Nonlinear Relationships

Bibliologue / by Veronique Greenwood /

In mathematician Steven Strogatz’s recent book, friendship and integrals collide, yielding a math story of unusual poignancy.

Long after Steven Strogatz had become a professor at MIT and Cornell, he was still writing letters to his high school calculus teacher, Don Joffray. The two started swapping word problems and logic puzzles when Strogatz, who studies chaotic systems, was in college. Their jovial mini-treatises form the body of Strogatz’s most recent book, The Calculus of Friendship.

But they always managed to keep a certain distance from each other, demonstrating, perhaps, the natural reserve for which mathematicians are known. Mr. Joffray, as Strogatz still refers to him, sent his congratulations on Strogatz’s engagement in a brief preface to a calculus problem, but Strogatz’s reply makes no acknowledgement. When he heard from secondhand sources that Joffray’s son had died, Strogatz wondered why Joffray never mentioned it, but never brought it up. After thirty years of omissions, it took a string of tragic events to bring their friendship into three dimensions.

Seed editor Veronique Greenwood spoke with Strogatz about the nature of friendship, why math isn’t about right and wrong, and how an elliptical swimming pool helped launch his teaching career.

How did this remarkable correspondence begin?

Steven Strogatz: When I was a junior in high school, I took calculus with Mr. Joffray. People often assume that he must have been my mentor, but I wouldn’t call him that. Mr. Joffray was not one of the three or four teachers that influenced me the most in high school. And in my senior year, I lost touch with him, because I had finished the available math courses and was working on my own.

So it was especially peculiar that I would happen to write to him, starting my freshman year of college. But something about my relationship with him worked—it kept on going. About once a year, I would write to him about something I’d learned in college that I thought he might like—a problem or a proof—and it went on like that for several years.

Things really got rolling when I was a junior in college, and he asked me a question that he didn’t know the answer to that had come up in one of his advanced placement courses. I could even tell you the question: He was imagining an elliptical pool with a 1-foot-wide border around the outside, and he wanted to know if the border would also be an ellipse. I mean, it’s certainly some kind of oval, but is it specifically the kind of curve that to a mathematician would qualify as an ellipse? Now it was no longer just me showing him things I was learning—he was actually asking for help. And, boy, I loved that! That was very exciting.

It was very generous of him. I wanted to teach—I always wanted to teach—but I was a kid, and I didn’t have anyone to teach! Who would sit there listening to me? Well, he would.

Seed: You describe one of your first math teachers at Princeton, a renowned topologist—“he was so shy that he slithered along the wall when he entered the lecture room, as if hoping to become invisible.” And you tell the joke—How can you tell if a mathematician’s an extrovert? He looks at your shoes when he’s talking to you.

SS: Ha! Yeah, that is one of our favorites.

Seed: But as you point out, mathematics can actually be a very social activity, as it was for you and Mr. Joffray in these letters.

SS: It sure can be. But it’s a very rare teacher who can make it that way. A lot of people hate math because their only experience with it in school was very authoritarian. People say all the time that math is about right and wrong, but that’s a very simplistic view of math. Even if two different answers are right, there might be one that’s better than the other, because it’s more elegant or produces more insight. Likewise, things that are wrong can be very illuminating.

Math is about playfulness. It’s a kind of art: It’s about expressing yourself. A lot of people don’t get that, including many teachers, who don’t really understand the subject. To them, it’s about following rules. You have to follow this algorithm or this procedure.

As a teacher, Mr. Joffray really had the spirit of mathematical inquiry, and I don’t know where he got it from. Maybe it was just that he is such a curious person in every way. When I was in high school, he was always outside looking at birds and thinking about where was the best place to stand when you kick a field goal.

Because the truth is, he wasn’t that great a mathematician. That didn’t matter. It gave you the feeling that you didn’t have be right all the time, because he wasn’t.

Seed: For a lapsed calculus student, where’s the best place to start working through the puzzles in this book? Is there a good entry point?

SS: That’s a good question. Because the book is structured chronologically, the first problem—a chase problem about four dogs—is pretty hard in terms of calculus.

You might want to start instead with a geometry problem, which you can find in the index of mathematical problems in the back. For instance, you can prove that the square root of two is irrational without using any calculus. I did try to make it so there were problems at every level, from simple logic puzzles all the way up to some pretty hairy calculus problems.

The idea is that anyone who has taken a year or two of calculus can follow just about anything in these letters. And there’s a reason for that—it’s what Mr. Joffray himself knew. He knew everything through high school calculus, and very little after that.

Seed: What was it like when you finally realized you wanted to meet with Mr. Joffray to talk about all the things you’d omitted in your letters?

SS: That was at a tough time in my life. My dad was dying—he was in tough shape from Parkinson’s disease, and he really wasn’t even himself after a while.

Mr. Joffray had been retired for a few years, and he was inundating me with letters, one every couple of weeks. I didn’t have time to respond. We had two little babies at that point. I remember feeling a lot of guilt about not being able to respond to all these letters piling up, and him saying in his letters, “Don’t apologize! It’s a special moment for me to be able to write to you.” He was always saying heartbreaking things like that, which just made me feel guiltier.

There was that, and then, as I said, my dad was disintegrating before my eyes. Then, suddenly, overnight, my brother died. He went into the hospital with a stomachache, and then he was dead.

If I were lying on a couch in a therapist’s office somewhere, what I’d say is that this might have something to do with the men in my life. It’s often complicated for men to talk about things with their fathers or brothers. My own father was an old-school sort of guy—a kind man, but not someone who talked easily about emotional stuff. Mr. Joffray never brought up emotional stuff either. He never mentioned anything about his son Marshall, who had died. And I didn’t ask.

When he sent me a letter with his condolences about my brother’s death, that was the triggering event. Why can’t I do this simple thing that he just did? I thought. Why don’t we talk about things?

When I went to go see him finally, I brought along a tape recorder. I didn’t want to miss anything when it got emotional. And you can hear the recording for yourself on WNYC’s “Radiolab” program, actually, in a show they did called “Numbers.” The third segment is me and Mr. Joffray.

Mr. Joffray is still alive—he’s 81. He has had two strokes, so he’s not quite what he used to be. And he’s so old-school—he doesn’t really understand why he’s on the cover of the book. He doesn’t see himself as somebody special. But for me, he really was.

Originally published January 26, 2010

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