As I wrote those words in yesterday's post I was watching an osprey cruise the shore in front of the house. Fish hawk, they call it here. It dipped and soared, hardly moving its wings, its keen eye fixed on the turquoise water below, waiting for a glint of sunlight from the scales of a silver fish. Flow. Flow. The bird, the air, the sea, the fish. Different at every moment. Always the same.
And somehow my thoughts drifted back more than half-a-century to...
Somewhere about the second or third week of every university calculus course, the student is introduced to the concept of the limiting value of a ratio both terms of which approach zero -- a concept crucial to all that follows. The definition was the most incomprehensible thing I had encountered in my life, a mess of mathematical gobbledegook. It was appended to the statement "These preliminary remarks should now enable us to understand..."
But, of course, I did not understand. I doubt if any first-year calculus student reaches this point in the course with understanding. Still, I was smart enough to know that if I sidestepped this initial hurdle I would never grasp what followed. So I beat my head against it for a week until the light bulb finally went on. I had figured out the concept of a limit.
The rest, as they say, was a piece of cake. The study of calculus became pure bliss, the neatest thing I encountered in college.
I took my final degree in physics, and physicists use calculus to express the laws of nature. But it was a funny sort of nature we studied in physics -- without ospreys, wind, sea, or fish. As the years passed I drifted into writing, and more or less forgot about calculus. But something ineradicable had been planted in my mind. Something about flow. About transformation. About continuous change.
Something about ospreys riding gifts of air.
The poet Marianne Moore wrote: "The power of the visible is the invisible." Calculus is about invisibles -- the infinite and the infinitesimal. That's what the cryptic definition of a limit is all about. A way to talk meaningfully about the unobservable instant. The thing we call a derivative in calculus is the calculable ratio of two numbers that separately vanish into nothingness, leaving behind something spooky but palpably real, like the grin of the Cheshire cat -- a rate of change, a measure of continuous flow. Calculus clicked when I made the connection between the grin and the cat.
Which brings me back to the osprey. As I watched that splendid bird riding gifts of air it occurred to me that I was observing the physical embodiment of those abstract differential equations I studied long ago. Calculus was invented as a language for describing continuous change in nature -- the glide, the dive, the soar, the flow. Watching the osprey I was an eavesdropper, listening in on nature's conversation with itself.