Why is math so important for physics?

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

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Episode Transcript

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
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