“The Beauty of Calculus,” a Lecture by Steven Strogatz

- Hello. Good afternoon, everyone. My name is Priya Natarajan, I'm an astrophysicist and faculty member in the Departments of Astronomy and Physics at Yale. I also serve currently as the director of the Franke Program in Science and the Humanities. I'm absolutely delighted to welcome you all to the Franke Distinguished Lecture Series. I would first like to thank the donors, Richard and Barbara Franke, whose generous support has made this possible. Before I introduce today's star speaker, it's been a real coup for us to be able to get him, to host him and hear him today. I need to make a few practical announcements. I'm required to give you all notice that this event is being recorded and photographed for educational archival and promotional purposes, including use in print on the internet and other forms of media. By attending this event today you agree to the possibility of your voice or likeness captured by these means and used for such purposes without compensation to you and hereby wave any related right of privacy or publicity. And I also want to announce, continuing first in the practical lane that immediately following the lecture, please do join us for a reception in room 108 to celebrate and to continue the conversation. Now to today's speaker, Steven Strogatz. Steve is a Jacob Gould Schurman Professor of Applied Mathematics at Cornell University. After graduating summa cum laude in mathematics from Princeton in 1980, Steve studied at Trinity College Cambridge, my alma mater, where he has a Marshall Scholar. He did his doctoral work in applied mathematics at Harvard, followed by a National Science Foundation post-doctoral fellowship, also at Harvard. From 1989 to 1994, he taught at MIT before joining the Cornell faculty in 1994, where he has been since. What is really remarkable about Steve is the originality that he brings to his broad set of research interests and mathematics. Early in his career he worked on a variety of problems in mathematical biology, which was a brand new field at the time. Studying the geometry of supercoil DNA, the dynamics of the human sleep/wake cycle, the topology of three dimensional chemical waves and the collective behavior of biological oscillators, such as swarms of synchronously flashing fireflies. In the 1990's, his work focused more into another exciting and challenging area of mathematics, nonlinear dynamics and chaos, as applied to physics, engineering and biology. And many of you probably have used his textbook at some point or the other. Several of these projects of his dealt with a couple dossiers such lasers, superconducting Josephson junctions and crickets that chirp in unison. And I have juxtaposed these things just to show you what the range of the systems to which he has been applying mathematical structures to. In each case, his research involved close collaborations with experimentalists. He also likes boldly branching out into new areas, often with students taking the lead. In the past few years this has lead him to such topics, fun topics, as the role of crowd synchronization in the wobbling of London's Millennium Bridge on its opening day, and the dynamics of structural balance in social systems. One of his seminal research contributions is a landmark 1998 nature paper on small world networks, that was coauthored with his former student, Duncan Watts. This is considered to be one of the most influential papers in network science and was, you know, it was the most highly sited paper on networks for decades as well as the sixth most highly sited paper on any topic in physics. It has now been sited more than 38,000 times according to Google Scholar. Steve has received numerous awards for his research, his teaching and his public communication. And enumerating all of them would take up too much time. So I will just mention two most recent awards. The Lewis Thomas Prize for Writing about Science, which honors the scientist as poet and the SIAM George Polya Prize for Mathematical Exposition. So one of the unique things about Steve in addition to his research work in mathematics, he's deeply devoted to the public dissemination of mathematics. He has been elected to many, many prestigious societies for both of these kinds of work, both his research work and the work he does in popularizing and making increasing mathematical literacy. He's a fellow of the Society of Industrial and Applied Mathematics, the American Academy of Arts and Sciences, the American Physical Society and the American Mathematical Society, and of course he's spoken at all the shi-shi venues, TED, Aspen Ideas Festival, et cetera, et cetera and he's a frequent guest on Radio Lab and Science Friday. And he is the author of Nonlinear Dynamics and Chaos, which is a 1994 textbook, which is the standard in the subject, and in 2009, starting in 2003 he started writing much more for the public as well as his research work. In 2003, he wrote a book called Sync, 2009 the Calculus of Friendship and 2012 book that most of you have probably heard about, The Joy of X, which incidentally has been translated into about 15 languages. His current book, Infinite Powers, will form the basis of his talk today. So in closing, I wanna quote the mathematician Georg Cantor who said "Mathematicians do not study pure mathematics "because it's useful. "They study it because it delights them "and they delight in it because it's beautiful." So without further ado, it's my pleasure to invite Steven Strogatz, who will be speaking to us today about the beauty of calculus. (audience applauding) - Thank you. - [Priya] Hi there, thanks for coming. - Thanks. - [Priya] Are you comfortable? - Yeah, I think. Welcome, thanks for coming. I'm very delighted to be here. I just drove down from Ithaca and it felt good to be back in Connecticut. I'm a Connecticut boy, grew up in Torrington. Just go right up Route 8. It's an interesting choice of quote, you mentioning Cantor's quote, because I'm an applied mathematician. So I don't know that I really agree with Cantor. I do delight in the usefulness of mathematics. (audience laughing) But I also certainly delight in the beauty, like all mathematicians. We have the exciting feature today that I don't have a a computer in front of me and we're gonna do this old school. The Dean of Science is here to help me go through my slides. (chuckling) So I will, I mean it may seem a little bit strange, but I'm gonna be saying next, next, next and he'll, he's gonna toggle through. We think this will work, so bear with us if it doesn't. But hopefully it will. We have not rehearsed. (chuckling) (audience laughing) Thank you Jeff for agreeing to do it. Okay, next. Should I say next? What's the coolest thing to say? Next. Okay, yeah next. All right, no, don't do next, well-- (audience laughing) So yeah, on this theme of calculus being useful, it's an unsung hero of our existence. It's everywhere and you may not realize it, but without calculus we wouldn't have radio, television, microwave ovens, yeah, you can kind of go through them fairly quickly because these are all just, we wouldn't have been able to put astronauts on the moon, unravel the human genome, we wouldn't have nuclear energy or nuclear weapons, we wouldn't have ultrasound for expectant mothers or GPS for lost travelers and we might not even have the Declaration of Independence. Now, some of these I realize may seem farfetched, I'd be happy to talk at the end if you want me to try to defend any of these. Some of the connections are tenuous, but I think I could make a case for everything I've just said there. So it's invisible, I mean most of us have no idea that calculus is a part of all of these things and many more, but it really is. Next. So the take home message from that is that calculus helped make the world modern. Now of course, it didn't do it on its own. It wasn't calculus that put people on the moon. You needed engineers, you need physicists, so it's a, think of it as like in a drama, you know, in a theater there are supporting players. This is, calculus is sort of a supporting player that you've seen in every movie or every play and you never really noticed that it was there, but it's a very important player in the drama. Next. Now, the question is, at least the first question, here's a branch of math that many people find sort of arcane and abstruse, certainly a lot of freshman taking freshman calculus would say, I don't see the point of calculus, both before they took the course and after they took the course. (audience laughing) (chuckling) And a lot of the students taking advanced placement calculus and I dare say many of their teachers would be hard pressed to say, why are we doing this exactly? You know, calculus is in such a big rush, there's so much to cover to get ready for the advanced placement test or for whatever, there's so many techniques to learn, so many theorems to be understood, so much jargon to absorb that the larger context is often excluded or just not even, you know, no one would even think of including the history, the applications, the philosophy, the connections to medicine and physics and sociology and everything else. But I wanna try to give you the big picture of calculus, and I don't assume that you've had it or that you even know what it is. Now, it could be that, you know, I don't know, you may be ringers out there. Maybe all of you are mathematicians, but I don't assume that. This lecture series as you may know, is supposed to be a meeting ground for the humanities and the sciences, and I'm very grateful to the Franke's for you know, for sponsoring this colloquium series. It's fantastic to bring the humanists and the scientists and the mathematicians together. So in that spirit I'm gonna be avoiding for the most part, technical math, because it's really not very realistic to do that if you've never thought about these things before. But I do wanna give you the flavor of some of the gorgeous arguments and reasoning and the stories of why calculus is so important. So, the question I'm asking there is you know, here's this arcane branch of math, how could it be that it helped to reshape civilization, and I'm making a little word game there because it grew out of geometry. It was originally about shapes and it reshaped the world. Okay, next. So in this spirit of humanities trying to find common ground with the sciences and with math, I wanna tell you a little story first of two men who met I'd say it was probably the late 1940s, early 1950s. So my parents would have recognized this gentleman, maybe a lot of you will not know who he is. Herman Wouk was a great novelist. He's actually still alive, he's 103 years old. But in my parent's generation he was the chronicler of World War II for a mass audience. And so he