There is a mystery at the heart of our universe a
Puzzle that so far no one has been able to solve it can be weird. Welcome to the world
If we can solve this mystery it will have profound consequences
For all of us and 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 have merely discovered I think discovered
Invented it's both. I have no idea
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 Hanna Frey 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
Human dissipation
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
You know, I think I can hear the neighborhood 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 rollercoaster
This ride wouldn't be possible without physics and engineering and at the heart of all of that science
It's math master oh my god
It's sobering to think how much we entrust our personal safety to matter
Without even realizing it
My rush of adrenaline
relies on someone's calculations of kinetic energy momentum
tensor strengths
coefficients of friction and much much more
Do you make me do
So pretty bluntly the modern world wouldn't exist without mathematics
he is hiding behind almost everything that's around us and
subtly imprinting almost everything that we now do
And yet it's invisible. It's
intangible
So where does mathematics come from where des 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. I
Don't 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 Faunus is like
And it doesn't matter whether it's called for cat fur fear 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 the 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
Where the maths is invented or discovered is something we can't agree on
It's just too extraordinary to think that
the mathematical truths and everything is sort of product entirely of our
Conventions and the human mind and 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 realize that it's a 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 and Swedish my mother-tongue that part we invent the baggage
The description the language of mathematics but the 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 only think one of the most beautiful is the nautilus shell
So there's a little creature that lives inside a and creates all these shapes and 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
Fourteen point five
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
Ninety nine point
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 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
3.2 - this number divided by this one gives
3.25 I think sorry I meant arithmetic isn't great
I think this number divided by this number then gives
3.24
and suddenly
The same number starts to appear around about 3.2 ish 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 show
I've got it right to one decimal place
bad
That will 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 three point two times the width of the turn before
And by repeating this very very simple mathematical rule. It can create this beautifully intricate
spiraled shell
clever author lists
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
May seem random, but they're all part of what's called the Fibonacci sequence
You start with the numbers 1 & 1 and from that point do you keep adding up at the last two numbers?
so one on one is -
1 + 2 is 3
2 + 3 is 5 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 spiraling 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 stair
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. I'm 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 pythagorean's were obsessed with numbers
They were people who believed that numbers were a gift from God and part of their fascination
Might have been thanks to their experiments with music
The pythagorean's 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 God's worlds that had been gifted to the pythagorean's
Mathematician and musician Ben sparks is fascinated by this age-old relationship between music and maths
Thank you for joining us on our with your beautiful cello there, okay, then you are
You're gonna have to explain this to me
Where does the math 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
Oh
Sounds lovely sounds lovely
Doesn't it and what they also notice is another no really related to that which is making if you make it wobble
Twice as fast and to do that you can make the string half the length. Okay, so you're putting your thing here why I guess
Pretty much. Is that not hard way along? Okay
What's weird about these two noses
They they sound kind of the same but they're definitely different and this is where the Greeks notice they we call it an octave
But if you play them together, does it sound nice?
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
Well exactly what the Greeks were thinking is like what can we find other notes that sound even nicer to get a more interesting together?
Can you play us? This is what they call a perfect fifth
What happens when you play those two together there
Very pleasant the first we're about to launch into a jig
Yeah 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 that even if we're not aware of the mathematics, right?
I mean, can you play a sort of really horrible harmony together, maybe like a semitone apart?
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?
I mean, this is very clear. You've got this this sort of string here
But what about and I don't know like the human voice, right?
Well, every noisy over here is things wobbling somehow whether it's your vocal cords
or a string or my vocal cords
Really? Have you never used your vocal cords for a bit of music. Can we try? Oh
I'm such a fine singer. Please. Don't make me do this. Okay, you got your earplugs in? Yeah
Let's try can you picture some note
I mean, there's something nice and low if you just do it to LA then now I've got a choice
Just like the cello it's the length of mine and Ben's vocal chords that's changing the pitch of these notes
So that was me singing a perfect fifth carats if I know love
You know, I think I can hear the neighborhood 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 young
Why else would these patterns exist?
