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SPEAKER: This is a production of Cornell University Library.
[APPLAUSE]
STEVEN STROGATZ: Thank you, everybody. Very encouraging to see such a nice turnout. Well, I'm here to talk to you about calculus today, and here we go.
Let me begin by trying to make the case for why calculus matters. Without calculus, we wouldn't have so many things that have made the world modern. We wouldn't have radio, television, or microwave ovens. We wouldn't have sent astronauts to the moon or unraveled the structure of DNA.
Let me just comment on those pictures up there. You may-- anyone know who I'm showing? Who's that on the left? Have you seen the movie Hidden Figures? Yes, you have. Good. So that is Katherine Johnson, the star of Hidden Figures, when she was younger. There she is.
She worked for NASA. Actually, she worked for the organization before it was called NASA, and used an outgrowth of calculus that we would call "differential equations" to help design the trajectories that put the astronauts on the moon and brought them back safely. That's a picture I really like. I tend to get choked up when I look at it.
You know she was unrecognized for so long for her contributions, but there's president-- I'm getting choked up again-- President Obama giving her the Presidential Medal of Freedom. She is still alive. She's over 100 years old now. Yes.
And someone who did not live as long, sadly, is this woman. That's Rosalind Franklin, who helped us figure out the structure of DNA with her work in X-ray crystallography. She made the most beautiful crystals of DNA that caused her to take this very famous X-ray diffraction picture that, for someone who is expert in this field, can recognize as revealing the helical structure of DNA.
This is the famous photograph that was shown to Jim Watson without her permission by her lab director, a person named Maurice Wilkins. And it gave Watson and Crick an essential clue that led them to the double helix. So it's quite terrible, really, that it was shown without her permission.
She died young of ovarian cancer. I think she might have been about 40 years old when she died. And she surely would have won the Nobel Prize, but you can't get the Nobel Prize posthumously, so it was given to the person who showed her photograph without her permission, to Maurice Wilkins. Not good. But I mean, not like Wilkins was such a monster. This isn't the time to go into that.
The point was that Rosalind used another outgrowth of calculus, Fourier analysis, which is essential for understanding the patterns that x-rays make after they bounce off of something like DNA molecules. So anyway, calculus in this more advanced form of differential equations or Fourier analysis, helped us do those two things that I just mentioned.
But also, just day to day, when we want to see pictures of our babies in utero-- ultrasound, other kinds of medical imaging couldn't, wouldn't exist without calculus. Trying to find your way home, use GPS, that is a wonder of calculus in so many different ways. So it's all over the place in our daily lives, more than we realize, sometimes not always for good. I mean, it gives us nuclear energy, but also gives us nuclear weapons.
And I should maybe clarify. I'm not saying calculus does all these things by itself. Obviously, you need nuclear engineering to have something like nuclear energy and nuclear weapons.
But I think you should view-- when I'm making these grandiose claims about what calculus has done for us, think of it as a supporting player in the great drama of science, that there are collaborations between scientists, engineers, mathematicians. And calculus is a sort of unsung hero that's in every movie but you just didn't notice it, because it's sort of that funny-looking-- it's not usually the star, but it's always there.
So that's the first claim, that just from this utilitarian point of view, calculus has helped make the world modern. But there's something a bit mysterious about that that I want to focus on, which is how could it be that an arcane branch of math, which really grew out of geometry-- it was originally about shapes-- how could that have helped reshaped civilization? And so the clue to how, I want to try to introduce the idea by talking about this gentleman, Herman Wouk.
Partly what I'm doing here is signaling that this is a talk meant for people like Herman Wouk. Herman Wouk did not know calculus. He was a novelist from my parents generation.
He's a person who helped tell the story of World War II back to the people who had just experienced World War II. So you're seeing a photograph of him there as a fairly young man. He wrote The Caine Mutiny, later made into a famous movie with Humphrey Bogart. People are talking about it a lot now in connection with our current politics, so you may want to learn about the Caine Mutiny if you don't know about it. I won't say more about that, either.
But so the winds of war-- I mean, at the time right after World War II in the '50s, Wouk wanted to write the grand novel of World War II. And he had this idea that he should do sort of the great American novel. It ended up ballooning into two novels that became The Winds of War and War and Remembrance, both very big bestsellers later made into movies.
But so in the course of doing research for that book or those books, he wanted to interview some of the scientists who had worked on the bomb. And he was told, go to Caltech. You will find people there. And he was in particular directed to a person named Richard Feynman.
So Feynman, who had been at Cornell and worked with one of our great physicists, Hans Beda, also both worked on the Manhattan Project, there's Feynman at Caltech in a picture from that era. You see it's only boys. They didn't have too many female students-- still don't, really.
But anyway, Feynman, they're smiling, very characteristic. He's a joker. Look at him. He's having the best time.
And he was a wise guy. He would break into safes at Los Alamos. He would play his bongos when he shouldn't be making noise. He likes to think of-- or did like to think of himself as a rascal and told many stories about himself.
Anyway, so he was a mischievous character. And he was the one that Wouk was sent to interview. And so they were talking about the bomb, and they found out pretty soon they had a lot in common-- two Jewish guys, one very observant in the case of Wouk, devout Orthodox jew.
In Feynman's case, totally opposite end of the spectrum-- very atheist, disrespectful, kind of, about religion. But culturally Jewish, and they're both from New York. So they had a lot to talk about there. And they were about the same age. They've both been in the war as young men.
Anyway, so they're kibitzing. And at the end of their conversation, Feynman says, so you really want to understand the bomb, I have to ask you something. Do you know calculus?
And Wouk, who as I say is a novelist, and a person in the humanities, admits no, I don't. I don't actually know calculus. Never studied it. And so Feynman says to him, well, you better learn it. It's the language god talks.
That became the title of a book that Herman Wouk wrote after his conversations with Feynman, partly inspired by their conversations, but much broader, actually. He went on to write about science. I don't know if you can see the subtitle from where you are, but it's a series of conversations about science and religion, and the tension between them, whether they can be reconciled, whether the sense of awe that scientists feel is anything like the sense of awe that deeply religious people feel at the wonders of the universe and the creation.
