All language subtitles for The Great Math Mystery PBS NOVA 720p hdtv x264w4f EN Sub_Subtitles01.ENG

y mission.

NARRATOR:
We live in an age
of astonishing advances.

MAN:
Descending at about .75 meters
per second.

NARRATOR:
Engineers can land
a car-size rover on Mars.

MAN:
Touchdown confirmed.

(cheering)

NARRATOR:
Physicists probe the essence
of all matter,

while we communicate wirelessly
on a vast worldwide network.

But underlying
all of these modern wonders

is something deep
and mysteriously powerful.

It's been called
the language of the universe,

and perhaps it's civilization's
greatest achievement.

Its name?

Mathematics.

But where does math come from?

And why in science
does it work so well?

MARIO LIVIO:
Albert Einstein wondered,

"How is it possible
that mathematics

does so well in explaining
the universe as we see it?"

NARRATOR:
Is mathematics even human?

There doesn't really
seem to be an upper limit

to the numerical abilities
of animals.

NARRATOR:
And is it the key to the cosmos?

MAX TEGMARK:
Our physical world

doesn't just have
some mathematical properties,

but it has only
mathematical properties.

NARRATOR:
"The Great Math Mystery,"
next on NOVA!

NARRATOR:
Human beings have always
is provlooked at natureowing:

and searched for patterns.

Eons ago, we gazed at the stars

and discovered patterns
we call constellations,

even coming to believe
they might control our destiny.

We've watched the days
turn to night and back to day,

and seasons
as they come and go,

and called that pattern "time."

We see symmetrical patterns
in the human body

and the tiger's stripes

and build those patterns
into what we create,

from art to our cities.

But what do patterns tell us?

Why should the spiral shape
of the nautilus shell

be so similar
to the spiral of a galaxy?

Or the spiral found in
a sliced open head of cabbage?

When scientists
seek to understand

the patterns of our world,

they often turn
to a powerful tool: mathematics.

They quantify their observations

and use mathematical techniques
to examine them,

hoping to discover
the underlying causes

of nature's rhythms
and regularities.

And it's worked,
revealing the secrets

behind the elliptical orbits
of the planets

to the electromagnetic waves
that connect our cell phones.

Mathematics has even
guided the way,

leading us right down

to the sub-atomic
building blocks of matter.

Which raises the question:
why does it work at all?

Is there an inherent
mathematical nature to reality?

Or is mathematics
all in our heads?

Mario Livio
is an astrophysicist

who wrestles
with these questions.

He's fascinated by the deep
and often mysterious connection

between mathematics
and the world.

MARIO LIVIO:
If you look at nature,
there are numbers all around us.

You know,
look at flowers, for example.

So there are many flowers

that have three petals
like this, or five like this.

Some of them may have 34 or 55.

These numbers occur very often.

NARRATOR:
These may sound like
random numbers,

but they're all part of what is
known as the Fibonacci sequence,

a series of numbers developed
by a 13th century mathematician.

You start with the numbers
one and one,

and from that point on,

you keep adding up
the last two numbers.

So one plus one is two,

now one plus two is three,

two plus three is five,

three plus five is eight,
and you keep going like this.

NARRATOR:
Today, hundreds of years later,

this seemingly arbitrary
progression of numbers

fascinates many,
who see in it clues

to everything from human beauty
to the stock market.

While most of those claims
remain unproven,

it is curious how evolution
seems to favor these numbers.

And as it turns out,

this sequence appears
quite frequently in nature.

NARRATOR:
Fibonacci numbers show up
in petal counts,

especially of daisies,
but that's just a start.

CHRISTOPHE GOLE:
Statistically,
the Fibonacci numbers

do appear a lot in botany.

For instance, if you look
at theottom of a pine cone,

you will see often spirals
in their scales.

You end up
counting those spirals,

you'll usually find
a Fibonacci number,

and then you will count
the spirals

going in the other direction

and you will find
an adjacent Fibonacci number.

NARRATOR:
The same is true of the seeds
on a sunflower head--

two sets of spirals.

And if you count the spirals
in each direction,

both are Fibonacci numbers.

While there are some theories

explaining
the Fibonacci-botany connection,

it still raises
some intriguing questions.

So do plants know math?

The short answer
to that is "No."

They don't need to know math.

In a very simple, geometric way,
they set up a little machine

that creates the Fibonacci
sequence in many cases.

