From Steele this morning, a metaphor for mathematics I hadn't heard before: mathematics is like an oil painting. Basically, people doing oil paintings start by making a very rough sketch of the painting and then progressively build up the details of the figures. (I've never painted in oil, so correct me if I'm wrong.)
Mathematics is similar. In research one only starts out with a vague idea of the result and then progressively refines it; in teaching one first gives a sketch of an argument and then comes back and fills in the details. Teaching was the context here; often in classes which depend on measure-theoretic probability, which this is, we first give a semi-formal proof of a result and only later come back and fill in the σ-fields, justify the magic words like "dominated convergence", and so on.
Compare perhaps Hackers and Painters by Paul Graham, which compares the two title groups.
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Forgive me, but which areas of learning and development aren't modeled by this analogy. Experience gives us a glimpse of the landscape and then it's up to us to choose to add detail,or not. Do you take the red pill, or the blue pill? Anyone serious about a topic, has made this decision, though I don't think we ever find out how deep the rabbit hole goes, we just run out of curiosity, or life.
My two favorite metaphors for mathematics are:
1) Finding a proof is like climbing a mountain. You struggle and struggle, slip and fall and have to climb back up, and it takes you weeks to finally find a safe route up to the top…at which point you see the nice smooth road going up the other side.
2) If I built a house the way I figure out a proof, I'd do the wiring, then put up the framework on the second floor, then wallpaper the kitchen, then pour the foundation.
The September 2008 issue of AMSTAT News (American Statistical Association) has an interesting article entitled "Math Is Music; Statistics Is Literature." The idea is that it is possible to have child prodigies in certain fields (music & math) but not in other fields (literature & statistics), and that this is because the latter fields require some life experience that no child (however gifted) could have. The Dungeons and Dragons analogy might be that literature and statistics require not only Intelligence but also Wisdom. Here's a PowerPoint presentation that one of the article's co-authors has presented on the topic.
And with oil painting (and research) sometimes an idea that seems so right in the mind just doesn't work once it's on the canvas. Scrapping it and starting over is hard, but the only way to recover in general. And the question of When is it Done? [Have I done enough here? Do I need more there?]
(As weldon points out, this isn't unique to research but I still like to think about the parallels in analogies. I love the two that intrinsicallyknotted shared!
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