[Epistemic status: preliminary. You're reading it almost as I think about it. Crossposted to my personal blog.]

[Prerequisites: a bit of cognitive science reading, a bit of familiarity with mathematics.]

This is a raw idea and a personal experience, related just in case they resonate.

In a particular model of cognitive science, there are two streams of thought that can run inside your brain. The top-down stream is something you're generating from inside your brain itself. The bottom-up stream is coming directly from sense data, from the physical world around you. The two streams interact (particularly in terms of prediction and surprise) to produce your model of the world.

Meditation at the novice level I practice it is almost always a matter of turning focus from the top-down process to the bottom-up process in order to build a concentration muscle.

But what if there's something for me to gain from focusing on a top-down process? (I don't have links, but I know I'm far from the first person to try this.)

I may be stretching the term a bit, but I think the most top-down process is mathematics. Penrose philosophy aside, doing mathematics is an act of generating things inside your head, with regard to a set of rules, but largely divorced from sense data.

I used to be a mathematician. I stopped being one a long time ago, and I sometimes struggle emotionally to do math (including at my very mathy machine learning job).

So what would it be like for me to take a few minutes every day to focus on a mathematical concept, such as symmetry groups, or Fourier transforms, or just the number 3? I would turn both top-down and bottom-up distractions away in favor of gently returning to a mathematical concept. If I did this for a few weeks, would I develop a new, different focus muscle?

As luck would have it, this thought occurred to me this morning on a long drive, and right afterwards, my wife asked if I wouldn't mind some alone time while she took a car nap.

So I put on some music (a normal part of math for me, but not a normal part of meditation) and gave it a try.

What I experienced was surprising: Joy with a capital J.

I chose to focus on the number 3. At first I just visualized 3 objects, which became 3 glowing green dots, which became the cyclic group of order 3, represented as {1, x, x²}. It took me a second to remember that x³ = 1 was the relation that made this group work, and as I multiplied different powers of x, I experienced quiet, stable joy. It's hard to describe; the best I can say is that I had a feeling of safety and a feeling of simplicity. I felt like I could do this as long as I wanted and stay happy doing it.

After a while I chose to think about 3 from another perspective. I told myself the Monty Hall problem (with its three doors and three objects). Since I hadn't thought about Monty Hall in a long time, I couldn't remember the solution. So I worked it out from scratch, and when I finished, I experienced a different joy: the thrill of re-discovery, like seeing the good part of a movie you've mostly forgotten.

At this point, my wife woke up, and I stopped. I think I meditated for about 8 minutes.

This was a worthwhile experience. I'm going to continue trying it and documenting the results.

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Huh, interesting. I just tried meditating on vector space duality for 5 minutes, and it was joyful in some sense but not quietly so - I experienced a lot of movement in my body, in a way that feels similar to reactions I've been having to really good music or beautiful visual scenes lately. Something was agonizing about it and I wanted to stop so I could get on with the business of living or something, but that might be mostly because I'm feeling under time pressure lately.

This is closer to what I expected for myself. Do you feel a similar pressure to move to the next activity when doing other types of meditation?

Edit: my being stuck in a car might have had something to do with it. Not much to move on to :-)

Yeah, sometimes. Looking back, I think I often don't set a strong enough intention to meditate.

I came to a similar conclusion a couple years ago, though I'm less certain of it now. I think you're correct that math is dual to the kinds of self-dialogue (or observation) that people normally have while meditating, but I think that's just a quirk of how people normally meditate.

When most people meditate, their thoughts are fairly free-form and unstructured. You can make them a bit more guided by top-down giving them some structure. When I do this, my meditative thoughts start coming in more structured bits, often through visuals.

You can do this with different structures. For example you can try imposing intention (as in "X intends to do Y", or "X intended to say Z") to your meditative thoughts.

I think this is closer to daydreaming than meditation. I won't contest that daydreaming about math is a great deal of fun.

Maybe you should do it with paper and a working utensils - I can't really do math without external memory, and other people including you are probably bad at it to.

I think I failed to communicate the point of the exercise; for me, an act of meditation (as distinct from an act of mathematics) is about trying to point your focus somewhere. The quality of the act of meditation is determined by the quality of the pointing, not the quality of the thing you do under focus. So if you're doing walking meditation, the important thing is not to walk well, it's to focus well on your walking.

I would agree that an act of mathematics is really best done with external memory of some kind. But it can be done well enough without (for instance, when working on a puzzle, or visualizing some kinds of topology problem). I worry that using external memory will interfere with the act of pointing focus, but I'll try it soonish and get back to you with my result.