En Passant

Jerry Randal Bauer (1992)

When he was about fourteen years old, my son Jeremy started consulting with me about what mathematics courses he should pursue in high school.   The standard stuff for college prep was geometry, algebra I and II, and trigonometry.   He was interested in architecture and thought he might become an architect.   He asked me about calculus.

"What is calculus?"

I told him that calculus is "how to cheat at arithmetic."   "All those things you were taught that you couldn't do, calculus is how to do them.   It's how to divide by zero."

"So what is eight divided by zero?"

"In general, any constant divided by zero is infinity.   We get this answer by `taking the limit.'   It goes something like this.   First, we ask `What is eight divided by one?'   We get the answer `eight.'   What is eight divided by point five -- sixteen.   We see that the smaller the denominator, the larger the result.   We can see where it is going; when the denominator is zero, the result is infinity.   We say, `The limit of eight divided by x, as x goes to zero, is infinity.'"

"That seems easy, and pretty obvious."

"That's because it is just a demonstration.   There are ways, acceptable to most mathematicians, of proving that.   In calculus, you learn to prove those things, not just demonstrate them.   Actually, the concept of `proof' comes with geometry, where you start with some information and with a sequence of steps, each legal within the context, show that some other information is either true or not true.   You prove or disprove the other information."

"I thought geometry was about parallel lines and stuff."

"That's the subject matter.   That is the imaginary world, with the information and the legal steps.   Points, lines, and angles are the chessmen, axioms are the rules of the game.   But they are not the game.   Once you've learned geometry, you will have learned more than how to prove triangles are similar.   You will have learned the rudiments of symbolic logic."

We went on with this for a while, glossing symbolic logic, algebra, analytic geometry, trigonometry, and some more calculus.   It finally hit him.

"Hey, this stuff isn't about numbers at all, is it?   It's about solving all kinds of problems.   It's how to think about things!   Numbers are just chessmen!"

I can still beat him at chess.   Maybe not much longer, though.