Category
📺
TVTranscript
00:00There is a mystery at the heart of our universe. A puzzle that so far no one
00:08has been able to solve. It's too weird. Welcome to my world. If we can solve this
00:15mystery it will have profound consequences for all of us. That mystery
00:21is why mathematical rules and patterns seem to infiltrate pretty much
00:26everything in the world around us.
00:30Many people have in fact described maths as the underlying language of the
00:37universe. But how did it get there? Even after thousands of years this question
00:45causes controversy. We still can't agree on what maths actually is or where it
00:51comes from. Is it something that's invented like a language or is it
00:56something that we have merely discovered? I think discovered. Invented.
01:00It's both. I have no idea. Oh my god! Why does any of this matter? Well maths
01:09underpins just about everything in our modern world. From computers and mobile
01:15phones to our understanding of human biology and our place in the universe.
01:22My name is Hannah Fry and I'm a mathematician. In this series I will explore how the greatest
01:30thinkers in history have tried to explain the origins of maths extraordinary power.
01:36I'm going to look at how in ancient times our ancestors thought maths was a gift from the gods.
01:48How in the 17th and 18th centuries we invented new mathematical systems and used them to create
01:54the scientific and industrial revolutions. And I'll reveal how in the 20th and 21st centuries
02:03radical new theories are forcing us to question once again everything we thought to be new about maths
02:12and the universe. The unexpected should be expected because why would reality down there bear any resemblance
02:19to reality up here. In this episode I go back to the time of the ancient Greeks to find out where our
02:29fascination with numbers started. You know I think I can hear the neighborhood cats screeching.
02:37And reveal why we're now looking for maths deep inside our brains.
02:53Our world is full of maths often in unusual places like this roller coaster.
03:05The thrill and excitement of this ride wouldn't be possible without physics and engineering
03:10and at the heart of all of that science, it's mathematics. Oh my god!
03:30It's sobering to think how much we entrust our personal safety to maths
03:35without even realising it. My rush of adrenaline relies on someone's calculations of kinetic energy,
03:49momentum, tensile strengths, coefficients of friction and much, much more.
03:57What do you make me do?
03:59To put it bluntly, the modern world wouldn't exist without mathematics. It is hiding behind almost
04:11everything that's around us and subtly influencing almost everything that we now do. And yet,
04:18it's invisible. It's intangible. So where does mathematics come from? Where do numbers live?
04:29It's a question that goes to the very heart of our world. We often think about numbers as something
04:45tied to objects like the number of fingers on one hand or the number of petals on a flower.
04:51This flower has got eight petals. If I take three away, then it will be left with just five.
05:02And it will look a lot less pretty. The petals are gone, but the number three still exists.
05:09The idea of three, or any other number for that matter, is still out there, even if we destroy
05:19the physical object. But you can't say that about everything.
05:26If pencils had never been invented, then the idea of a pencil wouldn't exist.
05:32But the idea of numbers would still exist. In every culture around the world, we all agree
05:43on what the concept of fourness is like. And it doesn't matter whether it's called four,
05:49four, quatre, vier, or even what the symbol looks like.
05:58With numbers, I can destroy the physical object, burn it to a crisp. But I can't destroy the idea of numbers.
06:06So here's the question I want to answer. Is it invented or discovered? Is there some magical
06:19parallel world somewhere where all mathematics lives? A place where you have fundamental truths
06:25that help us to understand the rules of science, helping us put man on the moon and to study the
06:31tiniest particles of the tiniest particles of the universe? Or is maths all in our minds? Is it just
06:39a figment of our imagination and intellect? Whether maths is invented or discovered is something we can't
06:49agree on. It's just too extraordinary to think that the mathematical truths and everything are sort of
06:57product entirely of our conventions and the human mind. I don't think we're that inventive.
07:04It sometimes feels like mathematics is discovered, especially when the work is going really well.
07:11And it feels like the equations are driving you forward. But then you take a step back and you
07:15realize that it's the human brain that's imposing these ideas, these patterns on the world. And from that
07:21perspective, it feels like mathematics is something that comes from us. The number five is called femme
07:29in Swedish, my mother tongue. That part we invent, the baggage, the description, the language of
07:37mathematics. But the structure itself, like the number five and the fact that it's two plus three,
07:42that's the part that we discover. There's virtually no part of our existence that isn't touched by maths.
