In the snowless flurry of activity around the "Holiday Season," I haven't had much time to work on my math. But derivatives are on my mind and I haven't forgotten anything. I have been trying to work out one of the proofs of a rule of derivatives, as detailed in the book, but have failed to understand one of the steps. The rule is, as the book states it, "The derivative of a sum equals the sum of the derivatives, and the derivative of a difference equals the difference of the derivatives." The proof is calculated (though I have unfortunately missed out on understanding it) so I must take it as true. But why? What would really convince me that this business about adding derivatives is true?

Usually, in any other learning discipline, they'd show me an example. And this happens here in the book, too. After proving it, they (the authors) show me examples of how this process works, using numbers and algebraic letter variables. This works, too, but I find it still apart from reality. What does this have to do with the "real world?" I suppose in pure mathematics, this is the wrong question to ask. Just prove it, put some numbers through it, and it is true. True forever, in ways that humanistic or poetic or religious truth can never be. Ask a mathematician, and he'll tell you. It's all hyper-true, provable even if human beings had never existed.

But if you ask an artist, the only thing my primitive visual-oriented brain can understand is, of course, a picture. Show me a picture, then. I take the example of the added functions from the book and put it into the graphing software on my computer. Then I take the f-prime function, the added derivatives of those added functions, and put that one on the same graph. Sure enough, as I ponder it, the second graph describes the progress of the slope of the first graph. Is this proof enough?

When I was doing second-year algebra, using my Cold War-era textbook, I sometimes solved problems not by calculating them through abstractly, but by finding as many solutions as I could by running different numbers through the equation. Eventually this would create a pattern, either in numbers or on a graph, that I could follow to get myself the answer. I used my calculator, which was anachronistic, since in 1958 they didn't have these things, but I wanted to find the answer, and with my futuristic machine I could find dozens of answers.

Evidently in mathematics this is called the method of "brute force," where you do as many calculations as possible to get yourself the pattern which will lead you to an answer. Calculus is supposed to relieve you of the need to do "brute force" mathematics. But this method, math by observation, is more like the learning experiences I am accustomed to, whether it be collecting quotes from relevant texts or noting the number of times a certain event happens after another event. Math by proof is unlike my usual way of learning. I can't prove by calculation exactly when the Prophet Zarathushtra lived, or why I should use a certain shade of blue-green in the center of my current painting. I can learn the proof, but I still don't know what to make of it.

Posted at 3:10 am | link