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The Calculus of Friendship

The Calculus of Friendship: What a Teacher and a Student Learned about Life while Corresponding about Math

Steven Strogatz
Copyright Date: 2009
Pages: 184
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  • Book Info
    The Calculus of Friendship
    Book Description:

    The Calculus of Friendshipis the story of an extraordinary connection between a teacher and a student, as chronicled through more than thirty years of letters between them. What makes their relationship unique is that it is based almost entirely on a shared love of calculus. For them, calculus is more than a branch of mathematics; it is a game they love playing together, a constant when all else is in flux. The teacher goes from the prime of his career to retirement, competes in whitewater kayaking at the international level, and loses a son. The student matures from high school math whiz to Ivy League professor, suffers the sudden death of a parent, and blunders into a marriage destined to fail. Yet through it all they take refuge in the haven of calculus--until a day comes when calculus is no longer enough.

    Like calculus itself,The Calculus of Friendshipis an exploration of change. It's about the transformation that takes place in a student's heart, as he and his teacher reverse roles, as they age, as they are buffeted by life itself. Written by a renowned teacher and communicator of mathematics,The Calculus of Friendshipis warm, intimate, and deeply moving. The most inspiring ideas of calculus, differential equations, and chaos theory are explained through metaphors, images, and anecdotes in a way that all readers will find beautiful, and even poignant. Math enthusiasts, from high school students to professionals, will delight in the offbeat problems and lucid explanations in the letters.

    For anyone whose life has been changed by a mentor,The Calculus of Friendshipwill be an unforgettable journey.

    eISBN: 978-1-4008-3088-6
    Subjects: Mathematics, History

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-viii)
  3. Prologue
    (pp. ix-xiv)

    For the past thirty years I’ve been corresponding with my high school calculus teacher, Mr. Don Joffray. During that time, he went from the prime of his career to retirement, competed in whitewater kayak at the international level, and lost a son. I matured from teenage math geek to Ivy League professor, suffered the sudden death of a parent, and blundered into a marriage destined to fail.

    What’s remarkable is not that any of this took place—such ups and downs are to be expected in three decades of life—but rather that so little of it is discussed in...

  4. Continuity (1974-75)
    (pp. 1-7)

    Calculus thrives on continuity. At its core is the assumption that things change smoothly, that everything is only infinitesimally different from what it was a moment before. Like a movie, calculus reimagines reality as a series of snapshots, and then recombines them, instant by instant, frame by frame, the succession of imperceptible changes creating an illusion of seamless flow.

    This way of understanding change has proven to be powerful beyond words—perhaps the greatest idea that humanity has ever had. Calculus enables us to travel to the moon, communicate at the speed of light, build bridges across miles of river,...

  5. Pursuit (1976)
    (pp. 8-12)

    Having finished the math courses offered by the school, I spent my senior year teaching myself, detached from Mr. Joffray. I’d sit alone for an hour every day in an empty classroom, reading a textbook on multivariable calculus or trying to rederive Huygens’s results about cycloidal pendulums.

    At other times I did a kind of research, often about chase problems. These problems of pursuit, as mathematicians call them, completely captivated me.

    The first one I ever heard of came from Mr. Joffray. It went like this: suppose a postman is trying to escape from a dog chasing him. The postman...

  6. Relativity (1977)
    (pp. 13-22)

    The theory of relativity is founded on empathy. Not empathy in the ordinary emotional sense; empathy in a rigorous scientific sense. The crucial idea is to imagine how things would appear to someone who’s moving in a different way than you are.

    At a time when it seemed absurd to claim that the earth moves around the sun, Galileo asked doubters to imagine being confined below deck in an enormous ship. There are no portholes in your cabin, no way to see the coastline passing by. If the sea is calm and the boat is gliding straight ahead at a...

  7. Irrationality (1978-79)
    (pp. 23-33)

    After my demoralizing encounter with linear algebra, I thought about switching from math to physics. Then in my sophomore year I had an inspirational teacher, Elias Stein, for complex analysis, and so I went back to majoring in math again.

    At some point during that year, my brother Ian had a talk with me. It’s hazy in my memory, but I picture us driving home for Thanksgiving. He was asking me about my future plans, what was I going to major in, etc., and then started trying to persuade me to take all the pre-med courses. Not this again—everyone...

  8. Shifts (1980-89)
    (pp. 34-41)

    Once I committed to pursuing math, life took me on a straight trajectory for the next nine years. These were the years of training. During that time Mr. Joffray and I seldom wrote to each other. There are just three items in the green Pendaflex folder from that period. But a few shifts were already starting to take place. In a letter dated December 16, 1980, I thanked Mr. Joffray for the first time:

    So till we meet again, be well and continue your great work—did I ever thank you for your encouragement? Well, I do so now—it...

  9. Proof on a Place Mat (March 1989)
    (pp. 42-70)

    Mathematicians aren’t known for their social skills. There’s an old joke in our community:

    Q: How can you tell if a mathematician is an extrovert?

    A: He looks atyourshoes when he’s talking to you.

