Steven Strogatz On The ‘Infinite Powers’ Of Calculus

IRA FLATOW: For the rest of the hour we’re going to be talking about, well, maybe something that you don’t find very poetic, but my next guest certainly does. And I’m talking about integrals, derivatives, infinite series. Are you getting the hint here? Bringing back any memories? 

I’m talking about calculus, calculus class. Pleasant memories I hope. Well, for me, not so much. I really struggled with Calculus 101, you know, maybe you did, too. 

But strangely, it was not until I got beyond the basics and into what you do with calculus– what calculus can do to appreciate how the world is ordered mathematically– that I actually began to appreciate its power and beauty. Elegance, as my math teacher would say. So I realized years ago how fascinating calculus could be, but not until I began reading a new book– Infinite Powers by my guest mathematician Steven Strogatz– did I appreciate its place in history and how it contributed to that history. 

Strogatz has his own definition of calculus, taking us back to ancient Greece, the Middle East, and the letter writing intrigue in the 17th century Europe. He describes how calculus underpins pretty much everything we know about the universe. Motion, gravitation, electromagnetism, quantum theory. He says we owe it all take calculus. There’s even a connection to the Beatles in there. 

We’ll get back to that rock group in a minute. Stephen Strogatz is Professor of Mathematics at Cornell, and author of Infinite Powers– How Calculus Reveals the Secrets of the Universe. We have an excerpt up on our website sciencefriday.com/calculus. 

Welcome back to Science Friday, Steven. 

STEVEN STROGATZ: Thanks very much, Ira. It’s great to be with you. 

IRA FLATOW: This is really a great– you know, if you can say is a page turner about a book about mathematics– this was a great feat. 

STEVEN STROGATZ: Well thanks very much, that’s wonderful to hear. 

IRA FLATOW: So tell us why you write a book about calculus. 

STEVEN STROGATZ: Oh, I’ve been in love with the subject since I was a teenager. And I feel like it’s just something beautiful that I would love to share with other people. I feel like it’s a gift. Especially because, as you mentioned, it sometimes gets a bad rap as being hard to understand, or it’s where some people feel like they hit the wall in their math education. 

So it has a bit of a rough reputation. And, you know, it makes me sad. Because you mentioned poetry. Yes, it’s absolutely the poetry of the universe. And I would like people to get that. 

See, not just– you know, not just lofty stuff like that, but that it’s really affecting everything we do in our everyday lives. From when we turn on our microwave ovens, to using our GPS to find our way home at night. That’s calculus. 

IRA FLATOW: Hm. And you frame the pillars of calculus as derivatives breaking things down, and integrals building things back up. Sort of a mathematical yin yang here. 

STEVEN STROGATZ: Yeah, you could look at it that way. I mean people who have taken calculus will know those words, jargon. Derivatives and integrals. But I would actually say it a little differently. I would say, what’s really going on, the heart and soul of calculus is the spirit of problem solving. That you take a really hard problem and break it into smaller pieces. 

Now anyone who’s ever solved problems knows that’s a good strategy. Breaking a hard problem into easier, smaller problems. But what’s so interesting– and an almost kind of maniacal about calculus– is it never stops. It chops a problem again, and again, and again, infinitely often, into infinitesimal bits. 

And that’s a mind-bending idea. That’s the great intellectual breakthrough of calculus, this strategic use of infinity to chop problems into their tiniest conceivable parts, where they become much easier. And then you solve those and then put them back together, so the– I’m sorry. 

IRA FLATOW: –and so you were writing that calculus, those tiny little, breaking it down into those tiny little bits goes way back further than we think that the Europeans invented. Goes back to the Greeks. 

STEVEN STROGATZ: Yes. It does. So you often hear it said that calculus was invented in the 1600s. But I would say that this definition of calculus I’m giving– this strategic use of infinity and infinitesimals– we can see it clearly happening in the work of Archimedes around 250 BC. 

He lived in Syracuse on the island of Sicily, and he gave us all those formulas that kids memorize for the SATs. You know, like, the area of a circle is pi r squared, where r is the radius. And pi is that amazing number that we just celebrated a few weeks ago. 