NARRATOR:
The mysterious connections

between the physical world
and mathematics run deep.

We all know the number pi
from geometry--

the ratio between
the circumference of a circle

and its diameter--
and that its decimal digits

go on forever
without a repeating pattern.

As of 2013,

it had been calculated out
to 12.1 trillion digits.

But somehow,
pi is a whole lot more.

Pi appears in a whole host
of other phenomena

which have,
at least on the face of it,

nothing to do
with circles or anything.

In particular, it appears in
probability theory quite a bit.

Suppose I take this needle.

So the length of the needle

is equal to the distance
between two lines

on this piece of paper.

And suppose I drop this needle
now on the paper.

NARRATOR:
Sometimes when you drop
the needle, it will cut a line,

and sometimes it drops
between the lines.

It turns out the probability

that the needle lands
so it cuts a line

is exactly two over pi,
or about...

...64%.

Now, what that means is that,
in principle,

I could drop this needle
millions of times.

I could count the times
when it crosses a line

and when it doesn't
cross a line,

and I could actually
even calculate pi

even though
there are no circles here,

no diameters of a circle,
nothing like that.

It's really amazing.

NARRATOR:
Since pi relates a round object,
a circle,

with a straight one,
its diameter,

it can show up
in the strangest of places.

Some see it in the meandering
path of rivers.

A river's actual length

as it winds its way
from its source to its mouth

compared to the direct distance
on average seems to be about pi.

Models for just about anything
involving waves

will have pi in them,
like those for light and sound.

Pi tells us which colors
should appear in a rainbow,

and how middle C should sound
on a piano.

Pi shows up in apples,

in the way cells grow
into spherical shapes,

or in the brightness
of a supernova.

One writer has suggested

it's like seeing pi
on a series of mountain peaks,

poking out
of a fog-shrouded valley.

We know there's a way
they're all connected,

but it's not always obvious how.

Pi is but one example

of a vast interconnected web
of mathematics

that seems to reveal

an often hidden and deep order
to our world.

Physicist Max Tegmark from MIT
thinks he knows why.

He sees similarities
between our world

and that of a computer game.

MAX TEGMARK:
If I were a character
in a computer game

that were so advanced
that I were actually conscious

and I started exploring
my video game world,

it would actually feel to me
like it was made

of real solid objects
made of physical stuff.

♪ ♪

Yet, if I started studying, as
the curious physicist that I am,

the properties of this stuff,

the equations
by which things move

and the equations that
give stuff its properties,

I would discover eventually

that all these properties
were mathematical:

the mathematical properties

that the programmer had
actually put into the software

that describes everything.

NARRATOR:
The laws of physics in a game--

like how an object floats,
bounces, or crashes--

are only mathematical rules
created by a programmer.

Ultimately, the entire
"universe" of a computer game

is just numbers and equations.

That's exactly what I perceive
in this reality, too,

as a physicist,

that the closer I look at things
that seem non-mathematical,

like my arm here
and my hand,

the more mathematical
it turns out to be.

Could it be that our world
also then

is really just as mathematical
as the computer game reality?

NARRATOR:
To Max, the software world
of a game isn't that different

from the physical world
we live in.

He thinks that mathematics works
so well to describe reality

because ultimately,
mathematics is all that it is.

There's nothing else.

Many of my physics colleagues

will say that mathematics
describes our physical reality

at least in some
approximate sense.

I go further and argue that it
actually is our physical reality

because I'm arguing that
our physical world

doesn't just have some
mathematical properties,

but it has only
mathematical properties.

NARRATOR:
Our physical reality is a bit
like a digital photograph,

according to Max.

The photo looks like the pond,

but as we move in
closer and closer,

we can see it is really
a field of pixels,

each represented
by three numbers

that specify the amount of red,
green and blue.

While the universe is vast
in its size and complexity,

requiring an unbelievably large
collection of numbers

to describe it,

Max sees its underlying
mathematical structure

as surprisingly simple.

It's just 32 numbers--

constants, like the masses
of elementary particles--

along with a handful
of mathematical equations,

the fundamental laws of physics.

And it all fits on a wall,

though there are still
some questions.

But even though we don't know

what exactly
is going to go here,

I am really confident that
what will go here

will be mathematical equations.

That everything
is ultimately mathematical.

