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 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 go back to the time of the Ancient Greeks
to find out where our fascination with numbers started...
# Laaaaaa! #
You know I think I can hear the neighbourhood cats screeching.
..and reveal why we're now looking for maths deep inside our brains.
Our world is full of maths, often in unusual places
like this roller-coaster.
The thrill and excitement of this ride wouldn't be possible
without physics and engineering.
And at the heart of all of that science...
..is mathematics. Oh, my God!
SHE SCREAMS
It's sobering to think how much we entrust our personal safety to maths...
..without even realising it.
My rush of adrenaline relies on someone's calculations
of kinetic energy, momentum, tensile strengths,
coefficients of friction, and much, much more.
Oh, my God!
What do you make me do?!
To put it bluntly, the modern world wouldn't exist without mathematics.
It is hiding behind almost everything that's around us
and subtly influencing almost everything that we now do.
And, yet, it's invisible.
It's intangible.
So, where does mathematics comes from? Where do numbers live?
It's a question that goes to the very heart of our world.
We often think about numbers as something tied to objects,
like the number of fingers on one hand or the number of petals on a flower.
This flower has got eight petals.
If I take three away, then it will be left with just five.
And it will look a lot less pretty.
The petals are gone, but the number three still exists.
The idea of three, or any other number for that matter,
is still out there, even if we destroy the physical object.
But you can't say that about everything.
If pencils had never been invented, then the idea of a pencil wouldn't exist.
But the idea of numbers would still exist.
In every culture around the world, we all agree on what the concept of fourness is like.
And it doesn't matter whether it's called four,
quatre, vier, or even what the symbol looks like.
With numbers, I can destroy the physical object,
burn it to a crisp, but I can't destroy the idea of numbers.
So, here's the question I want to answer,
is it invented or discovered?
Is there some magical parallel world somewhere
where all mathematics lives? A place where you have fundamental truths
that help us to understand rules of science,
helping us put man on the moon and to study the tiniest particles
of the universe? Or, is maths all in our minds?
Is it just a figment of our imagination and intellect?
Whether maths is invented or discovered
is something we can't agree on.
It's just too extraordinary to think that
the mathematical truths in everything
are a sort of product entirely
of our conventions in the human mind.
That's... I don't think we're that inventive.
It sometimes feels like mathematics is discovered,
especially when the work is going really well and it feels
like the equations are driving you forwards.
But then you take a step back, and you realise it's the human brain that's imposing these ideas,
these patterns, on the world and from that perspective,
it feels like mathematics is something that comes from us.
The number five is called fem in Swedish, my mother tongue.
That part we invent, the baggage, the description,
the language of mathematics.
But structure itself, like the number five and the fact
that it's two plus three, that's the part that we discover.
There's virtually no part of our existence
that isn't touched by maths.
So, if it is discovered, part of the fabric of the universe,
how can we unlock its secrets?
And if it's invented and all in our heads,
how far can our inventive brains take us?
I want to start with the discovered camp, those who say,
"Maths is all around us. You just need to know where to look."
Of all of the structures that you get in nature,
I think one of the most beautiful is the nautilus shell.
There's a little creature that lives inside here and creates all these shapes,
and it hops from one chamber to the next as it grows.
And this shell is just incredibly intricate,
and you might wonder how something so small can create
something quite so remarkable.
But, actually, there is a hidden pattern in here
that you can start to see when you measure these chambers.
So, that one is coming out at...
..14.5 mm.
And this one...
..on the same axis, is...
..46.7.
I'm measuring how wide the shell would have been as the nautilus grew.
I pick an angle and measure the inner chamber,
and then a second measurement to the outer rim.
99.5.
I do this three times for three different angles
until I have three sets of numbers.
When you look at them, I mean, they look pretty random, right?
It looks like there's no connection between them at all.
But looks can be deceptive,
because if you take each of these pairs of numbers
and divide one by the other, a very clear pattern starts to emerge.
So here, if we do this number divided by this,
we get 3.22.
This number divided by this one gives 3.25,
I think. Sorry, my mental arithmetic isn't great!
Eh, and this number divided by this number
then gives 3.24.
And suddenly, the same number starts to appear,
around about 3.2-ish.