07:51So if it is discovered, part of the fabric of the universe, how can we unlock its secrets? And if it's
07:58invented and all in our heads, how far can our inventive brains take us? I want to start with the
08:07discovered camp. Those who say maths is all around us. You just need to know where to look.
08:14Of all of the structures that you get in nature, I think one of the most beautiful is the Nautilus
08:22shell. So there's a little creature that lives inside here and creates all these shapes and it hops from
08:28one chamber to the next as it grows. And this shell is just incredibly intricate and you might wonder how
08:36something so small can create something quite so remarkable. But actually there is a hidden pattern
08:43in here that you can start to see when you measure these chambers.
08:48So that one is coming out at 14.5 millimeters.
09:00And this one on the same axis is
09:0646.7.
09:14I'm measuring how wide the shell would have been as the Nautilus grew.
09:18I pick an angle and measure the inner chamber and then a second measurement to the outer rim.
09:2499.5. I do this three times for three different angles until I have three sets of numbers.
09:34When you look at them, I mean they look pretty random, right? It looks like there's no connection
09:38between them at all. But looks can be deceptive because if you take each of these pairs of numbers
09:47and divide one by the other, a very clear pattern starts to emerge.
09:51So here if we do this number divided by this, we get 3.22. This number divided by this one
09:59gives 3.25, I think. Sorry, my mental arithmetic isn't great.
10:08This number divided by this number then gives 3.24. And suddenly the same number starts to appear,
10:20around about 3.2-ish. It doesn't matter where on the shell you measure,
10:25the ratio of the width of these chambers ends up being pretty much constant throughout the shell.
10:31I've got it right to one decimal place. Not bad. That will be down to my measuring skills,
10:38rather than the nautilus. What all of this means is that the nautilus
10:44is growing its shell at a constant rate. So every time it does a complete turn, it ends up sitting
10:51in a chamber that is around about 3.2 times the width of the turn before. And by repeating this
10:59very, very simple mathematical rule, it can create this beautifully intricate spiralled shell.
11:07Clever old nautilus.
11:09The nautilus isn't the only living thing that has a mathematical pattern hidden inside it.
11:16If you've ever counted the petals on a flower, you might have noticed something unusual.
11:22Some have three petals, some 5, some 8, some 13, but rarely any of the numbers in between.
11:36These numbers crop up time and time again. They seem random, but they're all part of what's called
11:44the Fibonacci sequence. You start with the numbers 1 and 1, and from that point you keep adding up the last
11:52two numbers. So 1 and 1 is 2, 1 and 2 is 3, 2 and 3 is 5, and so on.
12:04When looking at the number of petals in a flower, these numbers from the Fibonacci sequence keep
12:11appearing, but that's just the start. If you look at the head of a sunflower,
12:18you'll see the seeds are arranged in a spiralling pattern.
12:25Count the number of spirals in one direction and you will often find a Fibonacci number.
12:32Then if you count the spirals going in the opposite direction,
12:36you'll hit upon an adjacent Fibonacci number.
12:41Why do plants do this? Well, it turns out that this is the best way for the flower
12:47to space out its seeds so they don't get damaged.
12:53We find these spirals so intriguing we've worked hard to unlock their secrets.
13:00We've gotten very good at copying the patterns that we find in nature and using them to create
13:06things of great beauty, like this majestic staircase.
13:21Simple, glorious mathematical rules found hidden in nature doesn't seem to me like a coincidence.
13:29The mathematical patterns, once you spot them, do feel discovered. It's as if the maths is already out there,
13:40just waiting for you to find it.
13:42This fascination for finding hidden mathematical patterns is nothing new.
13:55Go back over 2000 years to the time of the ancient Greeks and you will find the philosopher Pythagoras
14:02and his followers were just as enthralled by the patterns they discovered.
14:09The Pythagoreans were obsessed with numbers. They were a people who believed
14:14that numbers were a gift from God. And part of their fascination might have been thanks
14:20to their experiments with music.
