Flavor: Breakthrough II: Sticking with stuckness.

(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. In the Breakthrough I post, I describe the spiral into stuckness. This post is about sustaining the stuck.

2. What is the emotional and logistical context?

I’m already feeling a little frustrated, at being stuck. I’ve probably been sitting in the same spot for two or three hours, without breaks, so there’s some physical discomfort that has started to make it into my awareness.

3. What thoughts are there?

Throughout the experience, there is a recurring mental volition to keep trying, in spite of the irritation. I’m stuck, confined to a small conceptual area, without many options for release. What I do next is carefully study this small area, getting an exquisite understanding of the shape of the line separating what makes sense and what doesn’t. I retrace my logic, looking for cracks that might let some light through.

But gradually the confinement and repetition dulls my senses, and I start to space out. Around this time, I stop writing as much and begin staring blankly into the air more and more. At this point, I stop being able to tell if I’m thinking at all. There’s some retracing of the shapes, but very gently, sort of a massaging.

And then, suddenly, I’ll decide I’ve had enough. I’m not getting anywhere, I’m wasting time, I’m not being efficient, I should take a break and come back to it. So I mentally (and usually physically) pack up and leave.

4. What quality of awareness?

Although when I start getting stuck the quality of my awareness can be one of many things, as I stick with the stuckness it converges to a common state. There is a feeling of mental strain, from the hard work of staying focused in spite of frustration. The physical and mental aches start to creep in from the periphery, and start to mildly affect my thought process and mood. The Teacher-in-my-mind presents itself more forcefully, to keep me on task, and there is a sustained conflict between the Teacher and the Student, who is struggling a little.

As mentioned in #3, eventually my thoughts submit to their confinement, and dullness starts to take over. My awareness becomes more diffuse but unrefined – I can’t tell if it is shallow or deep, if I’m thinking or just listening to the echoes of thoughts. There’s lots of blank stares and timeless absorption into dullness.

This is punctuated with jolts of something like desperation, a sort of “I can’t take this any more – something is going to break” cry for help from the Student, that the Teacher stifles.

5. What emotions?

Frustration, irritation, physical discomfort, emptiness.

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

Willingly or unwillingly, I often sustain stuckness for an hour or two after I first consider myself stuck. When I finally give up, there is a release. My thoughts leave math.

7. How frequent is this flavor?

About a third of my research sessions end this way. So maybe 1 – 3 times a week.

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

It’s usually not a choice – I have to rest and take a break from math for a little while. See Breakthrough III post.




Is it worth it? There’s a lot of time spent in this dull state, which doesn’t feel productive or efficient or nice. Is this a stupid mis-application of my will-power and masochism? Or is there a payoff?

Would 15 minutes of stuckness be enough? What’s going on in the depths of my mind, during that diffuse space-out period?



The jolts of desperation I feel, when I’ve glued myself to my chair and refuse to get up yet, are unsettling but also invigorating. They remind me of pre-breakthrough moments I’ve had while rock climbing, swimming, hiking, or meditating. I’ve pushed beyond what I thought was my limit, and 98 times out of 100, nothing breaks except some self-imposed mental chains.

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