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.