    But what most people don’t realize is that math itself is a very social activity. We mathematicians talk to each other incessantly. We bounce ideas off one another, cogitate together, and get stuck on the same problems together. When you’re working on something as hard as math, it helps to share it with someone who understands.

    Best of all is when we’re able to...

  10. The Monk and the Mountain (1989-90)
    (pp. 71-83)

    In the June 1961 Mathematical Games column inScientific American,Martin Gardner posed a riddle that has become a favorite in courses on the psychology of creativity:

    One morning, exactly at sunrise, a Buddhist monk began to climb a tall mountain. The narrow path, no more than a foot or two wide, spiraled around the mountain to a glittering temple at the summit.

    The monk ascended the path at varying rates of speed, stopping many times along the way to rest and to eat the dried fruit he carried with him. He reached the temple shortly before sunset. After several...

  11. Randomness (1990-91)
    (pp. 84-93)

    It was in the summer of 1990 that a red flag went up for my mother about Elisabeth. Elisabeth felt financially insecure in Boston and suggested that maybe we should move to Troy, New York. I’d been approached by Rensselaer Polytechnic Institute about a job there, the lure being early tenure. Elisabeth liked the idea because she’d grown up in that part of upstate New York and the lifestyle and affordability appealed to her.

    Well, that didn’t sit well with my mother. Not at all.

    Actually, Elisabeth had been rightly unhappy about many things for quite a while. She’d given...

  12. Infinity and Limits (1991)
    (pp. 94-106)

    In everyday language, infinity and limits sound like they contradict each other, but in calculus they are welded together as part of a single, overarching concept. In fact it is probably the most revolutionary concept in all of calculus, the one that distinguishes this branch of math from all that came before—algebra, geometry, and trigonometry.

    The great achievement of calculus is the domestication of infinity. The concept of infinity itself had long been avoided and even feared. The Greeks thought it made no sense. During the Renaissance, the Church forbade anyone to write about infinity—and Giordano Bruno was...

  13. Chaos (1992–95)
    (pp. 107-114)

    A dynamical system is anything that changes from moment to moment according to definite rules. For instance, a pendulum swinging back and forth obeys Newton’s laws of motion and the law of gravity. Using those rules and the techniques of calculus, physicists can add up all the moment-to-moment changes to deduce where the pendulum will be at any time and how fast it’ll be moving. The same is true for a planet orbiting the sun or a rocket flying to the moon. Other scientists have extended the reach of dynamical systems to describe the flow of traffic on the Internet,...

  14. Celebration (1996-99)
    (pp. 115-117)

    On April 22, 1999, I was invited to pay tribute to Joff at a special event. Called ‘‘An Evening in Celebration of Education,’’ it was a black-tie ceremony held in the Cathedral Church of Saint John the Divine in New York City. More than 100 alumni, faculty, parents, and friends gathered in that breathtaking space to celebrate teaching and to honor Joff in particular.

    It must have been a bittersweet moment for him. Each spring he’d confront the decision of whether to sign on for another year, and until now the answer had always been yes. As he once wrote...

  15. The Path of Quickest Descent (2000-2003)
    (pp. 118-127)

    Joff’s opening words when he resumed our correspondence were ‘‘Withdrawal from math teaching hasn’t been easy.’’

    Over the next several years he inundated me with letters, one after another without waiting for a reply. He’d devise his own math problems and show me his solutions, just like I had when I was his student. He’d write about nature and his vacation trips and tell me about his new friends, people I’d never met or heard of.

    Mostly I was silent and didn’t write back, too busy now with our first daughter. Then another. Never enough sleep. Helping Carole. Writing a...

  16. Bifurcation (2004)
    (pp. 128-139)

    Change can occur in varying degrees of violence and unpredictability. Accordingly, there are different kinds of mathematics for each.

    At the mildest extreme lie the orderly changes of a system obeying differential equations, gliding along according to laws of motion. Think of the planets orbiting the sun or the burbling of a brook. Here calculus is in its element. It is designed for systems that evolve according to definite rules.

    At the opposite extreme lies wild, irrational change. The kind that comes from senseless shocks to a system, external events bearing no logical relation to the system itself. The asteroid...

  17. Hero’s Formula (2005-Present)
    (pp. 140-154)

    Calculus began with Zeno’s paradoxes about time, motion, and change. The most famous one is about Achilles and the tortoise. Consider a race between the great warrior and a humble tortoise in which the creature is given a head start. By the time Achilles reaches the tortoise’s starting place, the tortoise has crawled a bit farther ahead. By the time Achilles gets there, the tortoise has moved a little more and hence is still in the lead. Thus, a swift runner can never overtake a slower one. Since common sense says otherwise, Zeno concludes that common sense is wrong. Change...

  18. Acknowledgments
    (pp. 155-156)
    Steven Strogatz
  19. Further Reading
    (pp. 157-160)
  20. Bibliography
    (pp. 161-162)
  21. Photography Credits
    (pp. 163-164)
  22. Index of Math Problems
    (pp. 165-166)