So formulas like that, or for the surface area of a sphere, the volume of a sphere– all of that, it’s taught in geometry class. But it’s really beyond geometry. Because geometry, before Archimedes, could not handle smoothly curved shapes– like circles and spheres– as far as finding– as far as measuring them. Finding their area, or the circumference, or their volume. 

You won’t find the formula, say, the formula pi r squared. That’s not in Euclid’s geometry. That had to wait for Archimedes and his incredibly ingenious use of infinity to find that formula. 

IRA FLATOW: I’m Ira Flatow. This is Science Friday from WNYC Studios. 

And then it was adopted years later. It took, what, over 1,000 years to come back. To a different– 

STEVEN STROGATZ: I put it closer to 2,000. 

IRA FLATOW: 2,000. Two millennia as you call it in the book. 

STEVEN STROGATZ: It’s really two millennia. I mean if you ask, who is the person who was most ahead of his or her time, I think it would be hard to beat Archimedes. He had the ideas of calculus. Like I say, 250 BC he was going strong. And then you don’t really see it with the same level of virtuosity until Isaac Newton in the 1660s. 

So that’s– I’d have to do the subtraction here– but something like 1900 years. 

IRA FLATOW: And you write that one of Archimedes letters actually brought a tear to your eye. 

STEVEN STROGATZ: Well, that could just be me but– 

IRA FLATOW: We’ll talk about it after the break, because we’ve got to take a break. I don’t want to let you get away without talking about it. 

STEVEN STROGATZ: OK, OK. 

IRA FLATOW: Read a little bit from your book. Talking with Steven Strogatz, Professor of Mathematics at Cornell University, and author of Infinite Powers– How Calculus Reveals the Secrets of the Universe. As I say, we have an excerpt from the book. sciencefriday.com/calculus. 

We’ll be back with more talk with Steven, so stay with us after the break. 

This is Science Friday. I’m Ira Flatow talking with mathematics professor Steven Strogatz. And his new book Infinite Powers. It’s all about calculus, how calculus reveals the secrets of the universe. And as I said at the opening, if you have had trouble with calculus like I did, this is an eye opener. 

You will really like reading and following up. Because I have spent the rest of my life trying to figuring out calculus. I think I’ve gotten there. 

Steven, let’s bring you back to Archimedes letters. You write, “there’s a sense of loneliness in Archimedes. He writes, I can’t do everything. I’m going to die at some point. So I want future generations to know what I know. I just find this very beautiful.” You feel like you’re talking to him. 

STEVEN STROGATZ: Well I do. And there’s especially a poignant letter that he wrote to another mathematician named Eratosthenes, who was one of the few people in the world who could understand what Archimedes was doing. Eratosthenes is someone we might have heard about in school as the person who figured out a way to measure the circumference of the Earth, that we now know to be around 24,000 miles. He figured that out. 

But so anyway, he was a really first rate mathematician himself. And Archimedes sent him this letter describing a certain method that he had used to find the answers to these incredibly hard geometry problems that nobody had been able to do ever, for centuries. And as he describes the method, he does two things that kind of choke me up. I have to admit. 

Which is, one, he admits that there– It’s like he exposes the soft underbelly of his method. He shares his private intuition. And most people in math are scared to do that, to show their soft side. But he shows Eratosthenes that there are some things about his method that aren’t completely rigorous. They might be a little unsound, but they give him the right answers. 

And then he goes on to say that he hopes that future generations will be able to use the method to find theorems that have not yet fallen to our share. Which is a strange expression, but I think what he’s trying to say is– there are things that I haven’t figured out. The theorems haven’t fallen to my share, I haven’t gotten them. But maybe somebody, somewhere in the future– you know, like sending a message in a bottle out into the vast oceans of time– somewhere in the future someone will learn what I’ve done and they’ll be able to solve these problems. 

And I just think, you know, here’s this person who is one of the all time great geniuses of the human race. And yet he feels the finiteness of his life against the infinity of mathematics. 

