Why is math so important for physics?

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Speaker 1: Sometimes I'll bump into a stranger, maybe on an airplane, and they'll ask me the inevitable question, what do you do for a living? When I say that I'm a physicist, I often get the reaction I hate physics so much math, And that makes me think, if it's the math you didn't like, then hey, hate the math, but you can still love the physics. But of course the two are closely linked. You can't love Shakespeare if you hate the English language, and that of course makes us wonder why math and physics are so intertwined. I mean, if people can actually enjoy Shakespeare and other languages, then it has something about it that's transcending the original English words. Is it possible for physics to transcend math or are they shackled to each other with math woven deeply into the fabric of physics. Hi, I'm Daniel. I'm a particle physicist and a professor at U c Irvine, and I'll admit that I love math. Some people find it confusing, but when I was a kid, I found it to be crisp and logical in a way that the rest of the world was sort of fuzzy and complicated. Like people For example, people are complicated and hard to understand when you're a kid. Are they going to be mean to you or nice to you if you sit next to them at lunch. It's hard to predict from one day to the next. But the rules of math were cast in iron two plus two equals four every day of the week, and if you know the rules, the answer follows. Math is reliable, it's predictable, and that's what led me to physics, the ability to use math to understand and predict the universe. And welcome to the podcast Daniel and Jorge ex Playing the Universe, where we do a deep dive into the rules of the universe, doing our best to reveal what science has uncovered in terms of the machinations of the universe, and laying out for you what science and scientists are still puzzling over. We tear the universe down to its smallest bits and put them back together to explain how things work and expose our remaining ignorance. And we can do that, thinks, in no small part to the power of math. Math underlies all of the stories that we tell ourselves about the universe. If you want to predict the path of a baseball, and you can use math to calculate its trajectory and tell you where it will land. Physics can predict the future, but the language and the machinery of it are all mathematical. There are times when our intuition fails us when the universe does things that don't align with our expectations, like in the case of quantum mechanics, which tries to describe tiny little objects that follow rules that seem alien to us, but they do follow rules, and those rules are mathematical described by equations. So when we've lost our intuition, we can close our intuitive eyes and just follow the math and trust that it will guide us to the right physical answer. On this podcast, we actually usually are trying to do the opposite, to avoid the mathematics, and that's partly because it's an audio program not well suited to equations or geometric sketches, but also because we are trying to feed your intuition about how the universe works, to strip away the opaqueness of math and make it all makes sense to you. And I'll admit that this is sometimes a struggle to accomplish. For some of the deeper concepts and physics like gauge invariants. The way I learned them is mathematical, and the way that I understand them is mathematical, which has become part of my intuition. So it's not always easy to know how to translate those concepts into pure intuition and talk about the without math, but you know, finding ways to talk about them intuitively has also led me to a deeper understanding of the ideas. That's one of the underappreciated joys of teaching. It forces you to strengthen your own knowledge. But to me, it raises a really interesting question. Is it possible actually to divorce physics from math? Is math truly the language of physics or is it just useful like a shorthand notation. Is math the language of the universe itself or is it just the way the humans like to think about it. So on today's episode, we'll be asking the question why is math so important for physics? A few weeks ago we talked to a philosopher of mathematics, Professor Mark cole Evan, about whether the universe was mathematical. Today, I've invited someone from the other side of the issue to join us. We'll talk today to a working mathematicians some and who spends his day's building mathematical tools and using them to describe the patterns and structures of the universe. So it's my pleasure to welcome Professor Steve Strogatz. He's the Jacob gould Sherman Professor of Applied Mathematics at Cornell University, having taught previously at M I T and having earned a PhD in math from Harvard, So those are some pretty impressive credentials. But he's also an expert in applying math to the real world, including understanding the math of firefly swarms, choruses of chirping crickets, and the wobbling of bridges. He's also a well known podcaster, host of the podcasts The Joy of X and The Joy of Why, both of which I highly recommend, as well as a prolific author. One of his recent books is Infinite Powers, How Calculus Reveals the Secrets of the Universe, the book I recently read and thoroughly enjoyed, and which inspired me to invite Steve on the podcast to talk about calculus, infinity, and the deep relationship between physics and math. Steve, Welcome to the podcast and thank you very much for joining us. Thank it's a lot, Daniel. It's a great pleasure to be with you. That's going to be fun. It's a treat to have you here as I've been very much enjoying listening to your podcast series and reading your book. And so I'd like to start by asking you a question I've heard you ask several of your guests about definitions. Your book is about calculus, a word that a lot of people have heard but might not really know what it means. Can you define for us what is calculus? Sure, let's try it in a sequence of definitions, and you could stop me when I get too detailed. So if I were giving it to you in one word, I would say, it's the mathematics of change. That's the keyword change. If we want to go a little more into it, it's the mathematics of continuous change, and especially things that are changing at a changing rate. So you say it's the mathematics of change. What exactly is changing there? Like if I just want to describe how a ball is moving through the air, what exactly is changing about the ball's motion? So in that case, what's changing is the position of the ball, or also possibly the speed of the ball. So your listeners will remember from high schoo Well algebra, we do problems about change and motion. That gets summed up in the mantra. Distance equals rate times time, and so that's motion at a steady speed or at a steady velocity. And you can handle that with algebra. It's just a matter of multiplication. Distance equals rate times time. The rate is the speed, and you're driving sixty miles an hour. For an hour, you're gonna go sixty miles. Okay, So in that case the distance is changing, position of the car on the highway is changing, but the speed of the car is not changing. We said it was a steady sixty miles an hour. And so at the time of Isaac Newton or even Johannes Kepler or Galileo, scientists started to become very interested in motion that was not just simple motion at a constant speed. You know, in connection with the things