Saint Sava (saint_sava) wrote,
Saint Sava

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The Tao of integration.

(posted elsewhere in response to an entry by johnmcelroy)

Buddhists say that before you study Zen, the mountains are mountains and the rivers are rivers; while you study Zen, the mountains are no longer mountains and the rivers no longer rivers. When you have grasped Zen, the mountains are again mountains and the rivers again rivers.

Math and Zen aren't the same, but there is a Zen aspect to math that is much different than the acquisition of a new language. A new language is largely a matter of acquiring fluency a word at a time, pausing briefly to arrange the pieces on the shelves of an already-built framework. However, learning math was like climbing to the top of an endless mesa: a plateau, then a satori that rises to another plateau, gaining elevation through a series of small but cumulative epiphanies.

Calculus was especially rich in sparks of enlightenment for me; and most of all so, the process of integration. Most any function can be differentiated, but most functions are at least somewhat resistant to integration; learning to handle these problematic integrals, first through substitution of variables, then through changes of coordinate, then through trigonometry, steadily gaining, through building insight, more tools to crack the toughest mathematical nuts. The path of satori continued on into differential equations and number theory, beginning to slow for me as I rose to my level of mathematical incompetence in Complex Analysis.

At first, the mathematics was mathematics: abstruse, impenetrable, the realm of people fundamentally more intelligent than I was. Halfway through Calc III, mathematics was a tool, now used frequently in other academic disciplines and having its own dialects in each; we were all overly careful and ridiculously
formal in our uses and interpretations of it, and we spoke as though we learn it from textbooks. By the time we were doing PDEs, though, mathematics had become again mathematics: scratches in a book that pointed at concepts we knew so intimately it was frequently unnecessary to do more than glance at the questions. Looking at an integral was an entirely different process: "Oh, this blows up. That goes to zero; ignore it. This thing here is irrelevant." As mature mathematicians, we had fully digested the culture-philosophy, and we could shoot from the hip with accuracy and confidence.

Those who studied for the sake of the exam generally didn't last long, considering studying a training -- being prepared against surprise. The ones who lasted did so for the sheer joy of mathematics, considering studying an education -- being prepared for surprise.
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