It doesn't matter where on the shell you measure,
the ratio of the width of these chambers ends up
being pretty much constant throughout the shell.
I've got it right to one decimal place. Not bad.
That'll be down to my measuring skills rather than the nautilus.
What all of this means is that the nautilus
is growing its shell at a constant rate.
So every time it does a complete turn, it ends up sitting
in a chamber that is around about 3.2 times the width
of the turn before.
And by repeating this very, very simple mathematical rule,
it can create this beautifully intricate spiralled shell.
Clever old nautilus!
The nautilus isn't the only living thing that has a mathematical pattern hidden inside it.
If you've ever counted the petals on a flower,
you might have noticed something unusual.
Some have three petals.
Some five.
Some eight.
Some 13.
But rarely any of the numbers in between.
These numbers crop up time and time again.
They seem random, but they're all part of what's called
the Fibonacci sequence.
You start with the numbers one and one, and from that point,
you keep adding up the last two numbers.
So one and one is two.
One and two is three.
Two and three is five, and so on.
When looking at the number of petals in a flower,
these numbers from the Fibonacci sequence keep appearing,
but that's just the start.
If you look at the head of a sunflower, you'll see
the seeds are arranged in a spiralling pattern.
Count the number of spirals in one direction,
and you will often find a Fibonacci number.
Then, if you count the spirals going in the opposite direction,
you'll hit upon an adjacent Fibonacci number.
Why do plants do this?
Well, it turns out that this is the best way for the flower
to space out its seeds so they don't get damaged.
We find these spirals so intriguing,
we've worked hard to unlock their secrets.
We've gotten very good at copying the patterns that we find in nature
and using them to create things of great beauty,
like this majestic staircase.
Simple, glorious mathematical rules found hidden in nature,
doesn't seem to me like a coincidence.
These mathematical patterns, once you spot them,
do feel discovered.
It's as if the maths is already out there
just waiting for you to find it.
This fascination for finding hidden mathematical patterns
is nothing new.
Go back over 2,000 years to the time of the Ancient Greeks,
and you will find the philosopher Pythagoras and his followers
were just as enthralled by the patterns they discovered.
The Pythagoreans were obsessed with numbers.
They were a people who believed numbers were a gift from God,
and part of their fascination might have been thanks
to their experiments with music.
The Pythagoreans discovered patterns that linked the sound
of beautiful music to the length of a vibrating string.
This, they believed, was no accident, but a window
into gods' worlds that had been gifted to the Pythagoreans.
Mathematician and musician Ben Sparks is fascinated
by this age-old relationship between music and maths.
THEY APPLAUD Well done!
Lovely. Thank you for joining us, Hannah, with your beautiful cello, there.
OK, Ben, you are, eh, you're going to have to explain this to me.
Where does the maths come in in making this instrument sound nice?
The wobbling is what's giving you the sound,
and if you make the string wobble, you hear a sound. So, maybe you could play your D string.
SHE PLAYS
Oh, that sounds lovely! It sounds lovely, doesn't it?
What they also noticed is another note really related to that,
which, if you make it wobble twice as fast, and to do that,
you can make the string half the length. OK.
So, you're putting your finger here -
well, I guess it pretty much is actually halfway along. OK.
SHE PLAYS NOTES
What's weird about these two notes is they sound kind of the same,
but they're definitely different, and this is what the Greeks noticed.
We call it an octave.
But if you play them together, does it sound nice?
SHE PLAYS CHORD
Hm! It was delightfully pleasant.
In the octave, the length of the vibrating string
creates a relationship, or ratio, of two to one.
So, that's if you chop it in half.
Are there other fractions that make it sound nice?
Exactly what the Greeks thought.
"Well, can we find other notes
"that sound even nicer together, more interesting together?"
Can you play us..? This is what they called a perfect fifth.
SHE PLAYS NOTE
What happens when you play those two together, then?
SHE PLAYS TWO NOTES
Very pleasant. Felt like you were about to launch into a jig there!
In a perfect fifth, the ratio of the vibrating string is three to two.
The high note is two thirds of the length of the low note.
What happens when you play a note that isn't one
of these neat fractions?
