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