IRA FLATOW: Interesting. You mentioned about how calculus was sort of, the roots of it, were sort of lost for two millennia and then brought back in Europe. But what was the key insight that Liebniz and Newton had that led to what we think of today as calculus? 

STEVEN STROGATZ: Well, so there’s this big 2,000 year interlude where algebra is being developed in the Middle East in Arabic countries, in Baghdad and in Cairo, and eventually it makes its way back to Europe in the, say around 1200, 1300. And once you get algebra fusing with old time classical geometry of the Greeks, in the work of people like Fermat and Descartes, that’s what really sets the stage for Newton and Liebniz, then, in the second half of the 1600s. Where they’re addressing old, still unsolved problems about curves. How to find tangent lines to curves, and things like that, that might not seem of much practical interest. But still, they’re of great interest in geometry. 

And Newton and Liebniz both figure out how to do that– not only finding tangent lines– but also areas under all kinds of exotic curves. And it all sounds a little bit pointless, maybe, until you realize that– as we do now– that you can draw a graph of any relationship between, let’s say, the level of a virus in a person’s blood who’s suffering from HIV. You can draw that as a curve on a graph. And so you can take something as disembodied as a curve, and use it to represent a life and death thing like the viral load in an HIV patient. 

And so with this enhanced understanding of how things change through graphical representations of that on these graphs that we nowadays draw, x versus y. Newton begins to figure out the laws of the universe. We hear about Newton’s laws of motion and laws of gravity, but ever since we’ve been drawing those same graphs and analyzing them using the techniques developed, now 350 years ago. 

IRA FLATOW: And let’s get into that viral load, because you talk about how calculus was key to determining the drug regimen, that cocktail of drugs that HIV patients must take. 

STEVEN STROGATZ: Absolutely. So, I mean, if we fast forward now to the 1980s, and we’ve got the AIDS epidemic. People are dying, there is no hope. It seems there’s nothing to do to– certainly a cure is nowhere in sight. Not even anything to make it a chronic illness. And it was a terrible epidemic, terrible plague, really. 

But once a new kind of drug became available called protease inhibitors in the 1990s, mathematicians working in collaboration with immunologists and doctors were able to figure out the answer to what had been a big mystery about HIV, which was– just to remind people– what would happen is, someone who get infected with the virus. They’d be sick for a little while, then they’d feel, you know, like they’re kind of getting better. And then they really wouldn’t show many symptoms for about 10 years. 

During that time it was unclear what’s going on. Is the virus lying dormant in the body, hiding out the way chicken pox can do, or what. So there was a question– should you give the patients the available treatment as soon as possible after they’re infected. The trouble with that idea being that they might become resistant to it, and then you’d have nothing to help them once they really got full blown AIDS. Or should you not treat them until they do show AIDS and then try to hit them with the drugs. 

And so it all depended on whether you thought the virus was lying dormant in the body or not. And what the Dr. David Ho, the clinician, and Alan Perelson, a mathematical immunologist realized– through the application of calculus to the data that David Ho was collecting with his collaborators– was that actually HIV was not hibernating during those 10 years in a patient’s body. It was in an unbelievably pitched, all out, raging battle with the person’s immune system, such that every day about somewhere between a billion and 10 billion new virus particles were being produced by the virus and being cleared out by the body’s immune system. And this is going on every second of every day for 10 years until the body eventually would get exhausted. 

And so then, you know, and then AIDS would set in. So the big insight was that if you gave one drug in the face of this very rapidly evolving virus, it wouldn’t be enough. It would develop resistance. Two drugs, also, the math showed wouldn’t be good enough. But three drugs would make it so difficult for HIV to do the three simultaneous mutations needed to escape the triple combination therapy, that it would give patients hope. 

And of course, that has turned out to be true. Nowadays HIV– for people who can afford or have access to the treatment– has become a chronic illness rather than a near certain death sentence. 

IRA FLATOW: I don’t think anybody ever heard that story before reading it in your book, you know. 