When notes aren't in these nice simple ratios,
we tend to notice it even if we're not aware of mathematics.
Could you play us a really horrible harmony together -
maybe like a semitone apart?
SHE PLAYS DISCORDANT NOTES
When the strings are not in a simple ratio,
the harmony sounds distinctly unpleasant.
The Greeks were obsessed with having simple ratios
describing the notes, so they get nice harmonious noises.
How does this work for other instruments? This is very clear,
you've got this sort of string here.
But what about, I don't know, the human voice?
Every noise you ever hear is things wobbling
somehow, whether it's your vocal cords in there or a string or...
Not my vocal cords? Well...really?
Have you never used your vocal cords for a bit of music?
Can we try? Oh, no!
I'm such a bad singer. Please, don't make me do this.
OK... Have you got your earplugs in?
Let's try. Can you pitch us a note? I mean, something nice and low.
If you just do it to "la" then I've got a chance to copy you. OK. All right. OK, OK.
SHE CLEARS THROAT
# Laaaaaa! #
Just like the cello, it's the length of mine and Ben's
vocal cords that's changing the pitch of these notes.
So, that was me singing a perfect fifth, a Chariots Of Fire note.
# Laaaa, laaaaa! #
You know, I think I can hear the neighbourhood cats screeching.
So, I think that's enough of that.
These patterns convinced the Ancient Greeks that they'd been gifted
a glimpse into this godly realm.
Why else would these patterns exist?
Pythagoras and his followers were in little doubt that maths
was just as real as the music was, and it was even neater
and more elegant that anything the human mind could conceive of.
The Pythagoreans were by no means the first people to use
some form of maths.
There's some evidence that marks found cut into bones
from the Upper Palaeolithic era 37,000 years ago,
were tally marks used for counting.
But it was the Pythagoreans who were the first to look for patterns.
It does feel to me as if maths is all around us and something
we discover, a fundamental part of the world we live in.
And yet, somehow, very strangely separate from it.
Trying to make sense of this apparent paradox
is at the heart of this battle about where maths really lives.
The philosopher Plato is one of the most important figures
of the Ancient Greek world.
But what he said about the origins of maths is still the basis
for what many mathematicians believe today.
He was fascinated by the geometric shapes that could be produced
by following the rules of mathematics,
rules that he believed came from God.
I'm going to try and draw a circle really, really carefully.
It takes me back to my school days, this.
Now, I'm not doing bad.
That's pretty good. But if you look really closely,
it's just not quite perfect, the circle.
But I'm not going to beat myself up about it because even if
I had access to the most accurate computer in the world,
the circle that it would draw still wouldn't be perfect.
Zoom in close enough, and any physical circle
will have bumps and imperfections.
That's because, according to Plato,
flawless circles don't exist in the real world.
He believed the perfect circle lives
in a divine world of perfect shapes,
a kind of mathematical heaven where all of maths can be found,
but only if you're a true believer.
He was convinced that everything in the cosmos could be represented
by five solid objects known as the Platonic solids.
So, the earth was the rock solid cube.
Fire was the very pointy tetrahedron.
And then, with eight triangular sides, air was the octahedron,
while the icosahedron, with its 20 triangular sides, represented water.
The last Platonic solid, the dodecahedron,
this one was supposed to encapsulate the entire universe.
It's the whole universe sitting in your hands there.
It's kind of a neat idea.
There's something special about the Platonic solids.
They're the only objects where every side is the same shape,
and there are only five.
Try as you might, you will never find another object
with these unique mathematical qualities.
All of these shapes, Plato believed, existed
in a world of perfect shapes
beyond the reach of us mere mortals,
a place we call the Platonic world.
I know that these ideas might seem they're a bit bonkers,
but there are actually quite a few people who believe them,
and those people come across as though they're sane.
Ooh!
My third most favourite mathematical structure,
the octahedron.
A-ha! The Platonic solids, I presume?
It's a dodecahedron! I love dodecahedra.
I have a misspent youth making models of polyhedra.
Oh, my goodness!
These are the Platonic solids!
Oh, guys!
OK. Very beautiful.
You know, at 67, this is Christmas.
Can I keep these two, please?
