A tale of two calculus books

To honor the fifth anniversary of Change is the Only Constant, here’s a conversation “from the vault” (i.e., the year 2019) between myself and Steve Strogatz.

When I began writing about calculus, I took for granted that my subject was not the sexiest. It lacked topicality. It lacked sparkle. Then again, I lack those things myself—and so, though I knew the risks (or thought I did), I chose to attend the Ball of New Books wearing an antique 17^th-century gown.

I never guessed the actual danger: the belle of the ball was already wearing my dress.

Cornell professor Steve Strogatz is the unofficial Mayor of Mathematics. He has written for the New York Times and New Yorker. He has appeared on Radiolab so many times that at some point they stopped introducing him: he’s just Uncle Steve, friendly neighborhood mathematician.

His book Infinite Powers, an ode to calculus’s deep history and diverse scientific applications, came out six months before my book, and landed on the New York Times bestseller list. Lucky for me, one of Steve’s infinite powers is his bottomless generosity, so he carved out two hours for an author-to-author heart-to-heart that swept from topic to topic:

• Is calculus really the language of the universe?

• Why is it so hard to write intellectual history?

• Which historical mathematicians were most the most sympathetic, and which were the most nasty-tempered?

• Why do obscure technical theorems dominate in the teaching of calculus?

• And what’s the right vision for the future of math education?

I was wrong in at least one respect: calculus is not passé. The subject’s popularity has quintupled in the last several decades. It is now taken by 1 in 4 students in the U.S. In our brawls and squabbles over math education, calculus can be found in the thick of the scrum.

The transcript of my chat with Steve follows. (It has been edited for brevity and clarity. Also, to add cartoons.)

1.
“Why Calculus?”

Ben Orlin: So why calculus? Why not graph theory, or linear algebra?

Steven Strogatz: My heart has always been with things that change smoothly. I like visualizing things flowing and moving. That’s my favorite way to think.

BO: And this way of thinking, you claim in the book, has unlocked miracles of technology.

SS: There’s also a claim that goes even farther: that calculus truly is the language of the universe. In his review, [historian] Michael Barany does not like that. Once I get into the mystical woo woo, it doesn’t work for him.

The problem is I believe it. I’ve always felt this.

I was enraptured by Einstein as a little boy—that’s why I went to Princeton. His awe about the universe, and the most incomprehensible thing being its comprehensibility—that business really means a lot to me. I think it cannot be dismissed as mysticism.

BO: That comes through in the book, in your reverential tone. Einstein’s message is that there’s something miraculous about mathematics and its power to speak to reality.

SS: For me, calculus embodies that power more than any other part of math.

BO: And you take an expansive view of what calculus is.

SS: Maybe I’m unusual in this. Your book focuses on integrals and derivatives. David Bressoud’s new book adds two other things: series and limits. I go much wider. Normally, Fourier analysis is not considered part of calculus, nor are differential equations. But they are to me. Really, I’m writing about “analysis” or “continuous mathematics,” but I don’t like that term.

BO: Well, I’m coming to calculus as a teacher. Bressoud comes from a pedagogical perspective, too. But you’re coming as a researcher. You’ve explored math across different fields. And you see a unity there.

SS: To me, calculus is so big.

2.
“Myths are Stickier Than Truth”

BO: Intellectual history is hard to tell. Ideas tend to develop in a nuanced, incremental way. But readers need a clear, coherent narrative. How did you find a balance?

SS: This was a problem for me as a pop writer. It’s one thing to be writing about math; now we’re talking about the minutiae of its history? My editor said, “People aren’t that interested.”

BO: Yeah, my first draft had too much history, too.

SS: I’m personally very interested in history of math, because I wanted to unlearn some of the folklore that we’re all brought up with—the fake stories, the mythology.

BO: Whereas my book propagates a lot of myths! I write about the apple falling on Newton’s head, the death of Archimedes, Einstein calling the cosmological constant “my greatest blunder,” and the founding of the city of Carthage—all of which may be apocryphal.

SS: But the mythology is real, in that we’ve all been told these stories. They take on a life of their own. You have someone like Sophie Germain, who was inspired by the story of Archimedes’ death, whether it’s true or not. So there’s this myth, yet it had a real impact on a real mathematician.

BO: As a teacher, I see kids light up when you tell those stories. Maybe one reason we lose the history over time is that we need folklore, too. It feeds something inside us. The problem is that the folklore lodges in that place in the brain where history would go, like carbon monoxide attaching instead of oxygen.

