Archive for May, 2011

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.

Flavor: Unintentioned immersion.

May 20, 2011

1. What’s going on mathematically?

Sometimes I can’t stop thinking about some idea or problem, call it X.

2. What is the emotional and logistical context?

It usually happens over the course of a relatively sleepless night. Sometimes because I drank too much coffee. Otherwise, there’s usually a strong emotional hook to it – I feel like I just can’t let go of the idea, even if I make an effort to distract myself.

3. What thoughts are there?

The thoughts are usually very repetitive. It feels like I’m retracing the same pathways over and over, digging deeper tracks, that may become ruts that I can’t get out of. It’s somewhat obsessive. If it happens during the day, it’s like having a song stuck in my head. But I only know snippets of lyrics, so I end up repeating those and not really fleshing out the song at all. I might try to connect the idea to others, to make some progress or get a new perspective, but this never really succeeds.

4. What quality of awareness?

It’s a very shallow awareness, constantly shifting from the periphery to the fore and back to the periphery. It’s like when you repeat one word or phrase over and over, until it stops meaning anything and disintegrates into a silly clump of letters and sounds. I guess it always has a acoustic, rhythmic quality.

5. What emotions?

If it happens during the day, there’s always a strong emotional component, that may be responsible for bringing it back over and over, but without much depth. Maybe X is a problem I’ve been trying to solve for a long time and think I’m close. Maybe I’m frustrated with X, which I’ve invested a lot of time and energy in, and I’m worried it’ll all have been a waste of time. Maybe X is an exciting new idea or breakthrough, that I’m super excited about.

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

Maybe I force a distraction, to take my mind off of it. Maybe I can’t handle it any more, so I sit down with pen and paper and books to try to make progress or get some movement out of the same tracks of thought. Maybe I eventually fall asleep.

7. How frequent is this flavor?

1-2 times a month.

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

Bad: sometimes I stick with the immersion, thinking it’ll bring some new insight. But really, it’s such a shallow state of mind that I’ve never gotten anything useful out of it. Good: force some interruption or distraction, and come back to X after a break.