These Platonic solids, to me, are a great example
of how mathematics is discovered rather than invented.
When Ancient Greeks discovered that this one existed,
they were free to invent the name of it.
They called it the dodecahedron.
But the pure dodecahedron itself,
it was always out there to be discovered.
I have this kind of Platonic view that there are triangles out there.
There are numbers, there are these circles I'm seeking to understand.
So, for me, they feel like quite tangible things.
They're all part of this mathematical landscape that I'm exploring.
But not everyone believes in this Platonic world
of mathematical truths.
I think that the Platonic world
is in the human head.
It's a figment of our imaginations.
I get that there are people who really buy into this other
realm of reality,
and, especially, if your days and nights are spent
thinking about it, investigating, researching this realm.
That doesn't mean that it's real.
Plato would have strongly disagreed.
He encouraged us to believe in this other world
where all of maths could be found, and not to be fooled into thinking
the world around us is all there is.
What we perceive as reality, he cautioned,
is no more than shadows cast on the walls of a cave.
Plato had a very lively and quite dark imagination.
To explain what he meant, he came up with an analogy
of a group of humans locked in a cave.
These people would have been imprisoned since childhood,
and they were shackled by their necks and their legs, and trapped,
staring at a blank wall directly in front of them.
In his mind's eye, Plato pictured a fire
burning high above the prisoners' heads.
But they have no idea it's there.
On top of the wall is a path along which all manner of people
and objects are travelling, but the only thing
the prisoners can see of them
is the shadows they cast down the wall.
Those shadows are the prisoners' reality.
According to Plato, we are no different to the prisoners
in the cave who mistake the shadows for reality.
If Plato is right, what does this mean for you and me?
Is what we think of as reality and maths just an illusion?
Are we living in Plato's cave, erm,
and just see the shadows?
It is not impossible that that is the case.
You know, we are maybe just all... We are just some simulation
in some world of some more intelligent being.
This is all possible.
I mean, if you think that there's some world of mathematical objects.
It's different from ours. It's not the physical world we live in,
But that doesn't make the physical world any less real.
So, I don't think there's anything to me in the philosophy of maths
that would force you to think that our world is an illusion of any kind.
Our senses evolved, really, for one purpose, survival.
But survival and the true nature of reality are two different subjects.
So, the fact that we have been able to survive by thinking
about the world one way does not, in any way, say
that that way of thinking about the world is truly what's happening out there.
Over 2,000 years ago, Plato took the geometry of shapes
as evidence of God's influence, ideas that were limited
to the senses and imagination.
Today, geometry is at the cutting edge of science.
New technologies have allowed us to look at the world
beyond our senses, and once again, it seems the natural world
really is written in the language of maths.
This is a model of a virus.
Straightaway, you notice its geometric shape.
Plato would have recognised this shape as one of the platonic solids.
If there's one person who understands geometry,
it's a mathematician.
Reidun Twarock is a professor of mathematics
at the University of York.
She's trying to work out how viruses use maths
to form their geometric shapes.
If you know that, you can find a way to stop them.
That's why Reidun and her colleagues have designed
a computer simulation that puts the mathematician
at the heart of the virus.
What we try to understand is how this virus forms,
and in order to do that, we will create the illusion of being
inside of the virus, in the position
where the genetic material normally is.
Reidun has discovered that the virus harnesses the power of maths
to build its shell in the quickest and most efficient way possible.
Armed with this knowledge, she's trying to find a way to stop viruses
such as hepatitis B, and even the common cold,
from developing in the first place.
Once you understand how this mechanism works,
you can turn tables on viruses and actually prevent that process.
That is what makes this research so exciting.
If you know the mathematics of how the virus forms its shell,
you can work out a way to disrupt it.
No shell, no virus, no infection.
Today, mathematicians like Reidun are joining the front line
in the fight against disease.
Far beyond the realm of human senses, it really does seem
like the universe somehow knows maths.
It really is amazing how often these patterns seem to crop up.
They're in plants, they're in marine life,
they're even in viruses.
There really is an awful lot that we can explore and exploit
using the mathematics that we have.
It does lend weight to the idea that there is some natural order
underpinning the world around us.