So anyway, it's an interesting book, and not very well known. I would recommend it to you. But in the course of that book, Wouk says something that I think you may be able to relate to if you're one of those people who also, like Wouk, said, well, I never actually did study calculus. I'm kind of aiming the talk at you. Like Herman Wouk is your surrogate. That's why I'm trying to get at here.
So Wouk says this in the book, admonished by Feynman about he better learn calculus, he thinks, OK, I'm going to try. So he says, "I picked up and skimmed some freshman texts in college bookstores, hoping to come across one that might help a mathematical ignoramus like me who had spent his college years in the humanities-- literature and philosophy-- in an adolescent quest for the meaning of existence, little knowing that calculus, which I had heard of as a difficult bore leading nowhere, was the language God talks."
So it's this interesting thing, that he thinks maybe he really does need to study calculus if he wants to understand the meaning of existence. Well anyway, so he then describes in the book his attempts to learn calculus. First of all, he picks up a book called Calculus Made Easy and finds it doesn't-- it's a famous book. And it didn't really work for him. It didn't help.
Then he tried hiring an Israeli tutor, a math tutor who he thought, maybe I'll learn some spoken Hebrew on the side, sharpen up my Hebrew, but also learn some calculus. Both hopes ran aground. That didn't pan out.
And then he took a high school-- he signed up for a high school calculus course, and sat there with the teenagers, and eventually started falling so far behind that he decided to quit. And as he was walking out, he said that the students clapped for him on the way out, which he compared to sympathy applause for a pitiful showbiz act. So I'm just making the point that even for someone with a sincere desire to learn calculus and why it matters, it's not so easy.
And so I wrote this book that we're here to talk about today, Infinite Powers, with the wish and the hope that maybe I could convey some of what is so marvelous about this truly world-changing thing, calculus, to people who either never studied it, or did study it and didn't see the point of it, which I know is true for many folks who took Advanced Placement Calculus in high school, or even, dare I say, students who take it here at Cornell. And there's a good excuse. In our defense, the excuse is there's a lot of stuff to cover.
You've probably seen these calculus books. They're more than 1,000 pages thick usually. And we have a lot to train the physicists and engineers of the future. We feel like we've got to get through this.
And there's no time to talk about the history, how it changed culture, the softer side of calculus. We don't have time for that. So it's not always easy to get the point. So that's what I want to try to give you today.
Let me go back, then, to this question of Feynman saying that this is the language god talks, which certainly makes it sound important. But what could he have possibly meant by that? Well, we don't really know, but I want to give you my take on what I think he might have meant.
Let me start with this picture. There's a picture of my daughter, Leah, my older daughter, when she was playing at her grandmother's apartment. It's a wintery day. It's dry. You can see what's happening.
She's fooling around. And she got some static electricity in her. And her hair is sticking up like that. And so I took a picture of her.
That's about all I really want to say about electricity. It's something that can make your hair stand up. Also, you can be probably familiar with the idea of magnets. You might have played with them as a little kid. You could sprinkle iron filings around them on a piece of paper, and they will make this amazing pattern showing that there is something invisible around the magnet that you can visualize with these iron filings.
There's this incredible patterned, what we would call the "magnetic field" around the magnet. And the idea that there's this invisible force field that you can then visualize with iron filings is bringing us to a bigger point, which is that in the 1800s, when people were trying to understand how electricity and magnetism really work, the key idea was to think about these invisible force fields around magnets or-- I mean, there's a lot more to electricity and magnetism than meets the eye. And so the idea of the field was a crucial thing that helped scientists start to untangle what's really going on with electricity and magnetism in a fundamental way.
So why am I bringing that up? Here's various physicists like Michael Faraday in England, or Andre-Marie Ampere, who gives us amperes, amps, I'm all amped up, that's Ampere. Ampere helped us also understand basic aspects of electricity and magnetism. And there was a collective effort of several scientists in the 1800s trying to figure out what is really going on with electricity and magnetism.
They discovered various laws of physics, as we would now put it, about these electric and magnetic fields. And so now you're wondering, where's calculus? Why am I telling you all this science?
Calculus starts to come into the story after the physicists have done their work discovering the laws. Those laws about electric and magnetic fields were encoded in the language of calculus by this gentleman, James Clerk Maxwell. That's a picture of him as a young man. He's more usually seen as an old man with a big beard and gray hair, but I sort of like this picture of him as a youngster.
Anyway, Maxwell took all these laws of Faraday, and Ampere, and others, and wrote them in this symbolic language of calculus. And the reason that that was such a pivotal thing is that once he had them in the form of calculus equations, he could start to manipulate them. And so you may have heard this phrase or term of art-- mathematicians, we talk about manipulating equations.
I want you to think of it as really literal. Like when you go to the masseuse, they start manipulating your back. They're massaging you or something. That's what it sort of feels like for us when we have equations.
We transform them. We knead them, with a k, knead them. I mean, we actually, we're working them, except not with our fingers and our hands but with logic. We might add something to both sides, or we might take the derivative of both sides. I mean, you don't need to know what that means.
The point is we do manipulations on them to transform them from one form into another. And why are we doing that? This is an exercise in logic. The math that we do with symbols is a way of carrying out a very long argument that would be too hard to hold in your head and too difficult, but it allows you to see implications.
So in this case, there were four laws of electricity and magnetism in their fields that were known, but nobody could extract their implications without calculus. Maxwell starts massaging them. And various things start to come out, some of which are not very enlightening.
But at some point, he gets a result that he recognizes because it reminds him of something else. To a mathematician, it's a very recognizable equation that is the same as the equation for a ripple on a pond, a wave spreading on the surface of water. And so Maxwell in that moment, and I wish I could be there in his-- I picture like a candlelit room, and it's like this amazing aha moment-- electricity and magnetism can dance together to make a wave of invisible energy.
And he calculates how fast would it move, and he finds it moves at a speed that he recognizes. It's exactly matching the speed of light. So in that moment, after thousands of years of wondering what is light exactly, humanity suddenly understood that light is a dancing, propagating electromagnetic wave, where the electric field regenerates the magnetic field, which regenerates the electric field. And they sort of bootstrap their way forward at the speed of light.
