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.

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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.

Flavor: Trying to explain math to a stranger.

June 19, 2011

1. What’s going on mathematically?

A non-mathematician asks me about math.

2. What is the emotional and logistical context?

For example, I’m on the bus or at a bar. I definitely don’t bring it up (and I’ve usually given up before I open my mouth). It’s sudden, and there’s not enough time. The context may be more or less conducive, we may be more or less willing and patient. This usually means I have between thirty seconds and three minutes before I lose their interest and attention.

3. What thoughts are there?

I have a few short prepared soundbites, that don’t take much thought (something about negative dimensions, or translating questions between algebra and topology, or looking for patterns…). I think to myself about sounding elitist and condescending, about the hopelessness of expressing any more than the briefest snapshot. I think about gracefully changing the subject.

But if the stranger and I are willing, I’ll try, until he or she loses interest. I’ll grasp for spontaneous metaphors, I’ll gesture a lot, I’ll come up with new soundbites. The more I know about my audience, the easier this is. I think about them, and what I know of them and their life experiences. Usually I can identify something and build a metaphor around it (“it’s like cooking, but with ideas”).

What people are mostly interested in is not the math, but what it’s like to do the math. This is easier, because math is just another human activity, and has the same emotions, stories, ups and downs, politics, tragedies, and jokes, as soccer or marketing.

4. What quality of awareness?

I’m aware of being vague and non-rigorous; the voice in my head keeps accusing me of lying. I’m aware of a disconnect between what I’m saying, what’s being heard, and what my stranger is thinking. You can really tell when you’ve lost someone, when they’re thinking about something else (any second they’re going to announce, “I always hated math” or ask, “Does this have something to do with string theory?”). Unless I’m really in the mood and the context is right, I’m also aware of not trying very hard, of having given up, of being shallow and distractible. On those rare occasions that we do connect, it is brilliant, inspired, freeing, expanding.

5. What emotions?

I feel proud to call myself a mathematician, proud to be cool, proud to dream the impossible dream. Wonder at the amazing thing that is math. Concern to not sound elitist or condescending. Sadness, frustration, and disappointment with the disconnect and failed attempts. That isolation used to make me feel special and feed my ego, but now it just makes me sad.

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

Six times out of ten it ends soon after the question, “Does this have anything to do with string theory/quantum physics?”, or we get snagged on questions about higher dimensions. If interest is lacking, usually I can redirect the conversation without too much permanent damage. Maybe once every ten times it ends up being really fun.

7. How frequent is this flavor?

Thirty second to three minute conversation: about three times a week.
Longer than three minute conversation: twice a month.

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

I try to be willing, to have fun, be humble, and cherish the times that my stranger becomes my friend. Is it wrong to want to be understood? I don’t know, but I don’t think mathematicians should get their hopes too high.

Flavor: Getting your hands dirty, to clear up confusion.

June 19, 2011

1. What’s going on mathematically?

I’m very confused about something, and feel the need to go back to basics, go back to the things I do understand. So I might work through some simple examples. Or I might just play around with the symbols and ideas, rearranging them and testing my understanding, trying to figure out what it is that I don’t get. The work becomes very hands-on, computational, involving lots of scribbling. I stop working on important questions (since “important” usually means “as complicated as humanly possible, maybe more”), and go back to small questions that have known answers – answers that I hopefully can find on my own.

Of course, hands-on computations are necessary for a lot of “important” questions, but this post isn’t about that.

2. What is the emotional and logistical context?

I may be a little distraught, feeling lost in abstraction and not sure where I got lost and how to get back. Deciding to stop and get my hands dirty requires that I’m feeling patient and have a generous amount of time.

3. What thoughts are there?

Getting my hands dirty means writing down a lot – writing down things I wouldn’t normally write down. Our symbols carry so much meaning, and can carry so much confusion. I start with what I know – definitions, basic properties and propositions – and try to be as clear and pedantic as possible. Then I build up ideas, hunting for the dark or fuzzy areas. I might write down some string of symbols and ask myself, “Does this make sense?,” “Is this true?,” “Could I prove this?”. My thoughts are playful and relatively basic. My thought process is methodical and steadily constructive.

When I find a dark or fuzzy area, I might test the extent of the confusion. Is this something I used to understand (“the stable module category”)? Has this always been a dark spot (“modular representation theory”)? Is this a dark doorway into a whole other world of darkness (“A^1 homotopy theory”)? If it’s the first type of confusion, I might try to clear it up then and there. If it’s the second type, I may write it down as a question to follow up on later. If it’s the third type, I’ll probably just leave it alone.

