Original subtitles

There is a mystery at the heart of our universe -

a puzzle that so far no-one has been able to solve.

I can't, it's too weird.

Welcome to my world!

If we can solve this mystery, it will have profound consequences

for all of us.

That mystery is why mathematical rules and patterns seem

to infiltrate pretty much everything in the world around us.

Many people have, in fact, described maths as the underlying

language of the universe.

But how did it get there?

Even after thousands of years, this question causes controversy.

We still can't agree on what maths actually is or where it comes from.

Is it something that's invented, like a language?

Or is it something that we've merely discovered?

I think discovered.

Invented.

It's both.

I have no idea.

Oh, my God!

Why does any of this matter?

Well, maths underpins just about everything

in our modern world, from computers and mobile phones

to our understanding of human biology

and our place in the universe.

My name is Hannah Fry and I'm a mathematician.

In this series, I will explore how the greatest thinkers in history

have tried to explain the origins of maths' extraordinary power.

You've ruined his equation!

I'm going to look at how, in ancient times,

our ancestors thought maths was a gift from the gods.

How in the 17th and 18th centuries, we invented new mathematical systems

and used them to create

the scientific and industrial revolutions.

And I'll reveal how, in the 20th and 21st centuries,

radical, new theories are forcing us to question once again everything

we thought we knew about maths and the universe.

The unexpected should be expected, because why would reality

down there bear any resemblance to reality up here?

In this episode, I discover how maths led Victorian scientists

into a world of invisible forces and particles that we cannot see.

Now, this couldn't be a coincidence.

And I reveal why the concept of infinity broke the rules

about where maths comes from.

I'm very tormented by infinity.

Is infinity real?

I do not know the answer to that question.

Our world is governed by the rules of science,

but science wouldn't work if it wasn't for a far deeper set

of rules - those of mathematics.

It predicts the movement of the planets and the ebb and flow

of the tides.

If you look hard enough at anything, you'll find mathematics

hiding underneath.

If maths is the language of the universe,

then where do numbers come from?

Before we learned that one plus one equals two,

the idea of one and two still existed.

The nature of oneness and twoness has always been there.

The concept of numbers is something universal.

All around the world and in every language,

we understand the idea of what one or two means

and this raises an intriguing question.

Is maths all in our heads?

Is it something that we've invented,

a language that we use to describe the universe?

Or is it an external, physical reality, something

that exists completely independently of us humans,

something that's just out there waiting to be discovered?

In ancient times, we were in awe of the power of maths.

Seen as a gift from the gods, it was considered pure and complete.

But, through the centuries, maths developed.

It wasn't complete, after all.

New areas and techniques have been invented.

And the more we explored science, the more it became obvious

that we couldn't just rely on simple experiments.

We needed a theory and, crucially, a mathematical description

to be able to understand the world around us.

Things that seem obvious at first often have a habit of melting away

when exposed to the rigour of experimentation.

The problem for humans is overriding our instinct

to trust our intuition.

Our senses aren't always the best guide to the truth.

The Greek philosopher Aristotle fell into this trap when he famously

declared that something heavy will fall quicker

than something that's light.

To him, it seemed blindingly obvious and for centuries,

nobody disagreed with him.

On the face of it, you might think that suggesting

that heavier objects fall faster than light objects was quite

a sensible idea.

After all, if you drop them at the same height,

the hammer lands first.

But a 16th-century scientist and mathematician called

Galileo Galilei had a different explanation.

He believed Aristotle had failed to consider something crucial.

The incredible fact is not that Aristotle was wrong,

but that his law of motion stood unchallenged for almost 2,000 years.

How could such a flawed idea survive for so long?

Well, to be fair, there are a few reasons.

You can see the hammer hitting the ground earlier than the feather.

The reason for that, of course, is air resistance,

and Galileo argued that if you dropped them in a vacuum,

they would land at exactly the same time.

To come up with this theory, Galileo imagined the idea

of a vacuum in which air resistance didn't exist,

and created a series of laws that describe the motion

of falling objects.

They completely overturned Aristotle's ideas.

Over 300 years after Galileo's prediction,

Apollo 15 astronaut David Scott gave the theory its most dramatic test.

Well, in my left hand I have a feather, in my right hand

a hammer,

and I'll drop the two of them here, and hopefully they'll hit

the ground at the same time.

