Play

I've gotten to the point where I don't think I can think unless I'm writing my thoughts out.

"Write, write, write," I exhort my students. "Write out your thoughts, in whatever form they take." "Play!" is a variant of this exhortation.

It's taken me over a decade and a half to realize that what I call "play," and what my wonderful mentor Jim Hagler no doubt called by this same word when he encouraged me with it during my own years as an undergraduate, is simply a mathematical version of low-stakes writing. "Play" can comprise words and numbers and notation, but it can also comprise visual elements such as pictures, charts, and graphs.

What purpose does "play" serve? The same purpose served by any writing-to-learn activity. I reminded my 280 students this morning that the very act of rendering concrete the abstract contents of our convoluted brains can help us make sense of those contents. The recursive processes involved in "play" (drawing, drafting, doodling, then reflecting, recollecting, and redrafting) help us organize our thoughts in ways we never could if we kept those thoughts confined.

Case in point? This afternoon I was working with one of the three math students whom I'm guiding in undergraduate research this term. Lulu's a terrifically intelligent student with whom I had the pleasure of working two years ago this term when she enrolled in my 280 course. She was easily one of the top two or three students in that large class, and since then she's distinguished herself by continued success in several upper-level math courses. Right now she's doing original work with me on the Möbius function of various subgroup and subgraph lattices, and she's making great progress so far. (Damn it...I've written so damned many rec letters this season that my blog posts are starting to sound like one...)

Today I was introducing her to the Deletion-Contraction algorithm for the recursive computation of chromatic polynomials. To illustrate the algorithm, I thought to use it to compute the chromatic polynomial of the cycle on 6 vertices, C₆. We began our work playfully, tinkering with the graphs which resulted from applying the algorithm to the initial graph. It soon became evident (through the simple stick charts which we drew) that we'd only find our answer if we knew something about the chromatic polynomial of P₆, the path on 6 vertices.

We let play push us forward. P₆, itself not the initial object of our investigation, soon led us further astray, as P₆ gave way to P₅, and this to P₄, and so on. A few computations (and a hand-wavy inductive argument) later, and we had a formula for the chromatic polynomial of P_n.

Play pushed us back, back to C₆. P₆ subdued, we turned to C₅, also intimately involved. But analogy (again recognized by the stick charts we'd drawn) allowed us to develop a pattern for this graph, too, and we soon realized a sort of inclusion/exclusion formula for the chromatic polynomial we sought.

Whether or not you follow the technical details above, I hope you can recognize this: the tinkering and tweaking we did on the blackboard in my office, including both playful formulaic experimentation and purposeful doodling, is precisely what helped us to develop the formula we sought. There is simply no way we could have arrived at that formula had we simply stared into space and "thought about" the graph C₆. Though the writing I was doing as Lulu and I worked this problem together helped me to communicate some new mathematical ideas to my young mentee, the writing also served the more critical role of making real my once-cluttered/now-clear thoughts.

Goethe famously said "I call architecture frozen music." Might writing merely be frozen thought?

I'll play with that.