4. What quality of awareness?

Prior to stopping to get my hands dirty in this way, I’ve probably been doing creative work at the edge of my understanding – this is chaotic, creative, and uncertain. But then I retreat from the edge, leave the clouds of abstraction and unbridled wonder, and land on the ground to sort things out. Getting my hands dirty is comforting and stable, staying mostly in precision and certainty. At first there’s still a good amount of perspective – as I move around and try to map out the areas of confusion, I avoid getting too wrapped up in any one murky area for too long. Then as I set about to clear up a particular fuzzy area, I switch to a narrative awareness, weaving together what I know, step by step, trying to build or rebuild solid bridges over the murky water. (Maybe having a bridge over the confusion is enough; maybe later I’ll roll out my big searchlight and try to penetrate the murk.)

I am exceptionally present when I get my hands dirty. The experience is relatively pointless – I’m not getting anywhere, mostly just retracing pathways that already exist and filling in one or two gaps. And the more “pointless” something is, the more present I am in the happening.

5. What emotions?

Playful, patient, curious, and pointless. It’s comforting to affirm what you know, but also scary to confront some of the dark areas. Fortunately, not all the darkness needs to be confronted by me personally.

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

I do always understand things a little better afterwards. And there will be a good list of questions to follow up on, ranging from simple (e.g. recheck a definition) to less simple (e.g. get a sense of how algebraic geometers use derived categories).

7. How frequent is this flavor?

This flavor is a good complement to the ever-expanding abstraction that is algebraic topology. Maybe once a week?

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

It feels great to return to the edge of my understanding with a reaffirmed core – things click and I see new connections. Unfortunately, sometimes I end up worrying about some murky area that is tangential, or start sinking after I hit the murky tip of a murky iceberg.

 


Connections:

Polya says: “If you can’t solve a problem, then there is an easier problem you can solve: find it.”

Flavor: Dull mind.

June 19, 2011

1. What’s going on mathematically?

The day after a really intense math day (8+ hours), I have a sort of refractory period of dullness.

2. What is the emotional and logistical context?

It’s after a very productive, or at least long, day. (The amount of sleep I get that night doesn’t seem to matter much.) So there’s lots of momentum, courage, enthusiasm, and a good research To Do list. I sit down to work, but my brain seems to mutiny.

This type of dull mind might also come after a decidedly mind-numbing experience, like a 30-hour bus ride, going for jury duty, or some sort of orientation.

3. What thoughts are there?

I lay out ideas before my mind, but nothing clicks, nothing goes anywhere. I stare, thoughtless, unsure what to do. It’s like trying to get a cat to play with a toy, when it just wants to lie still.

4. What quality of awareness?

There are all sorts of other unproductive mindsets. Sometimes my mind is too agitated, perhaps after (or before) some adventure or excitement. Sometimes I’m too emotionally unstable to do math. Sometimes I’m sleep-deprived, and my mind oscillates between agitation and dullness and I can’t steady it. Sometimes I’m just a little slow and groggy – but then I can usually get going, especially with caffeine.

This dullness is different; it won’t go away with coffee. I can’t just scrape the ice off the windshield and start driving, knowing that within ten minutes the car will be warmed up and I’ll be toasty. No, with this dullness the car won’t even start.

5. What emotions?

Maybe surprised, always bewildered. Mildly disappointed, but not too worried.

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

The dullness usually lasts for half a day, or a full day, and then I might be slow and groggy but can get going. Yoga or exercise won’t fix it. Meditation will fix it.

7. How frequent is this flavor?

Once or twice a month.

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

It’s best to recognize it, and go do something else. Or meditate and watch it pass. If I try to push through, I only get frustrated or make mistakes.

Flavor: Math anxiety – “I should be doing more math”.

June 19, 2011

1. What’s going on mathematically?

I haven’t done math in a while (a day or a few days), and I start to get a nagging feeling that I should.

2. What is the emotional and logistical context?

I’ve probably been distracted doing something fun, so I’m happy, or rested, or at least distracted. The nagging arrives on its own accord.

3. What thoughts are there?

The story of becoming a self-motivated learner and a self-directed researcher, is the story of internalizing the Teacher/Student archetype. I’ve successfully established a firm teacher and a curious student, coexisting within my mind. Not a day goes by without a discussion between them: What should I do? How should I do it? If the teacher isn’t strict enough, then nothing gets done. If the teacher is firm, or the student is procrastinating, then this slow nagging starts to build.