NARRATOR:
Max Tegmark's Matrix-like view

that mathematics
doesn't just describe reality

but is its essence
may sound radical,

but it has deep roots
in history...

going back to ancient Greece,

to the time of the philosopher
and mystic Pythagoras.

Stories say
he explored the affinity

between mathematics and music,

a relationship that resonates
to this day

in the work
of Esperanza Spalding,

an acclaimed jazz musician
who's studied music theory

and sees its parallel
in mathematics.

SPALDING:
I love the experience of math.

The part that I enjoy
about math

I get to experience
through music, too.

At the beginning,

you're studying
all the little equations,

but you get to have this
very visceral relationship

with the product
of those equations,

which is sound and music
and harmony and dissonance

and all that good stuff.

So I'm much better at music
than at math,

but I love math with a passion.

They're both just as much work.

They're both, you have to study
your... off.

Your head off,
study your head off.

(laughs)

NARRATOR:
The Ancient Greeks
found three relationships

between notes
especially pleasing.

Now we call them an octave,
a fifth, and a fourth.

An octave is easy to remember

because it's the first two notes
of "Somewhere Over the Rainbow."

♪ La, la. ♪

That's an octave-- "somewhere."

(plays notes)

A fifth sounds like this:

♪ La, la. ♪

Or the first two notes of
"Twinkle, Twinkle, Little Star."

(plays notes)

And a fourth sounds like:

♪ La, la ♪

(plays notes)

You can think of it
as the first two notes

of "Here Comes the Bride."

(plays notes)

NARRATOR:
In the sixth century BCE,

the Greek philosopher Pythagoras
is said to have discovered

that those beautiful
musical relationships

were also beautiful
mathematical relationships

by measuring the lengths
of the vibrating strings.

In an octave, the string lengths
create a ratio of two to one.

(plays notes)

In a fifth,
the ratio is three to two.

(plays notes)

And in a fourth,
it is four to three.

(plays notes)

Seeing a common pattern
throughout sound,

that could be
a big eye opener of saying,

"Well, if this exists in sound,

"and if it's true universally
through all sounds,

"this ratio could exist
universally everywhere, right?

And doesn't it?"

(playing a tune)

NARRATOR:
Pythagoreans worshipped the idea
of numbers.

The fact that simple ratios
produced harmonious sounds

was proof of a hidden order
in the natural world.

And that order
was made of numbers,

a profound insight that
mathematicians and scientists

continue to explore to this day.

In fact, there are plenty
of other physical phenomena

that follow simple ratios,
from the two-to-one ratio

of hydrogen atoms
to oxygen atoms in water

to the number of times the Moon
orbits the Earth

compared to its own rotation:
one to one.

Or that Mercury rotates
exactly three times

when it orbits the Sun twice,
a three-to-two ratio.

In Ancient Greece,
Pythagoras and his followers

had a profound effect on another
Greek philosopher, Plato,

whose ideas also resonate
to this day,

especially among mathematicians.

Plato believed that geometry
and mathematics

exist in their own ideal world.

So when we draw a circle
on a piece of paper,

this is not the real circle.

The real circle
is in that world,

and this is just
an approximation

of that real circle,

and the same
with all other shapes.

And Plato liked very much
these five solids,

the platonic solids
we call them today,

and he assigned each one of them
to one of the elements

that formed the world
as he saw it.

NARRATOR:
The stable cube was earth.

The tetrahedron with its pointy
corners was fire.

The mobile-looking octahedron
Plato thought of as air.

And the 20-sided icosahedron
was water.

And finally the dodecahedron,

this was the thing that
signified the cosmos as a whole.

NARRATOR:
So Plato's mathematical forms

were the ideal version
of the world around us,

and they existed
in their own realm.

And however bizarre
that may sound,

that mathematics exists
in its own world,

shaping the world we see,
it's an idea that to this day

many mathematicians
and scientists can relate to--

the sense they have
when they're doing math

that they're just
uncovering something

that's already out there.

I feel quite strongly
that mathematics is discovered

in my work as a mathematician.

It always feels to me
there is a thing out there

and I'm kind of trying
to find it

and understand it and touch it.

JAMES GATES:
As someone who actually
has had the pleasure

of making new mathematics,

it feels like there's something
there before you get to it.

If I have to choose,

I think it's more discovered
than invented

because I think
there's a reality

to what we study in mathematics.

When we do good mathematics,

we're discovering something
about the way our minds work

in interaction with the world.

