that nevertheless most mathematicians would, for the most part, agree on. You know it when you see it. When filling out results, the challenge is that at first you don’t see it. The results are incomplete, and your job is to complete them.
4. What quality of awareness?
Very fluid and open. I’m trying to step back and see the collection of results as a whole, possibly for the first time. It takes an open, flexible mind to see the best way to organize the ideas. Or rather, it’s more like the ideas self-organize into a natural flow, if I can only hold them all in my mind at once, in a big open awareness. Filling out the results means, while holding that big awareness, also noticing all the dark areas that need to be explored, or at least addressed, to complete the whole.
5. What emotions?
Manipulating my own results is always emotional. Often some hard-won proofs are subsumed, or irrelevant, or improved upon. Filling out results means identifying holes and small-minded reasoning, in other words, flaws. This is the phase when the ideas and proofs are depersonalized and objectified as much as possible, and it can be heart-wrenching.
Also, probing the aesthetics engages my emotions, as I try to decide when enough is enough, when results are interesting or complete rather than irrelevant or partial. In general, mathematical exposition is really an art form.
6. What does it resolve to, after how much time?
Personally I haven’t had many hugely climactic results. So after I’ve filled out the results I do have, there’s a feeling of having drawn an arbitrary line somewhere, and there are always copious new directions to pursue. See also the post on “Pulling together and writing up.”
7. How frequent is this flavor?
This happens in the lead-up to every paper, write-up, or research talk.
8. What are good/bad ways to change or follow it up?
I’m not very good at drawing a line and deciding enough is enough; I’m more inclined to keep proving and proving, until I really get a big tangled mess. Recognizing some necessary arbitrariness is healthy, then.
The aesthetic judgement part is not easy either, and I can get confused and frustrated. It’s helpful to step back and appreciate whatever nice flow of results is already there, and recognize that the sense for mathematical aesthetics is only grown slowly, through practice.