(The four posts in this series describe the steps in a breakthrough.)
1. What’s going on mathematically?
Sometimes it seems that, in order to have a big breakthrough, I need to have something big and thick to break through. These bursts of insight are always worth the effort, but, like walking a sawtooth function, playing this game means spending most of my time enduring the growing confusion and stuckness.
The spiral into stuckness is subtle. I’m working on some train of thought, but keep hitting dead ends or gaps in my understanding. I shift my perspective, try new examples, zoom out or zoom in, try to fill some gaps. At first this works, and I proceed a few more steps. But over time the dead ends and dark confusion surround me, and I’m totally stuck.
2. What is the emotional and logistical context?
This happens so often, in so many different contexts. I’m just sitting somewhere doing math. I could be happy or sad or relaxed or stressed. It could be morning or night, in a coffeeshop or in the back of my jeep or in a park.
3. What thoughts are there?
Over the course of several hours, I follow many lines of thought, sometimes striding through familiar territory, sometimes treading carefully around or through dark spots. Gradually the land becomes stranger and the visibility decreases. The dead ends and darkness become more and more frequent.
4. What quality of awareness?
Again, a wide range of possibilities here; it starts out as the generic research experience – I could be dull or sharp, open or muddy. There is an observer, the Teacher-in-my-mind, that watches as I venture into the new territory, keeps track of the growing confusion, where it is, what the fringes taste like. There is a mild awareness of where this may lead (see Breakthrough II-IV posts), but the Teacher allows the Student to innocently follow its curiosity.
5. What emotions?
As an explorer, I relish the new territory. When it becomes clear that I’m getting more and more stuck, I start to get a little frustrated. I start to notice things like hunger or body ache.
6. What does it resolve to, after how much time?
It’s possible that I’ll figure something out – I’ll successfully cross into the unknown and come back unscathed. But this amounts to avoiding getting stuck.
Sometimes when I’m stuck, I’ll give up immediately, perhaps deferring further progress until after I’ve talked with my advisor. But then I miss a potentially big breakthrough…
Sometimes I stick with it.
7. How frequent is this flavor?
Most research sessions end with me getting stuck to a greater or lesser extent, so this happens between 2 and 10 times a week.
8. What are good/bad ways to change or follow it up?
Thomas Edison is famous for saying, “Genius is 1% inspiration and 99% perspiration.” But he didn’t say where the inspiration shows up among the perspiration. For an inventor I imagine it’s close to the beginning – you have an insight, and then work hard to make it a reality. With math, the inspired idea seems to come more often at the end of a long period of hard work and stuckness.