Thursday, May 3, 2007

The Mathematician and The Plumber

The Aristocrats is a brilliant movie. On the one hand, it's about the nature and art of a good joke, and on the other hand, it's depraved.

Roger Ebert's review is also brilliant, in that eloquently Ebert way where you can agree with his analysis but disagree with his opinion. He breaks it down like this—there are two kinds of jokes: short jokes and long jokes. A short joke is like a good pick-up line. Because it has to end before your audience realizes that it's a joke, it packs everything in the punchline and whirls to that punch in a sentence or two.

But for a long joke, the punchline doesn't matter. It's just an artifact of the form of the joke, like some plastic-wrap packaging to be thrown away when you're done. The real joke is in the telling, in the construction, in the precariously tall tower of cards that you have to build up to get there. A good long joke makes sense during the telling, yet completely contradicts itself when you look at it from any other direction, like a Kurt Vonnegut novel.

The geometer Peter Doyle once told me one such long joke. I've condensed it here for easier reading:
Once upon a time there was a mathematician. His toilet was clogged. So he called the plumber. The plumber arrived later that evening, unclogged the toilet in 15 minutes, and handed the mathematician the bill. The mathematician looked at the bill and shouted: "Great scott! What a bill! You plumbers must make a fortune charging people this much. Do you mind if I ask how much you make?"

The plumber doesn't mind. He grins and names his salary.

The Mathematician: "That's more than I make. And it looks a lot easier than theoretical mathematics."

The Plumber: "Well, friend, you sound eager, so I'll give you a tip. The plumbing business is swell. And my foreman needs more men for the job. Tell him I sent you, and you can see for yourself what the work is like. One thing though: don't tell him you're a mathematician. He hates elitists, and he won't hire anyone with higher than an 8th grade education."

Well the mathematician was as serious as he claimed. The next day he did indeed go to see the plumber's foreman. By pretending to be a middle-school drop-out, he got the job easily. And lo! It was better than he had hoped. He found himself with better pay, fewer hours, and more respect than he ever had as a theoretical mathematician.

The mathematician worked many years as a happy plumber. The plumbing industry flourished, and one day the foreman decided that his plumbers really had to be at a 9th-grade level in order to remain competitive in this rapidly-growing field. He hired several private tutors, and required his plumbers to attend night classes.

On the first day of class, the math teacher was trying to gauge what his students knew. He singled out one plumber and asked, "Do you know the formula for the area of the circle?" Of course, the tutor had picked our mathematician, and like any good mathematician, he responded, "No. But I know how to derive it." "Derive it for me then," challenged the teacher.

So the mathematician went to the blackboard and began to compute the area of the circle as any good mathematician would. He wrote out the double integral with respect to x and y, computed the Jacobian with respect to r and theta so that he could perform a change of basis in terms of polar coordinates, then evaluated the double integral. And came to the solution -π*r2.

"Wait!" said the mathematician right before he announced his answer. "That can't be right!" He began to check his work, looking for where the negative got introduced. But he couldn't find his mistake. At last he threw up his hands, erased his computation, and started over again. He quickly set up the double integral, did the change of basis to polar coordinates, and simplified. And again he arrived at -π*r2 !

By now the mathematician was frantic. How could he have gotten such a ridiculous answer twice? Had his math skills really gotten so bad? He wildly looked to the teacher, but the teacher knew nothing of multi-variable calculus, and had no idea what was going on. Then he looked to the class...and noticed something quite odd.

The eyes of every plumber were fixed upon him! The class was anxiously, quietly trying to get his attention without alerting the teacher. Each one had his hands cupped around his mouth, and they were all whispering the same words over and over again in unison. Slowly, he leaned in to hear better. And this is what he heard:

"You forgot to take the absolute value of the Jacobian!"

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