Archive for the ‘flavors’ Category

Flavor: Using my math powers for evil.

August 18, 2011

1. What’s going on mathematically?

Very rarely, when I’m particularly frustrated or angry at some particularly illogical inconvenience, I’ll apply my powers of analytic reasoning with the intent to cause emotional harm.

For example, once while renting a car in Las Vegas, the salesman kept pressuring me to buy a full tank refill at their 10% discounted price. He was aggressive about it. I got flustered, and fired off a terse line of irrefutable-sounding mathematics, pointing out that this only made sense if I returned the car with less than about 10% of the tank remaining, which was not pragmatic. I said it with such damning certainty that he shut up immediately.

More recently, I tried to return a piece of video equipment that was checked out in my roommate’s name. They insisted that I couldn’t just drop it off – my roommate needed to check it back in, for legal reasons. Whereas a reasonable annoyed person would question the logic of this administrivia by lobbing a loose bundle of sense, I shot a dense two-sentence projectile with a certainty and precision that I almost made the poor attendant start crying.

2. What is the emotional and logistical context?

I have to be in a very, very bad mood, or very stressed out. The context is bureaucratic and supremely annoying.

3. What thoughts are there?

There’s the thought that what I’m about to say is aggressive and backed by a mild mental volition to cause harm. I don’t remember really choosing my words to be particularly concise, but they are, and it’s this logical conciseness that characterizes the experience. The logic wells up from my analytic intuition and mathematical training for direct and irrefutable communication.

4. What quality of awareness?

In Tibetan Buddhism, anger is understood as the neurotic manifestation of vajra, the state of clarity; the wisdom behind anger is clarity. And when I’m concocting my aggressively logical and concise retort, there is sharp clarity – the logic lays out clearly and my analytic mind finds the most powerful form in which to yield it.

Throughout the exchange I maintain an awareness of my mental willingness to cause some small level of harm. I’m aware of the small-mindedness of this.

5. What emotions?

It feels good to let myself be angry sometimes; I feel empowered and proud. But only briefly. Then I feel bad. Sometimes I’m amused at the whole thing, since it’s so rare for me to snap at someone like that.

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

I can never remember exactly what I said in the heat of the moment, only that it was uncommonly logical and concise. It sort of echoes in my mind, repeating itself in a more spread out form. I often end up pondering my bias towards rationality over other forms of intelligence and knowing.

7. How frequent is this flavor?

I can only remember doing this a few times. Other contexts: shredding apart the logic of unsuspecting Christians, or economists.

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

Apologies sometimes (spoken or unspoken), but not always.

Advertisements

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.

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.