SS: Yes! Because carbon monoxide has an extra binding strength.

BO: Exactly. Myths are stickier than truth.

SS: I’d never written a book where I was trying to be careful about the history before—not this careful. I kept wondering, “What am I trying to do? Am I trying to write about the big ideas of calculus? Or tell the history as a story? Or tell how it changed the world and emphasize the applications?”

BO: Did you feel like you had to subordinate one of those goals?

SS: A lot of history got sacrificed. I preferentially picked well-known western superstars. I probably did a disservice to other people. This is a point you’ve made. Calculus is not just the story of a few greats. You have a little diagram, a pie chart, with Leibniz and Newton.

BO: Oh yeah. It’s one of the doodles on the cover.

SS: I don’t know about the exact size of the pie, but I think that’s true.

3.
“Are You a Newton or a Leibniz?”

SS: I get a feeling you like Leibniz. Tell me what he means to you. How do you see him? How do you see Newton?

BO: I see them as the lone wolf and the consummate collaborator. What Newton wants is go home, sit down, and solve the problem. What Leibniz wants is to build an institute where everyone will come together, go to the chalkboard, and start talking, sharing, thinking… There’s something much more communal in Leibniz’s vision.

SS: I liked your personality quiz about them.

BO: Were you a Newton or a Leibniz?

SS: Oh, I was just enjoying the way you were structuring the quiz.

BO: I think you’re a Leibniz.

SS: Newton, for me, is very alien. There are contemporaries who say they never saw him laugh, in the whole time they knew him. He must have been a tough guy to be around. Whereas Leibniz I find so delightful. He’s funny. And he writes beautifully.

BO: I get suckered in by the ones who are good writers.

SS: He was so smart about everything. Yet people have never heard of him. My wife, who’s a generally educated person, says, “No, I’ve never heard of Leibniz.” Even in college philosophy, he’s treated as second rate, overshadowed by Spinoza or Descartes.

BO: Speaking of Descartes, your book totally changed my view of him.

SS: Everything I’ve read makes him seem mean, and selfish, and egomaniacal. He has a line about Pascal, who was a boy genius, doing fantastic stuff at 15. Pascal was interested in barometric pressure: can you have a vacuum? Is it possible to suck all the air out of something? And Descartes said of the boy, “The only vacuum is that between Mr. Pascal’s ears.”

BO: No! Descartes is what, twenty or thirty years older?

SS: Yes! You’re not supposed to punch down! I was telling my agent that I’d been teaching a history of math course. She said, “What did you learn?” I said, “Descartes was a jerk.” And she said, “Well, there’s your title.” But to write a book with that title, you need a certain attitude. I don’t have it.

BO: Yeah, I can’t see you writing a book with “jerk” in the title, unless it’s referring to the third derivative.

4.
“Infinity Didn’t Hurt Isaac Newton”

SS: I found your chapter on the Mean Value Theorem very interesting.

BO: I was nervous about that chapter—lots of mathematicians I admire have waxed poetic about the Mean Value Theorem. But I’ve become a skeptic. I take some swings at it. It doesn’t come up in Infinite Powers, does it?

SS: No. It’s not mentioned. Early on, I thought I was going to write about infinity as this dangerous, villainous thing. The way it’s often told—by pure mathematicians, anyway—is that there’s this wolf at the door, the wolf of infinity, who’s in danger of wrecking the whole enterprise.

BO: Yes. The idea that calculus was shaky and unproven until the 19^th century, when analysts finally put it on a sturdy foundation.

SS: But everybody’s written that book. And I realized that, in my experience as an applied mathematician, it’s a fake narrative. Infinity didn’t hurt Isaac Newton. It didn’t hurt Leibniz. It never really hurt anybody. As you say in your book, the Intermediate Value Theorem and the Mean Value Theorem are not big, exciting theorems for the beginner. They’re only interesting after you accept the game, which is, “What if we want to make calculus as logically solid as Euclidean geometry? What if we want to prove calculus from the ground up?”

BO: My undergraduate degree was all devoted to that vision of mathematics. The axiomatic rigor, the pure development… Looking back, it seems so narrow.