wrote these famous books, Winds of War, War and Remembrance novels. You probably have heard of the Caine Mutiny, Humphrey Bogart was in the movie. That's his first, I think it was maybe his first book, Pulitzer Prize winning book. So Wouk wanted to write a book about World War II. This was before, I mean that book, the book he had in mind grew into two big books. Those two books, Winds of War and War and Remembrance. But at the time he had this vision of I'm gonna write a great novel about World War II. He was in the war himself as a sailor, and he wanted in particular to interview some scientists who had worked on the Manhattan Project, on the building of the atomic bomb, and he was told that you could find people like this at Caltech. And if you go to Caltech, there's someone that you should meet, but he's difficult named Richard Feynman. So, let's have the next slide. There's Richard Feynman. You see him there in a classroom. Those are the days when it was all boys probably. Caltech is still quite, I think isn't it pretty bad gender ratio? I think so, but anyway, in those days probably 100%. But you know, he was a joker. You see him there playing the bongos, he used to break into safes at Los Alamos. He's written many books, chronicling his escapades. He loved to tell stories about himself. So he's a joker. He was also a great teacher and had a very deep understanding of physics and won a Nobel Prize for his work in an area called quantum electrodynamics, but anyways, so he is sitting there, when he had worked on the Manhattan Project he was very young, like maybe early 20's. So Wouk comes to see him, they start talking, they talk, they have a lot in common it turns out. They're about the same age, they're both Jewish, there's a difference you know, Feynman is quite an extreme atheist, Wouk is very devout and so they get to talking, they like each other and they're having fun back and forth, talking about things, arguing about science and religion, and then after they leave, just as Wouk is leaving, walking out the door Feynman says I have a question for you. Do you know calculus? And Wouk admits you know, no I don't. And Feynman says well you had better learn it, it's the language God talks. (audience laughing) So I wanna leave you with that thought right now. The language God talks. Next slide. That's a book then, that became the title of Herman Wouk's book about science and religion, sparked by his confrontation with, I mean if you wanna call it a confrontation, his meeting with Feynman. And it's a terrific book. It includes his conversations with Steven Weinberg and various other well known physicists. I don't know if you can read it, but the subtitle is on science and religion. So if you've ever wondered about trying to reconcile your faith with science or can they be reconciled? They certainly address a lot of the same deep mysteries. This could be an interesting book for you to look at. But anyway, so The Language God Talks, now Wouk as I say, did not know calculus and he tells a little story in the book about how you know, if this is the language God talks, maybe he should learn it. So he describes his efforts to go out and try to learn calculus and I take that as, sort of a metaphor for what I think we're trying to do here today. That here's someone from the humanities making a good faith effort to try to learn what we have to offer in math. So next slide. Wouk tells this little story of how, he says, he went to the bookstore, he skimmed some freshman textbooks hoping to come across one that might help a mathematical ignoramus like me who had spent his college years in the humanities, i.e. 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, he sets out to teach himself calculus. He tries reading books with titles like Calculus Made Easy, he hires and Israeli tutor to help him learn his Hebrew better at the same time learning some math, none of this working. He goes to, finally in desperation he goes to a high school and enrolls in the high school calculus class and after a few weeks he falls behind and gives up and leaves and as he's walking out the students clap for him, and he says it's like, you know, when there's a pitiful showbiz act, there is this concept of sympathy applause. (audience laughing) That's what he felt like. So he had this really hard, he could not manage to learn calculus and so, I wanna dedicate my book to Herman Wouk. I've never met him, I don't know him. I sent him a copy. He's in a home somewhere. I mean, he's very, very old. His wife recently died. I mean I think it's probably very hard for him at 103 right now. But, next slide. So, yeah so, Infinite Powers is my attempt to write for people like Herman Wouk, in the humanities. What is the reason that anyone would think that calculus is the language God talks. Why should, so my wife would tease me for saying it like this. Why should a normal person learn calculus? (chuckling) And by the way, I don't feel necessarily you need to learn calculus. This is a story about appreciating calculus. You don't have to learn it to be able to appreciate it. So first let me try to address this question of Feynman's cryptic comment, the language God talks. What did he mean? I wanna try to illustrate what I think he might have meant with an anecdote about electricity and magnetism. So, electricity, here's a picture of my daughter Lea, my older daughter, I have two daughters. But so when Lea was four years old, there she is at her grandmother's house and she's playing on the bed and it was winter and you know, static electricity made her hair do that and I took a picture of it. I mean, you've probably had similar experiences with static electricity. So electricity is something we all kind of know about from playing with static electricity, from the electricity that we can plug into in the wall. There's also magnetism, next slide, which you know, if you take a magnet, so there's a magnet with north and south poles. The north pole in red and if you sprinkle these so called iron filings, little slivers of iron on paper around the magnet and kind of shake it so that everything has a chance to move if it wants to, the filings form this amazing pattern around the magnet, which we take as indication that there's this invisible field of force around the magnet, the magnetic field, and these filings, these iron filings can help us visualize this amazing pattern of force. The directions that the magnetic field is pulling on things depending on where you are. So, you've got these concepts of electricity and magnetism, which in the 1800's when scientists first began to really try to understand them scientifically by doing experiments on you know, electrical currents, magnets and the relationship between them and here I'm thinking of people like Ampere and Faraday and Cowan, Lenz, I mean if you've taken freshman physics you will have heard these names. So this is a great era of discovering the laws of how magnets and electricity work and how they're interrelated. The key concept turned out to be the idea of a field. This invisible pattern of force around magnets and around current carrying wires and so on. And so Michael Faraday had the idea of the field, but he was not very mathematical and it was a later person named James Clerk Maxwell. Next slide. Here's a picture of Maxwell as a young man. Usually if you know Maxwell's pictures, he looks like an old man with a big beard and that was him later in life, but this is Maxwell as a young man. Scottish physicist, working in the mid 1800s and he realized that these fields that Faraday and others were talking about could be given a mathematical description. And when he tried to give a description of all the laws that had been discovered by Ampere, Faraday and others, he found that calculus was the perfect language for describing what had been discovered about electric and magnetic fields. But here's the part that gets spooky. I mean, where Feynman uses the term the language that God talks and you always hear people say, oh well math is a language. Yes, that's partly true, math is a language. But it's much more than a language and this analogy is missing something and this is the part that I consider a bit uncanny and I think it's what Feynman had in mind when he said it's the language, it's not just a language, it's the language God talks. In this sense, that when Maxwell looked at what he had written down, his equations that encoded all the known facts about electric and magnetic fields, he had done a translation from physics into the language of calculus, but then he could do more with calculus, which is he could start to operate on the equations, he could manipulate them. I mean in math we use that word, manipulate the equations. I want you to think about it literally. Like you go to the masseuse because you have a kink in your back. She will manipulate you, she will massage you, and she will work on you and you might start to relax and say something. And that's what happens to Maxwell's equations. As he starts manipulating them by adding one to another or in our language he takes the divergence of one or he takes the curl of another. He's doing mathematical manipulations on them, trying to get them to open up and talk to him and reveal their secrets, and he doesn't know what he's looking for. He just knows that he wants something to come out, and as he's manipulating, at some point he sees, and by the way, what does this mean really? Manipulating. You're transforming the equations from one form into another form that's logically mathematically equivalent. So in a way you're doing nothing. It's just one thing that's, I mean if you had the logic, if you could see the chain of reasoning in your mind, you wouldn't need the math. He's basically constructing a long argument. That's what the symbols are doing for him. I mean that's why we use this. It's not just a language, it's a system of reasoning. So he's doing his reasoning on the equations and at some point he recognizes a new equation that has come out because it's the same equation that describes the spread of ripples on a pond. It's the equation for how waves move, and except that this is a wave of electricity and magnetism. Where the electricity, the electric field is generating a magnetic field, which then regenerates the electric field, and they're doing this kind of dance, a pas de deux together dancing, propagating and he calculates. We're always interested in a wave, how fast does it travel? He plugs in the numbers and it turns out it propagates at the speed of light, which had been measured around that time for the first time. So this I think, must have been one of the greatest aha moments in human history. That in that moment, and I wish I could be there. I mean we all wish we could be there watching Maxwell as he