How about that? Mr Galileo was correct.

With no air resistance on the moon, the hammer and feather hit the lunar

surface at the same time.

A physical description of the world on its own isn't enough.

It has to go hand in hand with mathematics

before you can truly discover the nature of reality.

Galileo was incredibly impressed, for good reason, at the power

of mathematics to give us insights, to describe things that were

happening, to articulate the patterns that the human brain

is able to access.

It almost seems miraculous that some symbols on a piece of paper

can do that and, in that sense, it might lead you to think

that maths is the language of reality.

Galileo exclaimed that the world is a grand book written

in the language of mathematics.

I think the reason for this is that ultimately...

..the world is completely mathematical

and we're just discovering that, bit by bit.

He had this feeling that, by using mathematics,

he could get into these things which he wanted to be inevitable.

And mathematics gave him that certainty that things

are inevitable, so he was the first to understand that, in order

to explain phenomena, he needs to use mathematics.

Galileo's theories, though ahead of their time,

raised as many questions as they answered.

There appeared to be some kind of a force that was pulling objects

to the ground, but exactly what that force was

or how it worked remained a mystery.

Solving this mystery would take the genius

of a 17th-century Englishman.

His name was Isaac Newton.

I'm heading to north Wales to do something I'm not

entirely happy about.

So, my director was looking for a clever way to illustrate gravity,

and he came up with the bright idea to send me

down the fastest zip wire in the world.

Headfirst, as well.

I wasn't in that meeting.

I should have been in that meeting.

The same force that brought Newton's apple to the ground

is the thing that's going to be propelling me

towards a quarry.

Good luck! See you at the bottom. See you at the bottom.

What do I let myself in for?

But all this is nothing compared to how Newton performed

experiments on himself.

Newton totally believed that the path to true knowledge lay

in observation. So, rather than just read a book on optics, say,

he decided to experiment by poking a blunt needle

into his own eye.

Maybe don't try that one at home.

He wasn't going to take someone else's word for it.

He had to test these theories for himself.

As he began wrestling with bigger ideas, such as gravity,

only mathematics could help him find the answers.

When you think about it, gravity is actually quite

a strange beast.

It creates this invisible force of attraction between me

and everything around me, but one that's weak enough

that I can easily overcome it just by moving my own muscles.

Newton set out to find a way to describe this mysterious force.

Originally described in words, his law of gravity was later

written down in the form of an equation.

Now, don't be fooled by its simplicity

because this guy packs a real punch.

I'm using it to work out the force that will be acting on me

as I head down the zip wire.

To understand it, you need to know what all the letters stand for,

so, let's begin with F, the force.

Newton says that, between any two objects in the universe,

there is an attractive force, and this force depends on the mass

of those objects.

This capital M here, that's the mass of the earth,

and, then, slightly smaller, the little m, there, is me.

That little m is my mass and it's measured in kilograms.

There's also G, the gravitational constant,

which Newton knew had to exist, although he didn't know exactly

the size of it at the time,

and r, there, which is the distance between me and the centre

of the earth.

More generally, what this equation

is saying is that the bigger

the mass of your objects,

like planets, for example,

the bigger your force between them is going to be.

And the greater the distance between objects, the bigger

this r is, the weaker the force of gravity is going to be.

So what does Newton say the force of gravity will be on me?

So, if you plug in all of the numbers into this equation,

you could calculate the force on me as I travel down the wire.

It works out to be...

..736 and the unit is newtons.

Newton arrived at his now-famous formula after studying

centuries' worth of measurements from astronomers

that had gone before him.

His law of gravity not only explained why objects fall

to the ground -

it predicted the positions of every moon, planet or comet

anywhere in the cosmos.

That is one devastatingly powerful equation.

This was Newton's genius.

Once you've got a mathematical law, you can use it to apply to anything

- apples, planets and people.

And, if you can calculate exactly what that force will be,

it means that you can predict all kinds of other things,

like my terminal velocity as I travel down to the bottom.

So, er, let's put it to the test.

As the force of gravity pulls an object to the ground,

it reaches a maximum speed.

This is called its terminal velocity.

Before you can calculate this figure, there is a bunch

of things you need to consider, such as the gravity,

drag and friction along the cable.

Don't worry.

Time to put my faith in Newton...