My thoughts go back and forth between the teacher and student in my head, making To Do lists, lists of research questions, trying to decide when and where to work next. I’m not really *doing* math, just thinking about doing it. (The instant I sit down and start working, all the back and forth, and all the anxiety, vanishes.) When it’s bad, the nagging will grow into anxiety, which can become debilitating. I procrastinate, divert, find excuses.

But I also rationalize, and sometimes this is good. Sometimes I don’t really need to do more math, and the teacher just needs to take a chill pill.

4. What quality of awareness?

I’m avoiding, skirting around, hemming and hawing, hiding and lying to myself. As the anxiety grows, at first a distant ache and gradually coming more to the center, I might become fixated on it. Then I lose track of what’s reasonable – I might be super busy, with no sane way of fitting in math that day, but I can’t see this, I can only see my anxiety.

The nagging is unsettling in its persistence. It grows so steadily, getting louder and louder and eclipsing my present. I know that thinking about (or dwelling on) it only feeds it, and so I try to push it out of my awareness. This never works.

But sometimes there is a small breakdown, and a small breakthrough of awareness. I confront the truth of the moment, confront my expectations of myself. The struggling student asks forgiveness, the teacher backs off and forgives. With clarity, I can broker an agreement that both are happy with.

5. What emotions?

A persistent growth from nagging, to anxious, to panic – it’s just a question of when I can’t handle it any more and sit down to do math, or confront it and reassess my priorities. Nagging makes me impatient and irritable. I feel burdened with responsibility and seriousness. Anxiety gives me a sick, nauseous feeling.

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

If I sit down and start doing math, it all goes away in an instant. If I confront the anxiety and reassess my priorities, I’m usually left with an appreciation of the teacher’s effort. This anxiety is a necessary evil – it pushes me, fuels me, challenges me.

I’m bewildered that awareness of this fact – the power and utility of an internalized teacher/student dynamic – doesn’t prevent future anxiety attacks. My teacher doesn’t seem to mellow out; I still have the same swings of gut-wrenching anxiety and heart-wrenching reconciliation, as when I was fifteen.

7. How frequent is this flavor?

Since I started grad school five years ago, I don’t think I’ve gone more than 48 hours without the math teacher in my head chiming in. (It doesn’t help that I seem to have a whole panel of internal auditors, nagging me on a daily basis.) The anxiety that makes me nauseous happens about twice a month.

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

I don’t think the answer is to have the student always submit to the teacher, diligently setting to work every time the nagging gets to a certain level. I think the two must keep each other in check, and that might be why I let the anxiety grow until a breaking point.

Mathematics is infinite, and in my lifetime I’ll never know it all, or more than I tiny slice of it all. Given this fact, and the facts of the world in which a mathematical lifestyle are embedded – the ignorance, suffering, beauty, and wonder – how do you decide how much math you should do today?

 


Connections:

Alan Lightman has a book of short stories, called Einstein’s Dreams. One story imagines a world where people live forever. How do you live today, faced with eternity and infinity? Without any standard of “enough,” half the people race around doing, doing, and doing. The other half waste time and do nothing, putting everything off until tomorrow.

Flavor: Proving myself wrong, via counterexample.

May 20, 2011

1. What’s going on mathematically?

After working towards proving something for a while, I find a counterexample. This involves an insight, followed by a verification.

2. What is the emotional and logistical context?

These counterexamples usually show up suddenly. The most dramatic and surprising cases are after working towards a particular result for weeks, because my expectation is that I’ve been getting closer and closer to a complete proof. So counterexamples hit when I’m hopeful, maybe even overly idealistic.

3. What thoughts are there?

The initial insight is a surprising “Aha” moment, accentuated by the fact that most counterexamples have a simplicity and necessity that seems to stab directly into the essence of the problem. This is immediately followed by some concerned analysis of the situation – does this mean I just wasted three weeks? is there a way to fix it? But before a complete reassessment, there’s a careful verification, to prove that the counterexample is a counterexample, i.e. to prove myself wrong.

4. What quality of awareness?

It’s like the rug has been pulled out from underneath me. There’s a shock and surprise, grounded in certainty, that then trickles outward along logic pathways and finds a deserted city. Or worse, the city I thought I knew is now filled with people that speak a language I don’t understand. On a deep level, it’s an unsettled, shifting, almost paranoid wandering in this strange new city, searching for any familiar faces. But on the shallow level, there’s a sharp certainty and cleanness, as my proved counterexample resonates within itself.

These are the times when I’m most aware of the non-logical, heuristic, mysterious “intuition” I have built up about the math I do. I had a mathematical worldview in which Proposition X was true – this sense of the way things work guided me, helped me make sense of it all. But now that I have found a counterexample, it’s not just the statement of Proposition X, but the whole worldview, that needs to be adjusted.

