The Great Math Mystery (2015) | Full Documentary | NOVA

MAN: Roger, copy 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 looked at nature

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 the bottom 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 number 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."