you mentioned dropping a ball, the apocryphal or maybe true dropping the cannon ball off the leaning tower of pizza from Galileo, certainly Kepler with thinking about the motions of the planets. In all of those cases there were things that were changing. I mean we should also keep in mind with the planets. Another thing that can change is direction. So instead of motion in a straight line, if you have an orbit, then the direction of the planet as it's moving is changing. It's curving as it's going around the Sun. And so geometry is a big part of calculus too, when we start to deal with curved shapes as opposed to shapes made of straight lines or or planes. So in this case, do you feel historically like physics was in the lead or mathematics. I mean, people have been thinking about things that were moving and changing for thousands of years. But calculus is just a few hundred years old. Was it invented to solve a particularly difficult problem or did it appear in the minds of intelligent people and then allow us to solve problems that had been standing for thousands of years. That's interesting that you say it's only a few hundred years old. Most historians and certainly most scientists would say, yeah, calculus is from the middle sixteen hundreds, from Isaac Newton and Gottfried Wilhelm Leibnitz. But I don't personally I want to endorse that position because I think we can see if you want. I mean, I don't equibble about definitions, but there are definitely ideas of calculus. Almost two thousand years earlier in the work of Archimedes in Syracuse, in what was at the time the Greek Empire Um to fifty BC. Archimedes is calculating volumes of solids with curved faces, or also areas under a parabola. Or he's the one that gives us the volume of a sphere or the surface area of a sphere. Those are all calculus problems. We teach those today in calculus when we're teaching students about integrals, which is a generalization of the idea of area and volume. And so he's totally doing calculus in two fifty BC. In fact, he's doing the harder part of calculus, integral calculus. But we don't usually call it calculus because of I don't know why. Actually, I mean, I think it is calculus. So back to your question, what is calculus, I mean, another way of talking about it is it's the systematic use of infinity and infinitesimals to solve problems about curved shapes, about motion at a non constant speed, and about anything else that's changing in a non constant way. It could be amount of virus in your bloodstream if you have HIV. It could be a population you know, of the Earth going up. Any all of these things are grist for calculus. So why is it that infinity is such an important and powerful concept that lets us now tackle new problems that we couldn't tackle before. I want to think about it in terms of like the ball flying through the air, and you we're talking about something changing about its motion, but you're also referring to like calculating the volume of spheres. What's changing in that aspect? Why do we need infinity to help us tackle these problems? I mean, the sphere is not infinitely big, the ball is not moving infinitely fast. What exactly does infinity come into play? The main point is probably infinitesimals rather than infinity. So infinitesimals. Let us pretend that us fear is made up of flat pieces. May be easier to visualize with a circle. Some of your listeners will have probably played this game. If you put a bunch of dots on a circle and connect them with straight lines, it almost looks like a you know, it'll make a polygon for instance, you could picture putting four equally space points on a circle and connect them, they'll make a square. If you put eight, then you're making an octagon. You know, the more points you put, the more it starts to look like a circle. And from ancient times people had this intuition that a circle is kind of like an infinite polygon. It's got infinitely many corners, it's connected by sides that are infinitesimally small. Now that doesn't seem right because we think of a circle as perfectly smooth it doesn't have any corners at all. But in a certain sense, it's the limit of a polygon as you take more and more points on the polygon at the corners and more and more sides. And so that was the key insight that Archimedes had, that you could calculate the area of a circle the formula we all learn in high school I R squared. He's the first one to really prove that, and he did it by thinking in this calculus way, by looking at the limit of polygons. So similarly, in the case of Galileo, in the motion of say javelin or something thrown that's going to execute parabolic flight to make the problem easier. Gal Well, actually Galileo didn't really have this idea, but later in Newton we would think of the parabola as made up of infinitely many, infinitesimally small excursions along the path that are basically straight lines in the particle or the javelin is moving at a constant speed for that infinitesimal amount of time, So it breaks the problem down into something that we already know how to solve. Everything becomes distance equals right times time again, except only over an infinitesimal segment. So you solve a problem you can't solve by turning it into an infinite number of problems that you can solve. Bingo, you've really encapsulated the heart of calculus in that sense in infinite powers. I called that the infinity principle that to solve any difficult problem in evolving curve shapes or these complicated motions, if you reconceptualize it as an infinite number of smaller, simpler problems in which you have straight lines or motion at a constant speed, you can solve incredibly hard and important problems with this trick. The only problem is you have to somehow put all those infinite testimals back together again to reconstitute the original motion or the original shape. And that's the hard part of calculus. The subdivision part is easy. It's the reassembly part that's hard, right, And so it's fascinating to me that infinity sort of appears in two places there, one as you chop it into little pieces, and then again as you put it back together. And to me, it's fascinating because the infinity appears in only the intermediate stages. Like the ball doesn't have infinite velocity or infinite acceleration or infinite anything. But we've used infinity in calculating. It's very non infinite motion. And so it's fascinating to me that infinity is such a powerful mathematical tool, yet we don't actually observe it in nature really very often. Or some people might say, ever, that's really a great point you could. I mean, if we were doing this on video, your listeners would see me smiling. I really like that. It's almost like in those old cartoons with the you know, enter stage left and exit stage right, that infinity comes onto the stage and the infinitesimals, but only as you sort of say, like an apparatus. It it lets us solve the problem, but it's sort of not really there, it's not real. In physics, we're often seeing infinities in the final answer as a sign of failure writing particle physics, prediction of infinity is unphysical. You can't have infinite probabilities for some outcome and quantum mechanics, or you can't have an infinite force on a particle. And in general relativity we think of a prediction of a singularity infinite density as the breakdown of the physical theory. We try to avoid infinities. We hide them under renormalization whatever possible. So then my question to