Archive for June, 2011

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.

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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.