realizes what light is. I mean, humanity had known about light forever, but we never knew what it really was. Now we suddenly knew. Light is a electromagnetic wave. So, what was that that happened there that day? I mean, that was calculus as a language and as a system of reasoning and it was if Maxwell had tapped into something that is built into the structure of the universe. He wasn't just talking. I mean, he wasn't like being a poet. He was actually learning the language that the universe is speaking. So there's this prediction that electromagnetic waves exist, that they propagate at the speed light. Next slide. Yeah, click. And so 1860s, he predicts this. Within a few years, Heinrich Hertz measures electromagnetic waves. They're real, they do exist. Soon Tesla is using them to create the first radio systems and to do wireless transmission of energy, and then Marconi is sending messages across the Atlantic Ocean and you know, you have the birth of the telegraph and very soon after, click, we have radio, we have television, we have wireless. So, it's not correct to say that calculus created this, but calculus was indispensable. I mean, if it weren't for Maxwell and this calculus, all these things that we take for granted today, they may have still been discovered anyway, but maybe later or maybe not at all. So, that's what I think Feynman is talking about. That Maxwell's equations are telling us something about the universe, not just, it's not just a language. It's a language about something very deep. Next slide. So you may have seen these t-shirts. You know, this is a standard nerdy t-shirt you can get, and God said, and then there's Maxwell's equations, and then there was light. So, let's see, I don't have a pointer but, or do I? No, I don't think I do. Do I, no. All right, I'll do it like, no. All you, you don't need to be able to read that language, just to see that those symbols, what's going on the left where there's that upside down triangle, that's talking about electric and magnetic fields and how they change in space. How they change as you move from some point. Like think of these iron filings around the magnet. If you look at one of the filings and then move to the next one nearby, the direction may change a little bit. How exactly the field is changing in space is encoded by what's going on on the left. And then what's going on on the right is you might see that there's a symbol t in the denominator of one of those fractions. That t refers to time. It's talking about how the magnetic and electric fields are changing in time and how that is influenced by how they're changing in space. So there's this interaction between you know, things happening in space and time encoded in this and like I say, these four equations then imply the existence of electromagnetic waves, although it's not obvious, you have to be good at calculus and this particular part of calculus called differential equations, which is what these are. These are four differential equations. The word differential meaning that they express how things change from one point to another or from one moment to another. Those are differences in time or space. So that's the difference in differential. Okay, let's move on. So, next slide. Now, so the larger point though, I mean Maxwell is just supposed to illustrate this broader point. Why calculus matters, this is my claim. Sort of the thesis of the whole book. So if you wanna just get the punch line, they say it's good to give the punch line at the beginning 'cause people will space out and get tired. So I'm giving it to you right now. The rest you can start checking your email or whatever you need to do. But why does calculus matter? Here's the argument. First, because the laws of nature happen to be written in this particular language, in the language of calculus. Second, because calculus is more than a language, it's a system of reasoning that taps into something about the structure of the universe. It's as if the universe runs on calculus, if you want me to say it that way. I mean, I'm getting a little mystical here, but I think it's true in that we see this again and again. Not just with the laws of electricity and magnetism, but if you look at Einstein's laws about how gravity works encoded in general relatively, which I was just listening to in Priya's book, Mapping the Heavens, which I highly, highly recommend if you want a great book about astronomy and astrophysics. But, so Einstein's equations are differential equations. Newton's laws of motion and gravity, differential equations. The laws of how fluids move like air and water, differential equations. Everything under the sun that we understand in mathematical terms, we understand through differential equations, i.e. calculus. It is the language God talks. Next. By learning to speak this strange language, I mean it took humanity thousands of years to learn this language, but by learning to speak it and discovering that it is somehow tapping into this deep structure of the laws of nature. We've been able to use calculus, you know, in combination with science and technology, to remake the world. That's the argument. Now, how did this all happen? I mean we didn't start with Maxwell. Remember, Maxwell was at the time of the Civil War, that's 1860 something. We didn't begin with that. We began thousands of years earlier. It's commonly, you know, many books will tell you calculus was invented by Newton and Leibniz in the 1600s. I don't think that's the right point of view. I think that's way too parochial a view of what was really happening. I would much rather say we started calculus a few thousand years before that. Next slide. So I'll try to explain to why. And to do this let me try to personify calculus. I think it's the easiest way to think about it. Think of it as if it were a person. That is, not really a person of course, it's a subject, but when I say what calculus wants, I guess you could then if you're very literally minded you should hear that as, what the practitioners of calculus want. The people who do calculus, what is it that we want? The books, textbooks about calculus are like 1,000 pages long, they're very heavy, they weigh like cinder block weight. What is in there? What are we trying to do? It looks very complicated, but that's misleading. What calculus wants is simplicity. The reason it looks so bulky is it's tackling very hard, complicated problems, and it wants to make them simpler. Calculus is all about what do you do to make difficult problems easier and it has a grand strategy, and this strategy is buried under all the minutiae of derivatives and integrals for those of you who have taken calculus. You learned about trig substitutions and every other possible trick. That stuff is distracting. Here's the real point. What is calculus really about? I will now tell you. Next. Here's how it gets what it wants. It has one big idea and this idea runs like a theme, you know in a musical, theme and variations. There is one idea in calculus and if you get this one idea, which for some reason we never tell the students this idea, but everybody who knows calculus knows it. Jeff the mathematician will tell you, I'll check this on you, you tell me if you agree. This is the one big idea. The big idea is that we're gonna make problems easier by slicing them into smaller problems. Now that is an idea everybody has who knows how to solve problems. To make a hard problem easier, make it smaller. Break it into parts and work on the parts. That's an ancient idea. The crazy idea of calculus is don't stop. Do that forever. Do that at infinitum. Next. This is the big idea. I'm gonna call it the infinity principle. Nobody else calls it that, I'm just making it up. But this is what I think is the heart of calculus. Is that you can make problems easier by slicing 'em and slicing 'em and slicing 'em all the way to infinity, next. So you keep doing that forever and then what are you left with? You're left, next. You're left with infinitely many, infinitesimally tiny pieces and the philosophy is that those pieces, whether they're short pieces of a curve or little patches of a surface or an instance in time or tiny bits of material, whatever it is, that tiny thing is gonna be much more manageable than the original big thing. And so the strategy is a two staged strategy. First you cut the thing down to the smallest infinitesimal bits, next. You solve the problem for the tiny pieces. That turns out usually to be fairly easy. That's why you did this. Next. Then you have to put the answers back together to get the original whole. That tends to be hard, very hard. There's no free lunch. So putting the pieces back together is what makes calculus hard, but it turns out it will be easier than dealing with the original problem without this strategy. So this is the best thing we've ever thought of. It's one of the great ideas in human history. Up there with human rights, democracy, evolution, quantum mechanics. I mean, this one idea, the infinity principle has had as big of a consequential impact on the world as I think, just about any other idea. Next. So, the jargon. You've heard of differential calculus. That's what we call the operation of slicing and taking the tiny pieces. Calculating derivatives amounts to this. Putting them back together, that's the hard part. That's called integral calculus. And so if you've taken calculus you know that you learn derivatives before integrals, that's why. Because differential calculus is easier, you do that first. Now, there's a key assumption in this when we say that we're gonna chop problems into their tiniest bits, we're assuming that we can do that forever. This is the importance of being continuous. Next. The infinity principle says that it will work. I mean, this strategy will work only on those objects that can be infinitely subdivided, endlessly, okay? If that's not gonna work, oh, then you can't use calculus. So the calculus only works on these things that infinitely subdivisible. We call those things continuous. Meaning from the old Latin roots, what con plus tinerae would mean holding together. So a continuous thing is that which holds together in that sense. That it's all touching itself. So continuous objects are grist for calculus, next. But notice the creative fantasy. In case you think math is all very objective and rigorous, I want you to, don't wanna disabuse you of that idea. That's only half of math. That's the second half. The first half is creative fantasy. You have to have imagination and creativity. All math is like that. Then you tidy up and make things nice and logically pristine, but at the beginning it's wild fantasy, imagination and desire. And in this case, the desire is to pretend that the world is like this. Pretend that everything can be infinitely