you is in your mind are these infinities real? I mean, do they just exist in the intermediate steps of the mathematical methods we're using. Are they only in our minds? Are they half finished calculation? Or is this something real about the universe that calculus is capturing, This smooth and infinitely varying motion of a ball or changing of the velocity of a planet. Is infinity real? Is it part of our minds? Such a great deep question. I don't even know what I'm gonna say to the answer. I mean, I've been thinking about this for forty fifty years, and I still don't really know what the answer is. I I mean, the principal person in me, the philosophically tenable person, wants to say it's not real, it's a fiction, it's a useful device. Let's just try to make that argument first, before the more wild eyed person in me makes the counter argument. So the rational person would say, yeah, I mean, our from our best understanding of physics today, there are no infinitesimals. You can't subdivide matter arbitrarily finely. That's the whole concept of atoms, those things which are indivisible. You guys tell us, I say you. The physicists tell us that there is even a smallest amount of time and space that is referred to as the Planck scale, the smallest. You know, we don't really, of course, I believe understand how to unify quantum theory yet with general relativity. There are candidates. But anyway, the cool thing is that, just on dimensional grounds, if you look at the fundamental constants like the speed of light and planks constant that governs quantum phenomena, and Newton's gravitational constant for the strength of gravity, those can only be put together in one way to make a unit of length. That's the Planck length, and it's about ten to the minus thirty five meters, and that's sort of the smallest conceivable distance that has any physical meaning, wouldn't you say, whatever the theory ends up being, that's certainly an attempt to describe what might be the shortest distance. In my view, it's a not very clever attempt, but also the most clever attempt we have, and we have no better way to do it, and so this is the only thing. You know, we do this all the time in physics. We say, let's start with the most night idea and then try to build on it, and we're sort of still there with the shortest distance, you know. I mean, if you try to estimate, like how many candy bars a person eats in a year, just by combining various quantities with the right units, you might get an answer that's off by a factor a thousand, and that would feel like a pretty wrong answer in the case of the plant length, I think, which is sort of groping generally for where in the space that answer might be. But I think the point you're making is we have the sense that the universe is discreete and not continuous. Equantom mechanics tells us that you can't infinitely chop up the universe, and therefore mathematics of calculus that assumes that might not actually be describing what's happening in the universe. Fine, I mean, I take your point that we don't know that the plant length of tend to the minus thirty five is right. There could be factors of a thousand in one direction or another or more. I mean, we don't really know what the pre factor is, So okay, I accept that. Nevertheless, as you say, also from quantum theory, we have reason to think that nature is fundamentally discreet in every aspect, whether it's matter or space or time. And so if that turns out to be correct, that will mean that real numbers are not real. Real numbers are the things that we use in calculus all day long. There are numbers that have infinitely many digits after the decimal point, like pie. Right, people know, you can keep calculating and you'll never know all the digits of pie because there's infinitely many of them. Is that real? Like? In fact, you could ask that question about all of math, our circles real, you'd have to say no. Circles are not real either, because as you zoom in on them, you know what's there. It's all jiggily and there's fluctuations of the sub atomic particles. So there's no material circle in the real world. But nevertheless, going back to Plato or others, we can think about perfection in our minds. We can think about the concept of a perfect circle, and we can think about the concept of pie, and even the concept of infinity. And this is the uncanny part. These things are not real from the standpoint of physics, yet they give us our best understanding of the physical universe that we've achieved in as a species. And that's just a fact. I mean, that's just a historical fact. That calculus based on this fiction of infinitely subdivisible quantities works pretty darn well. While I was walking my dog this morning, I tried to figure out how many orders of magnitude if you tell us the universe is about ten to the twenty five meters big the visible universe, that's the estimate I looked according to When I asked Syria on my iPhone, she said, the visible universe tend to the twenty five ms, and the typical scale of a hydrogen atom is something like ten to the minus ten meters, So you've got thirty five orders of magnitude very well described by calculus, all the way from the Schroedinger equation at the lowest scale to general relativity at the highest scale, all built on calculus. So it's kind of capturing the truth. Okay, you couldn't. No, it's gonna start screwing up at the scale of quantum gravity, whatever that ends up being, we think. But I think that's a pretty good good notch in the belt of calculus that it works over such a vast range of scales. Absolutely, it's incredible because it powers not just you know, quantum field theory, which is full of integrals, but also general relativity and talking about you know, galaxies and black holes. All Right, I have a lot more questions about math and physics for our guest, but first let's take a quick break. Okay, we're back and we're talking with Professor Steve stroke Gats about why math is so important for physics. And it makes me think about the connection in physics and math of the concept of emergence. You know, some simple behavior at the scale of like me and you are a bowl of soup that neatly and compactly summarizes the almost infinite details going on underneath. I mean, even if you don't believe that a ball is infinitely divisible into bits, we know this is a huge number of bits with a huge number of details. But it's almost like the equations and the simplicity of calculus, the parabolic motion of that ball emerge somehow from all of these infinite testimals doing their bit together to tell a fairly simple story. And in physics, to me, it's something of a mystery, like why this happens? Why can we describe the universe in simple mathematical stories when we know that the details are crazy and gory in intense and calculus really wraps this up. So do you have an instinct or an intuition, or even a guest for why it's possible to wrangle these almost infinities into fairly simple stories that make sense to our human minds. That's another tough one. I don't mean to put you on this spot. Well, they're all great questions, they're so deep. My first instinct to that one after I don't know is Is it a question of the measuring apparatus meaning us that we happened to be macroscopic and so for us space has perceived as smooth, and time as smooth and so on. But if we were playing scale creatures, we wouldn't. Of course, we don't know what's going on down there, but under our current understanding, you