So why does that matter? Calculus extracted this prediction that light is propagating electromagnetic waves. And at that point, it's just a logical prediction of the laws that comes out of calculus. Does nature follow logic is really the question. I mean, why do we assume that it does?
Newton sort of taught us that. Nature behaves in accordance with the rules of our logic. It's amazing that you can take something that's true, manipulate it with logic, and out comes something else, and that other thing will be true.
That's crazy, but that's how it works. That's what allows theoretical predictions to work in science, especially in physics. So anyway, Maxwell predicted electromagnetic waves.
Just a short time later, a few decades later, they were measured, observed by Heinrich Hertz. And then a short time after that, a person named Tesla who we've all heard of now was able to use electromagnetic waves to do wireless transmission of energy. And then someone else we've heard of named Marconi thought, I could send signals across the ocean with this.
This is all a direct flow from Maxwell, and Faraday, and Ampere and all of it. And so as you see, calculus is a key player in this, in making the prediction that this new kind of way of transmitting energy would be possible. Once we looked for it, there it was.
So of course, there's a lot of ingenuity that went into the work of Marconi and Tesla. So again, I'm not saying Maxwell could do this all on his own. But he was an indispensable player in the story of how do we have wireless communication today? How do we have television and radio? It's all a key thing that-- I mean, calculus played a key part in all of this.
So some of you will know this t-shirt. It's a favorite of physics majors. "And God said," and then there's Maxwell's equations. And instead of God said, "Let there be light," and there was light, God said, "divergence of d equals rho, divergence of b equals 0," et cetera.
So there it is. Those four equations that I've been talking about, the encapsulation of these laws of physics, imply light. And those are all written in the language of calculus, a specific kind of calculus, vector calculus, but still calculus nonetheless.
So just to summarize where I've been going with this first part of the talk, the argument is that calculus matters because number one, it's the language God talks. Natures laws are written in this language of calculus. And it's not a triviality.
They're not written in the laws of algebra or the language of algebra. They're not written in geometry. Galileo thought that. I mean, Galileo was the first to make this analogy to language. He talks very movingly at one point about the great book of Nature, and that you can read the great book of Nature, but you have to know the language in which it is written. And he says it's shapes. You need to know about triangles and circles.
That's a start. Galileo didn't know calculus. It wasn't invented yet or discovered, whichever you think philosophically.
But in any case, the laws of Nature are not written in circles and triangles. They're actually written in the language of calculus, and more specifically, the language of differential equations. So we see this not just for electricity and magnetism, but the laws of gravity, the laws of how water moves, or air, or heat. Or even when we go to quantum mechanics inside the atom and even smaller scales, it's still written in the language of calculus and something called "Schrodinger's equation."
So it's amazing. Even in a realm of physics where Newton's physics breaks down, Newton's calculus still works. Kind of really incredible. So this is a big claim. Calculus matters because Nature's laws are written in that very particular language.
But more than that, it's very tempting to say calculus is just a language. Well, no, that's wrong. It's not just a language. It's much more than a language. It's also a system of reasoning.
That's what I'm talking about, this logical machine. You can feed stuff into it, and then it turns. And it spits out something else with the property that truth in gives truth out.
And it may be a new truth never discovered, never observed, but you can rely on it because somehow, it is tapping into the way the universe works. And somehow, its system of reasoning is faithful to the logic of the universe. It doesn't have to be like that.
And it's sort of incredible that a primate on our little, rocky, water-soaked planet could discover the logic of the universe. But we kind of did. So if that doesn't get the hairs on your neck standing up, I don't know what would.
It really is as close as we've come to discovering the secret of the universe. I really do believe that. But it was by learning to speak this strange language and discovering that it expresses the hidden logic of Nature that our primate species has been able to remake the world for good and for bad.
Now how does it do this? Let me try to talk about what calculus is. I haven't really said what it is. I want to give you a flavor in this next part. What is calculus and how does it work?
And I think it's helpful to do that by pretending that it's a person. In other words, I'm going to do some anthropomorphizing. I don't see anything wrong with that. It's often criticized.
I think it's very helpful, so I'm going to pretend calculus is an entity that wants something. If you're a very literal person, you should say calculus doesn't want anything. It's the practitioners of calculus who want things. OK, fine.
So what does calculus want? Calculus wants to make hard problems simpler. That's what it wants.
It may be surprising to you if you have taken calculus and you think, Jesus, this is so complicated. It is. But it has to look complicated because it's tackling complicated problems. And it wants to make complicated problems simpler. That's really what it wants-- simplicity.
And it has a strategy. And the strategy is it's going to slice a hard problem into smaller pieces. Now that in itself is not unique to calculus. Every good problem solver knows that to make a hard problem easier, break it down.
But calculus has a particular obsession with this idea and doesn't know when to stop. It is going to keep slicing and slicing, and never, ever stop. It keeps slicing ad infinitum. That's the unique idea in calculus. We're going to use the power of infinity.
So I'm going to call this the "infinity principle." It's a made-up term. You won't find it at any book other than mine. So it doesn't exist, but I think it's really the heart of calculus.
When trying to think, to explain this to the public, what is calculus all about, if you've taken it and someone asks you, you might say, it's about the study of limits, derivatives, integrals, and infinite series. OK, yeah, but that's not really what it's about.
What it's really about is the strategic use of infinity to solve hard problems. And all of those other buzzwords I just mentioned do that. It's infinity that's the key idea.
So specifically, the strategy is we're going to cut a problem of any type into infinitely many infinitesimally small pieces, and then solve the problem for those tiny pieces. That's going to turn out to be easy, usually, relatively easy, much easier than the original problem. The hard part is to put all the pieces back together again to study the question you were originally interested in.
So you can see that calculus has this double-barreled strategy-- cut and then rebuild. And that's why there are two halves to calculus that are the famous halves you've heard of. The slicing part is the part that we call "differential calculus." That looks at the differences between pieces.
The hard part is integrating everything back into the whole. That's the rebuilding part. That's integral calculus. And that is much harder.
Now this thing that I just described, this strategy of cutting and then rebuilding, only works if you can keep cutting forever. I say that's the strategy. You're going to keep slicing and slicing forever.
