I sometimes encounter a section of mathematics in which I cannot do anything but blindly learn a formula and then solve problems by plugging variables into it. The equations mean nothing to me, but I can figure out the pattern that is suggested to me by the worked-out problems in the book, and then solve the problem sets according to what the book tells me. This means that I am solving puzzles that have no meaning and are completely out of context. I am annoyed by this and think of it as a kind of "bad faith," in that I am doing things which have no meaning for me and in which I do not "believe."

This is what my current work in trigonometry is all about. A chapter in the BritBook introduced me to a form of equation characterized by y = a sin X + b cosX. The book shows me how to fiddle with this trigonometrically so that it becomes an equation in the form y = R sin (x + a). The book then not-so-helpfully says, "There are two main advantages in writing something like (this)…it enables you to solve equations easily, and to find the maximum and the minimum values of the function without further work."

Now this is where I get a bit uppity and out of line for a beginning math student. I think it would be rude to ask a professor this, but my question is, "What is all this for? And why do I need to find the maximum and minimum values of the function without further work? I'm already doing further work!" I am not sure this is the right attitude for a student like me to have. Since I took on this work for myself, with no hope of actually using it for anything practical like becoming an engineer or a real physicist, I should just, as another scientific blogmeister quoted in an unrelated context, "Shut up and calculate."

A similar situation arose some years ago when I was working my way through college algebra, using a book from 1958 which gave me the feeling of being in a time warp. One set of problems the time warp offered me involved deconstructing, completing the square, and finding a squared polynomial, all inside a square root sign. It was named as the "square root of a quadratic polynomial," and the book explained, "(it) will frequently be necessary in the integral calculus."

All of this goes under the heading of, "Do it now without understanding it or knowing its purpose, you will need it later on." Well, sometimes this gets a little old. Perhaps I should enjoy mathematics for the pure abstract patterns and the orderly cranking of its gears, but I am prosaic enough to want to know what it is for. What natural (or man-made) thing or process does this math describe? Is this too much to ask?

Finally one of my Friendly Experts came to my aid, when I sent a math rant e-mail to him about how I refused to go any further with this until I knew what it was for. (This is kind of funny, since no one is forcing me to make any progress in mathematics so I am rebelling against my own inner drive.) Fortunately this Expert is not an overburdened professor teaching me along with a class of 25 (or 200), so he was able to answer in some detail. It turns out that the sine + cosine formula I cited above has to do with describing waves and periodic phenomena, especially things like sound. With sound, the variables describe the amplitude (loudness), frequency (pitch) and phase shift (interaction with other sounds).

Well then, with sound I am on at least some familiar territory, since as I recounted some time ago, I spent plenty of time working with electronic sound generation and seeing the sine waves and traces on the screen of the oscilloscope. In fact, "amplitude" and "frequency" were part of our vocabulary in the electronic music studio of years past, long before those things were taken care of by computers and modern automated electronic sound manipulation. It's that old sine wave song again.

As for completing the square and achieving the square root of a quadratic polynomial, I'll look it up again when I need it. I take comfort in the fact that I can always go back and look up whatever I need in books or, if necessary, on websites. If and when I get to pondering the mathematics of periodic phenomena, I know where to go back to. But for now, I think it is finally time to move on.

Posted at 2:26 am | link