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

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.”

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