Since I am much involved with creating sufficient art product for the upcoming show, as well as re-arranging and cleaning almost my entire day job workplace, I don't have much time for math. But that doesn't mean I don't do any math. The dream of daringly riding the quantum waves of higher math and physics, to the industrial ambient sound of klystron engines, still has power for me. So I continue to work on calculus. I have just finished going through the Anton book's proof of the "quotient rule" of derivatives. This involves some of the same crisscross patterns I mentioned in a recent post, as well as one of the most common mathematical patterns in the universe (so it seems), a quantity squared, that is, multiplied by itself. Why would the Designer of mathematics be so square? If I ever meet Pythagoras, I'll ask him. He believed in re-incarnation, so for all we know, he might be somewhere right now, playing go in Matsusaka, Japan, or stirring vegetable soup in Chacabuco, Argentina.

The proofs of both the multiplication and the quotient rules involve lots of sorting and re-sorting of algebraic expressions. One way they (the provers? The Universe?) do this is by adding a seemingly arbitrary quantity which is then subtracted from the same line of expressions. This is proper math, because as long as you take something away after adding it, you're even. The introduction of this extra quantity helps the proof re-sort the expressions into the proper definition of a derivative, so they can be worked with. Without making a diagram of this, it's hard to describe but my Friendly Mathematicians and Scientists probably recognize what I'm talking about.

What intrigues me is that it does seem arbitrarily chosen, just for convenience's sake. It seems like a phantom piece of math that is here for a second and then gone. But that would be wrong. This process resembles that of a chemical catalyst, which adds nothing to the final product but enables the process to go on. The important thing, which reminds me of art and craft more than clockwork mathematics, is knowing just what quantity to throw in (and then remove) to get your proof to do what it's supposed to do. Being mostly self-taught, I missed that lesson in math class. Perhaps I am misinterpreting it anyway. But having worked through the proof, I feel that I am now worthy to do the set of derivative problems which await me in the next few pages.

Posted at 3:24 am | link