divided as much as you want. Why is it fantasy? Because that's not really correct. The world is not like that. We know today, depending who you talk to, but you know, this was a debate going back to the, you know, the ancients. When you think about atoms, what's the word atom mean? It means literally atom, it means uncuttable, right? The things, things that cannot be cut. The smallest bit of matter is an atom that cannot be cut any further. And so the Greeks, you know, in the time of Democritus, were arguing do atoms exist? Is the world grainy, is it made of tiny things? Or is it infinitely subdivisible forever? And not just matter, but space and time. Is there a smallest interval of time? Is there a smallest amount of space? This is a live question today. If you talk to the string theorists or people doing quantum gravity, they will tell you yes, there is a smallest thing. It's called the Planck scale, after Max Planck. There's the smallest unit of space. Something like 10 to the minus 35 meters. Way, way, way smaller than the smallest particle that we know. There's a smallest unit of time, which is the time it would take light to travel that distance and there's nothing smaller that we know of. So it is not true according to modern physics, that you can infinitely subdivide time and space and matter. Do you think we care about that in calculus? Yes and no. Okay, we wanna get things right, but we also want to make progress and so we're gonna pretend. That's what I mean by creative fantasy. We will pretend that the world is infinitely subdivisible. Next. So calculus then in my definition and essentially nobody else's, but I think it's really the heart of it, is calculus should be thought of as a use of the infinity principle to solve or to shed light on anything that's continuous. Whether it's a shape, an object, something moving, any kind of phenomenon, that's the strategy. And so, if I had to boil it down to what calculus has really been obsessed with, it's three things. Next. Oh, okay. The three are, curves and curved shapes. So geometry, but specially about things not made of straight lines and flat planes, but that are curved, in sinuous. That was a big problem in ancient geometry. Calculus came into solve that and that's where it became, dealing with curves. So that, let me go back, sorry. Let's just go back one. So yeah, so we're gonna start with curves, but calculus has, you know, the three things that it has focused on throughout its like 2,500 year history, curves, motion and change. So if you want a little mantra, that's your manta. Curves, motion and change. Some people like to define calculus as the mathematics of change. That works pretty well actually, because a curve you can think of as changing direction, and motion you could think of as changing position. So if you had to say it in one word, calculus is the mathematics of change. But specifically, continuous change. Never ending, ongoing change. Okay, next. So let's begin with curves. Now here's an ancient problem. Figuring out properties of a circle. You might think that's trivial, I learned that in high school geometry. It's not trivial, it's not easy to figure out. Like think of how weird the number pi is, right? Everybody is fascinated by pi with its infinitely many digits that don't show any pattern. Why is pi so weird? Because pi is a creature of calculus, it's not a create of geometry. The fact that it has these infinitely many digits that don't repeat and don't show a pattern is already a clue that there's something with the infinity principle at work here. You know, in high school you just memorize pi. Maybe you didn't think why is that the number? How does anyone calculate that number? You have to use calculus, that's how. So let me remind you, pi is defined as the ratio of the circumference of the circle to its diameter. Okay, circumference the distance around, diameter the distance across. But a big question, if that's the definition of pi, fine, that's one property of the circle. What about the space inside the circle? The area inside the circle? You may have memorized a formula for the SATs, pi r squared, the area of a circle. R is the radius, that's half the diameter. The distance from the center of the circle out to the edge there. Where does that formula pi r squared come from? That's a calculus result. You will not find it in Euclid by the way. I mean if you think it's Euclidean geometry, you don't understand. You can look at Euclid, you will not find pi r squared in Euclid. Euclid does say the area of a circle is proportional to the square of the radius, but he doesn't have pi. For him pi is not even a number. It's you know, he okay, so you have to wait till like two more generations for Archimedes and that's when we start to really begin calculus. He's the great maestro of the infinity principle, Archimedes. So let's see the next slide. Here's an argument. This is not really Archimedes' argument, it's close in spirit to his argument for calculating the area of a circle. But if you've never seen it, I wanna show it to you 'cause I think it's very dramatic and it's an example of what it feels like to have an aha moment in math. Okay, so what I'm doing there is I'm taking a circle and I've chopped it into four pieces. You could think of them as slices of pizza if you want. It's an abstract mathematical pizza and I'm rearranging those slices into this funny shape on the bottom which I'll refer to as a scalloped shape because you know, it's got these bulbus curves on the bottom. And I've put on the bottom pi r to indicate that the amount of crust, if you thought of it as a pizza, the length of the crust on the bottom is pi times the radius, why am I saying that? The whole crust is two pi times the radius, right? That's the circumference. Pi times the diameter, the diameter is two times the radius. So two pi r is the circumference. Half of the circumference, half of the curvy part is on the bottom, half is on the top. So it's pi r on the bottom and then the edge of the pizza, that straight piece, this piece, that was, that's just the radius of the pizza. So at the moment, the strategy is, if we could figure out the area of that thing, we would then know the area of the circle. The trouble is we've looked like we made the problem worse. That shape is harder than the circle. But what we're gonna try to do is somehow with the use of the infinity principle, change the circle into a shape whose area we know. That's the grand strategy. All right, so next slide. The thought is, maybe the reason that shape on the bottom was bad is we didn't take enough slices. So instead of taking four slices, what if you take eight and you arrange them like this? Can you see that the shape is getting better? It looks like it's trying to turn into some shape you recognize. Anyone wanna volunteer what shape it looks like it's trying to become? - [Student] Parallelogram. - A what? - Parallelogram. - It looks like it's trying to be a parallelogram, right? It's almost a parallelogram 'cause this side is parallel to that side. The bottom is not straight like it's supposed to be in a parallelogram, a little bit curvy. But here I wanna test your dexterity for a second. Jeff, can you toggle back and forth. Well what I mean is when you go to the optometrist, they will sometimes say better, worse, better, worse. Can you do that between the two slides? Better, worse. Which is better? Better. Okay, go one more forward. Better. Okay, that's with 16. That's better. Now can you see what's happening. Actually, keep your eye on the tilt of this side. Can we do that? Go back one. See it's more titled and more. So now going forward it's getting less tilted. If you did this infinitely often, you'd have infinitely many infinitesimally thin slices, but they would be standing upright and you would get that. Isn't that cool? And notice the whole time the bottom was always length, half the circle pi r, and this was always the radius. So now you know how to find the area of a rectangle. The length times the height. Uh yeah, pi r times r. That's pi r squared. That is basically the real reason the area of a circle is pi r squared, because it can be morphed into a rectangle with the help of the infinity principle. Now, if you're really a tight mathematician out there you know I've done some fudging here. Just relax. (audience laughing) This is basically correct, but you know, there's some loose moments in the argument. But remember what the venue is. Come on, give me a break, okay. (chuckling) But that's the gist of it. Okay, so that's an example of the power of infinity. That the, also notice that the shape becomes best at infinity. You know the optometrist test. Keep going, things get better at infinity. That's a key principle in calculus. Things are better at infinity. Okay, next. So as I said earlier, there are these three obsessions: curves, motion and change. And for the sake of accessibility I'm probably gonna mostly talk about curves, it's the least technical. But we can talk about how calculus has revolutionized our understanding on things that move on the Earth and in the heavens and anything that changes. Whether it's traffic on the highway, whether it's the level of HIV virus in the bloodstream of patients who are infected, I mean it's been applied to everything under the sun. Next. So back to curves just because I wanna show you one more masterpiece of Archimedes. To show you another little bit of how the infinity, this is actually more honestly what Archimedes did. Take a circle, you can make shapes with a circle. I mean they started with circles. That was the fundamentally first mysterious shape, and then you could make shapes like a cone, and if you slice a cone with planes of different tilts, you can make a shape like that called an ellipse or you can make a shape like this. If you slice parallel to the side of the cone you make a shape called a parabola. So originally these were thought of as sections of a cone and they're still called conic sections. Ellipses and parabolas and also hyperbolas, but let's not bother with them for now. So next. So one question that Archimedes wondered about, this is like 250 B.C., he's in Syracuse on the island of Sicily, part of the larger Greek empire. The Roman's would like to invade into, in fact they do invade and try to take over. Archimedes you know, takes some time away from his math to build war machines that according to legends, you know, Plutarch tells us in his history that Archimedes made these fearsome machines that could grab the Roman ships out of the ocean. Giant cranes lift them up, shake them so that the soldiers would, you know, the sailors would come out like if you were shaking sand out of your sandal, you know? I don't know if it's really true, but there's all kinds of stories