Well, I know that
because that's what I do.

I come to my office, I sit down
in front of my whiteboard

and I try and understand
that thing that's out there.

And every now and then,
I'm discovering a new bit of it.

That's exactly
what it feels like.

NARRATOR:
To many mathematicians,

it feels like math is discovered
rather than invented.

But is that just a feeling?

Could it be that mathematics

is purely a product
of the human brain?

Meet Shyam,
a bonafide math whiz.

MICHAEL O'BOYLE:
800 on the SAT Math.

That's pretty good.

And you took it
when you were how old?

Eleven.

Eleven.

Wow, that's, like,
a perfect score.

NARRATOR:
Where does Shyam's math genius
come from?

It turns out we can pinpoint it,
and it's all in his head.

Using fMRI, scientists
can scan Shyam's brain

as he answers math questions

to see which parts of the brain
receive more blood,

a sign they are hard at work.

MAN:
All right, Shyam,
we'll start about now.

Okay, buddy?

SHYAM:
Okay.

NARRATOR:
In images of Shyam's brain,

the parietal lobes glow
an especially bright crimson.

He is relying on parietal areas

to determine these
mathematical relationships.

That's characteristic of lots
of math-gifted types.

NARRATOR:
In tests similar to Shyam's,

kids who exhibit
high math performance

have five to six times
more neuron activation

than average kids
in these brain regions.

But is that the result of
teaching and intense practice?

Or are the foundations of math
built into our brains?

Scientists are looking
for the answer here,

at the Duke University
Lemur Center,

a 70-acre sanctuary
in North Carolina,

the largest one for rare and
endangered lemurs in the world.

Like all primates,
lemurs are related to humans

through a common ancestor

that lived as many as
65 million years ago.

Scientists believe lemurs

share many characteristics
with those earliest primates,

making them a window,
though a blurry one,

into our ancient past.

Got a choice here, Teres.

Come on up.

NARRATOR:
Duke Professor Liz Brannon

investigates how well lemurs,
like Teres here,

can compare quantities.

BRANNON:
Many different animals
choose larger food quantities.

So what is Teres doing?

What are all of these
different animals doing

when they compare
two quantities?

Well, clearly he's not using
verbal labels,

he's not using symbols.

We need to figure out whether
they can really use number,

pure number, as a cue.

NARRATOR:
To test how well Teres
can distinguish quantities,

he's been taught
a touch-screen computer game.

The red square starts a round.

If he touches it,
two squares appear

containing different numbers
of objects.

He's been trained

that if he chooses the box
with the fewest number...

(ringing)

...he'll get a reward,
a sugar pellet.

A wrong answer?

(buzzer)

We have to do a lot
to ensure

that they're really attending to
number and not something else.

NARRATOR:
To make sure the test animal
is reacting

to the number of objects
and not some other cue,

Liz varies the objects' size,
color, and shape.

She has conducted
thousands of trials

and shown that lemurs
and rhesus monkeys

can learn to pick
the right answer.

BRANNON:
Teres obviously
doesn't have language

and he doesn't have
any symbols for number.

So is he counting, is he doing
what a human child does

when they recite the numbers
one, two, three?

No.

And yet, what he seems
to be attending to

is the very abstract essence
of what a nuer is.

NARRATOR:
Lemurs and rhesus monkeys
aren't alone

in having this primitive
number sense.

Rats, pigeons, fish, raccoons,

insects, horses, and elephants

all show sensitivity
to quantity.

And so do human infants.

At her lab on the Duke campus,

Liz has tested babies
that were only six months old.

They'll look longer at a screen

with a changing number
of objects,

as long as the change
is obvious enough

to capture their attention.

Liz has also tested
college students,

asking them not to count,

but to respond
as quickly as they could

to a touch-screen test
comparing quantities.

The results?

About the same as lemurs
and rhesus monkeys.

BRANNON:
In fact, there are humans

who aren't as good
as our monkeys,

and others that are far better,

so there's a lot of variability
in human performance,

but in general, it looks
very similar to a monkey.

Substitute in the three,
you raise that to the four...

BRANNON:
Even without any
mathematical education,

even without learning
any number words or symbols,

we would still have,
all of us as humans,

a primitive number sense.

That fundamental ability
to perceive number

seems to be a very important
foundation,

and without it,
it's very questionable

as to whether we could ever
appreciate symbolic mathematics.