..and the fastest zip line in the world.

From my calculations, I reckon my terminal velocity

is going to be about 90mph.

I don't think I'm going to speak to this director again.

What am I doing?

Right.

Three, two, one...

Oh!

No!

Whoa!

That was actually really fun. OK.

OK, I also need to check my speed prediction.

Now, disclaimer - just before I came down,

they added some flags to the back of me,

just to slow me down, because the wind's picked up,

as you can probably hear.

So, I don't think I'm going to quite hit 90, but let's have a look here.

There is a big spike, there, on the graph and it says it's 41

seconds for one mile, which is about, what?

75mph, something like that?

Not bad, not bad.

For a back-of-the-envelope calculation, not bad.

The power of Newton's equation was that it could explain

and predict so much about the universe.

It allowed us to think of nature as ordered,

not just on Earth but throughout the cosmos.

The key breakthrough of Newton was that he had the audacity

to shatter this idea that Earth rules are different from heaven

rules, and the moon doesn't fall down because it's made

of heaven stuff,

and say, "Wait a minute, maybe all things obey

"the same physical laws."

His laws of force and of motion were not meant to merely apply in,

say, the heavenly realms or just on Earth.

They were meant to apply everywhere and the idea was the whole

of nature would really be captured by this single set of laws.

I mean, the fact that we can write equations and know how to power

a rocket and have it land on the moon

and come back, holy cow!

I mean, we take these things for granted, but think

about the power of equations to give us the trajectory

and figure out how to accomplish this incredible feat.

That is thrilling.

If evidence is needed to prove maths is discovered,

part of the fabric of reality, then, surely, this is it -

how could something we invented in our brain have the power

to reveal the workings of the universe?

And the extraordinary power of mathematics

wasn't just confined to the stars.

By the end of the 18th century, scientists and engineers

were using it to drive innovation on a grand scale -

what became known as the Industrial Revolution.

This changed everything.

People didn't just live and work in a field any more -

there was an explosion of growth in towns and cities,

as employment switched to factories.

And driving this entire revolution was the invention

of the steam engine.

The impact of this new technology was profound.

It opened up the country not just to people and goods,

but to ideas.

New ways of doing things were propelling us into

the age of the machine.

How fast does it go? 25mph maximum.

What are we doing now? About 15.

And yet your speedometer goes up to 100.

Yeah, not going there!

Behind all of this were the essential calculations

of the machine age - how strong the materials were,

how hot or cold something might get.

It was mathematics that was used to design faster

and more efficient machines.

So, how hot does it get in there?

In Fahrenheit, it goes to about 2,500 degrees.

Two and a half...? What's that in Celsius?

I'm not sure. I've got no idea.

Hot - very hot.

New skills were required in all of this,

so, whereas, before, you would have craftsmen using hand tools,

now you had people in factories operating machinery.

But there's also a sea change here in the way that we think.

It's a belief that, while the natural world might not be tamed,

it can at least be bent to our will.

The Industrial Revolution marked a major turning point in history.

From textiles to iron production and the spread of the railways,

almost every aspect of daily life

was influenced in some way.

And, at the heart of this revolution, was mathematics.

Now, this is a world that feels firmly rooted in reality.

We can trust the numbers and we know that they're not

going to let us down.

So, forget all of your airy-fairy, philosophical stuff here,

this is maths in action.

It's big, it's bold and, actually, it's pretty amazing.

Technological miracles were coming thick and fast.

Mathematics had given us a description

of how the world works.

It was driving our understanding forward.

But, also, it could hint at how seemingly separate things

could be connected.

By the 19th century, mathematicians and scientists

began to wonder what else was out there just waiting to be discovered.

They soon turned their attention to the invisible link

between electricity and magnetism.

Both had been known about for centuries,

from the raw power of lightning to navigation

by means of a ship's compass, but they'd always been thought of

as two very different things.

It was a working-class son of the Industrial Revolution,

Michael Faraday, who was the first person to see a connection

between the two.

I've come to the Royal Institution, to the place where Faraday

had his laboratory.

To see if electricity and magnetism were linked,

Faraday ran a series of experiments.

He took a wire that had electricity passing through it

and he watched as it moved the needle of a compass.

The electric wire and the magnetic needle weren't touching,

and yet one was having an effect on the other.

What was the connection?