about Archimedes. But we do know this because he wrote these treatises that we still have about how he found areas of curved shapes. So one was, what's the area of a shape like that? A so called parabolic segment. Next. This is amazing. Here's his strategy. To figure out the area of this curved shape, he is going to regard it as a sequence of shapes made of straight lines. So, you know, in Cubist painting you draw pictures of people or other things, Braque and Picasso, they're using all these rectilinear shapes. Archimedes is doing that in his vision. I think it's an amazing vision to see this curved shape as this combination of triangles. So what he does is he puts a big triangle inside and then there area two smaller triangles, lighter shade of gray, sitting on the side of the original big triangle, and then you can see there's a little bit of empty space left under the parabola. He would wedge in more triangles in there and keep doing that, and he's gonna exhaust all the area until it's all nothing but triangles. So that's his concept. But now which triangles exactly should he put in there? Next. Oh, I should say before I get to that. Here's the big punchline. He's gonna show that the area of that parabolic segment compared to the area of that first big triangle, they're in the ratio four to three. And that's not obvious, but that's what he's gonna show. The parabolic segment is 4/3 as big as the big triangle. And if you are interested in music you should think about the numbers four and three. Do you have any musical people here? Yale is great at music. You do music? Do you, do you, but I'm wondering, oh you do too. Wow, this is a musical family. Are you brother and sister? Yeah? Geez, but this is hard question I'm asking. Do you know anything about why four and three are related to music? You do? - Yes. - Why? - [Student] Isn't like the time signatures could be like three or four. - Oh, that's interesting three, four time or something? It could be a time signature. Uh, you have a thought? - [Student] Is it the ratio of frequencies for intervals. So, I don't, I know an octave is two to one. - Oh good, now you're on the right track. So and octave is two to one. Tell us more what you mean. What do you mean two to one for an octave? - [Student] Um, it's that-- - Like if I take two strings. Suppose I was playing a guitar. - Yeah. - And I put my finger on a fret halfway up-- - Yeah. - The string and I pluck one string and if it was the same, the same string but with the same tension and everything. - Yeah. - But not with my finger on the fret, the one that's long and the one that's half as long, they will sound an octave apart. - Yes. - Right? The one that's short would be an octave higher, twice the frequency. If the frequency, if you do this with strings but in the ratio four to three, Pythagoras and his people, legend according to legend, discovered laws of musical harmony. The three to two ratio makes, do you know? - [Student] So it makes a third. - So three to two would be a fifth, four to three would be called a fourth, I think. So and I don't know music, but my daughter, who's my younger daughter told me think about the Star Wars theme, that's a third. No, that's a... That's a fifth. (humming) That's, those are separated by a fifth sheet too. - [Students] Yes, that's correct. - Is that right? (audience laughing) But then she said, but here comes the bride. (humming) I didn't do it well. (humming) That's a fourth. That's four to three. So a four to three ratio, the point, okay. (chuckling) (audience laughing) The four to three ratio is something every self respecting Greek knew. Four to three was considered beautiful because it's connected to music and so, you could imagine how excited Archimedes would have been that the parabolic segment to the big triangle is in a four to three ratio. Okay, but why? Why is it four to three? So let's see the argument. He takes this triangle and now the way he constructs that triangle is he takes the line that defines the bottom on the segment, he slides it up until it's just touching the parabola at one point. That's in the jargon, the tangent line. Right, it'll be tangent to the parabola at one point. So if he, and that defines a unique point and then he builds the triangle touching that point. Then he's gonna use that parallel sliding trick to build new triangles. Next slide. So you see, then he builds those little triangles the same way. Slides the sides parallel till they touch at one point. Builds new triangles and what he can prove with the geometry that he knows and properties of the parabola that he knows, is that those new triangles will have 1/8 the area of the original big triangle. Not obvious, but he can prove that. And that, he shows that that rule is true at every stage. Whenever he creates a new triangle by this technique, it will always have 1/8 the area of the triangle it came from. And so if you total up the areas, so far we had one for the big triangle, 1/8 plus 1/8 makes a quarter, so we have one plus a quarter and then if you believe me about this rule that you're always gonna get a quarter of what you had before, you're led to next, an infinite series. Which is one plus a quarter plus a sixteenth plus dot, dot, dot. And that turns out to add up to 4/3. So if you've had high school algebra, you learned a formula for that, but Archimedes did not have high school algebra, because algebra is a product of the Middle East, right? I mean algebra is gonna be invented about like 800 years later in places like Baghdad. I mean there's a kind of geometric algebra that the Greeks know, but still for them geometry is the thing. So he does his algebra this way. When he calculates this infinite series he draws a picture which has four squares. Here's a square of size one in area. The whole picture is four units of area, right? Because it's four big squares. So that's the four in four thirds. Watch this, I don't know if you see what's going on. You take one that's one big square, then you take as quarter of that square, that's the thing marked a quarter. Then you take a quarter of that, that's the thing marked a sixteenth, and so the total of the gray is the infinite series he wants. But he says by inspection I can see that that's one third of the whole shape. Right? Because what is it? It's a big square plus a second sized square, plus a third sized square, but that's also copied over here. Big square, another one of size of quarter, the white ones. You have three copies of the same structure in this picture of size four. So 4/3 is occupied by the gray. So that's what we would call a proof without words, but you know what's really nice about Archimedes here too. Now speaking to the rigorous mathematicians for a second, is he actually calculates the error term correctly, he doesn't just say dot, dot, dot. I mean, you're taught, when you teach calculus properly you don't say dot, dot, dot. You say calculate the error term and make sure that the error term goes to zero, and the error term is that little tiny square up in the corner and he can now see exactly how big it is. You know, in other words, he can calculate the finite series, not just the infinite series. Okay. So anyway, that's an example of Archimedes ingenuity with the tools he has, which is geometry and ingenuity, but he doesn't have algebra, he doesn't even have decimals, right? Decimals are being created over in India. So this is a big world story, the story of calculus. It's not like, don't think of it as just a European Greek thing, it's not. But this part of the story is. Okay, next. So what about Archimedes today? This is another case of this invisible presence of calculus. You are using Archimedes all the time when you go to the movies and possibly when you go to the doctor, but you don't know it. So what am I talking about? You know how he built up that curved shape, the parabolic segment out of triangles. Well you can make a shape in computer graphics out of triangles that can approximate any smooth surface. So you can make a picture of a mannequins head by just triangulating it more and more finely and that is the technology that goes into something like the next slide. This kind of movie, Geri's Game, which you can find on YouTube. Don't do it right now, but, so Geri's Game was the first animated movie completely computer generated movie that had a human character that it was emotionally expressive. We had earlier movies like Toy Story, but that was, you know, those are toys. This is a real human being who acts like a person, except he's made of polygons. In fact, he's made of triangles, but millions of them. So you don't see the triangles, but they're there, and the people at Pixar who made this, you know, animate all those polygons. But how to make a polygon that, sorry, how to make a sequence of triangles that will make the shape that you want, an old man's face with his wrinkles under his eyes and so on, there's a lot of calculus in computing the right triangulation to make whatever you want. Next. So, you know, this is probably a movie you know better. Shrek telling Donkey that onions have layers and his little trumpet like ears, his round belly, these are all smooth surfaces that required millions of triangles. And so this is the technology that Pixar and Dreamworks and everybody uses nowadays. It's this Archimedean idea that you can represent any smooth surface with triangles. Next. Now here's a little more medically oriented example. If you just think it's kind of frivolous for kids movies, it's not. Here's a gentleman who you can see on the left. This is him before surgery. His eyes have been, you know, pixelated because they, for privacy, but anyway, this guy you can see his jaw is sticking out in a way that it's not just cosmetically unpleasing for him, but I think it was causing him medical problems, this malformation. And then there's a scan of his bones, taken in a medical scanner. So that's him before the surgery. Now, what doctors would like to do is figure out if they would cut out certain bones in his jaw, and then reattach everything surgically, you know, sow him back up, what's his face gonna look like after surgery? And the issue is that, there's a lot of other tissues than just the bones. Bones are pretty rigid, but there's soft tissues, the face is made of skin, there's all the tissues behind the face, there's cartilage, there's you know, a lot going on. Tendons, all kinds of things to think about. So, the branch of math that's involved here is, we want to make a mathematical model of all the soft tissues and bones in a person's head, such that when we change the conditions by cutting out bones and reconnecting things, there are gonna be many elastic forces. There are gonna be parts of the skin and cartilage and bone that are pulling on each