STEVEN STROGATZ: Well, if you’re on the inside people know. I mean, those of us that do mathematical biology have revered Alan Perelson for the work he did. And also another team did it, Martin Nowak and his students. Sebastian Bonhoeffer did it simultaneously. 

So it’s out there. But you know, it’s been so overshadowed by the success of the clinicians– and maybe I should say, just to be very clear. I hope it’s been clearer already. But to really drill it in, calculus didn’t solve HIV on its own, certainly. I mean this was very much a team effort with doctors and immunologists. But it did play an important supporting role. 

IRA FLATOW: Mm-hmm. Yeah and write through your book how under the radar calculus flies in all different aspects of our life. 

STEVEN STROGATZ: Well it does. I mean it’s interesting you use the word radar because that’s a perfect example. You know, I mean, radar was key to the British being able to fend off the Nazis in World War II during the Battle of Britain. And radar was, at that time, a state of the art technology. 

The idea that you could beam these radio waves off of airplanes and then reinterpret the signals as they bounced back to figure out where the planes were, and how far away they were, how fast they’re moving and all that. That was an outgrowth of calculus from work on the ideas that led to radio and television, and what today we’re using as wireless communication for cell phones. 

All of that came out of work in the late 1800s by Maxwell, the great Scottish physicist who first put together the laws of electricity and magnetism in the form of calculus equations that we call today Maxwell’s differential equations. Those equations predicted that electricity and magnetism could propagate as invisible energy at the speed of light, and that’s what radar is doing. 

IRA FLATOW: And you also talk about the role of infinity in the success of calculus. It’s one of your main theses, 

STEVEN STROGATZ: It is, it is. That’s why I call the book Infinite Powers. 

IRA FLATOW: Right. 

STEVEN STROGATZ: That inifinity is this key concept that’s very spooky for many people. I mean, my wife gets upset if I start talking about infinity. She says don’t, that gives me a headache. I don’t want to hear about that. 

You know, when people start thing about bottomless pits and– 

IRA FLATOW: Right. 

STEVEN STROGATZ: A lot of our worst nightmares are about infinity. But it’s this interesting beast, that if you can tame it, it’s a tremendously powerful tool for making sense of hard problems by, as I say, chopping them into infinitely many tiny parts. Which tiny parts– I mean to give you a visual of what I’m talking about, why infinity would be helpful. 

If you think about, say, something like a circle– which is round, obviously. But if you imagine looking at the edge of a circle, the rim under a microscope, and start zooming in– what looked curved will start to look increasingly straight. And that’s nice, because straight things are easier than curved things. 

And so you can kind of think of a circle as almost being like a polygon. In other words, take a square, then a hexagon, than an octagon, more and more sides. As you take more and more sides and make each side smaller and smaller, it starts to look like a circle. And that idea that a circle is well approximated by something having, almost having infinitely many infinitesimal sides, that turned out to be the key to understanding circles and other curved objects. 

IRA FLATOW: And you say that was one of the key reasons of how calculus got started, was that the geometry people knew that you could easily take base, and height, and whatever, and calculate a rectangle or a square, but a circle is this is smooth. It’s got waves, it’s a circle. Can we take that same idea and figure out how to work with curved objects? 

STEVEN STROGATZ: That was, it was a big insight. That it’s sort of a fantastic creative conception. I want to emphasize that point, that people sometimes think of math as, you know, very black or white and cut and dried. But it takes tremendous creativity– like poetry, like music, like any kind of art– to see something that is smooth and curved and imagine it as being made up of lots of little flat straight pieces. 

And today that’s a very practical idea. Because think of when your kids watch a movie like Shrek or Toy Story, and there are these animated characters running around on screen. When you see shrieks big bulging belly, or his little trumpety ears, those are made up of millions of polygons. 

Computer animators use Archimedes old idea of breaking curved things into lots of tiny flat jewel-like polygons. So it’s absolutely practical these ideas. We’re using them all the time today. 

IRA FLATOW: I’m Ira Flatow. This is Science Friday from WNYC Studios. Talking with Steven Strogatz, author of Infinite Powers– How Calculus Reveals the Secrets of the Universe. 