SS: For the inward-looking pure mathematician who wants his or her house to be in order, you need to do it. And we should be proud that people have figured out how. It’s extremely subtle and delicate. But it’s so exaggerated usually. It’s not the part of calculus that changed the world. It’s not the reason calculus is one of the greatest ideas of all time.

5.
“Math as a Liberal Art”

BO: Over the years, I’ve heard a lot of proposals to replace calculus with data science. Yet the trend is going the other way. Calculus is five times more popular in the U.S. than it was 30 or 40 years ago. What’s your feeling on all this?

SS: Who does our current system serve? A kid who’s going to become an engineer, a physicist. But most students would be better served by a data-rich, 21^st-century curriculum. That’s what Andrew Hacker was saying. Everybody jumped down his throat, because of the way he said it, which was objectionable. But his basic message is sort of true. So many kids fail out of community college because they can’t do algebra. There’s something absurd about that. And it’s usually phrased as calculus versus statistics, but there’s this third path that no one’s talking about.

BO: Math as a liberal art.

SS: Yes. I want Math 101, just like you can take Music 101, or Psych 101.

BO: For example, you’d see proof by induction on a chessboard.

SS: That stuff is really fun, and it’s truer to what math is. That’s what kills me. When Paul Lockhart says, “The problem with math in high school is that we don’t have any math,” he’s right. It’s desiccated. It’s a dried-out husk. People don’t get to conjecture, they don’t get to explore… But I’m preaching to the choir here. Let me hear your vision.

BO: I recently wrote a piece—I haven’t published it yet—called “The False Consensus of Math Reform.” The point is that, when people agree that math education needs to change, they’re overlooking a big disagreement on how to change it.

On the one hand, there’s this humanistic vision. Get students thinking, making conjectures. Jo Boaler says something to the effect of, “Give us any topic in math, and we can make it engaging and curiosity-provoking for students.” Leave the content the same, and reform the pedagogy.

Then there’s the other approach: a data-driven, 21^st-century version, where you learn spreadsheets or R or Python. Leave the pedagogy the same, and reform the content.

The problem is that these two approaches disagree on so much. For example, ask certain humanists about standardized testing, and you’ll hear, “Standardized testing is an abomination. It turns math into gruel that’s force-fed to students. Doesn’t matter what’s on your test; that format not going to lead to rich education.”

Whereas for the data-driven folks, standardized tests are a useful tool. You should just realign them to a more valuable set of skills.

I don’t know. There’s wisdom in both.

SS: I’m sure of that.

BO: I might prefer incremental reform on curriculum, with radical reform on pedagogy.

SS: I’m thinking of Finland—have you read Amanda Ripley’s book The Smartest Kids in the World? Finland made it so that only kids graduating in the top third of their college class could become teachers. They’re highly respected. And they turned around their country in math. So when we talk about pedagogical reform, I wonder if we’re really talking about getting different people to be teachers.

BO: That’s a third vision. Pipeline reform.

SS: You’re on the front lines of this. You’re a teacher right now.

BO: I still don’t know. The crisper the formulation of the ideology, the wronger it probably is. There’s also this disconnect among teachers, where our rhetoric is very progressive, and our practice is a bit more traditional.

SS: That’s me. When I teach differential equations in a 200-plus-person lecture, it’s not modern. It’s not active. It’s me holding forth, throwing in some stories about Euler—it’s what is mockingly called “the sage on the stage.” Some of them get fired up, and go watch Grant Sanderson on 3Blue1Brown, and have a life-changing experience. And a lot of the other kids think, “Hey, that was fun,” and give a high teaching evaluation. But you test them, and they can’t do anything. I could do better, and they could do better, if we could really slow down. Smaller class. Take more time. But we wouldn’t cover as much.

BO: “Covering” is a powerful enemy. It makes you cut a crucial 15 minutes off of everything. It favors quantity over quality.

SS: If you can help them fall in love with the subject—that’s the real goal. There’s so little I can teach, so little any of us can teach.

BO: As a student, I thought that was just a truism. “The most important thing is learning how to learn.” I was like, “Yeah, yeah. The most important thing is to learn the information that you’re teaching me.” But as a teacher, I know now that it’s true. It’s just hard.

Steven Strogatz is a math professor at Cornell University. His book is Infinite Powers: How Calculus Reveals the Secrets of the Universe.

Ben Orlin is a math teacher in Saint Paul, Minnesota. His book is Change is the Only Constant: The Wisdom of Calculus in a Madcap World.