Original subtitles

Roger, copy mission

We live in an age of astonishing advances

Descending at about 75 meters per second

Engineers can land a car-size rover on Mars

Touchdown confirmed

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?

Albert Einstein wondered,

"How is it possible that mathematics

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

Is mathematics even human?

There doesn't really seem to be an upper limit

to the numerical abilities of animals

And is it the key to the cosmos?

Our physical world

doesn't just have some mathematical properties,

but it has only mathematical properties

"The Great Math Mystery," next on NOVA!

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

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

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

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

Fibonacci numbers show up in petal counts,

especially of daisies, but that's just a start

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

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

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

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

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

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

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?

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

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

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

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

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"

A fifth sounds like this:

♪ La, la ♪

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

And a fourth sounds like:

♪ La, la ♪

You can think of it as the first two notes

of "Here Comes the Bride"

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

In a fifth, the ratio is three to two

And in a fourth, it is four to three

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?"

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

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

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

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

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

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

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

All right, Shyam, we'll start about now

Okay, buddy?

Okay

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

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

Duke Professor Liz Brannon

investigates how well lemurs, like Teres here,

can compare quantities

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

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

he'll get a reward, a sugar pellet

A wrong answer?

We have to do a lot to ensure

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

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

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

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

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

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

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

Roger, copy mission

Coming up on entry

In 2012, they landed a car-size rover

Descending at about 75 meters per second as expected

on Mars

Touchdown confirmed, we're safe on Mars

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

So Aristotle reasoned

that the rate at which things would fall

was proportional to their weight

Which seems reasonable

In fact, so reasonable,

the view held for nearly 2,000 years,

until challenged in the late 1500s

by Italian mathematician Galileo Galilei

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

According to Aristotle,

the bowling ball should fall over 15 times faster

than the bouncy ball

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

Dropping feathers and hammers is misleading,

thanks to air resistance

Well, in my left hand, I have a feather

In my right hand, a hammer

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

in 1971 during the Apollo 15 mission

And I'll drop the two of them here

How about that?

Mr. Galileo was correct

Little balls, soccer balls

So while counterintuitive

Vegetables!

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

Oh, yes!

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

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

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

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

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

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

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

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

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

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

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

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

Newton's groundbreaking insight was that the force

that sent the comet hurtling around the Sun

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

What's so impressive and so dramatic

is that a single mathematical law

would allow you to move throughout the universe

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

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

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

Some of the world's greatest minds have been amazed

by the way that math permeates the universe

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

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

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

WGBH, 89 7

Evidence of the amazing predictive power of mathematics

can be found all around us

I heard it took five Elvises to pull them apart

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

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

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

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

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

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

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

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

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"

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

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

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

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

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

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

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!

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

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

For us engineers,

we don't get paid to do things right;

we get paid to do things just right enough

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

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

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

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

We may have some idea to how all this works,

but not the complete answer

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