So far, it does feel like the idea that maths is discovered
is leading the charge.
But perhaps we've been looking for patterns in the wrong places.
If it's all in our heads, then the brain feels
like a good place to look.
Is there evidence in there of maths being an invention
of the human mind?
I've got a real treat in store for me today.
I am heading over to UCL, the university that I work at,
where some colleagues are going to scan my brain
and see which bits of it are working whenever I do mathematics.
Neuroscientist Professor Fred Dick is going to place me
inside an FMRI scanner.
If you'd just put your feet up, please.
He'll measure my brain activity by tracking where the blood flows
when I'm answering questions ranging from language to maths.
OK, how was that? All right. Excellent.
If my brain treats the mathematical problems in the same way
as any other problem,
then it suggests there's nothing special about maths.
It's the same as any other language.
A clue, perhaps, that it's an invention.
I'll have ten seconds to think about each question.
I don't need to answer out loud -
I just have to work out the answer in my head.
OK, Hannah, how was that? Good.
Some of those questions were really hard!
Well, we didn't want you to relax in there, really.
I've answered all the questions to the best of my ability.
After a few hours of processing,
Prof Sophie Scott has my results.
Is that my brain? That's your brain.
Let me make sure I understand what I'm seeing, then. OK.
So this is like you've cut my brain in half... Yes.
..and I've got the left-hand side there. Is that right? Yes.
And the right-hand side is that.
So, it's like you've chopped my head down the middle
and then split it out.
Exactly.
So what you can see here, Hannah, is the pattern of activity
in your brain when you're hearing straightforward language,
and here you can see, in the left hemisphere,
very classic language areas activated.
The bright yellow areas are where there's increased blood flow,
an indication that the neurons in the left-hand side of my brain
are working harder.
This is a side of the brain that we know is linked to language.
Compare that to the right-hand side of my brain where there's hardly
any yellow areas, which means there's far less activity taking place.
So, can we see maths, please?
Oh, hold on.
And look at that. This whole bit here. Yeah,
and also down at the bottom there.
This time, when I'm thinking about maths, there ARE yellow areas
in the right-hand side of my brain.
This is very different to the lack of activity seen
when I was thinking about language.
These scans reveal there seems to be a place in our brains
where maths lives.
What we're definitely able to say is this is not just the meanings of the words that you were reading.
We're not just looking at you thinking about the meaning of words.
You're seeing something that does seem to be qualitatively different for the maths.
Maths is real. Maths is real. At least in my head. Yes.
THEY LAUGH
I tell you what really struck me about that conversation with Sophie
just then, is that...
..it doesn't matter whether you're doing two plus two equals four
or whether you're answering these much higher-level maths questions.
It's the same bit of your brain that's doing the grunt work.
It's not the same thing that does words or language.
You're seeing these problems, and you're manipulating them
in your mind.
Research with similar experiments shows it's broadly the same
for all of us.
In your brain and mine, there is a specific place
where we do maths.
But this doesn't prove that maths is something we discover.
It could still be an invention - just one that we learn at school.
To get to the bottom of this question, I need some
volunteers who've never had a maths lesson in their lives.
We need to just put one of those on there. Very nice!
That worked very nicely, didn't it?
Yeah, you liked that, did you?
Dr Sam Wass is an experimental psychologist
at the University of East London.
Helping him with some experiments are six-month-old Ira
and Leo, who's just under a year.
To begin with, each child is placed in a room where they're shown
a series of images.
Sam uses a battery of tests to analyse how they react
to different situations.
The first experiment uses eye-tracking technology
to see how the baby follows the movement of a piggy puppet.
Was that good?
So, here we can see a feed out of what the child is looking at,
and those two red dots are where the baby is looking.
What we're presenting is a puppet that jumps up and then disappears,
and it jumps up and disappears two times in a row and then it stops.
We present this same sequence again and again.
And as the baby watches it again and again,
their looking times, the amount of attention that they're paying
to the screen diminishes. And that tells us that the child has learned this sequence.
Now, instead of popping up twice, as expected,
the puppet appears three times.
Does the child notice a difference between the two-ness of it
popping up twice in a row and the three-ness of it popping up three times?