Faraday looked deeper.

What he did was to take a magnet like this one

and a roll of copper wire wrapped around a cylinder like this,

and then to pass one through the other very quickly like this.

The wire surrounds the outside of the cylinder, so the magnet

can't come into contact with it.

And that's really all there is to it.

There's nothing more complicated than that.

The wire never touches the magnet and, yet, as you can see

from these LEDs - probably not the originals -

that is enough to generate electricity.

Faraday realised there had to be some kind of invisible

force working behind the scenes

and he had a clever idea of how to make it visible.

What you do is you take a permanent magnet and you place some paper

on top of it, and then take some iron filings

and sprinkle them on top.

Now, this, I think, is one of the most memorable

experiments that you do at school.

And I can remember that moment where you see

the invisible force field that's created by the magnet.

As the iron filings fall onto the paper,

they line up with the magnet's field lines.

Now, this is just two-dimensional here, but actually these lines

are three-dimensional.

They come out and they warp and curve and wrap around

the entire magnet.

That's pretty cool, isn't it?

It's pretty cool.

Faraday's iron filings experiments revealed the existence

of an invisible field stretching out into space.

He could see the lines of the force, but he was an experimentalist

and lacked a complete mathematical description of what was going on.

As a result, many of his contemporaries dismissed his

ideas as fanciful.

It was the Scottish scientist James Clerk Maxwell

who took Faraday's ideas

and came up with a mathematical way to link

electricity and magnetism.

Drawing from the observations of previous scientists,

Maxwell distilled electricity and magnetism down

into four equations that worked for nearly every situation.

The symbols themselves aren't important to the story -

the key point is that Maxwell spotted a gap.

The mathematics was telling him there was something missing

in this last equation.

He realised there has to be another term in this equation,

one that looks like this.

And essentially what it's saying is that if an electric field is moving,

then a magnetic field will wrap itself around it.

And it's mirrored by this equation up here,

which says that if a magnetic field is moving, an electric field

will wrap itself around it.

With this missing piece in place, suddenly everything fitted together.

Mathematics had led Maxwell to see the bigger picture.

electricity to magnetism, magnetism to electricity,

back and forth from one to the other.

Using only mathematical ideas, Maxwell had found the evidence

to prove that electricity and magnetism were

inextricably linked.

Together, electricity and magnetism formed what he called

an electromagnetic field.

This helped explain so much.

The equations perfectly described what Faraday

had seen in his experiments.

But Maxwell didn't stop there.

He showed how these field lines could move in time with each other,

creating electromagnetic waves.

By playing around with these equations, Maxwell could calculate

the speed of this wave and it came out to be about

300,000 kilometres a second.

That wasn't a random number.

That was a number that Maxwell knew very well because it was the same

as the speed of light in a vacuum.

Now, this couldn't be a coincidence.

You don't really get coincidences like that in the universe.

There was only one possible explanation -

light had to be an electromagnetic wave.

Maxwell's discoveries were genuinely revolutionary.

He'd given us a unified theory for electricity and magnetism

and, as an added bonus, an explanation of light itself.

For the first time, an electric field,

a magnetic field and light could all be explained

using a single theory.

The elegance and simplicity of this solution was breathtaking.

Surely, nothing the human mind could conceive of would ever

be capable of thinking up something so sublime -

equations that reveal new truths about the universe.

It feels very much as if this answer was always out there.

It just needed someone who thought differently to discover it.

It's quite uncanny how mathematics has again and again predicted

new things in the physical world that we weren't even looking for.

You come up with novel predictions.

You come up with ideas that there should be structures

in the world that you haven't yet discovered

and, then, on inquiry, you discover those to be real.

That's really extraordinary.

I can tell you from my personal experience,

it is shocking, not just surprising but shocking,

that mathematics makes predictions about the world around us.

The Ancient Greeks found intriguing patterns in nature which seemed

to follow the rules of maths.

Then Newton showed us how mathematical equations

had the power to predict the movement of the planets,

revealing an ordered universe.

By the 19th century, the formidable power of maths

allowed Maxwell to unify electricity and magnetism.

It seemed inconceivable that maths could be anything other

than something we discover.

But then something happened that turned this worldview

on its head.

There was a new way to look at maths.

Someone had invented a different way of doing things.