Pythagoras and his followers were in little doubt the maths
Was just as real as the music was and it was even neater and more
elegant than anything the human mind could conceive of
The pythagorean's 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 Paleolithic era
37,000 years ago. We're tally marks used for counting
But it was the pythagorean's 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
When I try and draw a circle really really carefully
Takes me back to my school days this now not doing bad
That's pretty good, but if you look really closely
It's just not quite perfect to the circle
but I'm not gonna beat myself up about this 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
Say the earth was the rock solid cue
fire was the very pointy tetrahedron and then with eight triangular sides air was the
Octahedron while the icosahedron there was twenty triangular sides represented water
The Lance brittonic Sollers the dodecahedron. This one was supposed to encapsulate the entire
Universe, it's the whole universe sitting any more 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 like 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
My third favorite favorite mathematical structure the
octahedron
AHA the Platonic solids I presume
It's a very heated. I love dodecahedra. I have a misspent youth making
Models of polyhedra. Oh my goodness
These are the Platonic solids Oh guys
Okay, very beautiful, you know at 67 mrs. 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 double key. He drew himself
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 that 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
Imaginations I get that that there are people who really buy into this other realm of reality
And especially if your days and nights are spent thinking about and investigating and 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 of 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
Entrapped staring 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 traveling
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 and and just see see shadows
It is not impossible that that is the case
You know, we are maybe just all we're 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 that there's anything to me in 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 straight away you notice it's geometric shape
Plato would have recognized this shape as one of the Platonic solids
If there's one person who understands geometry, it's a mathematician
Ride into a rock 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 ridin and her colleagues have designed a computer
simulation that puts the mathematician at the heart of the virus
but we try to understand is how this virus forms and
We taught her to do that
We will create the illusion of being inside of the virus in the position where the genetic material normally is
Ryden has discovered that the virus harness is 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 the 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 the way to disrupt it
No, shell no virus no infection
Today mathematicians like rydym 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 there in plants there in marine life there 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 of Iowa cap
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
He'll measure my brain activity by tracking where the blood flows when I'm answering questions
ranging from language to maths
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
Okay, hello, how is that
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
Professor Sophie Scott has my results. It's not my brain. That's your brain
Let me make sure I understand what I'm seeing it. Okay, so this is like you've cut my brain in half
Yes, and I've got the left hand side there. Yes, and the right hand size 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 and you're hearing the straightfoward 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
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. This is real at least in my head
At a what really struck me about that conversation and safe it is then it's that
It doesn't matter whether you're doing
Two plus two equals four or whether you're answering these much higher level math 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 math lesson in their lives
Dr. Sam wass is an experimental psychologist at the University of East London
Helping him with some experiments are six months 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 analyze 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. Is 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 the same sequence again and again and as the baby watches it again
and again
They're looking times the amount of attention that they're paying to the screen diminishes and that tells us that the Trotters to learn this sequence
Now instead of popping up twice as expected the puppet appears three times
Does the child notice the difference between the tunas of it popping up twice in a row and the three nests 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 are did the 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 and that there was a really really strong
provocative finding yeah this idea that the concept of mathematics and the basics of mathematics rules might be
Hardwired my DNA in our genetic code
This research isn't conclusive but it does suggest we all come pre-programmed 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 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 gonna 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 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 300 BC 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 flip through you can see the kind of things that you cleared was considering
So here it says that you can draw a straight line between any two points, which it's blindingly obvious
And here it says that all right angles are the same
And these are quite simple concepts
But I think that 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 is 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 feel decidedly shaky
One thing we know about languages is that they never stand still they're constantly evolving to meet the challenges of a changing world
Forty-nine
Wien
let's go for another tricky one for
Centuries the language of mass was thought to be fixed and unchangeable
That is until something was found to be missing
Most exactly is zero
Zero means nothing if you've got zero
flowers, you've got no flowers and if you guys zero stop
Zero or something. You've 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 seventh 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 only dear?
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
suspicions
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
Mcm-l
VIII 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 place zero could change the values of the numbers around it
Think of the difference between eleven and a hundred and one
Although the concept of zero might have been created elsewhere
It was in India that zero started to be accepted as a proper number
This is a page from the Indian back Shawnee manuscripts 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 a thousand years Indian mathematicians worked happily with zero while their Western counterparts
Piled on with the Roman numerals that was until Italian mathematician Fibonacci
recognized its potential
Now he'd been educated in North Africa. So he'd seen this number system working firsthand
Zero is a placeholder
Signifying the absence of a value a 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
Realizing all this Fibonacci champions the new number and brought it to the attention of Western Europe
Zero wasn't something that we discovered it so much as something that was created as part of a new language to describe
Numbers and that's not to say 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
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
two times two equals four
three times three equals nine 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 math 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
Square it then the answer is always going to be positive
If plus two squared gives me plus four and
Minus two squared gives me plus four. What do I have to square to get minus four?
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
Meat I a number we simply made up
Not everyone was keying it became known as an imaginary number a deliberately chosen
Derogatory term to scoff at its existence
It turns out AI 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 are beached airy-fairy the imaginary numbers that we just made up
But if you've ever flown in an aircraft, you've already trucks they're your life to these strange numbers
Airports air traffic controllers here rely on radar to keep everything moving safely and quickly
Once you can busy definitely need Raider, 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 develop eration
Imaginary numbers allow us to track planes in real time without them. We never would have been able to use radar in our skies
Did you know folks father? Nice try dad? They just told you to text you a Papa
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 creates new numbers and they work then perhaps I've 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 notice that as you dig a piece of earth out to make a hole
There's something an indication that 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 emerging 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 it appeared and then disappeared
The opened one book of poems and at random
Cost the opening line of one, which read what road shall I kiss you 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 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 ensley 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 alone
it's Sinan's like an incredibly simple idea but actually
It revolutionized 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 characterizes 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 do was to question
Accepted wisdom at 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
And 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. There's 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 whether
All more evidence our maths being a discovery
Now this could 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 where the math 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 code at UK forward slash maths and follow the links to the Open University
And
Hannah returns next Wednesday at 9:00 in the meantime. She's with Adam Rutherford
Investigating everyday mysteries from pain thresholds to deja vu the curious cases of Rutherford and Frey on iPlayer Radio
Tomorrow here on BBC four Brian Cox marveling at our place in the great scheme of things human universe a date
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