And if it does, then that tells us that the child understands
the difference between two and three.
These tests reveal that the child is surprised
when the puppet appears more often.
When larger scale experiments were carried out by researchers
in the US, the results suggested that infants DO have a sense of quantity.
So, this research is really important because it's suggested
that even infants as young as five months old can do the basics
of addition and subtraction.
They know the difference between one plus one equals two
and one plus one equals one, which is an incorrect conclusion.
That, though, is a really, really strong, provocative finding.
This idea that the concept of mathematics and the basics
of mathematics rules might be hard-wired, might be innate
our genetic code.
This research isn't conclusive, but it does suggest we all come
preprogrammed to do maths.
Some argue that we evolved this maths part of our brains
to discover the world of mathematical truths.
The evidence for maths being discovered is compelling.
We've found patterns in nature, the latest technology has uncovered
startling patterns in viruses,
and scans reveal there's a part of our brains where maths lives.
But this question is too important to leave the evidence here
and move on.
If it IS discovered, if it lives in this other world,
can we trust what it's telling us?
How do we know that our idea of numbers is right?
How do we know someone isn't just going to come along at some point and say, "Well, actually,
"you've got that completely wrong, and one plus one doesn't equal two after all"?
How do we know that we can rely on the maths that we take for granted?
What you need to be sure of is your foundations.
If they're shaky, then all of your carefully constructed ideas
come crashing down.
And there was one mathematician who understood this only too well.
His name was Euclid.
Around 300BC in Alexandria,
he wrote one of the most famous and important books of all time -
The Elements.
He was trying to go right back to the beginning to find
the smallest elements on which you can build the vast,
gigantic structure of mathematics.
If you have a little flick through, you can see the kind of things
that Euclid was considering.
So, here it says that you can draw a straight line between
any two points, which seems blindingly obvious.
And here, it says that all right angles are the same.
Now, these are quite simple concepts, but I think they really illustrate
just how exhaustive Euclid had to be to build the foundations
for what was to come.
He took statements like these, which mathematicians assumed were true,
and put them to the test.
He then set out to prove a whole host of other theories
based on these fundamental building blocks.
This was really the first time that someone had written down
formal proofs for mathematical assumptions.
Now, mathematical proof isn't like scientific proof
or proof in a court of law.
There's no room for reasonable doubt here.
Instead, if something is true mathematically once,
then it is true forever.
And that is why this book is so important.
It's the reason why Euclid's Elements is still relevant today.
Every page within it is as true now as it ever was.
And from that point of view, it really does feel like we're tapping into a world that already exists.
One that's just out there waiting to be discovered.
Unless, of course, you throw a spanner in the works,
change the language of maths, and invent a better way
of doing things.
Suddenly, this rock solid world of God-given truths
might fill decidedly shaky.
Uno... ALL: Dos, tres, cuatro, cinco.
Well done. Good counting.
Muy bien!
One thing we know about languages is that they never stand still.
They're constantly evolving to meet the challenges
of a changing world.
47? Cuarenta y siete.
49? Cuarenta y nueve.
Muy bien!
Let's go for another tricky one.
For centuries, the language of maths was thought
to be fixed and unchangeable.
That is, until something was found to be missing.
It is the number zero.
What exactly is zero?
A zero means nothing.
If you've got zero flowers, you've got no flowers.
And if you've got zero of something, you have got nothing.
So, you can't really do anything with the zero.
I don't really use it when I'm counting in numbers.
Before the 7th century, neither did anyone else.
Though people have always understood the concept of having nothing,
the concept of zero is relatively new.
We had numbers and could count, but zero didn't exist.
If you think about it for long enough,
zero is actually quite a strange concept.
It's almost as though the absence of anything becomes something.
Is it just a number or an idea?
And how can something with no value have quite so much power?
It's not exactly clear who first thought of zero.
It might have originated in China or India.
What we do know is that zero arrived in Europe from the Middle East
at about the same time as the Christian Crusades
against Islam, when ideas coming out of the Arab world
were often met with suspicion.
The West already had a numerical system - Roman numerals.
They did the job, but were a bit unwieldy.
For example, the number 1958 is written as
MCMLVIII.