Since the days of the Greek mathematician Euclid more than 2,000

years ago, right angles and parallel lines,

the kind we learned at school, have been the bedrock

upon which all of geometry and our understanding

of space is built.

But, in the 19th century, mathematicians started to wonder

whether everything really was as it seemed,

or whether there was the possibility of something a bit weird

going on behind the scenes.

You can see it with games like Pac-Man.

What kind of a shape is the Pac-Man universe?

Your instinctive answer might be a square

and you'd be right, sort of.

For instance, if this little pink character exits to the left,

it will re-enter on the right...

..which actually makes this universe...

..more of a cylinder.

What's more, in other, similar games, you can exit

out of the top and re-enter at the bottom,

which means that these two loose ends have to bend around

and connect up to one another.

It's a bit of a strange idea to get your head around,

but these kind of computer games are not played on a square.

They are played on a doughnut.

Once you move from a flat square to another shape,

you can't take it for granted that geometry will follow the rules

you've always expected it to.

Behind the scenes, there can be something else

going on entirely.

But, hold on to your hats, because this is all about

to get much weirder.

Consider for a moment a traditional geometric view

of the world.

Imagine there are four coloured courtyards.

What would happen when I leave one of the courtyards?

If the world was as Euclid says it is and everything worked

normally, if I turned left four times, I would eventually get back

to where I started.

I've left the yellow courtyard.

I've gone through orange, red and blue and I'm back in yellow again.

Nothing controversial here.

But who says there has to be four courtyards next to each other?

What if you got back to where you started after turning

left only three times?

But, hang on, I hear you cry, that's impossible -

except it's not if you're living on a cube.

Begin on this side, turn once, turn twice,

turn three times

and you're back where you started.

No longer was there only one description of space.

By changing the rules, you could now choose a different

type of geometry.

It turns out there's many different ways to think about space.

It would be very much like if somebody discovered

Piccadilly Circus by taking a left turn where they had always taken

a right turn before.

People hadn't even thought that there could be a distinction

between the physical space and the mathematical space

that Euclid had studied with his axioms.

Cos all of a sudden Euclidean geometry just looks like one way

of describing a space and, in fact, you know,

it happens to be a good one for describing the space

we're sitting in right now,

not such a good one for describing space on astronomical scales,

it turns out.

So, it's a little bit like a game.

Namely, I teach you the rules of chess and we play chess.

I change the rules and we play a different game but we still can

play a game.

So, that was the feeling, that maybe it is all,

you know, depending on which set of actions you choose,

you can get a new type of mathematics.

But hang on a minute.

If we can just make up a new type of geometry,

then perhaps I've got this wrong.

Maybe maths IS something we invent, after all.

With this new-found freedom mathematicians began exploring

ever more abstract ideas, the most intriguing

of which was the notion of infinity.

Can everybody show me the sign that we are going to be using

to solve this problem?

Off you go.

From an early age, we all have an idea of what infinity is,

but it's hard to pin down.

Our minds aren't built to wrap themselves around the concept

of something that is completely endless and boundless.

And that makes describing exactly what infinity is pretty tricky.

It's a number that keeps on going and never stops.

The biggest number I can think of is 99 billion.

400.

Googolplex.

There's nothing bigger than infinity

because that's the biggest number that you could, um,

that you could possibly need.

I'm very tormented by infinity.

I have a love/hate relationship with infinity.

I love using it when I teach courses at MIT cos

it makes things so easy to derive and prove.

But, in my gut, I know there is no actual infinity,

it's just a convenient approximation.

Is infinity real?

It's about as real as

the number one or the number zero.

It's a concept.

It's a useful concept in describing a certain set of elements

and, in that sense, yes, it's real.

I think it's fair to say that nobody in the laboratory is ever

going to have a dial that registers infinity,

that measures infinity.

We're never going to literally count to infinity.

We can approach it,

but, from that point of view, I don't think we're ever

going to embrace it the way that we embrace tables and chairs

and finite objects.

It's only by definition we can't go there,

you can't get there.

Try and get closer to infinity and it always stays

just as far away!

You might imagine that something as abstract as infinity

is not very useful.

But, in reality, infinity offers a way to solve problems

that previously would have seemed impossible.

If you wanted to know the distance between the UK and New York,

you could try and use a ruler on a globe like this.

