Archive for December, 2012

Flavor: Wandering around (math) for fun.

December 23, 2012

1. What’s going on mathematically?

I’m surveying some new mathematical terrain, mostly out of curiosity and for fun.

As a child and young student, I remember falling in love with math by exploring, and following my curiosity. Once math becomes a vocation, there’s more at stake and one is less free to just flitter about dabbling and seeking math pleasure. But occasionally, I do it anyways. This flavor of math is the grown-up analog of the childhood experience.

2. What is the emotional and logistical context?

There’s time. Perhaps I’m going to a conference, and want to know more about the talks, the speakers, and the content. Maybe there’s a seminar I’m about to attend, and the title and abstract are intriguing but slightly out of my area. Or maybe I don’t have an excuse, and am just curious about something.

3. What thoughts are there?

I use the internet, and my resources are usually: Wikipedia, papers on the arXiv, personal websites of mathematicians, and the Mathematics Genealogy site.

Like a detective, I start with a few terms on Wikipedia. I follow leads, to read about the people behind the ideas (when and where they studied, from whom; where they are now; who else they work with; what else they’ve done). I try to reconstruct a history of the development of the ideas, while piecing together the theory itself. (I find chronology a very useful tool for unpacking knowledge.)

Certain words, terms, and theorems keep appearing, and I try to pin down precise definitions, as well as get a gist of main properties and consequences. Gradually, a rough map of ideas comes together. It is superficial, to a larger or smaller extent, depending on how familiar I am already with the ideas.

The exploration is not completely aimless, unlike the times I start reading the news about Syria, then end up scouring the internet for information about the millions of pounds of chemical weapons the US has dumped in the ocean, then reading about underwater landslides and the currents they produce, and giant squids, etc, etc… I stick to math, and stick to the rough topic I started with, and if necessary I abandon an intriguing thread and return to the starting point.

4. What quality of awareness?

A productive session has a good dose of self-awareness, to keep pulling myself back to the central idea(s). There are periods of absorption, when I try to let my curiosity lead me, as unfettered as possible. These alternate with stepping back, checking in, closing some browser tabs, and returning to somewhere familiar, to head off in a different direction.

There are also good periods of spacing out. After skimming one too many papers, my eyes glaze over and my mind detaches from the reading, to go out on its own adventure. I catch myself staring off into space, like I’m listening to the new ideas resonate within the context of everything else that I know.

5. What emotions?

Getting lost in curiosity is delightful. There’s a bittersweet tug every time I stop myself and return to the central idea(s). The space-out periods are also pleasant, the pure joy of absorbed thought. I think there is value in this type of unconscious percolation, although it is hard to quantify or describe.

There’s a feeling of irresponsibility, that is enjoyable. As I piece together an understanding like a detective, a narrative emerges. I can enjoy the characters and what they’ve gone through, without feeling accountable. It’s like watching a movie of mathematics. I can relate. It is meaningful to hear the human story, and understand the conceptual inter-relations. Even if the content is different in substance, analysis instead of algebra, say, the architecture of the content is still logical and rigorous, and the humans responsible for the development are still members of the same community. I relish the taste of this mathematical essence.

Interestingly, there’s always a gradual trend within me towards anxiety, that I watch and try to keep in check. Inevitably, I’m confronted with the vastness of Math That I Will Never Know, or with things I used to know but have forgotten, or with a long list of articles that I really should read and understand. Traveling quickly and superficially around the mathematical terrain is frightening, just like traveling quickly through any unfamiliar terrain for too long. What’s tragic and somewhat fascinating is watching this anxiety (which I never had during the childhood periods of mathematical exploration) sneak up on me, every time.

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

There’s really no concrete end, since one can always go farther or deeper. So I just decide to stop at some point. Usually after an hour or a few hours. Most times, I will have found some papers that I genuinely want or ought to revisit in more depth, and these get saved as PDFs and put in a certain spot.

7. How frequent is this flavor?

Maybe once or twice a month. I ought to do it more often.

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

In the best cases, I even write out a quick outline or sketch of the ideas that I’ve mapped out; this helps me internalize the exploration, and remember some of what I’ve learned. This could be an outline, a chronology, or an idea map.

This flavor is a great sort of intermission, to working more seriously on some project. So afterwards, hopefully, my mind is relaxed and I feel happy, and I can dive back in to “work.”

In the worst cases, the anxiety catches me and I can’t get perspective on it, so I’m left in a sort of panic, facing the overwhelming deluge of information.

Flavor: Pondering.

December 23, 2012

1. What’s going on mathematically?

I decide to try to see (mathematical) things differently. It’s a relaxed, playful attempt to play with math. Similar to an “intentioned immersion” but looser and less aggressive.

2. What is the emotional and logistical context?

Usually I’m riding on public transit, commuting, in a car, waiting for something, or taking a long walk where I won’t be interrupted. I’m alone. I won’t be able to work very rigorously, so I have low expectations of what will happen; still, there’s an intention to engage math.

3. What thoughts are there?

There’s a range of pondering modes, or games you can play. Here are a few.

Sometimes I take a specific object or idea and try to visualize it. I try to describe its essence. I imagine seeing this image or description for the first time, and see what questions arise.

Sometimes I juxtapose two things via a nonsensical premise, and see what happens.

Sometimes I imagine that I can turn any statement I want into a true statement, like I had a magic truth wand. What would be the most beautiful situation in a given context? What do I really want to be true? And then what are the relevant hypotheses?

Sometimes I construct an internal dialog with an imaginary colleague, or pretend I’m teaching a class.

Again, the basic idea is to play. Rearrange, experiment, start again… but in a light way.

Usually I don’t write, or don’t write very much, so my lines of reasoning can’t get very far. If the game I was playing was like a chess game, then I don’t try to strategize out four moves ahead, but rather just imagine different possible vague developments in the layout of the board.

4. What quality of awareness?

It takes effort, to stay focused on my pondering, and to think in ways I’m not used to. There is a strange juxtaposition of the mathematical headspace and the real world. My perception and awareness jumps back and forth, usually in a disjoint way, between these two realities. So my hearing of the ambient surroundings cuts in and out, perhaps alternating with the sound of an internal mathematical narrative. Or my spacial sense of where I am cuts in and out. Or my visual perception of what’s in front of me gets interrupted by mathematical imagery.

There have been a few times when the two realities sort of coalesced. Rather than a juxtaposition, or cutting back and forth between the two, there was a synthesis. I don’t quite know how to describe these experiences, except as a kind of integration of mind and body.

5. What emotions?

It’s usually fun, and sometimes funny. Sometimes it’s too effortful though, because I’m too distracted. It never feels like I’m getting anywhere, and so it often feels pointless and dumb. But there’s a pleasant feeling that comes with bringing some cognitive depth into a somewhat numbing situation like riding the metro or waiting on line.

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

Often I do get some new idea (of questionable worth). Or a new perspective, or a new question. I think I need about 15 minutes of stable pondering, if I’m going to settle into it and get anywhere.

7. How frequent is this flavor?

Mathematical pondering doesn’t come naturally to me, but I feel like it should. (I ponder other things much more effortlessly.) Maybe once a week, unless I intentionally practice it.

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

I always think that pondering will become easier, or more fruitful, eventually. It seems like the transitions from being in the real world to being in the math headspace, and vice versa, should become easier and easier to traverse. It seems like there’s a lot of insight in being able to enter the math headspace in different ways, at different times, and so pondering is a way of practicing this transition.

I try to write down any insights I gain from pondering, no matter how vague. But other times I give up pondering because it’s hard and doesn’t feel productive, or even fun.