Flavor: Breakthrough IV: Insight.

(The four posts in this series describe the steps in a breakthrough.)

1. What’s going on mathematically?

I worked and got stuck, then stuck with being stuck for a while, then gave up and took a rest. Every so often, during or right after rest, there will be a flash of insight that makes everything, or at least many things, perfectly clear.

2. What is the emotional and logistical context?

I’m relaxed, and walking or biking usually.

3. What thoughts are there?

I’m not thinking about the material I was just banging my head against; I’m usually either not thinking, or lightly thinking about some scheduling. But then there’s a POW or a CLICK or an echoing silence, and I have the answer I was looking for. Usually the ideas have shifted and resolved themselves; I’m now seeing them from a slightly different perspective than while I was working so hard on them. But because I know every in and out of their shapes, it’s immediately clear that this new resolution of the problem is absolutely correct and certain. Or at least it seems that way; I don’t spend time verifying the logic, as I normally would.

Sometimes this one breakthrough leads to a very quick cascade of breakthroughs, as my mind pursues the consequences. With a surge of happiness, I usually just leave the insight alone, to do what it will; I won’t bother to sit down and write anything out. There’s no doubt, no concern of losing the insight. I go back to doing what I was before it happened. This all takes about five minutes.

4. What quality of awareness?

There’s a relaxed effortlessness, that observes the process without taking ownership of it. It’s a beautiful moment of clarity, and I’m aware of gratitude and appreciation, sort of gawking in awe, like watching a flock of birds flying through the sky. The moment is precious and very real, undivided, complete, and yes, effortless. The thoughts feel like a release of a great conceptual tension, but the awareness is more like a bubbling through of some perfect moment that was always already here.

5. What emotions?

Oh, it feels good. I’m happy, and I smile. I feel fearless affirmation.

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

I’ll let it linger, without pursuing it intellectually. After all, sometimes these insights are wrong, and why ruin it too quickly? Often I’ll look for someone to tell. But then I just go back to doing whatever I was planning to do, before it happened.

7. How frequent is this flavor?

Maybe once every month. They end up being wrong about 50% of the time, but that percentage is going down slowly.

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

It’s nice to enjoy it. I never feel the need or desire to sit down to work through and verify anything immediately, but eventually (after a few hours, or the next day) I’ll do this. Actually, trying to write it out too soon seems counterproductive – after the initial insight, the breakthrough needs to settle in gently. It seems best to gently observe this settling, rather than forcefully dissect it and linearize it.



The biggest question for me is, given that I’ve noticed and to some extent understand this four-part process of breakthrough, can I induce it? You’d think that, during my unpleasant sticking with stuckness phase, I’d have the foresight to think, “I just need to stay stuck a little longer, then go take some rest, and then, POW, it’s going to hit me…” But I never seem to have that thought. It seems like approaching the process with such an intention would somehow ruin it.

Perhaps the Teacher-in-my-mind is aware of the possibility of a breakthrough, and therefore is so strict with the Student-in-my-mind, but the Teacher also shields this possibility from my conscious gaze. It would be like the construction and living out of a vivid dream – one part of my mind has already woven the story and set the scene, and another part is an unsuspecting character within the story, who gets surprised by things it should already know.

Or maybe my mind is just so dull when I’m stuck, that I really don’t have the self-awareness to recognize that I’m in step two of a well-established four-step process.



There are many overlaps between this four-step process of breakthrough I’ve just described, and Jacques Hadamard’s analysis of the process of invention in mathematics. He questioned many mathematicians of his time, and concluded that, in spite of personal variation, there seemed to be a common pattern. His book describing his theory is fascinating and one-of-a-kind.

Hadamard was inspired by a famous quote of Poincare, describing a breakthrough he had while stepping onto a bus. It seems Poincare may have been the first mathematician to carefully describe what may be a somewhat universal math experience.


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