wouldn't have the concept of space time as smooth. So, as you say it emerges, it appears smooth only at our scale, or well down to even atomic scales, but it's still twenty five orders of magnitude bigger than the Planck scale to go to the hydrogen atoms diameter. So yeah, it might just be where you know, all those jitters, their quantum gravitational jitters are invisible, they get smoothed out, or they start to look smooth, spacetime emerges. Right. That's the latest talk that spacetime as a as a manifold in the jargon of differential geometry as this smooth structure. That's another fiction. That's an emergent property of something about quantum fluctuations, maybe having to do with entanglement. Anyway, I don't know about that stuff. You probably have other guests who could address that better. But yeah, so we're probably studying the emergent theory that just the way thermodynamics works well, even though statistical mechanics is the deeper theory. Calculus and all of smooth classical physics, and even the smooth parts of quantum physics, say the Schrodinger equation or the direct equation, those things are emergent. But I guess your question was why does emergence work so well? That's something about a different branch of math, that's about statistics, about laws of large numbers. And it's a very fortunate accident. I mean, maybe it's not an accident. Maybe we couldn't exist as intelligent creatures except at that scale. If we were these hypothetical quantum gravitational scale creatures at the Planck scale, we'd be so jiggili it would be hard to keep a thought in our heads. You know what I mean. I'm being simply, We wouldn't have heads. Podcast episodes would be tend of the minus thirty four seconds long. So I guess I'm giving an anthropic principle style argument here, aren't I. But it's hard to answer these deep questions. It is hard and to meet. It really goes to the heart of these questions about whether we are describing the universe as it is or just our view of it, and whether our view of it is somehow human centric in a way that we can't unravel and can't peeled back because we only have our human view, and the appearance of calculus and like short simple stories to me like are an interesting clue to grab onto. So let me steer us back the other direction. Because we've sort of described calculus as a useful fiction. We've said it's a handy tool for doing calculations, but as you say, it comes on the stage and disappears before the answer is revealed. And yet it is really really powerful, right, Calculus and math and general is sometimes described as like being unreasonably effective in your book. I really like this line you wrote. You said, what fascinates me as an applied mathematician is the push and pull between the real world around us and the ideal world in our heads. Phenomena out there guide the mathematical questions we ask. Conversely, the math we imagine sometimes foreshadows what actually happens out there in reality. When it does, the effect is uncanny and later you wrote it's eerie that calculus can mimic nature, So well, can you elaborate on that a little bit? Why do you think math is so good at describing the universe if it's just sort of a fiction in our minds? And why do we then describe it as unreasonable or uncanny or eerie when that happens, why are we surprised by that? Well? Should we stipulate that we we believe in all this? I mean, is it worth going into any case studies of the eerie effectiveness? Or do you think we should just assume that we know it? Please give us some examples. Well, okay, yeah, let's talk about a few. Because I did feel myself recoiling a bit. I felt like you were almost verging toward a kind of circular reasoning claim that we as human beings can only think a certain way or perceive certain things, and so it all kind of comes out tidy because of our own limitations, like we're convincing ourselves. I don't think you were saying that exactly, but if if some people heard it like that, I would have to push back on that. Because of the concept of prediction, we use our math, we use our all of our scientific laws and observations to make predictions of things we haven't seen before or haven't measured. And there's no circular aspect to what we predict. Either nature does what we predict or it doesn't. And there's been plenty of cases of you know, flagists on theories and kinds of other things that turned out to be wrong. So science is done in good faith. We make predictions and they sometimes come out wrong. And yeah, I mean there are old people who will hang onto the theory after they should have given it up. But it's a self correcting enterprise, it really is, I think over the long run. So I don't think there's any circularity happening here. And you know, for me, the eerie examples are things like, you know, take Maxwell, James Clerk, Maxwell, who has these empirical laws from people like amp Here and Faraday and uh, I don't know, Bo Savar, all these laws that we learned about in electricity and magnetism courses for what happens with magnets with electric currents and circuits and stuff. So these laws then could be rewritten in a certain mathematical language, and Maxwell did that using the language at the time which was called quaternions, but nowadays we would use vectors vector calculus, and he saw certain things in those laws that looked a little contradictory to him. That led him to introduce a new concept, the displacement current. And when he put that in the known laws and started cranking the math, medical crank, just manipulating the equations. Now in the world of pure idealization, in the world of calculus, he saw that those equations predicted something, which is that electric fields and magnetic fields could move through empty space. Although for him it was the ether, but nowadays we would say empty space in this kind of dance with the electric field changing and generating a magnetic field that would change and regenerate the electric field, and the whole thing would propagate at a certain speed, governed by an equation that in calculus we call the wave equation. So he's predicting electromagnetic waves those were not known. That's a prediction, and his math gives him a prediction for the speed of those waves, and when he calculates it using the known physics, it comes out to be the speed of light. So it's one of the biggest a haa moments in the history of humanity. That light is an electromagnetic wave, and Maxwell's the first to realize that, and it turns out it's right. You know, years later his predictions get checked out in the lab Hurts measures there really are electromagnetic waves, and pretty soon after that, Marconi and Tesla are building telegraphs and we've got wireless communication across the ocean and all this stuff is real. But it was born out of calculus combined with physics. Let's be clear. It's not calculus on its own. It's calculus supplemented, not supplement I mean, calculus is more like the supporting player. The real stars are Michael Faraday, An amp Here and the rest. But their laws of nature have these logical implications that lead to predictions that turn out to be right. And so what's uncanny there is that nature is obeying logic that's not necessary, right, This is puny primate logic. This is us. We're not the best imaginable thing under the sun, but our logic somehow is enough to make these predictions. And you know, there's countless examples of this. So but maybe that Maxwell one makes the point. I'm