5. What emotions?

Of course, I usually feel disappointed and frustrated, depending on the severity of the situation. At worst, it can devolve into fatigue and meaninglessness. (I’m fortunate that the most time I’ve thrown away on a false proposition is 2.5 weeks; I’m sure it gets much worse than that.) There’s also an undeniable sense of finality, that comes with proving any result – “at least now I know for sure.” It is a very strange feeling, to prove yourself wrong. This certainty is a feeling I almost only get from math, and for some reason I feel it more strongly when I’ve been proven wrong than when I’ve been proven right.

I’ll usually take a break from the problem for a bit, and then I feel some revulsion towards it. Maybe it’s a feeling of being betrayed, but I don’t want anything to do with the question. This goes away soon, though.

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

A good mathematician would say that in every counterexample there’s new ideas to follow up. Maybe I just need to tweak my hypotheses; maybe the counterexample is pointing towards the essence of what’s going on; maybe the fact that Proposition X fails is a “good” thing, that e.g. allows for more interesting behavior. I can usually start to pick up the pieces after a few hours.

7. How frequent is this flavor?

Oh, I’m such a bad research mathematician, this happens way too much.

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

Bad: take it personally and get discouraged. Good: take a deep breath and get to work picking up the pieces. Mathematical intuition isn’t built overnight, and without surprises math would be boring.

Season: Pulling together and writing up.

May 20, 2011

1. What mathematical activities? What level of rigor?

After a period of creative research (of days, weeks, months,…), it’s necessary to consolidate and pull together results. This involves very carefully retracing steps, chronologically, and lining up ideas and proofs. Initially, results are scattered throughout my notes; or proofs haven’t been written down; or some statements are wrong, or outdated, or improved upon.

Because the original path to the result is almost always not the most direct, everything must be restructured. The goal here is to present ideas and proofs in the conventional form, an explanation to a particular audience. So there is a pure logic component, of lining up arguments correctly, but also a conversational component, as I decide how much detail to include, how much exposition, how much rigor.

This task is relatively easy and straightforward. It can feel administrative at times, for example when compiling a list of references. Virtually all math is type-set in Tex, so writing up involves hours and hours of typing Tex code, which is not very intellectually gripping.

I try to keep a running list of random ideas or questions that pop into my head as I’m writing something up. But I won’t pursue these until I’ve finished, since switching back and forth seems to make the writing up process less efficient.

2. What relevant interactions with other mathematicians?

This is maybe the most independent extended math experience I know. I might need to check some work with someone else, but presumably at this stage I’ve already solidified the results. It’s helpful to ask for tips on Tex syntax. I might have someone check that I’ve included the right amount of justification and exposition for my target audience. When submitting a paper, there is a well-established process of refereeing, which involves recursive feedback and reworking, and this can drag out past any self-contained “writing up” experience.

3. How does it feel, what is the mood?

Pulling things together can be affirming, and satisfying. It feels good to solidify knowledge. Writing up can be relaxing, or mildly frustrating. It’s unnerving when I find a mistake I made a long time ago, and have to fix it.

4. What state of mind? stable vs. chaotic? focused vs. dispersed?

Pulling together feels easy, methodical, and uniquely compartmentalized. I only need to worry about one proof or handful of ideas at a time, and can safely ignore the periphery. Of course, I try to stay open to the occasional random new idea or question, but I intentionally stop my mind from wandering too much from the concrete task at hand. Writing up results is conversational and performative – my mind traces through the ideas as though I were explaining them out loud, in real time.

This kind of math is also relatively easy to turn on and off. Sometimes while doing it my mind wanders from math, and I get lost in a daydream. It’s maybe the closest that math comes to being a “day job.”

5. What type of self-reflection during the experience, and did it help?

As mentioned above, I try to keep a balance between capturing any possibly-valuable peripheral thoughts, and not getting too distracted from finishing the write-up. So I allow myself to use the restructuring and reviewing as an opportunity to gain perspective, but this perspective only comes if I keep some distance and don’t get wrapped up in following new leads. Maintaining this balance requires some self-reflection. In fact, it seems that the better I’m attuned to this balance, the closer I can get to simultaneously maximizing perspective and efficiency.

6. An everyday metaphor for the experience?

Pulling together and writing up is just like washing dishes. The goal is to sanitize all the mess of discovery, and to dry off any trace of the restructuring. We present a stack of dry, clean, glistening ideas, full of order and necessity, untouched by humans. These spotless ideas are complete in themselves, but sit ready to be used and rearranged as vessels and tools for someone else’s new mess.