You'd have some trouble because, of course, the world is round

and curves, unlike straight lines, are quite tricky to measure.

Good luck in geography class with a globe and a measuring stick.

But what if rather than just using one ruler,

you use two much smaller rulers

and you use how they overlap to wrap around the curve of the earth?

Now, by doing that you're not going to get the exact distance

between London and New York, but you're going to get a much

better approximation for it.

And you can imagine the more and more rulers

that you use, the better they will wrap around the curve

of the globe and the better an approximation you'll end up with.

So here's the key idea.

If you zoom in enough on any curve, it will start to look straight.

And if you have an infinite number of teeny, tiny rulers,

you can perfectly measure the length of any curve just by adding

up all of those straight lines.

It's only by harnessing the power of infinity

that any this is possible.

OK, so why should you care?

Well, it's not just the Earth that's got curves.

Everything from the movement of satellites in the sky,

to the rise and fall of the stock market,

to understanding how our human behaviour changes over time,

all of them rely on this idea of infinity.

Relying on an idea we don't really understand isn't something

that sits comfortably with mathematicians.

In 1924, the renowned German mathematician David Hilbert created

a famous thought experiment to try and help explain infinity.

He did it by imagining a large hotel.

But this was no ordinary hotel.

It had an infinite number of rooms.

Hi. Hiya.

Can I have a room for tonight, please?

Sorry, madam, we're fully booked tonight.

Oh, you haven't got any rooms at all?

Unfortunately not, sorry. Oh.

Hilbert wondered what would happen if all the rooms were full

and a guest like me turned up.

Would there be room for one more in the infinite hotel?

So, today I've turned up and the place is fully booked.

They're saying they haven't got any rooms at all,

whatsoever.

I've tried asking them if they know who I am,

but, apparently, they're not familiar with my back catalogue

of extremely niche online maths videos,

if you can believe it.

Even in a hotel with an infinite number of rooms,

there's a problem.

The manager can't just put me in the last room

because, in an infinite hotel, there is no last room.

So, if the hotel is full, how do I find a bed for the night?

All we have to do is politely ask the person staying in room one

to move along into room two.

The person in room two to move to room three.

Three to four.

Four to five.

And so on.

As there's no last room, if you move everyone along by one

room number, every guest has somewhere to sleep.

And that leaves room one for me.

Even if the hotel is full, a room can always be found.

That's because infinity plus one, is still infinity.

So there's always room at the infinity hotel

because you can always add on an extra room at the beginning

to make infinity just that little bit bigger.

And, if my friend wants to come and stay, too,

well, infinity plus two is still infinity,

which is perfect for a girls' weekend away.

I told you it was weird.

That's the thing about infinity.

It's a very slippery beast.

There was one mathematician who set out to tame the infinite beast.

His name was Georg Cantor

and the question he wanted to answer sounded deceptively simple -

how big is infinity?

With that one, simple question, Cantor would start a revolution,

one that would have a profound effect on the foundations

of mathematics.

I've come to Halle in Germany.

It was here that Cantor taught in the city university.

For him, infinity was the key that opened the door

to a new mathematical landscape.

I don't know about you, but I find it quite hard to picture

in my head the size of something like our solar system,

or our galaxy, the Milky Way.

These distances are so big that they defy our imagination.

But each of these things scales into insignificance.

They are infinitesimally small when compared

to the vastness of infinity.

While the idea of infinity was known to the ancient Greeks,

some of Cantor's contemporaries saw it as an offshoot of maths

rather than anything worth understanding in its own right.

This wasn't good enough for Cantor.

If our knowledge of the world is built on infinity,

he said, we can't just accept it,

we have to understand it.

To get a handle on infinity, take a look

at these two sets of numbers.

Let's imagine that, along here, you've got all the natural numbers,

the counting numbers. So, one, two, three,

four, five, six, seven,

eight and so on.

Now, there's going to be an infinite number of these.

Now, next to it, let's put the even numbers.

So, two, four, six, eight, and so on.

On the surface of it, it looks like this infinity

will be bigger than that one.

As both of these lines will go on forever,

it seems obvious that the infinity of one,

two, three, four will be bigger than the infinity of the even

numbers, two, four, six, eight.

After all, there's only half as many of those.

But, actually, if you shuffle all of these along,

they actually match up rather nicely.