wondering if you know anymore about the history of it, because I've heard this story about Maxwell's AHA moment, and I wonder historically was their own moment. It's such an incredible realization. I'd like to imagine that he was sitting there by lantern light at his desk and it all clicked together and he had this epiphany where like he saw the universe in a way nobody had ever seen him before. Do you know if there was such a moment for sort of a gradual coming together? And I don't know. I want to know that too, And it's funny. I have the same fantasy image of the lantern, the little hovel in Scotland, So I don't know. I think it is known. I think this is another point that history of science is such a rich and detailed and often non logical thing. Like we we you and I are telling ourselves this story a certain way, and I just told it a certain way, and I don't really know what I'm talking about. For instance, I read somewhere fairly recently that he knew that light was going to turn out to be an electromagnetic wave before his math showed him that just on dimensional grounds. You earlier, we're talking about the Planck scale, and you know the cancelation of units and stuff or not cancelation, but you can get arguments based on dimensional just by monkeying around with units. I think he knew that me not an epsilon, not these properties of the vacuum having to do with its magnetic and elect trical properties, that they could be combined in a certain way to make a speed. And I think he did that calculation about a decade before he actually derived the wave equation. Wow, it would be delicious to understand the history of that a little bit better. But I love the argument you're making here that essentially the math guided the physics. That he saw something that wasn't symmetric, that looked imbalanced mathematically, and he patched it up just because of his mathematical intuition, and the physics sort of followed suit. That that was a better description of the universe because mathematically it hung together more crisply than the previous ideas. That the math really did guide us to truth about the universe. Is that the core of the argument. Yeah, And the part that the spooky is, Look who's behind it. It's it's this creature that has evolved on this planet in an ordinary galaxy, you know. I mean, it's not like we have godlike intelligence. The thing that so so spooky is we're so bounded in our understanding. We can understand so much through the help of this crazy fictional thing that involves infinity. It's almost like we're in the sweet spot for pleasure in doing science and math. If we were much smarter than we are, we wouldn't be surprised. Everything would be trivial, Like playing tic tac toe is not interesting for someone who understands it, and so grown ups don't play tic tac toe for fun because it's boring. And if we were just a bit smarter, physics and math might be boring in the same way. But it's fun for us because we're in this place where we're not as stupid as a lobster. I mean, a lobster is not inventing calculus. We're at this happy, resonant place where we're smart enough to get it but sort of stupid enough to be surprised all the time. It's amazing. It's a really fun game, but it's also teaching us things about the universe, which is incredible. As I was hearing you talk about that Scottish mathematician. I was reminded of another Scotsman more than a century later, Peter Higgs, who made sort of a similar realization. He was looking at the mathematics of not just electromagnetism, but electromagnetism and the weak force, how they clicked together, and realizing there was a missing piece and predicting the existence of a field we now call the Higgs field. So you know, maybe it's something in the water in Scotland. Well, it's another great example because it took a long time for that prediction to be checked in the lab and tremendous effort and cost from great teams of physicists and engineers at large Hadron Collider is that right, Yeah, detected? So I didn't have to be there. And some things aren't there, right, like supersymmetry. Is this other beautiful set of ideas that so far has not turned out to be in the experimental data. Maybe in the future. But I'm just saying that this is a very honest enterprise in science. It's not circular reasoning. It's not like we're convincing ourselves. We're really doing fair play. And the universe either does what we imagine or not, and frequently and uncannily it does if we use calculus and and feed in. I mean, that's the other thing. Like you'll hear people say calculus is a language or math is the language of science. Partly true, but it's much more than that. Math and calculus in particular are a calculating machine. They're a logical prosthesis. I mean, there's something which lets us take our logic again puny primate logic, and strengthen it by introducing symbols and letting us do logical manipulations, like you know, solving equations. That kind of thing is a big extension to what we can hold in our heads. That's why we have paper and pencil. You can make these arguments much more elaborate than you could have easily held. Like think of algebra. Before we had symbols and it was all verbal. It was a much weaker thing. So now we just shove these symbols around on paper according to certain rules, and out we get predictions for electromagnetic waves or the existence of the Higgs particle. I find that very uncanny. I just get I don't know how else to say it to me. It's the spooky, est and most profound thing there is that this works. And I want to emphasize in case people are saying this is just math. Why do I harp on calculus so much? I really do think calculus has a singular place in the landscape of math, in that the laws of nature are written in a subdialect of math. It's calculus. And even there it's the particular part of calculus we call differential equations. So from F equals M A and Newton to Einstein's general relativity to Schroedinger's you know, wave equation, those are all differential equations. So it's not like we're using combinatorics or some other part of discrete math that is not the language of the universe. Sorry, maybe it is at the smallest scale, Okay, maybe it will turn out the combinatorics is the answer to the Planck scale stuff, and calculus is just this emergent, smoothed out version of what's really going on, which is combinatorics if we get down to the bottom. But for the thirty five orders of magnitude that we've done science on so far, it's calculus. Baby, Well, I had in my own aha moment as a junior in quantum physics, seeing the prediction of properties of the electron muan how to tend decimal places, and then seeing the experiments which verify those predictions digit after digit after digit, and feeling for the first time that maybe math wasn't just a description of what was happening out there in our language, but it really was the essential underlying machinery of the universe itself, that the universe was using these laws, that we weren't describing them but revealing them somehow. And I know that's, you know, it's a philosophical position. But I had this moment as an undergrad of feeling this, and I thought of that moment when I read this passage in your book. You wrote, quote, the results are there waiting for us. They have been inherent in the figures all along. We are not inventing them like Bob Dylan or Tony Morrison. We are not creating music or novels that never existed before. We are discovering facts