NARRATOR:
The building blocks
of mathematics

may be preprogrammed
into our brains,

part of the basic toolkit
for survival,

like our ability to recognize
patterns and shapes

or our sense of time.

From that point of view,
on this foundation,

we've erected one
of the greatest inventions

of human culture:

mathematics.

But the mystery remains.

If it is "all in our heads,"
why has math been so effective?

Through science, technology,
and engineering,

it's transformed the planet,

even allowing us to go
into the beyond.

As in the work here, at NASA's
Jet Propulsion Laboratory

in Pasadena, California.

MAN:
Roger, copy mission.

Coming up on entry.

NARRATOR:
In 2012, they landed
a car-size rover...

MAN:
Descending at about .75 meters
per second as expected.

NARRATOR:
...on Mars.

MAN:
Touchdown confirmed,
we're safe on Mars.

(cheering)

NARRATOR:
Adam Steltzner
was the lead engineer

on the team that designed
the landing system.

Their work depended
on a groundbreaking discovery

from the Renaissance

that turned mathematics
into the language of science:

the law of falling bodies.

The ancient Greek philosopher
Aristotle

taught that heavier objects
fall faster than lighter ones--

an idea that, on the surface,
makes sense.

Even this surface:
the Mars yard,

where they test the rovers
at JPL.

ADAM STELTZNER:
So Aristotle reasoned

that the rate at which things
would fall

was proportional
to their weight.

Which seems reasonable.

NARRATOR:
In fact, so reasonable,

the view held
for nearly 2,000 years,

until challenged
in the late 1500s

by Italian mathematician
Galileo Galilei.

STELTZNER:
Legend has it that Galileo

dropped two different weight
cannonballs

from the Leaning Tower of Pisa.

Well, we're not in Pisa,
we don't have cannonballs,

but we do have a bowling ball
and a bouncy ball.

Let's weigh them.

First, we weigh
the bowling ball.

It weighs 15 pounds.

And the bouncy ball?

It weighs hardly anything.

Let's drop them.

NARRATOR:
According to Aristotle,

the bowling ball should fall
over 15 times faster

than the bouncy ball.

STELTZNER:
Well, they seem to fall
at the same rate.

This isn't that high, though.

Maybe we should drop them
from higher.

So Ed is 20 feet in the air
up there.

Let's see if the balls fall
at the same rate.

Ready?

Three, two, one, drop!

Galileo was right.

Aristotle, you lose.

NARRATOR:
Dropping feathers and hammers
is misleading,

thanks to air resistance.

DAVID SCOTT:
Well, in my left hand,
I have a feather.

In my right hand, a hammer...

NARRATOR:
A fact demonstrated on the Moon,
where there is no air,

in 1971
during the Apollo 15 mission.

SCOTT:
And I'll drop
the two of them here.

How about that?

Mr. Galileo was correct.

STELTZNER:
Little balls, soccer balls...

NARRATOR:
So while counterintuitive...

STELTZNER:
Vegetables!

NARRATOR:
...if you take the air
out of the equation,

everything falls
at the same rate,

even Aristotle.

But what really
interested Galileo

was that an object
dropped at one height

didn't take twice as long
to drop from twice as high;

it accelerated.

But how do you measure that?

Everything is happening so fast.

STELTZNER:
Oh, yes!

NARRATOR:
Galileo came up
with an ingenious solution.

He built a ramp,
an inclined plane,

to slow the falling motion down
so he could measure it.

STELTZNER:
So we're going to use this ramp

to find the relationship
between distance and time.

For time, I'll use
an arbitrary unit: a Galileo.

One Galileo.

NARRATOR:
The length of the ramp
that the ball rolls

during one Galileo
becomes one unit of distance.

So we've gone
one unit of distance

in one unit of time.

Now let's try it
for a two-count.

One Galileo, two Galileo.

NARRATOR:
In two units of time,

the ball has rolled
four units of distance.

Now let's see how far it goes
in three Galileos.

One Galileo, two Galileo,
three Galileo.

NARRATOR:
In three units of time,

the ball has gone
nine units of distance.

So there it is.

There's a mathematical
relationship here

between time and distance.

NARRATOR:
Galileo's inspired use of a ramp

had shown falling objects
follow mathematical laws.

The distance the ball traveled

is directly proportional
to the square of the time.