other with this new configuration, and things are gonna shift. Obviously that's the whole point of doing the surgery. So, people have made these elasticity models. Now, okay, elasticity. Elasticity is a branch of engineering and applied math that uses calculus. If you take a course in elasticity in the engineering department at Yale or anywhere else, you will be doing tensor calculus, you'll be doing partial differential equations. It's calculus, okay? It's been applied everywhere. And so, I'm trying to say that when this poor guy had his face redesigned, the doctors were able to tell him what his new face would look like. The computer model said it would be that third image from the left. They did the surgery and that's his new face. And you can see that the prediction basically got it right. Now this has been quantified. I'd refer to the study in the book. You can check the original paper if you wanna see the data. But I mean, with calculus and computer modeling and lots of careful scanning and all this other stuff, you can essentially build like a flight simulator for surgeons. The surgeons can do the study because you know it's very serious to cut someone's bones. You don't wanna do it wrong. You're not gonna get a second chance. So they can practice on virtual patients before they do the real thing, with the help of calculus and computer modeling. So that's an example, but now what does it have to do with Archimedes? Well, when I talk about this elasticity model, what is the model? Next slide. It's the kind of thing that is built on triangulation, this old Archimedean idea, except that instead of just triangulating the face, I don't know if you can really see, but they've gone behind the face and they have three dimensional analogs of triangles that are tetrahedra, that are modeling the soft tissues and they have different stiffness's and elastic properties, and so they have this gigantic model of billions of polygons and simplices, tetrahedra and other things, so that they can figure out all the forces and how everything's gonna rearrange itself after you do whatever you're gonna do with the cutting of bones. So this is just to give you a sense of how today, through computers and calculus, we are living Archimedes legacy all over the place. Okay, next. All right, so I don't remember quite when we started. I'm just about ready to wrap up. I'll give you, can I have like five more minutes? I just wanna talk a little bit about what you might think of as calculus, because everything I did so far probably doesn't look like the calculus you had. I understand that. And that's because you know, calculus was going on for thousands of years before algebra came along. The calculus you learn nowadays is all algebra, all formulas. It's a tremendously powerful thing using symbols that came from the East, that came from India and the Islamic world and gradually made its way into Europe at the beginning of the Middle Ages. So like 1200, 1300, you start getting algebra in Europe. And algebra then collides with geometry, and that's when differential calculus is soon born in the middle of you know, like say the 1600s or so. But so 1800 years after Archimedes, algebra and geometry collide, but before Newton and Leibniz, you have people like Fermat and Descartes creating the subject of analytic geometry. This is his fusion of taking a curve. Like here I'm showing a parabola just like Archimedes was studying, but whereas Archimedes thinks of it as a section of a cone, now to Descartes and Fermat, it's an equation. Like y equals x squared. The relation between these symbols and this very familiar picture that we think of Cartesian coordinates with x and y axis, all the classical curves of the Greeks can now be thought of as equations. And so you have this great, you can play two ways with them. You can visualize 'em or you can work on them as equations. So analytic geometry is a great breakthrough that then sets the stage for calculus in that. Now there's all kinds of new curves you can make. You can write down an equation, there's a new curve. It doesn't have to be section of a cone. You can do anything. You can create a whole jungle of new curves and start asking questions about them. What's the area under the curve? Or we saw how Archimedes used a tangent line in his constructions. What's a tangent line gonna be like, you know, now that we're doing it with algebra? So that became a big question for Descartes and Fermat. And I just wanted to give you intuition about how to think about tangent lines. Imagine a microscope and I'm gonna zoom in on this point. So let's see the next slide. If I zoom in, you don't see what looks like a parabola anymore, but I'm now just sort of, this is the part in my microscope field of view. The curve has gotten straighter. Sort of like what we saw when we were doing that Archimedean pizza proof, right? And the bottom got flatter and flatter. This curve is gonna start getting straighter. Next slide, if we zoom in more, it really looks very straight. I mean, you can see from the numbers, .5, .505. We have zoomed in a lot. So under great magnification, a curve starts to look like something made of straight pieces, and this is the great idea behind differential calculus, that you can approximate curvy things with straight things. Or what in the jargon we would say local linearization. But it just means that you can think of a curve as made of lots of straight pieces. But they have to be infinitesimal and um, this is again, the infinity principle. To deal a problem of a tangent line to a curve, defined by any equation, we can solve it by zooming in enough and then figure out slopes of straight lines. So I wanna end with something that is more modern. Still, you know, Fermat and Descartes, curves were just curves. They still wanted to do geometry. But nowadays we think of curves as meaningful about the world. Curves tell us about stock prices going up and down, they show your blood pressure, your heart rate. You know, everything can be graphed. And so for us, curves are now visual representations of certain kinds of data. So let's see the next slide. They can represent in particular, things that are moving and changing. The dynamic world that calculus describes began with the study of things that could be represented as curves. So just to give you a taste of what that's like. Let me look at the next slide. I wanna just remind you of something. So, don't start this video yet. This is a video taken in 2008. It was at the Olympics when Usain Bolt was running the 100 meters. So, not that long ago, only 11 years ago. Usain Bolt is a sprinter from Jamaica who doesn't look like a sprinter. He's six foot five, very tall, very gangly. He was also very mischievous. A joker, he loves to joke around. And he never really ran the 100 meters. That was not his race. He was known as a 200 meter, actually he used to run the 400 meters, a longer race and he was great at that because he wasn't so great at starting, but once he got going he was very fast. And but he sort of wanted to try the 100 meters and his coaches said you're never gonna be any good. You're not a, you don't look like a 100 meter sprinter. Those guys are short and muscular, and he said I could, you know, I could get muscular. So they started pumping him up, he got more muscular. He had five races in the 100 meters competitively at this time that he was entering the Olympics. So think of him as a beginner. Okay, he's only done it five times in competition. But he was already able to contend with the best in the world. And so, I guess there's nothing to do but just. Oh, are we gonna have audio? I guess we'll find out. Let's see if this will work. Because it's sort of fun to hear the announcer. Um, let's try. Go ahead. No, I guess not. Well anyway, you can see the runners are getting themselves ready and watch what happens. (muffled announcer talking) Now did you see what he did at the end? Where he put his arms down? That is not normal. He's coasting. He's so far ahead he's coasting, that he's, in fact he's slapping his chest. (audience laughing) I mean, you saw there was this whole other group of runners, the world's best runners with clear daylight between him and them and he's fooling around going like that, sorry. He's banging his own chest. Excuse me. So and that's like I say, he was not being disrespectful, he's just mischievous, he's celebrating a little bit. He's so far ahead. So there was a question, let's see the next slide. I mean that shows you at the end of the race. (audience laughing) And, (chuckling). You won't be able to see it from where you are, but if you look very, very closely, I can tell you that that's his shoe lace. He has an untied shoe lace. (audience laughing) So he's way, way ahead and the question among sports aficionados was if he had run hard, what time, he set a new world record of I think his was 9.69, how fast could he have run if he had really run all the way to the end? So that's a calculus question that we could answer. So let's see the next slide. This is, the way that they record performances like this is that there're detectors every 10 meters. And so we know where, at what time he crossed 10 meters, 20 meters, 30 meters down the track. So we have his split times, those are the dots. We don't have his position in between the dots, but with calculus we can fit a curve in an optimal way to dots and you know, what's the smooth curve that has zero velocity. 