November 15, 2022 • 56 mins

Daniel talks to mathematician Steve Strogatz about why calculus seems to describe the Universe so well.

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Speaker 1 (00:08):

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

(00:29):

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

(00:51):

shackled to each other with math woven deeply into the

fabric of physics. Hi, I'm Daniel. I'm a particle physicist

(01:14):

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

(01:34):

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

(01:55):

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

(02:17):

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

(02:39):

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

(03:03):

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

(03:25):

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

(03:46):

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.

(04:08):

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.

(04:29):

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,

(04:50):

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

(05:11):

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

(05:31):

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

(05:53):

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.

(06:14):

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,

(06:36):

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

(06:56):

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.

(07:17):

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,

(07:38):

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

(08:00):

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

(08:20):

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

(08:43):

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

(09:05):

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.

(09:28):

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,

(09:52):

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

(10:13):

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,

(10:34):

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

(10:58):

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

(11:20):

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

(11:40):

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

(12:01):

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

(12:21):

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

(12:42):

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

(13:03):

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

(13:23):

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

(13:44):

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

(14:07):

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

(14:27):

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

(14:47):

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

(15:10):

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

(15:32):

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

(15:52):

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

(16:12):

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

(16:35):

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

(16:55):

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

(17:16):

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

(17:38):

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

(18:01):

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

(18:22):

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

(18:44):

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

(19:04):

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

(19:27):

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,

(19:47):

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.

(20:09):

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

(20:30):

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

(20:51):

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

(21:14):

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.

(21:37):

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,

(22:00):

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

(22:21):

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

(22:42):

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

(23:03):

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,

(23:26):

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

(23:48):

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

(24:08):

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

(24:30):

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,

(24:51):

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

(25:13):

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

(25:33):

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

(25:56):

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

(26:19):

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

(26:42):

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,

(27:05):

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

(27:26):

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.

(27:46):

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

(28:07):

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

(28:28):

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

(28:49):

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

(29:10):

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

(29:30):

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

(29:51):

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

(30:12):

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

(30:34):

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

(30:56):

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.

(31:17):

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.

(31:40):

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

(32:02):

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

(32:26):

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

(32:47):

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.

(33:08):

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.

(33:30):

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

(33:50):

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

(34:12):

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,

(34:33):

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

(34:54):

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

(35:15):

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

(35:35):

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

(35:56):

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,

(36:17):

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

(36:39):

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

(37:01):

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

(37:24):

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.

(37:45):

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

(38:08):

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

(38:32):

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

(38:53):

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

(39:14):

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.

(39:34):

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

(39:55):

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

(40:18):

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

(40:40):

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,

(41:01):

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

(41:23):

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

(41:44):

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

(42:06):

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

(42:27):

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

(42:47):

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

(43:09):

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

(43:32):

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

(43:55):

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

(44:17):

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

(44:38):

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

(45:00):

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,

(45:22):

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

(45:43):

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.

(46:06):

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

(46:27):

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,

(46:50):

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

(47:14):

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

(47:35):

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

(47:55):

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

(48:16):

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

(48:37):

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

(49:00):

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

(49:20):

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.

(49:42):

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

(50:04):

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

(50:26):

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

(50:48):

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

(51:11):

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

(51:35):

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

(51:58):

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

(52:19):

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

(52:42):

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

(53:04):

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

(53:26):

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

(53:47):

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

(54:08):

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

(54:30):

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

(54:52):

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

(55:15):

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.

(55:35):

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

(55:58):

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

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

(00:29):

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

(00:51):

shackled to each other with math woven deeply into the

fabric of physics. Hi, I'm Daniel. I'm a particle physicist

(01:14):

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

(01:34):

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

(01:55):

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

(02:17):

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

(02:39):

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

(03:03):

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

(03:25):

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

(03:46):

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.

(04:08):

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.

(04:29):

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,

(04:50):

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

(05:11):

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

(05:31):

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

(05:53):

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.

(06:14):

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,

(06:36):

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

(06:56):

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.

(07:17):

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,

(07:38):

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

(08:00):

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

(08:20):

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

(08:43):

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

(09:05):

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.

(09:28):

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,

(09:52):

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

(10:13):

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,

(10:34):

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

(10:58):

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

(11:20):

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

(11:40):

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

(12:01):

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

(12:21):

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

(12:42):

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

(13:03):

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

(13:23):

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

(13:44):

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

(14:07):

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

(14:27):

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

(14:47):

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

(15:10):

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

(15:32):

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

(15:52):

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

(16:12):

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

(16:35):

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

(16:55):

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

(17:16):

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

(17:38):

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

(18:01):

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

(18:22):

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

(18:44):

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

(19:04):

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

(19:27):

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,

(19:47):

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.

(20:09):

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

(20:30):

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

(20:51):

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

(21:14):

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.

(21:37):

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,

(22:00):

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

(22:21):

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

(22:42):

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

(23:03):

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,

(23:26):

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

(23:48):

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

(24:08):

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

(24:30):

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,

(24:51):

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

(25:13):

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

(25:33):

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

(25:56):

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

(26:19):

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

(26:42):

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,

(27:05):

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

(27:26):

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.

(27:46):

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

(28:07):

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

(28:28):

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

(28:49):

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

(29:10):

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

(29:30):

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

(29:51):

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

(30:12):

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

(30:34):

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

(30:56):

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.

(31:17):

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.

(31:40):

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

(32:02):

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

(32:26):

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

(32:47):

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.

(33:08):

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.

(33:30):

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

(33:50):

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

(34:12):

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,

(34:33):

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

(34:54):

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

(35:15):

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

(35:35):

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

(35:56):

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,

(36:17):

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

(36:39):

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

(37:01):

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

(37:24):

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.

(37:45):

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

(38:08):

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

(38:32):

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

(38:53):

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

(39:14):

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.

(39:34):

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

(39:55):

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

(40:18):

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

(40:40):

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,

(41:01):

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

(41:23):

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

(41:44):

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

(42:06):

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

(42:27):

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

(42:47):

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

(43:09):

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

(43:32):

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

(43:55):

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

(44:17):

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

(44:38):

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

(45:00):

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,

(45:22):

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

(45:43):

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.

(46:06):

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

(46:27):

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,

(46:50):

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

(47:14):

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

(47:35):

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

(47:55):

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

(48:16):

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

(48:37):

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

(49:00):

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

(49:20):

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.

(49:42):

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

(50:04):

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

(50:26):

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

(50:48):

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

(51:11):

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

(51:35):

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

(51:58):

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

(52:19):

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

(52:42):

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

(53:04):

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

(53:26):

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

(53:47):

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

(54:08):

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

(54:30):

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

(54:52):

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

(55:15):

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.

(55:35):

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

(55:58):

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

Daniel Whiteson

Jorge Cham

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