Nova (1974–…): Season 42, Episode 7 - The Great Math Mystery - full transcript
NOVA leads viewers on a mathematical mystery tour -- a provocative exploration of math's astonishing power across the centuries. We discover math's signature in the swirl of a nautilus shell, the whirlpool of a galaxy and the spir...
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"
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"