That mathematical relationship
that Galileo observed

is a mathematical expression
of the physics of our universe.

NARRATOR:
Galileo's centuries-old

mathematical observation
about falling objects

remains just as valid today.

It's the same mathematical
expression that we can use

to understand how objects
might fall here on Earth,

roll down a ramp.

It's even a relationship
that we used

to land the Curiosity rover
on the surface of Mars.

That's the power of mathematics.

NARRATOR:
Galileo's insight was profound.

Mathematics could be used
as a tool

to uncover and discover
the hidden rules of our world.

He later wrote,

"The universe is written
in the language of mathematics."

Math is really the language

in which we understand
the universe.

We don't know why it's the case

that the laws of physics
and the universe

follows mathematical models,
but it does seem to be the case.

NARRATOR:
While Galileo turned
mathematical equations

into laws of science,

it was another man,
born the same year Galileo died,

who took that to new heights
that crossed the heavens.

His name was Isaac Newton.

He worked here at Trinity
College in Cambridge, England.

SIMON SCHAFFER:
Newton cultivated the reputation

of being a solitary genius,

and here in the bowling green
of Trinity College,

it was said that
he would walk meditatively

up and down the paths,
absentmindedly drawing

mathematical diagrams
in the gravel,

and the fellows were instructed,
or so it was said,

not to disturb him,

not to clear up the gravel
after he'd passed,

in case they inadvertently
wiped out

some major scientific
or mathematical discovery.

NARRATOR:
In 1687, Newton published a book

that would become a landmark
in the history of science.

Today, it is known simply
as the "Principia."

In it, Newton
gathered observations

from around the world

and used mathematics
to explain them--

for instance, that of a comet
seen widely in the fall of 1680.

SCHAFFER:
He gathers data worldwide

in order to construct
the comet's path.

So for November the 19th,
he begins with an observation

made in Cambridge in England
at 4:30 a.m.

by a certain young person,

and then at 5:00 in the morning
at Boston in New England.

So what Newton does
is to accumulate numbers

made by observers
spread right across the globe

in order to construct

an unprecedentedly
accurate calculation

of how this great comet
moved through the sky.

NARRATOR:
Newton's groundbreaking insight
was that the force

that sent the comet
hurtling around the Sun...

(cannon fire)

...was the same force

that brought cannonballs
back to Earth.

It was the force behind
Galileo's law of falling bodies,

and it even held the planets
in their orbits.

He called the force gravity,
and described it precisely

in a surprisingly
simple equation

that explains how two masses
attract each other,

whether here on Earth
or in the heavens above.

SCHAFFER:
What's so impressive
and so dramatic

is that a single
mathematical law

would allow you to move
throughout the universe.

NARRATOR:
Today, we can even witness it
at work beyond the Milky Way.

This is a picture
of two galaxies

that are actually being drawn
together in a merger.

This is how galaxies
build themselves.

Right.

NARRATOR:
Mario Livio is on the team

working with the images
from the Hubble Space Telescope.

For decades,
scientists have used Hubble

to gaze far beyond
our solar system,

even beyond the stars
of our galaxy.

It's shown us the distant
gas clouds of nebulae

and vast numbers of galaxies
wheeling in the heavens

billions of light-years away.

And what those images show

is that throughout
the visible universe,

as far as the telescope can see,

the law of gravity
still applies.

LIVIO:
You know, Newton wrote
these laws

of gravity and of motion

based on things
happening on Earth,

and the planets in
the solar system and so on,

but these same laws,
these very same laws

apply to all these
distant galaxies

and, you know, shape them,

and everything about them--
how they form, how they move--

is controlled by those
same mathematical laws.

NARRATOR:
Some of the world's
greatest minds have been amazed

by the way that math
permeates the universe.

LIVIO:
Albert Einstein, he wondered,

he said, "How is it possible
that mathematics,"

which is, he thought,
a product of human thought,

"Does so well in explaining
the universe as we see it?"

And Nobel laureate in physics
Eugene Wigner

coined this phrase:

"The unreasonable effectiveness
of mathematics."

He said the fact
that mathematics

can really describe
the universe so well,

in particular physical laws,

is a gift that we neither
understand nor deserve.

NARRATOR:
In physics,

examples of that "unreasonable
effectiveness" abound.

When nearly 200 years ago

the planet Uranus
was seen to go off track,

scientists trusted the math

and calculated it was being
pulled by another unseen planet.

