Flavor: Intentioned immersion.

1. What’s going on mathematically?

I make a decision to immerse myself in math for a certain period of time. By this I mean, specifically, I establish the intention to keep my mind focused on a particular problem or question or idea, call it X, with as little deviation as possible. If and when my thoughts wander, I try to notice as quickly as possible, and return to the contemplation of X.

2. What is the emotional and logistical context?

This really involves putting everything else aside, for a fixed amount of time, and requires the right setting. The other threads in my life need to be relatively stable, so I can neglect them temporarily without grave consequence. Most of the time is spent sitting, working or thinking, but I will often go for a walk or ride public transportation. When I do, I often get lost. I also usually end up eating poorly, sleeping poorly, and coming across as inconsiderate and disconnected.

3. What thoughts are there?

Because the immersion sustains my focus on topic X, I end up going much deeper, and seeing it from new perspectives. Establishing such an intention provides a nurturing space for the math to blossom, protected from the harsh intrusions of everyday thought processes. I just remind myself, “Now is not the time for me to think about the class I’m teaching, or the email I should write; they can wait; now is the time to nurture idea X.” The immersion reveals that I have certain common ways of approaching math ideas, a sort of toolbox (e.g. working through proofs that relate to X; writing down what I know about X; reading about a related, parallel idea and trying to transfer to X). Only by hanging in there, and gently returning my awareness to X, over and over again, do I get a chance to practice the less common, less established tools (e.g. imagining explaining X to a colleague, imagining possible conjectures and working each backwards a few steps). Going out in public is especially insightful (and a little trippy); the juxtaposition of everyday stimuli and a sustained mindfulness of X, will necessarily present new ways of looking at X. A lot of this time feels unproductive and un-mathematical, but in a sense that’s the whole point: don’t worry about being productive or mathematical, just keep X in mind and see what comes up. Careful documentation (scribbling down *any* new questions or new directions that pop up) is essential.

4. What quality of awareness?

A lot of the immersion is spent doing math as I would normally do it (sitting down with pen, paper, books). But what makes the immersion a unique experience is the transitions between work sessions, when you would normally not be thinking about math. The immersion is similar to a “mindfulness” meditation, except that it’s a little more goal-oriented (goal: to understand X better). So, as in mindfulness meditation, you are basically establishing a witness within yourself that keeps watching, making sure you’re thinking about X. When your mind wanders (and it always will, eventually), the witness must notice and step in, to return your focus to X. You are continually “starting again”, and I think the mathematical usefulness of this experience comes from this. According to Krishnamurti, one way to see something truly as it is, with a fresh, new perspective, is to just keep looking at it. Once your old ways of seeing get stale, and you get bored, you keep looking. And keep looking. And then, through the boredom of the stale perspective, you break through to see things that you hadn’t noticed before.

5. What emotions?

When I do these immersions, I take them very seriously. Every minute I’m thinking about X. Maybe it’s necessary to decide when and what to eat, but I only let myself be briefly distracted. (If I’m generous I’ll suspend the immersion for 15 minutes to eat, but otherwise I hardly taste the food.) So I necessarily neglect my body and environment, and this can be taxing. I usually get quite physically sore, and the social disconnection can be a little unnerving. Gradually a strain builds on my awareness; sometimes I fall into some existential confusion. But if I’m generous and allow breaks for eating, some relaxation, or meditation, then these immersions can be quite fun and profound.

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

Ending the immersion is a conscious decision, after a predetermined amount of time. I’ve done it for an afternoon, or for a whole day. Once I did it for a whole week. Afterwards, I let myself relax and not think about math for a little while. But then there are usually all sorts of new ideas to continue pursuing.

7. How frequent is this flavor?

Not very frequent. I’ve only done it a handful of times.

8. What are good/bad ways to change or follow it up?

Good: wait a little while and then jump back in, using any new insights you gained. Bad: wait too long.

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