So, one goes with two, two goes with four,

three goes with six and so on and so on.

Neither of these lists are ever going to run out.

As each list of numbers never stops, every counting number can always

find an even number to pair up with.

As a result, both infinite lists of numbers have to be the same size.

We know this is true because we can count them.

I know that seems like a bit of a strange idea,

but just go with me on this for a second.

Because you can start at the beginning and work your way up,

counting as you go.

First number, second number, the third number,

and so on and so on.

Now, it's true, you would have to carry on counting forever,

but you could be sure that you wouldn't miss

any of the numbers as you went.

Even though the infinity of the counting numbers looks bigger

than the infinity of the even numbers,

they're actually the same size.

Next, Cantor tried something different.

He set out to count all the numbers between 0 and 1.

Where is the most sensible place to begin?

Is it 0.1?

Well, no.

Because 0.01 is smaller.

It can't be 0.01 either,

because 0.001 is smaller still.

And 0.0001 is smaller still.

Wherever you try and start,

I can always find another number to squish in.

And that means there is no sensible place to start.

However hard you try,

you can't count up the number of numbers between 0 and 1.

This infinity is uncountable.

Cantor's disturbing conclusion

was that some infinities are bigger than others.

The sheer audacity of his work

set off a quiet revolution in the world of mathematics.

If Cantor thought that his work was going to be welcomed

with open arms, then he was to be sorely disappointed.

He was attacked on all sides by his academic colleagues.

They called him a scientific charlatan

and a corrupter of the youth.

And some even tried to sabotage the publication of his works.

Could it be that Cantor's ideas on infinity

were merely a product of his own imagination?

Something he invented?

His work on infinity consumed every waking minute.

In May of 1884, he suffered a nervous breakdown.

Eventually, he was brought here to the Nervenklinik in Halle,

a psychiatric hospital.

How did Cantor's desire to tame the infinite impact on his illness?

I am meeting the hospital's director, Dr Frank Pillmann.

This, for example, is a case note from 1907.

"Mania, an acute episode of circular psychosis."

This is what we would today call bipolar disorder.

There are some people who have suggested that sort of,

you know, that the struggle that he was having with his mental health

was exacerbated by his fight

to try and find these answers around infinity. What's your opinion?

I would feel that the intellectual occupation

with mathematical theories is nothing

that makes you prone to get a psychiatric illness.

As far as we know about his personality,

he has always been described

as a very ambitious person,

certainly creative.

Of course, he tried to solve some unsolvable problems.

But I think that's the life of every mathematician!

That's probably true!

Probably true.

The struggle!

The struggle with very difficult problems.

This is a memorial to Cantor.

He was feared by his critics

because he dared to question their assumptions

of conventional mathematics.

His work on infinity was crucial

for building more complex mathematical ideas

than we rely on today.

This is where mathematics starts to stray much more

into the realms of the abstract.

Infinities, bigger infinities,

countable and uncountable infinities.

These are not things that you tend to find in the physical world.

So, is it all just a product of our intellect and imagination?

Is this mathematics invented?

Certainly, when you just take the basic concept of infinity,

it's meant to be the biggest possible thing, right?

And then someone tells you that there's lots of infinities.

So, it's certainly a very puzzling concept, but it's an essential one.

It's an essential feature of huge numbers of mathematical systems.

Insofar that mathematics arises as an interaction

between reality and conscious, rational minds

and that's what creates mathematics,

I would say infinity is real in that sense.

If you ask me is it real in actual reality?

I do not know the answer to that question,

nor do I know how to find the answer to that question.

Some people find that emotionally disturbing,

this idea that reality is bigger than we thought.

I actually find it kind of liberating.

I think it would be rather claustrophobic

if our reality was really small.

Maths has taken us from a time when we could spot patterns in nature

to being able to describe the invisible forces

that form the structure of the cosmos.

To probe this hidden world,

we've invented mathematical tools and equations.

Maths has quietly, almost invisibly,

revolutionised the way we understand our place in the universe.

Today, the argument about whether maths is invented or discovered

is much more than a philosophical debate.

This is where it gets real.

This jumble of pipes and wires looks chaotic,

but it's at the cutting edge of science.

If the researchers here succeed in their goal,

they'll have found the answer to the world's energy needs.

A form of power that's clean, renewable and free.