14:39The Pythagoreans discovered patterns that linked the sound of beautiful music to the length of a
14:45vibrating string. This, they believed, was no accident, but a window into God's world that had been gifted
14:53to the Pythagoreans. Mathematician and musician Ben Sparks is fascinated by this age-old relationship
15:01between music and maths.
15:03Well done. Thank you for joining us, Hannah, with your beautiful cello there.
15:18OK, Ben, you are, you're going to have to explain this to me. Where does the maths come in in making
15:24this instrument sound nice? The wobbling is what's giving you the sound and if you make the string wobble,
15:29you hear a sound. So maybe you could play your D string.
15:33Oh, sounds lovely. Sounds lovely, doesn't it? And what they also notice is another note really
15:38related to that, which is if you make it wobble twice as fast, and to do that, you can make the
15:43string half the length. OK, so you're putting your finger here, well, I guess pretty much is actually,
15:49not halfway along. OK.
15:54What's weird about these two notes is they sound kind of the same, but they're definitely different,
15:58and this is what the Greeks notice. We call it an octave.
16:01But if you play them together, does it sound nice?
16:08Delightfully pleasant.
16:11In the octave, the length of the vibrating string creates a relationship or ratio of two to one.
16:19So that's if you chop it in half. Are there other fractions that make it sound nice?
16:24Well, exactly what the Greeks were thinking. It was like, well, can we find other notes that
16:26sound even nicer together, more interesting together? Can you play a, this is what they call a perfect fifth.
16:31What happens when you play those two together then?
16:39Very pleasant. I feel like we're about to launch into a jig there.
16:42In a perfect fifth, the ratio of the vibrating string is three to two. The high note is two-thirds
16:51of the length of the low note. What happens when you play a note that isn't one of these neat fractions?
16:58When notes aren't in these nice, simple ratios, we tend to notice it even if we're not aware of the
17:03mathematics, right? I mean, could you play a really horrible harmony together? Maybe like a semitone apart?
17:09When the strings are not in a simple ratio, the harmony sounds distinctly unpleasant.
17:15The Greeks were obsessed with having simple ratios describing the notes, so they get nice harmonious noises.
17:22How does this work for other instruments? I mean, this is very clear, you've got this sort of string here,
17:26but what about, I don't know, like the human voice? Right, well, every noise you ever hear is things wobbling
17:31somehow, whether it's your vocal chords, or a string, or a... Not my vocal chord!
17:35Well, really, have you never used your vocal chords for a bit of music? Can we try?
17:39Oh no! I'm such a bad singer, please don't make me do this.
17:43Okay. Have you got your earplugs in?
17:47Let's try, can you pitch us a note? I mean, there's something nice and low,
17:50if you just do it to la, then I've got a chance and a copy.
17:53Okay, all right, okay, okay.
18:00Just like the cello, it's the length of mine and Ben's vocal chords that's changing the pitch of these notes.
18:10So that was me singing a perfect fifth.
18:12Aha!
18:13There's Jarrett's a fire note, la, la!
18:16You know, I think I can hear the neighborhood cats screeching, so I think that's enough of that.
18:22These patterns convinced the ancient Greeks that they'd been gifted a glimpse into this godly realm.
18:30Why else would these patterns exist?
18:32Pythagoras and his followers were in little doubt. The maths was just as real as the music was.
18:42And it was even neater and more elegant than anything the human mind could conceive of.
18:48The Pythagoreans were by no means the first people to use some form of maths.
18:55There's some evidence that marks found cut into bones from the upper paleolithic era 37,000 years ago,
19:03where tally marks used for counting.
19:07But it was the Pythagoreans who were the first to look for patterns.
19:14It does feel to me as if maths is all around us and something we discover,
19:19a fundamental part of the world we live in, and yet somehow very strangely separate from it.
19:27Trying to make sense of this apparent paradox is at the heart of this battle about where maths really lives.
19:35The philosopher Plato is one of the most important figures of the ancient Greek world.
19:44But what he said about the origins of maths is still the basis for what many mathematicians believe today.
19:54He was fascinated by the geometric shapes that could be produced by following the rules of mathematics.
20:01The rules that he believed came from God.