that already exist. And as I was reading your book, I was wondering, you know, Steve a realist or is he not a realist? And I sort of went back and forth a few times since I read these passages. Oh really, was I not clear where I stand on that? Well, that's earlier what I was saying with these two people. I guess I didn't make the argument for both sides that there's the chicken hearted person in me who is the one that thinks it's just a language and it's just you know. But in my heart, I think it's what you're calling realism, which is that the universe isn't just described by calculus. The universe actually runs on calculus. I really do, in my heart of hearts, think that, and I don't know why that would be true. I think the answer could be, again, some kind of anthropic argument that that a universe that doesn't run on math in some way is such a disorganized, tiggle dy piggledy universe that it can't support life intelligent enough to ask questions. So I sort of think just the fact that we exist and we're here pondering it tells you the universe has to obey a certain amount of orderliness, and calculus is going to come up in such universes. So it's not the most convincing argument. I don't like that argument, but that's the best I can do. I mean, obviously you could give a theological argument that God knew calculus better than anybody and chose to make a universe that runs on calculus. Okay, if that satisfies you, then that's you could use that argument. But to me, that just raises a lot more questions. But I don't have the answers to why was it designed this way or built this way, or why did it evolved to be this way. I don't have any idea, But yeah, that's interesting. I mean that ten digit example you give from quantum field theory, from quantum electrodynamics, that's really the poster child for the claim that the universe is running on math, and that we happen to have stumbled across that math. That's also fun to think about that. Just think of the story. There's Archimedes in Syracuse pondering circles and spheres to fifty b C and he's stumbling across the math that turns out to describe sub atomic particles like nuance. Ultimately, a few thousand years later, it's that same math and he wasn't thinking about that. It's really spooky that that should work, but it did. It is pretty amazing. All right, I'm really excited about these topics. But let's take another quick break. Okay, we are here talking with Professor Steve's drogats about why math and physics are so closely intertwined. In your episode of The Joy of Y, you interviewed Kevin Buzzard, and mathematician, and he described math as a single player puzzle game. And I was actually expecting you to object a little bit because it makes it sound like math is just this game we play. It's fun to use to describe the universe, but not actually fundamentally important. It makes it sound like checkers or chess, you know, just a game that we invented rather than something physical and true. Uh, well, that might be me As a podcast host. I probably ought to push back more, and maybe it would make for a lively or discussion. I try to be even handed, unfair to the guest. And there is an aspect of math that is game playing, especially in pure math, and that's not to be sneezed at. Just playing games for the intellectual pleasure of playing games and the fun and the curiosity of how does the game turn out or what happens if I change the rules in this way or that way. That's all part of the scientific enterprise as well as the mathematical enterprise, and it's a healthy one for one thing, I mean, if you want to be utilitarian about it. A lot of great discoveries in science have come from playing games like that. You know. You could think about all those centuries that we thought Euclidean geometry was the one true geometry, and then people started playing games and ask, well, what if we don't have the parallel postulate. What if we allow you know, infinitely many parallel lines to a given line, or what if we say there are no parallel lines to a given line, you know, through a specific point. Well, then you invent hyperbolic geometry and curved spherical or elliptic geometry. Those are games for a few hundred years until it turns out the universe uses them. In Einstein's work, so you could say, let the people play the games, because it's going to turn out that the universe is going to use them and they'll be very practical, you know. Or similarly, games about prime numbers have led to the way that we can do encryption on the internet for all of our financial transactions or for keeping secrets. So game playing is not to be sneezed out on utilitarian grounds. It turns out it's often very practical and useful maybe a few centuries later, but I I wouldn't want to just make the utilitarian argument. It's also part of the human spirit. Just be curious for the sake of curiosity, and it may never turn out to be useful, and that's okay. That's what makes it good to be alive for me. One way to make this question less philosophical and more concrete is to think about aliens. I know it makes people snicker to talk about aliens, but instead of asking, you know, is math universal or is it cultural, which is a question philosophers have been chewing on for millennia without making that much progress. I wonder like, if technological scientific aliens arrive, it's a question we're actually going to have to face whether they do math. We asked Noam Chomsky about it on the podcast recently, you know, how do you get started talking to aliens in that scenario? And he went with the math. He said we should start with arithmetic because one plus one equals to everywhere, and he was suggesting that any intelligent being in the universe is going to end up being mathematical, which is essentially making the argument that math is not just human right, that it's part of the universe itself. So, I don't know if you've given this question any thought. What do you think if aliens arrive? Are you volunteering to be one of the envoys? Should we send our mathematicians to talk to the aliens? I've seen too many Twilight Zone episodes. I know how this turns out, but I've seen the ending of that one. Well, it's probably the best suggestion there is to that that math would be the most universal possible language in this scenario. I'm not totally convinced that they would know about one plus one, because you could make up stories about intelligent life based on plasma or fluid dynamics where they don't have discrete particles, so they don't really have one plus one. Maybe it's all continuum for them and they would rather talk about calculus rather than I'm hammering again on the discreete stuff, but no, I mean, basically, if the point is that we would could communicate through math better than any other way, yeah, maybe, so I tend to think they would have to have some version of math, or they couldn't have built their rocket ships or teleportation and devices, or however they got here. I think they have to have math. I do think the math is inherent in the universe. I like the quote you gave earlier. There's a psychological dimension to this that I want to bring up, which is that there's what philosophers talk about, and I like philosophy, but there's also what working physicists and mathematicians feel when they're doing math or making discoveries. It really feels like the results are out there waiting for you. I mean, maybe it's a fiction, maybe it's a psychological self deception, but it's very