Let y stand for yogurt and f in this case for lactobacillus acidophilus and related fermentation agents. If x = milk, then y = f(x), that is, "yogurt = cultured milk." In between spoonfuls of the y-stuff I contemplate eternal truths, at least those of mathematical form. So far I have tasted the "constant function" derivative rule, and the "exponent" or "power" rule. Currently I'm learning the rule of multiplying by a constant. As the book puts it, "Let c be a constant. If f is differentiable at x, then so is cf, and (cf)'(x) = cf'(x).…in words, 'a constant factor c can be moved through a derivative sign.'"

And how do you know this? Because the book gives you a proof of it. Plain old algebra multiplies functions by a constant all the time, but I never quite realized what was going on there. The proof multiplies the function by some constant number, then cranks the calculus clockwork to show you that the derivative function is also multiplied. Then they show some of this in action with sample numbers and functions.

I've never been that familiar with proofs. I've done 'em in geometry, big long jobs, but have only done the simplest kind in algebra. Evidently a proof done correctly establishes the truth of a theorem. And a proof is sort of like a step by step process to show how one thing becomes another in a legal and legitimate way. If all truth were provable by proofs, life would be so much easier. In fact, some scientists and mathematicians actually believe that everything should be established by some sort of proof, and if it can't be proved by these logical methods, then it isn't true. Unfortunately, I can't prove the aesthetic derivative of my art, or anyone else's.

What it looks like to my color-addled brain is that a colorful constant, which is not c but "carmine" or "cerulean," is multiplied into a colorless function. And then if you put the function on your graphic palette and take its derivative, then not only will the colorful constant dye the whole derivative, but the exponent, should there be one, will also enter into the color mix, and alter the color of the whole coefficient. Multiplication will add in more colors. Sometimes you get a nice new shade, but other times you get a muddy brown or grey. Maybe this is why artists aren't good at math. But graphs and lines help me understand. If a derivative is a slope, then it can be multiplied by some number and still have that slope, at any possible tangent. At least this is what it looks like to me. Because I am an architectural artist, I can keep my perspective.

Posted at 3:27 am | link