I've come here today to the Culham Centre For Fusion Energy

where a group of people are trying to do something rather remarkable.

They're taking a mathematical description of reality

and trying to bend it to their will,

harnessing the power of a star

and using it to change humanity's future.

Controlling the power of a star such as our sun is,

as you might imagine, incredibly difficult.

The Sun is one, giant, hot ball of gas called a plasma.

Its heat is generated when atoms of hydrogen inside this plasma

collide with each other very quickly,

releasing vast amounts of energy.

The challenge is to recreate that reaction down here on Earth

and the first step is to form plasma.

Within this shape, they're trying to recreate

the conditions that you find in the inside of the Sun

and hold that plasma in place

while it reaches temperatures of up to 200 million Celsius.

This doughnut-shaped space is called a tokamak.

The most difficult part of this whole process

is ensuring the plasma remains stable.

If part of it touches the walls,

the plasma cools and the reaction stops.

Trying to prevent that from happening is the job

of Dr Anthony Shaw.

The difficulty is that, at 200 million degrees,

you get quite a lot of extra effects coming in.

It gets turbulent, like the churning sea.

There are various currents and turbulences and tides

and all these things that make the behaviour of it very tricky

to understand and if you don't account for the right things

at the right time, it'll do what it wants instead of what we want.

Driving this behaviour are lots of subatomic reactions

that no-one has ever seen.

The only reason we believe they exist is down to maths.

Anthony and his colleagues are using maths to try and predict

how these invisible particles will behave inside the plasma.

So, here we have a photograph

that was taken inside the tokamak.

You can see the hydrogen plasma here just glowing around the edges

and they've overlaid a photograph of the structure

just so you can see roughly where it's hitting.

For comparison, there is also a simulation of this,

a mathematical simulation.

And, on this one, you can see very clearly these little lines,

they're called filaments.

This is where wisps of plasma go out and touch the side.

Now, this one is purely mathematical,

but what the physicists do is make comparisons between the two

to see how well their mathematical version

matches up to what really happened.

And, if you put these two side by side,

you can see how well the mathematical version matches up

with what's really happened.

By comparing the simulation of how the plasma is predicted to behave

to what actually happened, it becomes clear

that the mathematical model accurately predicted

where the plasma would break down.

Now, the reason why this is important is because

there is no limit, really, to the number of mathematical simulations

you can run, but once you get them matching up to reality,

once you know that your mathematical version is an accurate reflection

of what's happening inside,

that is the first step to being able to control your plasma.

Nuclear fusion holds out the promise

of almost unlimited supplies of clean energy.

If they can ever solve this problem,

the answer will lie in mathematics

and its ability to describe an invisible world

of subatomic particles and forces.

The only way you know what's happening inside that plasma

is by using mathematics.

It's the maths that tells you how all of this works.

In trying to replicate what's happening inside a star,

we're pushing the boundaries of what science and maths is capable of.

But we've been doing research in this area for decades,

we've had the equations for even longer, and, yet,

we're still not quite getting perfectly and neatly to the answer.

If there are these gaps around the edges,

if there are limits to how far the maths can take us,

then how can it be discovered?

Maybe it is just an invention after all.

So, where have we got to with our investigation

of mathematics so far?

Well, Newton came along with his fundamental laws of gravity

that led to these incredibly powerful equations

that can precisely predict the movement of planets in the universe.

But they're not quite perfect.

But then you have Cantor and his amazing ideas

about different sizes of infinities

and maybe maths starts to go down a slightly different path

and, the more you go down that road,

the more it starts to feel like mathematics is invented.

Next time, things get even weirder

as the logic of maths starts to break down...

There's a bit of a paradox here.

Who shaves the barber?

And we take another giant leap forward...

Hey!

Amazing!

..as mathematics redefines the nature of space and time.

Einstein completely upended our understanding of space, time,

matter, energy and kind of what else is there to the nature of reality.

I mean, how did he think of that?

Our world is becoming stranger than we realise.

And there may even be multiple versions of it.

Mathematically speaking, in an infinite universe,

everything that's possible has to happen somewhere.

If we trust the maths,

then where it's taking us is somewhere truly bizarre.

Explore more about the magic and mystery of mathematics

and how it impacts our everyday life.

Just go to bbc.co.uk/maths

and follow the links to the Open University.