profound and it goes way back. Archimedes says it two thousand years ago. He says that the things he discovers about the sphere are not his inventions, they're inherent in the figures themselves. So he expresses very clearly a philosophy of math, which I find kind of heartwarming because it makes me feel like I'm having a conversation with this person thousands of years ago, and that he's feeling some of the same things I'm feeling as a mathematician today. And also that he's very humble that he doesn't know how to solve certain problems, and he just says he hopes his methods will help future generations solve the things that he cannot figure out. I think it is an important lesson there. I mean, we can talk about math is inherent to the universe, but also there's a human aspect to it. I mean, we really appreciate the beauty of math the way we appreciate the beauty of a gorgeous view from the top of a mountain. Some of my favorite bits in your book are when you write very elegantly about your appreciation for understanding something and seeing things come together, and making these connections with ancient mathematicians and knowing that you have this joy in common with them. I mean, I think a lot of people often see math portrayed is like cold and crisp and rigorous, But in your book you write about the creativity necessary to play this game. You said, quote rigor comes second. Math is creative. Why do you think that is that we find beauty in math? Is it the same reason we find beauty in nature? Is it necessary that we would have found math to be beautiful? Is it possible we could have evolved and all found math to be like a horrible chore, even if it is useful. Well, this sensation of beauty is not universal. There are people who don't have much patience for the kind of talk who are still good mathematicians. There are a lot of reasons to love math. Some people do love the beauty of it. Some, you know, like the human struggle. Some like the social aspect that you get to do it with your friends and think about it together and you can surprise each other. Some people like the competitive aspect. I'm smarter than the other person because I figured out. Beauty is one side. I think there's a tendency to go on a little too much about beauty, especially because it can be very exclusionary. People who aren't seeing math as beautiful are even more excluded when they don't get what's beautiful about it. You know that it can be um a kind of cudgel or a gate keeping a bit of language. So I know that when we harp on about beauty or trying to make the subject appealing to people and say, hey, it's just like music. You like music, you should like math maybe, but you have to be very sensitive to helping a person appreciate the beauty. I'm reminded of like opera, where I don't get opera. When I hear opera, it sounds like a lot of hysterical carrying on, and I just think, you know, get over yourself. But I see other people weeping from it, and they understand it, so it's beautiful to them. It's very profound and emotional, and I feel like I'm missing something but I'm not getting it, so you know, I actually there is this one commercial for wine. I think it was Ernest and Julio Galla where they're seeing somebody singing Omeo Bambino Carro and it's so beautiful even I got it, okay, But other than that, I mostly don't get opera. But anyway, my point here, silly point is um, you know, as educators or as communicators, like through your podcast or the one that I try to do, or when I write books, I want to be careful about this beauty argument. There there are a lot of ways into our subjects. Like I'm trying to press every possible button, so I might hit somebody's button at a given time. Well, let's talk a little bit about the button of creativity. Some folks feel like math and science and physics are a different kind of intellectual venture than things like music or art. But there's a creative side to science where and to math and to tellect. We do sometimes feel like you're playing a game. You wrote in your book. Mathematicians don't come up with proofs. First comes intuition, and rigor comes later. Can you talk a little bit about the element of creativity that's involved in your work specifically? Oh, well, before I say anything about my work specifically, I do appreciate you bringing up that point, because in math, especially in high school geometry, we're taught the proof has to be rigorous, it has to follow logic. You're sometimes teachers will even have students right out statements in the left column and reasons in the right column. There's a point to that to help young students learn how to get organized in their thinking and construct logical arguments. And so that is definitely a part of math. Mathematicians are very proud of being able to have absolute proof in a way that scientists cannot. Right, sciences get revised as more information comes in, But in math, the theorems that were proven thousands of years ago are still true, and the proofs if they were correct back then, they're still correct. Some of us like this absolute nature of the subject, but that's only half of the story. And how do you come up with the proof in the first place, or how do you dream up what theorem you're even trying to prove? Those things are more akin to music and poetry and art and other creative parts of of human activity. I mean, you have to have imagination, and you have to dream, and you have to have wishful hopes. All that kind of stuff is a big part of math, and anyone who does math or physics or any other part of science knows all of that. I mean, when you're doing it, you're still a person. You still have dreams and hopes. So I don't know why we don't teach that more. I mean you learn it when you're in as an apprentice, as a young scientist or mathematician, and you're in the lab. You feel it. You all want something, But we don't do a great job in our textbooks or are lecturing and conveying that. And I think that's why a lot of people, you know, they might think it's a cold subject that wouldn't hold any appeal to them, But once they get in the lab or actually do some math, they'll see it's just like anything else, that it's really fun and occupies your whole human spirit. I want to talk a little bit about accessibility of math. You tell a story in your book about a novelist who received the advis city he wanted to write about physics. He needed to understand calculus, but as a non technical person, he was unable to find his way in, even going so far as to audit a high school class. And you say that your book is for people like him who want to understand the ideas and the beauty of math but can't otherwise find their way in. Do you think there's a wide bread appetite for this? Do you think if we taught math differently, it might have more supporters and you might less often find people on airplanes who go I hated math in high school. Yes, unequivocal yes to that question. There is a hunger for it. I know it as a fact. I've done the experiment. The New York Times back in opinion page of all things, not the science page, but the opinion page. Editor David Shipley asked me to write a series of columns about math, starting with preschool math about numbers and going as far as I could up to grad school level topics for his readers, for a curious person who like the kind of person who would read the opinion page, but who like him, fell off the math train somewhere, you know, just didn't see the point of it, didn't like it anymore, or