And no matter where you place, say, the letter C,
it will always represent the number 100.
It was good for its time, but times change,
and a better system was needed.
Zero was different.
Where you placed zero could change the values of the numbers around it.
Think of the difference between 11 and 101.
Although the concept of zero might've been created elsewhere,
it was in India that zero started to be accepted as a proper number.
This is a page from the Indian Bakhshali manuscript
from around AD 225,
which shows the dots above the characters representing zero.
This is the earliest known use of the symbol zero
that we know today.
For almost 1,000 years, Indian mathematicians worked happily
with zero, while their Western counterparts ploughed on
with the Roman numerals.
That was until Italian mathematician Fibonacci
recognised its potential.
Now, he'd been educated in North Africa,
so he'd seen this number system working first-hand.
Zero is a placeholder signifying the absence of a value.
Zero is also a number in its own right.
It allowed you to write down numbers and manipulate them
much more quickly and easily than Roman numerals.
Realising all this, Fibonacci championed the new number
and brought it to the attention of Western Europe.
Zero wasn't something that we discovered,
so much as something that was created as part
of a new language to describe numbers.
That's not to say it isn't useful -
the whole of modern technology is literally built on ones and zeros.
But, suddenly, maths feels like something we've come up with.
Something we've invented.
We needed a more user-friendly numerical system,
so someone came up with the clever idea of zero.
Not a gift from the gods, but a smart way to make
counting more convenient.
This is intriguing evidence that maths might be invented after all,
a product of our intellect and imagination.
Once the idea of zero had been widely accepted,
mathematicians could relax.
All conceivable numbers lay out on a single line
with no holes and no gaps to speak of.
Over here you have the positive numbers.
One sheep, two donkeys,
the kind of stuff you find in real life.
And in the other direction, all the negative numbers.
It's a bit trickier to imagine what negative one sheep looks like.
SHEEP BAAS
The number line stretches out in both directions all the way to infinity,
and zero sits proudly in the middle.
Everything was well in Numberland... Or was it?
This is where it all starts to get a bit strange,
because there are some numbers that are simply weird.
There are some fundamental rules of maths that you learned at school.
2 x 2 = 4.
3 x 3 = 9.
A positive number multiplied by itself
equals another positive number.
Nothing controversial so far.
Curiously, a negative number multiplied by itself
also gives a positive number.
Why is that?
Well, this is not a maths lecture, so let's just accept it
as a fact and move on.
In fact, if you take any number and multiply it by itself or square it,
then the answer is always going to be positive.
If +2 squared gives me +4,
and -2 squared gives me +4, what do I have to square to get -4?
But it's a question without an answer.
There is no number that when multiplied by itself
gives a negative answer.
That is unless you invent your own.
Meet "i", a number we simply made up.
Not everyone was keen. It became known as an "imaginary number",
a deliberately chosen derogatory term to scoff at its existence.
It turns out "i" is really useful,
especially when it comes to simplifying problems with things
like electricity or wireless technologies, things that otherwise
would seem impossible to solve.
Essentially, if you're working with waves, you will use "i".
This imaginary number broke all the rules.
It didn't come from this world of ethereal numbers,
it wasn't God-given.
It was very definitely invented.
If you can have one imaginary number,
why can't you have two or three
or infinitely many of them?
Why can't you have negative imaginary numbers, as well?
Why can't, in fact, imaginary numbers have their very own number line?
Exactly the same as the real one, just on a different axis.
The number line isn't a single line at all.
Numbers are two-dimensional.
You might think this all sounds a bit airy-fairy.
Imaginary numbers that we just made up.
But if you've ever flown in an aircraft,
you've already trusted your life to these strange numbers.
AIR TRAFFIC RADIO CHATTER
At Gatwick Airport, air traffic controllers here rely on radar
to keep everything moving safely and quickly.
Once we get busy, we definitely need radar.
So the busier the tower, the busier the operation,
you need radar.
Radar works by sending out radio waves and examining that part
of the signal that's reflected back.
The complex equations that allow us to filter out the correct signal
from other conflicting frequencies is heavily dependent
on imaginary numbers.
In this case, separating out moving objects like planes from flocks
of birds or stationary objects.