And only certain things allow for that. They have to be, in the jargon, "continuous," from Latin words-- "con" like together and tenere, to hold, so stuff that holds together is continuous. You probably have an intuition what does it mean to be continuous and unbroken.
That's what the infinity principle applies to things that are continuous, in the sense that they can be sliced and sliced, finer and finer, and it will never stop. So calculus is about that, but I want you to notice the creative fantasy involved in this idea. Such things don't really exist. The abstraction of studying a continuum is a fiction. There's nothing in the universe that we know of that is actually continuous.
So let me, before I go through that, why do I say that? Imagine you are a physicist, and I just come in here and say, I'm going to just keep thinking about things that can be cut finer and finer forever. Physicists would say, we don't believe in that. There are atoms.
The word "atom," which I suppose comes from Greek. Yeah, you're nodding. Can you tell us? Do you know Greek or enough to say what an atom means?
AUDIENCE: [INAUDIBLE]
STEVEN STROGATZ: Yeah, so "a" for not, and "tom" like tomography, or a-tom is uncuttable. So an atom is a thing that cannot be cut. It's so small that it's done. You can't cut it any finer.
So the existence of atoms, these indivisible particles, was this ancient idea from Democratus and others in ancient Greece. But the idea that the world is built of uncuttable entities is the atomic theory. Now we don't believe atoms are uncuttable anymore. We think there are smaller things-- quarks, whatever, other subatomic particles.
But even if you talk to people at the far extremes of physics today, they would say, we think there are smaller scales than that. There's the realm where gravity and quantum theory will be reconciled someday. That hasn't happened yet, but when we have a theory of quantum gravity, space itself will start to look discontinuous.
There's a scale predicted by quantum gravity-- you're wondering how can there be anything predicted by quantum gravity if it doesn't exist yet? We know that there are certain numbers that will definitely come into any future theory. The speed of light, because it governs something about the structure of space, Planck's constant because it governs the strength of quantum effects, and a number called big G which governs the strength of gravity in the universe.
And there's only one way you can combine big G, the speed of light, and Planck's constant to make a length. There's only one way. That was figured out in around 1900 by Max Planck, and today we call it the "Planck scale." And the Planck scale is a length about 10 to the minus 35 centimeters, which is way, way, way smaller than the smallest particle we've ever measured. It's incredibly small, obviously, very small.
And there's also a smallest time. The time it would take light to traverse that distance, the Planck time, is 10 to the minus 43 seconds or something like that. So these are unfathomably tiny distances and times, but there's every reason, if you believe in modern physics, to say that space and time will lose their continuity below that distance. It doesn't even make sense to speak of a distance shorter than that.
So you might think, well, I don't care about that. That's very small compared to me. I'm not going to-- like what's that, I guess it's Annie Hall where Woody Allen, you know his mother, says-- he's very depressed. And the psychiatrist, says what are you so depressed about? The universe is expanding is the answer. And the mother says, well, Brooklyn is not expanding.
So anyway, maybe we shouldn't care about the Planck length. I mean, it doesn't really affect us at our scale. But it's not that far. I mean it's only-- if you are measuring things in digits, if we're at a scale of kind of 1, it's only 43 digits away from us. And there are people who have memorized pi to a lot more than 43 digits. People know hundreds or thousands of digits of pi in contests.
So here's a way you could put it-- the whole universe, if you measured it in units of the Planck length, the diameter of the whole universe is only 10 to the 60th Planck length, which would mean you could express everything in the universe in terms of distance with just 60 digits, which is not that much, in a way. So what I'm saying is, things are very far from continuous. And yet calculus is going to pretend that they are.
And this incredible artistic fiction has given us all the successes of calculus. So I find that kind of fun, that we have something in common with art here, that we're not being strictly scientific when we imagine things are continuous. So anyway, creative fantasy, long riff about that.
So the infinity principle, then, gives me my definition of calculus that is essentially shared by nobody, but I think it's a good definition, which is that calculus, in my terms, would be the use of the infinity principle to shed light on anything that is continuous, any shape, any object, any emotion, any process, any phenomenon that you can approximate as continuous, you can study with calculus. So let me give you a little taste of how this would work. What are we really talking about here?
Here's an age-old problem-- find the area inside a circle. So I'm just reminding you that the circumference is the distance around the circle. The diameter is the distance across the circle at the widest point going through the middle. The radius would be half of the diameter. That's the distance from the center out to the circle.
And you probably learned a formula for your SATs. The area of a circle is pi r squared. We all heard that at some time in our lives, r being the radius, pi being this mysterious number that has infinitely many digits. It's already a signal that there's some calculus going on.
Pi is not really part of geometry. Pi is actually part of calculus, turns out. And that's why your geometry teacher can't really explain pi to you very well, because it's coming later in the calculus part of the your-- well, they can't. It's not the fault of the teacher. It's a very deep idea that doesn't really belong in geometry.
In fact, you won't find pi in Euclid's geometry. In the elements, pi is not in there. He just says that the area of a circle compared to the square of the radius is the same for all circles. That's how he would phrase. He doesn't have any concept of pi.
But anyway, we do. And I want to tell you about this. Where do you get this formula, pi r squared? So the argument I'm going to show you is essentially Archimedes' argument from around 250 BC.
And it's one of the earliest calculus arguments. It's not exactly Archimedes' argument, but it's close. if you're curious, I could tell you how it's a little different from what he actually did, but here's the flavor.
What I'm going to do is take a circle. And I'm just reminding you what the radius is, the distance from the center out to the circle. And the strategy is we're going to chop up the circle into pieces, and try to rearrange the pieces into a shape whose area we can figure out.
In other words, I can't figure out the area of the circle, because imagine I'm in ancient Babylon, or Egypt, or Japan. This was an unknown thing at that time. People didn't know the exact area of the circle.
How could we figure it out? So we're going to rearrange the pieces into a shape that's better. Now, this shape doesn't seem better. In fact, it looks like we're making the problem worse. I mean, what's the area of that thing? It looks like some kind of slug. I don't know.
So it's like in drama. The protagonist gets into trouble. And now we're interested. What's going to happen.
