1. What’s going on mathematically?
I decide to try to see (mathematical) things differently. It’s a relaxed, playful attempt to play with math. Similar to an “intentioned immersion” but looser and less aggressive.
2. What is the emotional and logistical context?
Usually I’m riding on public transit, commuting, in a car, waiting for something, or taking a long walk where I won’t be interrupted. I’m alone. I won’t be able to work very rigorously, so I have low expectations of what will happen; still, there’s an intention to engage math.
3. What thoughts are there?
There’s a range of pondering modes, or games you can play. Here are a few.
Sometimes I take a specific object or idea and try to visualize it. I try to describe its essence. I imagine seeing this image or description for the first time, and see what questions arise.
Sometimes I juxtapose two things via a nonsensical premise, and see what happens.
Sometimes I imagine that I can turn any statement I want into a true statement, like I had a magic truth wand. What would be the most beautiful situation in a given context? What do I really want to be true? And then what are the relevant hypotheses?
Sometimes I construct an internal dialog with an imaginary colleague, or pretend I’m teaching a class.
Again, the basic idea is to play. Rearrange, experiment, start again… but in a light way.
Usually I don’t write, or don’t write very much, so my lines of reasoning can’t get very far. If the game I was playing was like a chess game, then I don’t try to strategize out four moves ahead, but rather just imagine different possible vague developments in the layout of the board.
4. What quality of awareness?
It takes effort, to stay focused on my pondering, and to think in ways I’m not used to. There is a strange juxtaposition of the mathematical headspace and the real world. My perception and awareness jumps back and forth, usually in a disjoint way, between these two realities. So my hearing of the ambient surroundings cuts in and out, perhaps alternating with the sound of an internal mathematical narrative. Or my spacial sense of where I am cuts in and out. Or my visual perception of what’s in front of me gets interrupted by mathematical imagery.
There have been a few times when the two realities sort of coalesced. Rather than a juxtaposition, or cutting back and forth between the two, there was a synthesis. I don’t quite know how to describe these experiences, except as a kind of integration of mind and body.
5. What emotions?
It’s usually fun, and sometimes funny. Sometimes it’s too effortful though, because I’m too distracted. It never feels like I’m getting anywhere, and so it often feels pointless and dumb. But there’s a pleasant feeling that comes with bringing some cognitive depth into a somewhat numbing situation like riding the metro or waiting on line.
6. What does it resolve to, after how much time?
Often I do get some new idea (of questionable worth). Or a new perspective, or a new question. I think I need about 15 minutes of stable pondering, if I’m going to settle into it and get anywhere.
7. How frequent is this flavor?
Mathematical pondering doesn’t come naturally to me, but I feel like it should. (I ponder other things much more effortlessly.) Maybe once a week, unless I intentionally practice it.
8. What are good/bad ways to change or follow it up?
I always think that pondering will become easier, or more fruitful, eventually. It seems like the transitions from being in the real world to being in the math headspace, and vice versa, should become easier and easier to traverse. It seems like there’s a lot of insight in being able to enter the math headspace in different ways, at different times, and so pondering is a way of practicing this transition.
I try to write down any insights I gain from pondering, no matter how vague. But other times I give up pondering because it’s hard and doesn’t feel productive, or even fun.