And I actually think that Oscars have gone to those mathematicians who’ve created these polygons, the ideas of smoothing out– creating the animation figures. 

STEVEN STROGATZ: That’s true, right. There have been– in the old days, they would use millions of polygons per movie. I think when the movie Avatar was made that was all completely computer generated, they were in the billions of polygons. 

IRA FLATOW: Is there something about calculus that you don’t know that you’d like to know more about? I mean you write about everything in this book. 

STEVEN STROGATZ: Well that’s the question I was not expecting. I mean I’m stuck with that. I can’t say no. I mean, of course there are things in calculus I don’t know. 

And calculus has gone on– I should say, it’s sort of like the gateway to all of modern math. You know, we’ve talked about its applications in the universe and in the real world. But for mathematicians calculus is just the first step. 

That’s why it’s– of course we teach in freshman year to all of modern math. And so that includes subjects with names that sound like they’re part of calculus. Differential geometry, integral equations, analytic number theory. Those are all calculus words. 

And they all have to do with our now understanding that math is this giant web of ideas, all interconnected. Rather than– a lot of people think of math as a tower. You know, that you have to build up and go higher and higher. But it’s really not a tower, it’s a web. And we now know that through calculus. 

IRA FLATOW: Now I can’t get away without– I teased at the top of the program about the connection of this little band from Liverpool called The Beatles and calculus. Fill us in on that connection. 

STEVEN STROGATZ: Well all right. So after X-rays were discovered, it was known, at first, you could use X-rays to look for broken bones or you could use them on teeth. So for hard structures in the body, like bones and teeth, X-rays were great. 

But if you tried to look at a person’s brain– like, say, to look for a brain tumor, or a hemorrhage, or a blood clot– X-rays we’re really no good because the brain would just look like a big cloudy mass, just gray nothingness. So when it was proposed in the 1960s by a couple of– there was one engineer who came up with the idea and one physicist– that maybe if, instead of just shooting X-rays from a single direction– if you did many different directions of firing the X-rays, and then use calculus to help you figure out how to recombine the information obtained from all those different directional shots, maybe you could see brain tumors. Maybe you could see blood clots. 

And this is what we nowadays call CAT scanning. And the interesting thing about it, in connection with The Beatles, is that one of the first companies to take a leap on this wacky idea was a company called EMI, which was Electric and Music Industries. Or maybe it’s electronic. 

Anyway, EMI in England had all this money lying around because they happened to sign The Beatles. And so they were flush with money, and they took a stab on this crazy idea of CAT scanning, which later won a Nobel Prize and revolutionized medicine. 

IRA FLATOW: And we have The Beatles albums the thank for– 

STEVEN STROGATZ: Thank you, Beatles, yes. Thank you, guys from Liverpool. 

IRA FLATOW: You know and as your book says, there’s just so many connections that calculus is connected to so many different things. And to me as someone who loves the history of science, I love the applications of calculus, because I learned that in college and I really appreciated calculus like I couldn’t before. But the history that you have in the book, how far back– thinking about calculus, going back to the Greeks, and then that hiatus it went through. 

I just love that, of understanding how, really people are involved in it. And then the great figures you have, Descartes and all those other people in there. I want to thank you very much for taking the time to write that book, and for taking time to be with us today, Steven. 

STEVEN STROGATZ: Well thank you, Ira. I guess we’re probably out of time. But I just want to inject one last point, which is that there are several women in the story, too. We’ve only talked about men so far. But women mathematicians started to become recognized and important players by the 1800s. And so if you want to hear what half of humanity did to contribute to this, they’re in there, too. 

IRA FLATOW: Great point, great point. I’m glad you brought it up. Steven Strogatz, Professor of Mathematics at Cornell. Author of Infinite Powers– How Calculus Reveals the Secrets of the Universe. Excerpt on our website, sciencefriday.com/calculus. Good luck with the book Steven. 

STEVEN STROGATZ: Thank you Ira.

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