20:07I'm going to try and draw a circle really, really carefully.
20:12Takes me back to my school days, this.
20:15Now, I'm not doing bad.
20:19That's pretty good.
20:20But if you look really closely, it's just not quite perfect, the circle.
20:25But I'm not going to beat myself up about this, because even if I had access
20:29to the most accurate computer in the world, the circle that it would draw still wouldn't be perfect.
20:37Zoom in close enough and any physical circle will have bumps and imperfections.
20:43That's because, according to Plato, flawless circles don't exist in the real world.
20:49He believed the perfect circle lives in a divine world of perfect shapes.
20:55A kind of mathematical heaven, where all of maths can be found, but only if you're a true believer.
21:07He was convinced that everything in the cosmos could be represented by five solid objects, known as the platonic solids.
21:16So the earth was the rock solid cube. Fire was the very pointy tetrahedron.
21:25And then, with eight triangular sides, air was the octahedron, while the icosahedron, with its 20 triangular sides, represented water.
21:35The last platonic solid, the dodecahedron, this one was supposed to encapsulate the entire universe.
21:43It's the whole universe sitting in your hands there. It's kind of a neat idea.
21:49There's something special about the platonic solids. They're the only objects where every side is the same shape.
21:57And there are only five. Try as you might, you will never find another object with these unique mathematical qualities.
22:09All of these shapes, Plato believed, existed in a world of perfect shapes, beyond the reach of us mere mortals.
22:18A place we call the platonic world.
22:23I know that these ideas might seem like they're a bit bonkers.
22:26But there are actually quite a few people who believe them.
22:29And those people come across as though they're sane.
22:35Oh, my third most favorite mathematical structure, the octahedron.
22:43Dodecahedron. Uh-huh. The platonic solids, I presume.
22:49Is it dodecahedron? I love dodecahedron. I have a misspent youth making models of polyhedron.
22:56Oh, my goodness. These are the platonic solids. Oh, guys.
23:06Okay, very beautiful.
23:08You know, at 67, this is Christmas.
23:11Can I keep these two, please?
23:15These platonic solids, to me, are a great example of how mathematics is discovered
23:20rather than invented. Because when ancient Greeks discovered that this one existed,
23:25they were free to invent the name of it. They called it the dodecahedron. But the pure dodecahedron
23:31itself, it was always out there to be discovered. I have this kind of platonic view that there are
23:36triangles out there. There are numbers. There are these circles that I'm seeking to understand.
23:41So for me, they feel like quite tangible things. They're all part of this mathematical landscape
23:45that I'm exploring.
23:47But not everyone believes in this platonic world of mathematical truths.
23:52I think that the platonic world is in the human head. It's a figment of our imaginations.
24:00I get that there are people who really buy into this other realm of reality. And especially if your
24:08days and nights are spent thinking about and investigating and researching this realm,
24:14that doesn't mean that it's real. Plato would have strongly disagreed. He encouraged us to believe
24:22in this other world, where all of maths could be found, and not to be fooled into thinking the world
24:29around us is all there is. What we perceive as reality, he cautioned, is no more than shadows cast on
24:38the walls of a cave. Plato had a very lively and quite dark imagination. To explain what he meant,
24:49he came up with an analogy of a group of humans locked in a cave. These people would have been
24:55in prison since childhood, and they were shackled by their necks and their legs, and trapped, staring
25:02at a blank wall directly in front of them. In his mind's eye, Plato pictured a fire burning high above
25:14the prisoners' heads. But they have no idea it's there. On top of the wall is a path along which all manner
25:25of people and objects are travelling. But the only thing the prisoners can see of them is the shadows
25:32they cast down the wall. Those shadows are the prisoners' reality.
25:41According to Plato, we're known different to the prisoners in the cave who mistake the shadows for
25:48reality. If Plato is right, what does this mean for you and me? Is what we think of as reality and maths
25:58just an illusion? Are we living in Plato's cave and just see some shadows? It is not impossible that that
26:09is the case. You know, we are maybe just all, we're just some simulation in some world of some more
26:17intelligent being. This is all possible. I mean, if you think that there's some world of mathematical
26:23objects, it's different from ours. It's not the physical world we live in. But that doesn't make
26:27the physical world any less real. So I don't think that there's anything to me in philosophy of maths
26:33that would force you to think that our world is an illusion of any kind. Our senses evolved really for
26:39one purpose, survival. But survival and the true nature of reality are two different subjects. So
26:47the fact that we have been able to survive by thinking about the world one way does not in any
26:52way say that that way of thinking about the world is truly what's happening out there.