That's what they say, right? In playwriting, you're supposed to put your hero up the tree and then find a way to get the person down. So now the hero's up the tree.
We've got this weird shape, but there are some things we do know about it. Half of the shape is these-- I mean, look at the bottom here. Those two curvy arcs at the bottom are half the original circle's circumference. So you probably also memorized circumference is 2 pi times the radius.
So half of that, pi times the radius, is these two curves. And then the other half up there would be another pi r. And then here's r that's-- think of this as pizza slices, if you want. So there's the radius.
So I don't know what the area of that shape is, but the thing is that I didn't take enough slices. Remember the strategy, the infinity principle, is keep dividing. So let's see if it gets better.
That's with eight slices. And now I've rearranged them in the same sort of up-down type of way. And you can sort of see it's trying to become something simpler. It's still got the curvy bottom. That's still pi r. It's still got this radius on that tilted side.
But let me play the game with you that the optometrist plays when you go in to have your eyes checked. And then they try to say, is this better or worse? So I can do that with my little mouse here. Better or better, better, better?
So which one is better? Now things I want you to notice are, I think this one is better because this shape looks a little more recognizable. Let me give you one more lens. Better? Watch-- this, this.
So that's to 16 slices. Now do you see what's happening? That shape is trying to become-- what would you say it's trying to become? It's trying to become a rectangle.
I mean, it almost looks like a parallelogram right now. It's not quite a parallelogram because it has these scalloped bumps on the top and the bottom, but notice that the bumps are getting less pronounced. That's very bumpy, less, less, trying to flatten out.
And also, look at the tilt of the radius-- very tilted, less tilted, almost standing straight up. So can you now imagine if I did this infinitely often, so I now have infinitely many infinitesimally thin pizza slices ready to see? What you're going to see, you would see ba-bam. Don't you agree? That's a reasonable-- now that's a little bit of fudging there, so the mathematicians are a little uncomfortable.
So that's where it's not exactly a proof, but that can be made into a proof. The essential idea is captured by what I just showed, that when you take the limit of this out to infinity, you would really get a rectangle. And it's still got the pi r down here that it always had, and still got the r, except now we know how to find the area because it's a rectangle.
It's just the length times the height. So pi r times r, that's pi r squared. And that really is the essence of where the formula comes from. It's from this. It's really creative use of rearranging the circle to make a rectangle.
So if that gave you a little aha moment, that's it. That's why we like to do math. That's the pleasure. That's the exhilaration of math, is sudden feelings of insight when you get to the truth of something. So that's a creative example, a very elementary example, of the infinity principle in action.
Now, Archimedes' legacy today is that we do this all the time. You may not notice it, but when we see computer graphics, like in a movie, animated movies, those are not smooth-- like here's a picture of a mannequin's head, but it's built out of triangles. Just like I built the circle out of those pizza slices, we today use triangles to make smooth shapes. And the more triangles you use, the closer you can get to a realistic-looking, smooth surface.
So this was the kind of technique that gave rise to the first-- it's not that long ago, but this was the first time anyone made an animated film completely in a computer, with no human models or anything. And they were trying to make-- you know what that was. They're trying to make a character that has realistic-looking emotions, like this gentleman Geri.
You can find this on YouTube if you want to see the movie, but don't do it right now. But anyway, Geri has wrinkles and an expressive face. And he looks like a real person.
But then, went to even greater length of artistry and comedy in Shrek. There's Shrek explaining to Donkey that onions have layers. And you see there's the onion. Yeah, I love that movie.
So anyway, but his little, trumpety ears, big belly-- all these smooth surfaces are made of literally millions of polygons, triangles, that the animators-- I guess this is still Pixar. Anyway, all these smooth shapes are made of shapes that have straight lines and angles, just like we could convert the round circle into a straight-edged rectangle. So I guess what I'm saying is that you can see the spirit of Archimedes approximating something curvy with something straight in modern computer graphics.
But we also have very practical uses in medicine. Here's a gentleman who had a congenital-- he has a malformation in his chin. I don't know if you could-- I mean, maybe it's obvious to you. His chin is not really where it should be.
And this is a computer model of his bones in his head, his skull. They put him in a scanner. They could see what his head was like before they did the surgery.
Now, what's the issue? The surgeon has to decide what bones to cut out of the person's head. And then they're going to sew him back up. And they hope that after he heals, that he will look the way he's supposed to, the way he wants to look, the way that would be healthier to look.
Without a computer and without calculus, it's just guesswork what's this patient going to look like. But now we have algorithms, with the help of engineers, biomedical engineers, using calculus to model the elasticity of the soft tissues in the head, of the bones, of ligaments, blood vessels. all of that is built into these giant computer models that led to predictions about if we do the surgery the right way, his face afterward will look like this.
And then this is the actual post-operative result. And you can see that they sort of know what they're doing. This is a kind of flight simulator for surgery. Just like we have simulators for pilots, you can now do simulations that help craniofacial surgeons.
And again, it's using calculus. What they had to do was make a model of the patient's head, his face and the soft tissues inside. And you can maybe see there's lots of triangles here. That's modeling the skin.
But there's also shapes inside his head. These are three-dimensional analogs of triangles. These are tetrahedra, like pyramids.
So they make a model where they build up all the tissues in the computer, with the different elastic strengths, and constants, and so on, and then anticipate if this stuff is cut out and then sewn back together, after all the tissues pull on each other, what's the equilibrium shape of his head going to be? And that's how it's done. It's just incredible. So to me, this is-- most people say this is a wonder of bioengineering or medicine, but it's calculus under the hood that's making it work.
So let me shift gears now. I've been focusing so much on shapes and things like faces, or computer graphics, or circles. So that's the geometric side of calculus.
But really nowadays, we think of calculus-- if you had to say it in one sentence, most people would say calculus is the study of change. It's the study of how things can change, how they change, especially continuously in time or space. So it's all about dynamics and change.
And so that allows us to describe motion, but also more abstract kinds of change. It could be changing levels of virus in the bloodstream of a patient with HIV. It could be changes of electric potential in the nerve impulse propagating down someone's axons in their brain or in their body.