Imaginary numbers are a very efficient tool to be able
to manipulate radio waves.
Imaginary numbers are fundamental to the operation.
Imaginary numbers allow us to track planes in real time.
Without them, we never would've been able to use radar in our skies.
AIR TRAFFIC RADIO CHATTER
When I started this investigation going back to the time
of the Ancient Greeks, it did seem like maths
could only be discovered.
There were too many coincidences, too many mathematical patterns
popping up all over the place.
But if we can invent the rules and create new numbers
and they work, then perhaps I got it wrong.
Maybe maths IS invented after all.
The concept of zero or negative numbers or complex numbers
or imaginary numbers, they caused great consternation
to the cultures that first invented or encountered them.
There are some conjectures that zero came because someone constructing
noticed that as you dig a piece of earth out to make a hole,
there's something, an indentation left there, that should have a name.
Zero kind of maybe came from that observation.
The power of mathematics lies in the way its language and symbols
have allowed us to manipulate the world.
But this was a world that followed the rules of God and the Church.
By the 17th century, a new breed of intellectual
was merging, not afraid to challenge authority.
There was one man who dared to question all of the philosophical
and scientific assumptions that had gone before.
This was someone who was trying to promote a new way of thinking,
using reason, experimentation and observation.
This was the young Frenchman called Rene Descartes.
It was while in a restless sleep in 1619 that Descartes experienced
a series of dreams that would change his life - and mathematics.
The first two could be better described as nightmares.
But the third dream...
The third dream was intriguing.
As his eyes scanned the room, he saw books on the bedroom table
that appeared and then disappeared.
He opened one book of poems and at random caught the opening line
of one, which read, "What road shall I pursue in life?"
Then someone appeared out of thin air and recited another verse,
saying simply, "What is and is not?"
As with dreams, it's all about the interpretation
you place upon them.
In Descartes's case, the effect of these dreams was profound.
He was convinced that the dreams were pointing him
in a single direction,
bringing together the whole of human knowledge by the means of reason.
He was nothing if not ambitious, but his genius led to
perhaps one of the greatest advances ever in the field of mathematics.
As with so many brilliant ideas, it was deceptively simple.
Let's say that I'm meeting a friend for a coffee.
Now, I'm standing at the end of Endsleigh Gardens,
and they are somewhere over on Gordon Street.
It's very easy for me to work out how to get there.
All I need to do is go on a map and check the route.
In this case, three streets down and one along.
It sounds like an incredibly simple idea,
but, actually, it revolutionised mathematics.
He showed that a pair of numbers can uniquely determine the position
of a point in space.
It sounds trivial, but this was just the start.
It gets more interesting when you apply this idea to curves.
As this point moves around a circle its coordinates change,
and we can write down an equation that precisely and uniquely
characterises this circle.
For the first time, shapes could be described by a formula.
By uniting the language of numbers and equations and symbols
with shapes, Descartes was able to expand the horizons
of mathematics, thus laying the foundations
for the modern scientific world.
What Descartes and the other trailblazers like him did
was to question the accepted wisdom of the time.
They thought differently, and the result was that they delivered
monumental breakthroughs for our understanding of the universe.
Descartes lived in a time when many philosophers backed up
their arguments with appeals to God.
But Descartes preferred to place his trust in the power
of human logic and maths.
He believed all ideas should have their foundations
in experience and reason rather than tradition and authority.
It still feels like maths belongs to a discovered world,
but after Descartes, it's a world that is increasingly devoid
of a divine influence.
We started this episode with just one question -
is mathematics invented or discovered?
And, based on the evidence so far, I'm leaning quite heavily
towards discovered, because it doesn't seem to me to be possible
that something so all-encompassing could be the product
of the human mind alone.
Next time, I see how new mathematical systems
allowed Newton to create his laws of gravity...
..and even started describing the existence of things
we didn't know were there.
All more evidence for maths being a discovery.
Now, this couldn't be a coincidence.
But I'm forced to think again when I confront one of
the strangest mathematical concepts there is -
infinity.
And it makes the question of whether maths is invented or discovered
a lot more difficult to answer.
What makes our world work the way that it does?
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.