Archive for July, 2011

Flavor: Breakthrough I: Getting stuck.

July 22, 2011

(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?

See Breakthrough II post.

 


Connections:

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.

Flavor: Breakthrough II: Sticking with stuckness.

July 22, 2011

(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.

 

 


Questions:

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?

 


Connections:

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.

Flavor: Breakthrough III: Rest.

July 22, 2011

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

1. What’s going on mathematically?

I’ve just pushed myself to the limit, getting myself very stuck and staying there for a while. I decide to take a break.

2. What is the emotional and logistical context?

I’m confused and somewhat irritated, that things haven’t been working out. I’ve probably just packed up my math and am walking or biking somewhere. I’m not engaging any new stimulus – not talking, or doing other work. Although I might eat something.

3. What thoughts are there?

Almost no math thoughts, not even echoes. I’ve sealed them off. I’m probably thinking about what’s next on my schedule.

4. What quality of awareness?

Still pretty spaced out, but not quite stunned and sublime, because of the low-level trauma and lack of resolution. But quickly my mind is refreshed, like I’ve opened the windows to let in some fresh air. As I re-engage the world, I have that always-new feeling, like when you take a slow, deep inhale.

5. What emotions?

My frustration goes away as quickly as any physical aches that have developed; as soon as I stretch my legs and get fresh air, I start to feel really good.

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

I usually allow myself to be distracted by some other engagements, e.g. socializing or being active. Sometimes I get hit with a breakthrough.

7. How frequent is this flavor?

After every time I stick with stuckness. So, 1 -3 times a week.

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

The best thing to do seems to just relax and rest. Giving some buffer time before engaging with people or other activities seems to increase the chance of a breakthrough (see Breakthrough IV post).

 


Connections:

I think of Thomas Edison again, and his legendary napping regiment.

Flavor: Breakthrough IV: Insight.

July 22, 2011

(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.

 


Questions:

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.

 


Connections:

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.