So there's a connection, though, which is when we want to visualize change, we draw pictures using curves. We draw graphs. That's why your teacher taught you about the x-y plane in high school.
We use our pattern recognition abilities to leverage geometry to think about change. And so that's the connection, how geometry-- now calculus goes from being something about smooth shapes to something about smooth change more generally. So to try to clarify what I'm talking about there, let me give you this example of something involving motion, which is the 100-meter dash at Beijing Olympics.
So this is now 11 years ago. Usain Bolt-- we've all heard of Usain Bolt. He, at the time, though, had never really made his name in the 100-meter dash. He was known more for running the 200 meters, a longer race.
But he thought maybe he could do well at the 100. And he begged his coaches to let him try it. I don't know how much you know about him. He's from Jamaica. He's very tall, 6 foot 5, not a normal shape of a sprinter, very gangly, tall.
He's also a joker. He's mischievous, likes to play practical jokes on people. He's very light-hearted.
And anyway, so he had run competitively only five times to run the 100 meter. I mean, he was a beginner, more or less, at that distance. But in Beijing, he made it to the finals. And I just want you to watch what happened that night back in August.
[VIDEO PLAYBACK]
- And so back to 2008 and the Olympic Games in Beijing, the final of the men's 100 meters. Usain was ready to do his thing.
[GUNSHOT]
- And they're off. Great start by Richard Thompson in lane 4 from Trinidad. And here comes Usain Bolt. He's pulling away. Bolt's looking back. You don't look back in 100 meters. 9.68 is a world record for Usain Bolt.
- 9.68, a new world record, the fastest man in the world. That celebration on the line told the world he could run faster. And soon, he would. But it was the Olympic gold medal he really wanted.
Do you see what he did there at the end?
[END PLAYBACK]
He was so far ahead that he had time to slap himself on the chest. He also put his hands down. Look at this-- those other guys are running really hard. They're really fast.
And he's fooling around. He's putting his hands down. He broke his stride. He's slapping himself on the chest. And he still broke the world record.
So the question is, everyone wondered, what if he had run seriously? How fast could he have run? So let's ask that question as a calculus question.
Let's analyze his position versus time down the track. So what I'm showing on the vertical axis is his distance down the track from 0 to 100 meters. And they happened to have cameras every 10 meters on the track. So then they could see, and also timers.
So they know when he crossed 10 meters, or 20 meters, or 30 meters. So they have those split times every 10 meters. And so those are the 10 dots on the picture.
I have connected the dots with a smooth curve, but we don't really have the data in between the dots. So it's some guesswork there. But here's what goes into the guessing-- we know that at the beginning of the race, he's not moving. He's in the blocks, crouched. His velocity is 0.
So what that means is that at the initial time, he's starting at zero time and zero distance down the track. But you see how the curve is flat there? It's coming out with a flat slope and then it gets steeper.
The steepness tells you the speed. That's the slope of the distance versus time is the speed at a given instant. And so initially, it's flat, because he's not moving initially.
So I fit a reasonable smooth curve that starts with zero slope through the dots. And statisticians have various ways of doing this. So you could do it various ways. They'll all give more or less the same result.
So having fit a smooth curve through there, then I look at the slope at every point to measure the velocity. And get this-- so here's his velocity as a function of time. And you'll notice something that you saw already with your eye. You see what's happening there?
He's picking up speed. He's going up to 12 point something meters per second. That's like-- I think at top speed, he was going about 27 miles an hour, if that's more meaningful to you. It's pretty unbelievable.
And then he starts goofing around. And that's that. In the last 20 meters or so, he dropped off when he was fooling around.
So we could try to extrapolate and figure out how fast he would have crossed the 100 meter mark if he had just kept going, like he probably could have, like this. But instead, we don't have to do that. We can just wait another few months for the World Championship in Berlin.
Because a few months later, he's in Berlin. And this time, scientists come out with laser guns, like the type that they would use to catch a speeder, like a radar gun for someone driving too fast. So they had all the runners wear reflective dots on their back, and then somebody is pointing a laser gun at each sprinter.
And then they run down the track. And then they could really get very good resolution data. 100 times a second they can measure his speed and his position on the track.
And so watch what happened when they did that. This is the data from Berlin. By the way, he won again. He also set a world record.
And I think it was about, what, 9.68 in Beijing? I think it was 9.58 or 9 in Berlin. Does anyone know? Is that still a world record? It might be. I don't know. But anyway, it's very fast.
And look at the interesting data for his speed. It's sort of qualitatively what we had before, except do you notice how wiggly the data is? The data is the wiggly curve. What do you think all those wiggles are?
AUDIENCE: [INAUDIBLE]
STEVEN STROGATZ: His stride, exactly. It's picking up the difference in speed when he puts his foot down when he lands, and then when he launches off and he's airborne. You can actually see the speed throughout the stride. He I think runs something like 43 strides when he does 100 meters. Everyone else runs 47, because he's tall.
So he just-- I might be off. It's something. He's got fewer strides than most people. But whatever, you could count the number of wiggles, and it's something like 43 or 41.
Anyway, meanwhile, though, there's all this wiggly stuff that we're not that interested in. OK, yes, we can see his stride. But what I find kind of philosophically meaningful and a larger message here is that sometimes going to very high resolution is not what you want, because in fact, all we really want is his trend line, which we could have done if we had only measured a few times.
We didn't have to use a laser gun to see sort of his average trend. And so I think there is a larger lesson here, going back to this idea that calculus has this creative fantasy built into it, that although it's only an approximation to the way we think reality really works, somehow calculus is getting at the essence of the way change happens in our world.
And so I'd just like to end with this thought coming from the world of art, that just as Picasso said that art is a lie that makes us realize truth, I think you could say something similar for calculus. So let me leave you with that. I'm happy to take any questions.
[APPLAUSE]
So I think we have good ceiling microphones here, so anyone who wants to talk can fire away. You want to-- I saw you, but why don't you go first.
AUDIENCE: So I don't have a question. I just wanted to thank you. I didn't really go to high school, and my education got me into high school got me into here was listening to you on Radiolab.
