Archive for November, 2012

Flavor: Natural inspiration.

November 28, 2012

1. What’s going on mathematically?

Mathematics is overlaid onto a nature experience. Nature seems to present a vague metaphor, that inspires some mathematical insight.

2. What is the emotional and logistical context?

I’m out in nature. Maybe walking, sitting and looking, or traveling. I’m relaxed, and not particularly thinking about math at first, except for some pondering perhaps.

3. What thoughts are there?

It starts with an acute focus on the natural experience. Maybe watching and listening to wind blowing in trees, or looking at clouds, or looking at interference patterns of raindrops in puddles. I’m absorbed in the sights, sounds, smell; it’s a full experience.

As the scope of my awareness expands (see #4), I start to overlap the nature experience with some math idea. This is usually not an explicit direct metaphor (maybe it would be for a physicist, or a mathematician studying geometry or dynamical systems; the math I do is too abstract, too structural to ever really correspond to anything in physical reality). Instead it’s a loose structural metaphor, a vague feeling like nature is presenting me with the secret I’ve been looking for. Often I have the thought that I’m looking at, or experiencing, the Answer, I just don’t quite see how to interpret it.

Mathematicians are only as good as their imagery and metaphors, which allow them to fasten abstract ideas to commonplace intuition. In these moments of natural inspiration, it feels like I’m getting hints of new images, metaphors, and structures on which to hang my math ideas.

4. What quality of awareness?

First there’s a strong focus on physical sensations, and then absorption into those sensations. This brings an expansion of awareness to a totality, engaging all the senses in one singular experience of the moment. Bringing in the math requires a small conscious intention to do so, to bring some math ideas into my awareness. And then I just hold the two flavors simultaneously: the natural experience and the math ideas. Sometimes this juxtaposition takes effort, sometimes it seems effortless.

5. What emotions?

At the start I’m usually relaxed and happy, then become more happy and slightly less relaxed, and towards the end I just feel peaceful, and a feeling like everything is going to be okay.

6. What does it resolve to, after how much time?

There’s usually no concrete insight, just a sort of spreading out and a new perspective. I hold the juxtaposition for five minutes to 45 minutes (if I’m on a hike, for example), then let go of it. I usually leave it and do non-math things afterwards.

7. How frequent is this flavor?

About once every 2 months.

8. What are good/bad ways to change or follow it up?
As I said, there’s usually no specific insight or answers that arise. Just a vague feeling of fresh perspective and harmony. I think it’s a mistake to try to squeeze some insight out of these vague, almost subconscious, mental interactions. It’s also a mistake to try to hold onto the delicious peace that accompanies them. Better to not try to understand everything, and enjoy the moment as it passes. I’ve always trusted, blindly perhaps, that these experiences make me a better mathematician.

Flavor: Filling out results.

November 28, 2012

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.