When I was in high school, I went through an experimental phase of essay writing. The standard essay form was too dry, too monotonous to express everything passing through me and percolating in my thoughts as I read great books. I wrote meditations, poems, and letters, strange apings of the books I was reading. I could not respond with my full self unless I expanded the form.

Something of this idea persists in me. It seems to me still that listening, reading, and participating in art should be active, dynamic. I would go as far as to say that most creators seek just such an audience that will respond and delve with their whole beings. Many musicians, for example, want to get their audience dancing. In dance, we listen to music and respond with our entire body. There is no substitute to this in kind of active participation from the listener.

When reading, the best thing is to underline, take notes, and write in response. The difference between written, articulated reflections on an essay or story and a casual reading is vast.

This got me thinking: what about math? How do we engage deeply with a powerful mathematical idea? One of the most inspiring answers to this question comes from Paul Halmos:

Don’t just read it; fight it!
Ask your own questions, look for your own examples, discover your own proofs.
Is the hypothesis necessary?
Is the converse true?
What happens in the classical special case?
What about the degenerate cases?
Where does the proof use the hypothesis?

~ Paul R. Halmos

Abstruse Goose, not happy with simply reading the quote, responded with his own drawn evocation:

I like Halmos’ quote. Being comprehensive as a listener, participating as an audience, feels like one of the great gifts you can give to the creator of the work. This is what’s required to make a serious student of mathematics (or literature, or music, or science) out of you: don’t be complacent, don’t be shallow. Dig in, get your hands dirty, and test it, stretch it, and make it your own. This is also, to me, a wonderful exception in our world increasingly seduced by superficial experience. Math is impenetrable to the superficial attempt; as an art form, it demands deep study. This presents a barrier, of course, but I also think it gives us inspiration. Depth is refreshing, enlivening. Mathematics also gives us a great equalizer in this respect: when King Ptolemy asked Euclid for a shortcut to learn math, Euclid is said to have replied “there is no Royal Road to geometry.”

I’m still chewing on all this, but I’d like to end here with (and this seems connected to me, at least tangentially) a brief video response to those who say that “writing about music is like dancing about architecture.” This is the Leah Stein Dance Company, founded by the woman who taught me contact improvisation in college. Their mission: dancing about architecture.