Eigenmumbles

Joe was a good kid. You’d have liked him if he were in your class: he worked hard, he cared about the material, he tolerated the bad jokes. And he got it, mostly. He knew how to write a proof, and could turn out code that mostly worked most of the time. He mostly made life better for everyone in the class, professors and students alike.

But like most of us, Joe had a flaw. He was in love with the Right Answer. Given the choice between a one-line argument to show $x$ is between $2.5$ and $3$ or a ten-page argument showing $x = \exp(-\pi/2)/\phi$, Joe would pull out a pad and start filling pages with algebra. Never mind that all that mattered for the application at hand was that $x < 10$! Equalities made Joe comfortable; inequalities filled him with a vague dread. In times of stress or uncertainty, he fled back to the land of equations, even if the roads through that land were treacherous and unlikely to lead to a satisfactory end.

Joe, of course, didn’t see his inability to approximate as a problem. He was very proud of his ability to get the Right Answer, and was always a bit condescending toward friends who had better taste and trained in the art of back-of-the-envelope estimation. Fortunately, he was well socialized, if a bit arrogant, and so mostly managed not to be insufferable when he was invited to cocktail parties, even when the topic turned to things technical – as it almost invariably will when there are enough scientists and engineers gathered in one place.

After a couple years, Joe graduated with good grades and a respectable degree, then joined Teach For America. I understand that he fell in love with teaching high school math, and continued on with it after his stint with Teach For America ended. He’s married now and has a kid, which makes me feel old; he seemed so young himself, not that long ago! I’d mostly not thought about him for years until a few days ago, when he wrote me an email saying he was in town and asking if we might go have a coffee.

“So,” he said as we sat down at the cafe, “I’ve decided that you were right about something way back when.” I replied that I hoped I was right about more than one thing, but asked which thing he had in mind. “You were right,” he replied, “when you said that there’s some value in approximation even when an exact answer is available.” I was surprised to hear those words coming out of Joe’s mouth, and asked what inspired the revelation, and he told me that one dreary day when he was sitting with his little girl, she asked him where the sun goes when it rains. He told her that the sun was still there, and but that the clouds get in the way, and she said: “does that mean the sun doesn’t like to play in the rain?”

“Well,” he said, “that’s not really right, but maybe it’s a good approximately in the right direction.”

“Daddy,” she replied, “I like approximately better.”