STEVEN STROGATZ: What? Wait a second, you must-- I hope you had a better education than that. It sounds like you did. Congratulations.
AUDIENCE: [INAUDIBLE]
STEVEN STROGATZ: Oh, thank you. How nice. I have not heard that one before. Fantastic. Yes, please.
AUDIENCE: Two questions, one I think is simple and one maybe is also simple, both about pi. The first was just how did you get pi r on the slices?
STEVEN STROGATZ: Right, the question was, why did I know that the curvy bottom, when I was drawing those funny shapes that I called scalloped or sort of like a slug, why was it pi r on the bottom? So the definition of pi is the ratio between the circumference and the diameter. And the diameter is 2 times the radius, 2 radii to make a diameter.
So there's a formula, c over times-- I don't have a, oh, I guess I do. I won't be in the visibility of the camera. Can I be? Yeah? All right.
So c over 2r is how we define pi. So that means that pi r, which is what I kept drawing, is 1/2 of the circumference. And so what I was saying is that when I have a shape that looks like that, that's half of the circumference.
Because like when I had four slices, I put two of the curves on the bottom. So that's all. Pi r is half the circumference. Half of it was on the bottom and half was on the top.
AUDIENCE: Yes, so that leads into the second question.
STEVEN STROGATZ: Yeah, second question.
AUDIENCE: So why wouldn't the Euclideans have thought about the ratio of this circumference to the diameter?
STEVEN STROGATZ: They did, right.
AUDIENCE: Wouldn't they say, that's an unusual relationship?
STEVEN STROGATZ: So certainly Euclid and people before Euclid had considered the ratio of circumference of a circle to its either its radius or its diameter. They just didn't consider that ratio to be a number. That's hard for us to think about.
It's interesting. Different cultures at different times don't think the same way we do. The Greeks loved proportions. They were very comfortable with the idea of proportions. We know that from their work in architecture and art, as well as math.
So they didn't really think of the circumference divided by the diameter as a number, because for them, numbers meant whole numbers or ratios of whole numbers. And pi is not that. Pi is irrational.
It's actually worse than irrational. Maybe you know what I'm talking about. It's transcendental. But it cannot be expressed in terms of whole numbers. And so for them, they thought that shapes were more powerful than numbers, that proportions, because the existence of irrational numbers caused trouble for them, being descendants of Pythagoras, where whole numbers were the essence.
So yes, they could phrase things about shapes and other relationships in terms of proportion, but they didn't regard pi as a number. They also didn't think of square root of 2 as a number. So they called it a magnitude, and it was a different-- had a different philosophical status in their mind. OK, yes.
AUDIENCE: I'm also a fan of calculus.
STEVEN STROGATZ: Oh, good.
AUDIENCE: I know several places in physics where calculus leads to problems, singularities, point charges, we know, where kind of fake differential equations have unique solutions. But quantum mechanics tells us that oftentimes, there aren't unique paths forward for time evolution. So I'm curious, a lot of physicists get upset about these failures of calculus. I'm curious what your vision of the future of calculus is.
STEVEN STROGATZ: That's a very probing question you've asked. So given that calculus isn't going to keep working, especially in the quantum realm-- we're already seeing certain kinds of breakdowns-- what's the future of calculus? Well, I don't know.
I think its days may be numbered. It's given us tremendous insight for 2,500 years, and we're still using it when we tune the weights of our neural networks, when we do deep learning and artificial intelligence, to get those AIs to do the extraordinary things that they're doing, and even more extraordinary that they're going to do in the future. We do use some calculus to adjust them in their learning algorithms.
But I have a feeling calculus might start to play a subordinate role to artificial intelligence. I don't know. It's usually a bad bet to bet against calculus. It's been good for a long time, so maybe we'll find ways of extending it to sort of a kind of quantum, souped-up version of-- we already have stochastic calculus that can deal with noise.
So I've been exaggerating when I say it's only about things that are smooth. There are generalizations of calculus that allow random jitters. And there may be a quantum calculus of the future. There may already be. I don't know enough about quantum theory to say, but that's probably the future, the merging of what we now know about quantum effects with calculus.
I don't see how calculus can survive quantum gravity, for the reasons I said. Because space itself and time will become so foamy and indescribably weird that I don't see how calculus, with its emphasis on smoothness, can still work. But maybe it will evolve into something else that works. I don't really know. So yes, please.
AUDIENCE: So thinking about Usain Bolt [INAUDIBLE], when I think about rates of change, I think about the calculus that was discovered or invented by Leibniz and Newton, I was wondering, were there any other civilizations found a way of describing that prior to them? Did the Arabic mathematicians have a way of dealing with that kind of problem?
STEVEN STROGATZ: Right, right. So one of the things that I do try to do in the book is talk about calculus cross-culturally. The story is usually told as very European, very masculine, and certainly Newton and Leibniz get a lot of credit in the 1600s.
But as I've already said, I see precursors of calculus in Archimedes almost 2,000 years earlier. But in India, there is a culture in Kerala where there were great astronomers who, hundreds of years before, literally about 200 or 300 years before Newton and Leibniz, had infinite series for the sine and cosine that they were using in astronomy, and compiling very accurate astronomical tables about the motions of the planets. So you could say, and some people have said, that calculus really was, in that form, discovered in India.
I don't think that's right, because I think calculus is really discovered in ancient Greece. But it's really all-- it's like world heritage. We see signs of it in, as you mentioned, in Arabic culture, too.
There were great mathematicians in Baghdad and Cairo, in Islam, who did things that built on Archimedes or went beyond Archimedes. So yeah, we really should avoid the eurocentric view of it. I mean, I think calculus sort of culminated in Europe, but the roots of it are in algebra and in geometry outside of Europe. Persia, too, very important.
So yeah, good point. As far as the masculine side, I did try hard to include some of the contributions of some of the great female mathematicians and tell stories about at least four of them. But there are many more. In the end, actually, I sort of ran out of gas while writing the book.
There was a deadline so I didn't get to some of the people I wanted to talk about, like Euler, and Riemann, and Cauchy. That is, I couldn't get out of the 1600s. There was a lot that happened in the 1600s.