'Cause we know when he began he wasn't running, he's in the block. You have to start with zero velocity. Zero velocity would mean this curve has to have a flat tangent. It has to, the slope tells you how fast he's running. When it gets steeper, he's moving faster. So you can see at the beginning he's not moving that fast, but then he gets faster and faster. Now what we could do with this having fit the curve through there, we can then calculate the slope at each point using what I just said. You zoom in as if with a microscope and then record the slopes at every point, and if you do that you get the next slide which shows his velocity as a function of time in meters per second. You see he speeds up and then somewhere after like about eight seconds he's very visibly slowing down, which we saw by just watching him as he's goofing around at the end. So you can answer questions like what was his maximum speed? You can just read that off the graph. When did he achieve his maximum speed and so on. But I haven't actually done the calculation of how fast could he have gone if he kept running hard, because actually we don't need to do it. The next, a few months later there was the world championships in Berlin and then he didn't fool around. So I don't have the video of that, although it's on YouTube, you could watch it if you want. But people won't, thought that something might happen in Berlin, so they went with radar guns like the type that police would use to you know, for catching a speeding car. So they put little reflectors on the back of every runner, they're aiming their lasers at the runners, and these things could measure data, you know, like 100 times a second or something like that. And so the next slide shows his speed as a function of time, as detected by the radar gun at Berlin, and you'll notice some interesting thing. His instantaneous speed is that wiggly curve, and the average that's sort of going through the wiggles is shown dotted. Do you have any idea why there are all those wiggles? Oh, I'll take another young, you're so fast, you should be on Jeopardy. You wanna tell, what do you think the wiggles are? - [Student] The time it takes-- - The what? - [Student] Because the time it takes for his feet to leave the ground. - Time for his feet to leave the ground. Hello. (applauding) Yeah, you're seeing his individual strides. You're actually seeing the fact that when, when his, I mean what is running? Running is a series of leapings and landings, right? And when he lands he's slowing down a little bit, and when he's leaping he's going a little faster. So that is all being resolved in that wiggly curve and you'll, I think there should be 41 'cause it's known that he takes 41 steps every time he runs the 100. Everybody else does 44, but he's tall. So he always does 41 steps, 41 strides and what I find interesting about this picture is that we don't care about the wiggles. Having measured them, we're not interested in it, I mean they're sort of interesting. But if we really, I mean what we really want is the trend, and I think that there's a kinda metaphor here that, if you start getting too precise, sometimes you're picking up information you're not that interested in. And so, this has been a recurrent theme in the story of calculus. That like I said, it has creative fantasies in it where it ignores certain things and focuses on others. And it seems to me that this is a case where you know, we could have just drawn a smooth curve through his data and we would have gotten really just as much information as this more microscopic view. So yes it's true, if we start going very microscopic, calculus may break down, but maybe we don't care. Maybe we've really gotten the essence by looking at this bigger picture. So this was said better than I'm saying it now by Picasso. Can I have the next slide? Yeah, "Art is a lie that makes us realize truth." Right, I mean no picture is really realistic, and yet if it's great art, it often captures the essence of the truth. So I would like to suggest that calculus has done the same thing for us in science. It's a kind of a lie that has helped us realize the truth. Thank you. (audience applauding) - Thank you. - No problem. - Thank you so much for a fantastic talk. So I hope you're gonna be open-- - I'm open, I'm here. - To take questions. Okay. - I'm happy to take any talks or criticisms. Compliments, you name it. Uh, okay, yes. - [Student] Uh, I'd love to hear about the connection to the Declaration-- - Declaration, yeah sure. That is a teaser, isn't it? Okay, here's what the story I tell in the book Infinite Powers. But, it's not my idea. This is an idea that goes back to the historian of science I. Bernard Cohen, a great Newton scholar. So, Cohen points out that the founding fathers in the U.S., all you know, these enlightenment thinkers, were very much interested in Newton and Euclid and logical reasoning. So in particular, think about the structure of the preamble in the Declaration, right? There's this famous ringing line, "We hold these truths to be self evident. "That all men are created equal," et cetera. This, it's an interesting phrase. We hold these truths to be self evident. Where does that come from? If you look at Euclid, so Euclid who gave us the first geometry textbook begins with the self evident truths. In math we call them the axioms. All right, the axioms are the things that are supposed to be self evident. You just accept them, of course they're true. And then starting with the self evident truths and commonly agreed upon rules of reasoning, you erect this edifice of propositions and theorems, building on the axioms. Now if you look at the rhetorical structure of the Declaration, it has a Euclidean structure. Because now why, that is it starts we hold these truths to be self evident, then the axioms are listed, the right to you know, pursuit of happiness and all that, and then there's an argument and then there's an inescapable conclusion, a theorem, which is essentially that the colonies have the right to separate themselves from the tyrant king. So, why would Jefferson have written this Euclidean document? I call it Euclidean, why? I mean 'cause rhetorically it goes back to Euclid. But if you look at Newton in The Principia, so Euclid is writing about geometry, but Newton is using this geometrical style of argumentation to write about the world, because the system of the world where he begins with the, he begins with axioms, which are his laws of motion, and then using logic derives how comets move, how the planets move, the tides. I mean, in other words, if you wanna make an argument that no one can argue with, to the thinkers of the enlightenment, the best way you could do it was to use a geometrical style self evident truths first, theorems later. And I'm not making this up. I mean, you can see it Spinoza. So in Spinoza's ethics, the actual title of Spinoza's ethics book is Ethics Demonstrated in Geometrical Order. He tries to derive the rules of ethics with geometrical reasoning. That's considered irrefutable. So, I really think it's no accident, and like I say, but I. Bernard Cohen had the idea first, that the Declaration is this quintessential enlightenment document, and just to top it off, we know that Jefferson had a Newton, Euclid fetish. So, for instance, he had a death mask of Newton. That's the kind of thing people used to do, right? Someone is dead, you make a big wax or I don't know what, you make some kind of impression of their face and then it's like whoa, cool, I have Newton's face. So, Jefferson had one of those. And he also wrote to John Adams back when they were no longer presidents are were just kinda shooting the breeze in 1812, I think it was. He's writing to John Adams and he says, you know, like I'm really happy to be done with politics and he says I've given up newspaper, I've given up reading newspapers for Herodotus and Thucydides, Newton and Euclid, and I find myself much the happier. So, so that's the argument. But those, we also see Jefferson using Newtonian principles like he designs a plow that will cut the soil in the optimal way. He uses a branch of math called calculus of variations. What's the correct shape of a plow to have minimum resistance as it's going through the dirt and to put the, you know 'cause remember he's interested in agriculture. To plow as much dirt as fast as you can, as easily as you can. He doesn't know calculus of variations, Jefferson, but he knows that that's what he needs to know, and he gets the help of a professor at Penn to help him design this optimal plow, which you can still see down in Monticello, is that how you pronounce it? Cello, yeah. What, have you been there? - Monticello. - Monticello, have you been? - In Charlottesville? - Yeah, Jefferson's old place, yeah. Yeah, yeah, it's there. Anyway, yes. So, you have a question too or comment, no? - Oh yes, yes, is this on? Hi. So you talked a little bit about how calculus slash geometry and algebra were developed separately, and I've noticed that I guess, quantum mechanics comes to mind, but you can look at it in either kind of a differential calculus way or a linear algebra way, with the calculus being more intuitive, but computationally more difficult, and the algebra being less intuitive but computationally easier, and I was wondering if you could kind of speak to the relationship between the two sides. - I think I missed one word early in what you said. Are you asking for the relationship between algebra and geometry and the two ways of thinking? Was that the idea? - Yeah, yeah. I think that's about it. - Oh, okay. I mean I feel like I missed a key word. But, that's, okay-- - Yeah, that's what it is. - That's the gist of the question? - Yeah. - Well yeah, I mean they sort of represent two styles of thinking, right? That algebra has this virtue of being very systematic, you can, as I said earlier you could massage equations, manipulate symbols. You in a way don't have to think. Like you don't have to have insight to do algebra. You just have to do the bookkeeping correctly, and that's one of the great strengths. Actually, Leibniz when he's doing calculus, his version of calculus says and actually he's the one who gives us the world calculus. He says you know, what the virtue of my calculus, at first he calls it my calculus, my way of thinking, my style of reasoning. Newton doesn't refer to it as calculus by the way. He calls it fluxions. He's interested in the world in flux. So but anyway, so Leibniz says something like the virtue of my calculus is that it frees the mind, frees the imagination. You don't have to spend any effort thinking, you're just, it's like knitting. You know, you just move the symbols around, it's all good. Whereas geometry, although it's extremely intuitive and visual, requires tremendous ingenuity and you know, this is partly why it takes 2,000 years after Archimedes for anyone to really advance calculus much beyond what he did, because he was such a towering genius that, and there were no tools for everyone else, you know? I mean if you couldn't reason like Archimedes geometrically or invent, I'm exaggerating because there were things that were done after him, but progress was very slow when it was just geometry. So algebra really did speed things up. But I mean to me, algebra is quite sterile. I find algebra is not about anything. Whereas geometry is about something, it's just very hard. And so when you put geometry and algebra together, now you're really cooking and that's what happened in the 1600s. One thing I feel a little bad about and I wanna, I can't really properly correct it, but I just wanna show you that I'm woke and I mean, I really am, I'm aware of this, is that women are a part of the story and so are people in India and China and Japan and in the Mayan civilization. I mean there's a lot of calculus being done around the world. I mean, with women, honestly, it's only sort of around the 1800s that women start to be allowed to even go to university and hear lectures and stuff. But as soon as they're able, you get great people. Sophie Germain and Sofia Kovalevskaya, and certainly in the 20th