And so they discovered Neptune.

Mathematics
had accurately predicted

a previously unknown planet.

SAVAS DIMOPOULOS:
If you formulate
a question properly,

mathematics gives you
the answer.

It's like having a servant

that is far more capable
than you are.

So you tell it "Do this,"

and if you say it nicely,
then it'll do it

and it will carry you
all the way to the truth,

to the final answer.

RADIO HOST:
WGBH, 89.7.

NARRATOR:
Evidence of the amazing
predictive power of mathematics

can be found all around us.

I heard it took five Elvises
to pull them apart.

NARRATOR:
Television, radio,
your cell phone, satellites,

the baby monitor, Wi-Fi,
your garage door opener, GPS,

and yes, even maybe
your TV's remote.

All of these use invisible waves
of energy to communicate,

and no one even knew
they existed

until the work of James Maxwell,

a Scottish mathematical
physicist.

In the 1860s,
he published a set of equations

that explained how electricity
and magnetism were related--

how each could generate
the other.

The equations also made
a startling prediction.

Together, electricity
and magnetism

could produce waves of energy

that would travel through space
at the speed of light:

electromagnetic waves.

ROGER PENROSE:
Maxwell's theory gave us

these radio waves, X-rays,

these things which were simply
not known about at all.

So the theory had a scope,
which was extraordinary.

NARRATOR:
Almost immediately, people
set out to find the waves

predicted by Maxwell's
equations.

What must have seemed
the least promising attempt

to harness them is made here,
in northern Italy,

in the attic of a family home

by 20-year-old
Guglielmo Marconi.

His process starts
with a series of sparks.

(buzzing)

The burst of electricity creates
a momentary magnetic field,

which in turn generates
a momentary electric field,

which creates
another magnetic field.

The energy cycles
between the two,

propagating
an electromagnetic wave.

(buzzing)

Marconi gets his system
to work inside,

but then he scales up.

Over a few weeks, he builds
a big antenna beside the house

to amplify the waves
coming from his spark generator.

Then he asks his brother
and an assistant

to carry a receiver
across the estate

to the far side
of a nearby hill.

They also have a shotgun,

which they will fire if they
manage to pick up the signal.

(buzzing)

(buzzing)

(gunshot)

And it works.

The signal has been detected

even though the receiver
is now hidden behind a hill.

At over a mile,

it is the farthest
transmission to date.

In fewer than ten years,

Marconi will be sending radio
signals across the Atlantic.

In fact, when the Titanic sinks
in 1912,

he'll be personally credited
with saving many lives

because his onboard equipment
allowed the distress signal

to be transmitted.

Thanks to the predictions
of Maxwell's equations,

Marconi could harness
a hidden part of our world,

ushering in the era
of wireless communication.

(voices on radio overlapping)

Since Maxwell and Marconi,

evidence of the predictive power
of mathematics has only grown,

especially
in the world of physics.

100 years ago,
we barely knew atoms existed.

It took experiments
to reveal their components:

the electron, the proton,
and the neutron.

But when physicists
wanted to go deeper,

mathematics began
to lead the way,

ultimately revealing a zoo
of elementary particles,

discoveries that continue
to this day here at CERN,

the European organization
for nuclear research

in Geneva, Switzerland.

These days, they're most famous
for their Large Hadron Collider,

a circular particle accelerator
about 17 miles around,

built deep underground.

This $10 billion project,
decades in the making,

had a well-publicized goal:
the search

for one of the most fundamental
building blocks of the universe.

A subatomic particle

mathematically predicted
to exist nearly 50 years earlier

by Robert Brout and Francois
Englert working in Belgium

and Peter Higgs in Scotland.

TEGMARK:
Peter Higgs sat down

with the most advanced
physics equations we had

and calculated and calculated

and made this audacious
prediction:

if we built the most
sophisticated machines

humans have ever built

and used it
to smash particles together

near the speed of light
in a certain way

that we would then discover
a new particle.

You know, if this math
was really accurate.

NARRATOR:
The discovery
of the Higgs particle

would be proof
of the Higgs field,

a cosmic molasses that gives
the stuff of our world mass--

what we usually experience
as weight.

Without mass, everything would
travel at the speed of light

and would never combine
to form atoms.

That makes the Higgs field

such a fundamental part
of physics

that the Higgs particle
gained the nickname

"The God Particle."