found it hard or repelling in some way, repulsive anyway. So I tried to write for that audience. And there was a big audience, and they liked it, and they were very grateful and appreciative, and I their comments, you know, because on the Internet people can talk back, and they did. And of course there were some people who talked back saying they had a better explanation or they think I got it wrong. But for the most part, there was a big audience that was very grateful and said things like, you know, I wish you were my high school teacher. I wish math was taught this way. Why wasn't it taught this way? And that's a good question. Why isn't it taught in a way that engages people more? You know, it's complicated about the story of education in the United States. There's a lot of demands on teachers to get their students to learn certain things that the government requires or wants people to learn by a certain age. There's all the pressure of getting into college. I mean, there's a million things. Also. You think of the position of teachers in our society. How much reverence is or is not accorded to the profession of teaching at the elementary or high school level, So what people are attracted to it, how much teachers are paid. I mean, there's a million things we could talk about that we don't have time to talk about, but for all kinds of reasons, we aren't teaching math in the optimal way. Well, I certainly appreciate your efforts to translate some of these d ideas and the historical stories of mathematics. Even as a physicist who thinks about math all day long, I certainly gain and benefit from your efforts, and I think a very wide group of people do as well. And I want to ask you a personal question about why you're a little bit unusual. I mean, there are people who write for the public about science and math, but you're also somebody who's doing that. You're actively researching, you're publishing papers. You're an academic and you're participating in these studies yourself. What is that like for you, sort of living in both worlds. Is your academic intellectual professorial community supportive of this or do you have to sort of push back against trends that encourage you not to spend your time doing this kind of outreach. That's a happy story. Actually, the community is pretty supportive, I would say, And I'd be curious your own take on this too, because you you must be encountering it. A lot of us fear that if we go into public communication of physics or math, that some of our colleagues would think we're getting soft, or we're selling out, or we're pandering or dumbing it down or whatever. And that seems to be mostly a misplaced fear. If colleagues do feel that way, they've been polite enough to not tell me so that I appreciate that. But but mostly people seem to take it in in a good spirit, like, you know, thanks for trying to do this. It's difficult and it's worth trying to do, and the public certainly seems to appreciate it. But no, I haven't found much resistance or even antagonism from colleagues about it. It's also very much fun for me as a perpetual student. I learn a lot from interviewing guests on the podcast in fields I don't know anything about. I talked to people about inflammation or the origin of life or whatever on this joy of Y podcast. So I'm constantly in school. You know, for anyone out there who's had this feeling like, now that I know so much more, I wish I could be a student at this age. I was so busy. I was so young and had all those hormones raging, and I had so many things on my mind. Now that I'm old and I can think straight anyway, I'm just saying that it's fun for me as a student to be able to do this, and I think it actually helps my research too. It's giving me a broader perspective. I'm thinking about questions that never occurred to me before. So no, I think it's all to the good. I had the same feeling. I really appreciate the license to explore topics I wouldn't otherwise feel like I had time to dig into and to educate myself about them to a level where I feel comfortable explaining them in intuitive terms. It's a lot of fun. I really feel like it's broadened my understand ending. But let me ask you one more, maybe even more pointed question. What would be your advice to a young person whose career isn't as well established as yours but is excited about outreach, you know, maybe a postdoc or a graduate student. Would you recommend that they not participate in that and focus on their academics until they're better established, or is this something you think we should be encouraging in young people as well. That's a hard one because realistically, I don't think it will really help a person's chance in the academic life at a young stage. It's not the answer I want to give, but I think it is the honest answer. That the culture of the academic world for a person who wants to become a professor is such that you have to focus on research, depending what kind of place you want to work at. So if you're working at a place that considers itself a research powerhouse or aspires to be one, then you've got to focus on your research and there wouldn't be much benefit to doing outreach work. Honestly, I mean the priorities are first research, second teaching, third service, of which outreachesnsidered one of aspect of service. So yeah, don't do it for that reason. Now that's not to say you shouldn't do it. There are people who decide why should I be a professor? I can make money supporting myself on YouTube um, and there are fantastic streamers on YouTube. I mean, think of Grant Anderson on three Blue One Brown, who's producing some of the best math explanations on the planet through his wizardly use of of computer graphics and his brilliant pedagogy. I mean, that guy would be the best teacher at any university where he was a professor. But he's chosen not to be a professor, at least not yet. And he's reaching millions or tens of millions of people. So I'm not sure someone with those aspirations needs to be an academic. You know, there is an ecosystem only in recent years where you can actually thrive and do really good work for humanity, as he and a bunch of other people are doing so. I guess I would say for a person who wants to do that, if you're going to do it in the academic setting, get tenure first, do your research, and then you know, go wild. But if you're doing it outside of the academic world, you could make money creating companies that do it. You may have to get lucky, like say con Academy teaching math and science to the world. But what a great service Salman Khan has provided too. So there's there's a lot of possibilities today. All right, great, Well, thanks very much for coming on the podcast and talking with us about an incredible breath of topics, from the beauty of math to communicating with aliens to advice for young researchers. Really appreciate your frank and open conversation. Well, thank you, Daniel. This is a really great pleasure for me and I'm very grateful to you for having me on the show. So you see that the question of why math is so important for physics is a difficult one to answer, even for a physicist and a mathematician talking about it for almost an hour. Hope you enjoyed that conversation. Tune in next time. Thanks for listening, and remember that Daniel and Jorge explained The Universe is a reduction of I Heart Radio or More podcast from my heart Radio, visit the i heart Radio app, Apple podcasts, or wherever you listen to your favorite shows. H