century we've got lots of tremendous women mathematicians all over the world. So, although I've told it a certain kinda dead white male European way, I don't mean it that way. (chuckling) And it's not, that is not an accurate picture of what really happened. So I do try to tell it the right way in the book. Including Katherine Johnson, you know who we all know now from Hidden Figures, you know who helped put astronauts on the moon and bring them back safely using calculus of course. Yes? - [Student] Um, just very quickly. Thank you so much and also if the internet had been around when Wouk was looking, he would have found out that in 1910, Silvanus Thompson wrote Calculus Made Easy. - He did know, he looked at that book. - Oh he did? - He absolutely looked at that book. Yeah, yeah, that's one of the books he looked at. It's a famous book, Calculus Made Easy. That's the title. No, he looked and found it. I'm sorry if I meant it, it sounded like I said he looked for it, but didn't find it. No, he did find it. He read Calculus Made Easy by Thompson and many people say that that is still the best book to learn calculus from. So, you may, if you're interested you may wanna look at that book. It is quite funny and quite skillful pedagogically, but it's sort of an old book. It's 100 years old and I think it's you know, it probably has some deficiencies, but for whatever reason it didn't really work for Herman Wouk. - All right so. - Uh, yes? Oh, wait, okay. - Question. - Who's next? Oh there you are, hey. - So I have two related fast questions. First, thank you for the talk. - Sure. - I hope even like high school students or first year college students get exposed to calculus this way. And my first take as the first idea you were talking about that nature speaks calculus. - Yeah. - [Student] What, how would you think if we reframed the saying that calculus is our only tool to measure only what we can measure and only what's quantifiable, because that sounded to me that it eliminates so many things in life that are not quantifiable and they're very abstract and they insist in nature. - Uh huh. - And relatively, which is very fast and related, my second question is you seem to embrace modernism to a great extend in the talk, which is understandable because we're talking about calculus. But if you can have a brief answer to that, like what's your take on post-modernist thought that question the legitity and the privileged tone of modernism? - Uh huh, okay let's see if I can handle those. So um, the first question is interesting. Let's see if I can rephrase it. So you're saying that calculus does well at measuring, at quantifying the things we can measure. - To some extent. - To some extent, but the things we can't measure or whatever, doesn't have anything to say about those. And I think I probably agree with you. Do I? I'm not sure, maybe. I mean because the state of not being able to measure something is often a temporary state, right? Sometimes something can't be measured now, but can be measured later and my claim would be that when it can be measured, but maybe your claim too, would be that once we can measure it, calculus will turn out to be just the right thing for it. Would you agree with that? - [Student] Not really because-- - Not really. - [Student] Yes, 'cause I believe there are effective measurements of some are just approximate. Like because they are just promises of what we then test only what we decipher-- - Uh huh. - [Student] It just is the little speck of the world. It's part we don't logically do-- - So yeah, so there are things that we can't measure. Do they effect us in anyway, these other things? - Of course. - Okay. - [Student] You can't measure religion technically-- - Right, so things like that. Uh huh. Yeah, right. So calculus of course would be hard pressed to say anything about that. So I think I accept that first point. I mean, just that there's something, I wonder if though, if I'm agreeing too quickly because well... Just that there's some, towards the end of the book I mention one thing that I regard as very spooky to me, that makes think, but I don't think it will refute or even contradict what you're saying because it's still in the realm of what can be measured. I'm just thinking of a case where there are certain properties of subatomic particles, of electrons that have been measured and accurately predicted to eight decimal places with this Feynman's part of physics, quantum electrodynamics. That we can understand something in the gyromagnetic ratio of the electron can be calculated to eight digits after the decimal point. Why am I making a big deal about that? I mean, that's that is so unbelievably accurate, but it wasn't built into the theory. We didn't know it would agree. It does agree to eight digits. Now you're not saying it's some kind of like, equivalent to faith or social construction or are you? I mean I'm saying it can't possibly be because it works so well and we didn't know that it would work that well ahead of time. That to me it's capturing something that I wanna call the truth or close to the truth. I wouldn't say it's the end all be all truth, because truth is always provisional, but it's a very close to the truth, and if you wanna tell me you don't believe there is such a thing as truth, then I think we really do disagree. But I don't hear you saying that. I think you're saying there's some domains where we have sort of, I almost hear you saying the boring domains, the things we can measure. Or you're not saying that, you don't wanna go that far? - [Student] I do computational biology and-- - Okay, okay. You're just raising the issue. - [Student] It's a little perspective to what I'm saying, that just prove it has a quantification. - Yeah, well certainly the things that can't be quantified I don't see what calculus would have to say about them. Because as I say, it's all about this principle of regarding problems. It applies to problems that can be viewed as continuous in some way, blah, blah, blah. So for religion or whatever, I don't think it will have anything to say about that. But then you asked about modernism, and I don't feel well enough equipped to even know what that, I don't know enough about post-modernism to know how to respond to that. Um, okay, maybe I should try someone else's comment or question. Yes? - [Student] Um, okay, so if calculus is the language of God are-- - Let's hold on there. Someone is gonna try to pass a microphone over to you. - [Student] Oh I'm fine just speaking louder. - That's okay, everyone will be happy if you just talk in there. - [Student] Yeah. Um, you mentioned the title of the book that calculus is the language of God, right? - This is what Feynman said. - Okay. - I personally don't believe in God in case you're curious. - Oh, okay. - But okay anyway. - Well then, okay this is-- - But go ahead, you can ask, I can pretend. - And if God, thank you. (audience laughing) If God, if the reason for our feelings is a result of God, you know, everything is permitted through God, then calculus should be able to measure anything. - Oh, let's see. - Or, Feynman-- - You're saying is calculus-- - Or if Feynman didn't know what he was talking about and calculus really isn't the language of God. Or which one is it? - So wait, is this a comment or question? You're saying. - It's more of just like a rhetorical question, just as a thought-- - That if calculus is God's language and since-- - Yeah. - God should be all powerful in most conceptions in God-- - Then anything measurable. - Then of course. - Could be like. - Yeah, well a couple of big if statements there. - Yeah. - Yeah, I don't know. I don't know, I don't, okay so the word God gets used a lot in different ways, right? Einstein talks about God, but he was careful to say not the God who cares about you know, whether you've been naughty or nice. It's the God, the creator of the universe God, who makes the laws of nature, et cetera. Or maybe just the laws themself. Like he always used to say, I'm talking about Spinoza's God. Meaning the God that is nature, not the God, not a personal God who cares about sin and things like that. So that, if we just mean nature. I mean it's still a big mystery. I do feel this is a mystery that's running throughout the story as I tell it. That why can math, I do actually, I didn't really raise this question clearly. Let me try to raise it now. Why does math work? In, yeah okay, let's suppose we restrict ourself to the domain you're talking about. The domain of things that can be measured. It's still a question, why does nature obey logic? In particular, human logic. That's, it's not obvious to me that that should be possible. We're not the best thing imaginable. We're you know, a few generations removed from primates, and we have logic that can get the gyromagnetic ratio of the electron to eight digits. That's weird that our puny logic matches what really goes on in this universe this well. It didn't, I don't know why that worked. Why does it work? This is the question that Eugene Wigner famously asked. You know, of the unreasonable effectiveness of mathematics. And lots of people have wondered about this. Some would say it's because God made the world very orderly. Others say, if the world were very disorderly, we would not be able to evolve to even our puny level of intelligence. Like only universes that support enough order to allow intelligent life, those are the only universes we can observe. So it's sort of, that's the anthropic view of why the universe makes sense. (muffed talking) Yes. - [Priya] Gonna be the very last question. - Oh okay, who? Oh yes, please go ahead. - [Student] Um so I was like, so I've been wondering, like I researched about the black hole picture. Like I wanted to, like I was wondering did calculus contribute to that like on a big level? 'Cause I know that calculus contributes to stuff, but like, does it contribute like greatly to the black hole picture or? - To the black hole picture? Do you mind talking about that? We have like one of the world's experts right here. She actually helped make the black hole picture. I think it's fair to say. Well you tell us. - Yes, yes absolutely. - Yes. - So calculus is really fundamental because as you saw there were curves that needed to be computed to be put together. That's a great question and probably a great place to stop while we go to the reception. Thank you everyone. - Oh, the reception. Yes, please join us at the reception. - Thank you Steve for a fantastic talk. - Yes, thank you. (audience applauding) (light music)

Channel: YaleUniversity

Views: 174,149

Rating: 4.8410597 out of 5

Keywords: calculus, mathematics, Feynman, Strogatz, STEM, beauty, universe, Yale, Franke

Id: 1r6893ga_So

Length: 88min 55sec (5335 seconds)

Published: Wed May 01 2019