And so the book is not perfect. I would like another crack at it, but there are other things to do. But anyway, yes, definitely contributions outside of Newton and Leibniz, big ones.
OK, wow, any other questions? Yes please.
AUDIENCE: [INAUDIBLE] like building a house or doing a recipe, or whatever, you can use algebra, geometry. But in everyday life, I don't use calculus.
STEVEN STROGATZ: That's probably right. Yeah. It's an interesting comment, that we do certainly use arithmetic all the time in making a recipe. And now you've got more people coming than you thought, you need to double the recipe, so we use proportions.
We use a lot of arithmetic all the time. No one would doubt that arithmetic is relevant. Not too many of us use the Pythagoras theorem, but maybe if you were a surveyor, or a carpenter, or something.
I mean, even geometry and algebra you'd be hard pressed. I'm a little surprised you said you use algebra in your daily life. You really do?
AUDIENCE: Like 2x in a recipe.
STEVEN STROGATZ: Yeah, 2x, kind of glorified arithmetic. I mean, I think you raise an interesting point, which is, do we want to make the case for math as you're going to use math in your real life? I think that's fake, and most people know that's fake, that you can lead a very comfortable nice life never doing math beyond arithmetic.
And that's just a fact. And those of us that teach math ought to just recognize that, because everyone else recognizes it. And so that can't be the reason for learning math.
I mean, OK, sure, if you're going to be a quantitative person in finance, or you're going to do radiology, or you're going to be a physicist, obviously those people need math. Everyone knows that. But for most people in the rest of the walks of life, do they really need math? No, they don't. Honestly, they don't.
But you don't need to read Shakespeare. You don't need to read Toni Morrison. You don't need to learn how to play soccer. You don't need-- you know what I mean?
There are things that make life worth living, that you don't strictly speaking need, but you sure want them, don't you? And I think calculus is one of those. And all of math is one of those enriching, marvelous things that makes your life better outside of utilitarian need.
And so I kind of want us to teach more of that. Instead of math to just train future engineers and scientists, why can't we have Math 101? Like we have Psychology 101. I'm not going to be a psychologist, but I want to learn the basics of psychology, or Music 101 so I can learn how Beethoven is different from Schumann or something.
How come we don't do that in math? We should. I think I want math in there as part of world culture, which it is, and yet we're in such a rush to train the future technicians that we don't.
You see where I'm going. So I think we're doing a bad job of getting across why math really matters to, like, 80% of the people who have to take it. And so maybe we can work on that. Yes.
AUDIENCE: Well, on those lines, you started by talking about pattern recognition. And we do the use pattern recognition in things beyond math, beyond science.
STEVEN STROGATZ: That's a good point.
AUDIENCE: Language learning.
STEVEN STROGATZ: True.
AUDIENCE: If you recognize that the verb endings are going to be the same in the first person, second-- I teach Spanish. If you can recognize that this word order will have this effect, and this change in this word order will have this other effect, I mean, it may not be strictly math, but it is certainly pattern recognition.
STEVEN STROGATZ: Right. So that's an interesting argument. So a standard argument, and a good, powerful argument for why math, is the math helps you develop your pattern recognition abilities, your recognition of structure in general, your logical skills, your patience, willingness to solve problems, resourcefulness, tenacity. All of that is true.
But I don't buy it, because that's not unique to math. That is, we could also be teaching chess, or we could teach philosophical argumentation, or we could teach anything that requires-- we could teach sudoku. And I'm not saying we shouldn't.
If your goal was to teach people how to recognize patterns, and how to think creatively, and how to be tenacious, math is not the only game in town. There are a lot of other things that would do that-- debate. So like you say, language, linguistics-- linguistics can be very quantitative.
So anyway I don't think we can-- you know what argument I really hate, is that math is like calisthenics, that math will make you stronger. You'll build up your brain. Like who wants to do push-ups?
Come on, are you comparing my subject to push-ups? Some people do. I mean, you get that from some high school teachers. Of course we know that math is worthless, but it will make you stronger, and it's good to be strong.
Come on, no. Math has its own proper place, just like poetry, and Spanish, and everything else, as content. Yes, I agree that the things that-- the way your body and your brain change when you do math are valuable outside of math.
So there's a lot of reasons you could-- OK, Kate, you wanted to throw something in? Oh OK, and Mary, you're going to cut us off. Yeah. OK, go ahead with your thought.
AUDIENCE: Yeah, I was just thinking back to, so am I really going to be using calculus in my daily life? Would you support the argument that maybe when we use our phone to get home by GPS, that that GPS is built on--
STEVEN STROGATZ: You're using it there, sure. You don't know you're using it.
AUDIENCE: So maybe we don't know we're using calculus, but maybe we're benefiting from it.
STEVEN STROGATZ: You are totally. I think the gentleman-- you would agree that you're benefiting from it without knowing it, or would you not? I mean, you're using calculus all day long. You just don't know it.
AUDIENCE: Only specialists use it.
STEVEN STROGATZ: The specialists that built your phone or that built your GPS, they were using it in the sense you're using the word "using." You're relying on it unknowingly. But I think-- I mean, your question is reasonable. Like in my active daily life, though I want to use it, I don't ever have opportunity to use it.
And I think you're right. You don't really. And yet so what? Because it's there. Like I say, it's under the hood. You needed it. You just don't know that you needed it.
Anyway, it's an interesting question. What's the real justification? All right. I see that we better stop. Thank you again.
[APPLAUSE]
SPEAKER: This has been a production of Cornell University Library.
Calculus, the mathematical study of continuous change, underpins some of the most fundamental and miraculous achievements of humankind, from determining the area of a circle to enabling innovations in modern medicine, computing, and space travel. It can also illuminate the patterns of the universe, according to Steven Strogatz.
In a Chats in the Stacks talk presented at Mann Library, Strogatz discusses his new book, “Infinite Powers: How Calculus Reveals the Secrets of the Universe” (Houghton Mifflin Harcourt), which was listed as a New York Times Best Seller this year. The Jacob Gould Schurman Professor of Applied Mathematics and Stephen H. Weiss Presidential Fellow, Strogatz charts the thrilling multi-millennial history of calculus that’s filled with brutal competitions and glorious discoveries.