The dish-washing process is narrative (to me), relaxing, and mechanical. I can let my mind wander, to some extent. There are definitely more efficient and less efficient ways to do it.

7. An example of a good day and a bad day?

A good day ends with a few new pages of nice, clean Texed math. On a bad day, I find a gap or hole, and can’t fix it.

8. What did you do when you were stuck?

Getting stuck might mean finding a gap in some argument; this needs to be fixed. Or it might mean that I lost or can’t find some proof, so I have to reprove it. Or it might be that I don’t know the Tex syntax for the symbol I want, which means I have to hunt through Tex documentation.

9. When and why did it end?

It ends when the results typed up and pretty.


Connections:

Terry Tao has advice on writing mathematics.

Flavor: Intentioned immersion.

May 20, 2011

1. What’s going on mathematically?

I make a decision to immerse myself in math for a certain period of time. By this I mean, specifically, I establish the intention to keep my mind focused on a particular problem or question or idea, call it X, with as little deviation as possible. If and when my thoughts wander, I try to notice as quickly as possible, and return to the contemplation of X.

2. What is the emotional and logistical context?

This really involves putting everything else aside, for a fixed amount of time, and requires the right setting. The other threads in my life need to be relatively stable, so I can neglect them temporarily without grave consequence. Most of the time is spent sitting, working or thinking, but I will often go for a walk or ride public transportation. When I do, I often get lost. I also usually end up eating poorly, sleeping poorly, and coming across as inconsiderate and disconnected.

3. What thoughts are there?

Because the immersion sustains my focus on topic X, I end up going much deeper, and seeing it from new perspectives. Establishing such an intention provides a nurturing space for the math to blossom, protected from the harsh intrusions of everyday thought processes. I just remind myself, “Now is not the time for me to think about the class I’m teaching, or the email I should write; they can wait; now is the time to nurture idea X.” The immersion reveals that I have certain common ways of approaching math ideas, a sort of toolbox (e.g. working through proofs that relate to X; writing down what I know about X; reading about a related, parallel idea and trying to transfer to X). Only by hanging in there, and gently returning my awareness to X, over and over again, do I get a chance to practice the less common, less established tools (e.g. imagining explaining X to a colleague, imagining possible conjectures and working each backwards a few steps). Going out in public is especially insightful (and a little trippy); the juxtaposition of everyday stimuli and a sustained mindfulness of X, will necessarily present new ways of looking at X. A lot of this time feels unproductive and un-mathematical, but in a sense that’s the whole point: don’t worry about being productive or mathematical, just keep X in mind and see what comes up. Careful documentation (scribbling down *any* new questions or new directions that pop up) is essential.

4. What quality of awareness?

A lot of the immersion is spent doing math as I would normally do it (sitting down with pen, paper, books). But what makes the immersion a unique experience is the transitions between work sessions, when you would normally not be thinking about math. The immersion is similar to a “mindfulness” meditation, except that it’s a little more goal-oriented (goal: to understand X better). So, as in mindfulness meditation, you are basically establishing a witness within yourself that keeps watching, making sure you’re thinking about X. When your mind wanders (and it always will, eventually), the witness must notice and step in, to return your focus to X. You are continually “starting again”, and I think the mathematical usefulness of this experience comes from this. According to Krishnamurti, one way to see something truly as it is, with a fresh, new perspective, is to just keep looking at it. Once your old ways of seeing get stale, and you get bored, you keep looking. And keep looking. And then, through the boredom of the stale perspective, you break through to see things that you hadn’t noticed before.

5. What emotions?

When I do these immersions, I take them very seriously. Every minute I’m thinking about X. Maybe it’s necessary to decide when and what to eat, but I only let myself be briefly distracted. (If I’m generous I’ll suspend the immersion for 15 minutes to eat, but otherwise I hardly taste the food.) So I necessarily neglect my body and environment, and this can be taxing. I usually get quite physically sore, and the social disconnection can be a little unnerving. Gradually a strain builds on my awareness; sometimes I fall into some existential confusion. But if I’m generous and allow breaks for eating, some relaxation, or meditation, then these immersions can be quite fun and profound.

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

Ending the immersion is a conscious decision, after a predetermined amount of time. I’ve done it for an afternoon, or for a whole day. Once I did it for a whole week. Afterwards, I let myself relax and not think about math for a little while. But then there are usually all sorts of new ideas to continue pursuing.

7. How frequent is this flavor?

Not very frequent. I’ve only done it a handful of times.

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

Good: wait a little while and then jump back in, using any new insights you gained. Bad: wait too long.