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