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 You