(cheering)

In 2012, experiments at CERN

confirmed the existence
of the Higgs particle,

making the work of Peter Higgs

and his colleagues
decades earlier

one of the greatest predictions
ever made.

And we built it and it worked,

and he got a free trip
to Stockholm.

(applause)

LIVIO:
Here, you have
mathematical theories

which make
very definitive predictions

about the possible existence

of some fundamental particles
of nature,

and believe it or not,
they make these huge experiments

and they actually discover
the particles

that have been predicted
mathematically.

I mean,
this is just amazing to me.

ANDREW LANKFORD:
Why does this work?

How can mathematics
be so powerful?

Is mathematics, you know,
a truth of nature,

or does it have something to do

with the way we as humans
perceive nature?

To me, this is just
a fascinating puzzle.

I don't know the answer.

NARRATOR:
In physics, mathematics has had
a long string of successes.

But is it really
"unreasonably effective"?

Not everyone thinks so.

I think it's an illusion,

because I think what's happened

is that people have chosen
to build physics, for example,

using the mathematics
that has been practiced,

has developed historically,

and then they're looking
at everything,

they're choosing to study things
which are amenable to study

using the mathematics
that happens to have arisen.

But actually, there is a whole
vast ocean of other things

that are really quite
inaccessible to those methods.

NARRATOR:
With the success of mathematical
models in physics,

it's easy to overlook
where they don't work that well.

Like in weather forecasting.

There's a reason meteorologists
predict the weather

for the coming week,

but not much further out
than that.

In a longer forecast,
small errors grow into big ones.

Daily weather is just
too complex and chaotic

for precise modeling.

And it's not alone.

So is the behavior of water
boiling on a stove,

or the stock market,

or the interaction of neurons
in the brain,

much of human psychology,

and parts of biology.

DEREK ABBOTT:
Biological systems,

economic systems,

it gets very difficult to model
those systems with math.

We have extreme difficulty
with that.

So I do not see math
as unreasonably effective.

I see it as reasonably
ineffective.

NARRATOR:
Perhaps no one
is as keenly aware

of the power and limitations
of mathematics

as those who use it
to design and make things:

engineers.

Look at that wheel!

NARRATOR:
In their work,
the elegance of math

meets the messiness of reality,
and practicality rules the day.

Mathematics
and perhaps mathematicians

deal in the domain
of the absolute,

and engineers live in the domain
of the approximate.

We are fundamentally interested
in the practical.

And so frequently, we make
approximations, we cut corners.

We omit terms and equations

to get things
that are simple enough

to suit our purposes
and to meet our needs.

NARRATOR:
Many of our greatest
engineering achievements

were built using
mathematical shortcuts:

simplified equations
that approximate an answer,

trading some precision
for practicality.

And for engineers,
"approximate" is close enough.

Close enough
to take you to Mars.

STELTZNER:
For us engineers,

we don't get paid
to do things right;

we get paid to do things
just right enough.

NARRATOR:
Many physicists
see an uncanny accuracy

in the way mathematics
can reveal

the secrets of the universe,

making it seem to be
an inherent part of nature.

Meanwhile, engineers in practice
have to sacrifice

the precision of mathematics
to keep it useful,

making it seem more like
an imperfect tool

of our own invention.

So which is mathematics?

A discovered part
of the universe?

Or a very human invention?

Maybe it's both.

LIVIO:
What I think about mathematics

is that it is an intricate
combination

of inventions and discoveries.

So for example, take something
like natural numbers:

one, two, three, four, five,
etcetera.

I think what happened

was that people were looking
at many things, for example,

and seeing that
there are two eyes, you know,

two breasts, two hands,
you know, and so on.

And after some time,

they abstracted from all of that
the number two.

NARRATOR:
According to Mario, "two"
became an invented concept,

as did all the other
natural numbers.

But then people discovered
that these numbers

have all kinds
of intricate relationships.

Those were discoveries.

We invented the concept,
but then discovered

the relations
among the different concepts.

NARRATOR:
So is this the answer?

That math is both invented
and discovered?

This is one of those questions
where it's both.

Yes, it feels like
it's already there,

but yes, it's something
that comes out of our deep,

creative nature as human beings.

NARRATOR:
We may have some idea
to how all this works,

but not the complete answer.

In the end, it remains
"The Great Math Mystery."

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