26:59Over 2000 years ago, Plato took the geometry of shapes as evidence of God's influence.
27:06ideas that were limited to the senses and imagination.
27:13Today, geometry is at the cutting edge of science. New technologies have allowed us to look at the
27:20world beyond our senses. And once again, it seems the natural world really is written in the language of
27:28maths.
27:28This is a model of a virus. Straight away, you notice its geometric shape. Plato would have
27:39recognized this shape as one of the platonic solids. If there's one person who understands geometry,
27:47it's a mathematician. Raiden Twarock is a professor of mathematics at the University of York.
27:53She's trying to work out how viruses use maths to form their geometric shapes. If you know that,
28:02you can find a way to stop them. That's why Raiden and her colleagues have designed a computer
28:09simulation that puts the mathematician at the heart of the virus.
28:14What we try to understand is how this virus forms. And if we got it to do that, we will create the
28:23illusion of being inside of the virus in the position where the genetic material normally is.
28:29Raiden has discovered that the virus harnesses the power of maths to build its shell in the quickest and
28:36most efficient way possible. Armed with this knowledge, she's trying to find a way to stop viruses such as
28:44hepatitis B and even the common cold from developing in the first place.
28:51Once you understand how this mechanism works, you can turn tables on viruses and actually prevent that process.
28:58That is what makes this research so exciting. If you know the mathematics of how the virus forms its shell,
29:11you can work out the way to disrupt it. No shell, no virus, no infection.
29:20Today, mathematicians like Raiden are joining the front line in the fight against disease.
29:28It really does seem like the universe somehow knows maths. It really is amazing how often these
29:42patterns seem to crop up. They're in plants, they're in marine life, they're even in viruses.
29:49There really is an awful lot that we can explore and exploit using the mathematics that we have.
29:55It does lend weight to the idea that there is some natural order underpinning the world around us.
30:07So far, it does feel like the idea that maths is discovered is leading the charge.
30:13But perhaps we've been looking for patterns in the wrong places. If it's all in our heads,
30:20then the brain feels like a good place to look. Is there evidence in there of maths being an invention of the human mind?
30:32I've got a real treat in store for me today. I am heading over to UCL, the university that I work at,
30:39where some colleagues are going to scan my brain and see which bits of it are working whenever I do mathematics.
30:47Neuroscientist Professor Fred Dick is going to place me inside an fMRI scanner.
31:02He'll measure my brain activity by tracking where the blood flows when I'm answering questions ranging
31:11from language to maths. If my brain treats the mathematical problems in the same way as any other
31:21problem, then it suggests there's nothing special about maths. It's the same as any other language.
31:28A clue, perhaps, that it's an invention. I'll have 10 seconds to think about each question.
31:36I don't need to answer out loud. I just have to work out the answer in my head.
31:48Okay, Hannah, how was that? Good. Some of those questions are really hard.
31:53Well, we didn't want you to relax in there, really.
31:54I've answered all the questions to the best of my ability. After a few hours of processing,
32:01Professor Sophie Scott has my results. Is that my brain? That's your brain.
32:07Let me make sure I understand what I'm seeing then. Okay, so this is like you've cut my brain in half.
32:12Yes. And I've got the left hand side there, is that right? Yes. And the right hand side is that.
32:18So it's like you've chopped my head down the middle and then split it out. Exactly.
32:25So what you can see here, Hannah, is the pattern of activity in your brain when you're hearing
32:29straightforward language. And here you can see, in the left hemisphere, very classic language areas
32:36activated. The bright yellow areas are where there's increased blood flow, an indication that the
32:43neurons in the left hand side of my brain are working harder. This is a side of the brain that we know is
32:52linked to language. Compare that to the right hand side of my brain where there's hardly any yellow areas,
33:01which means there's far less activity taking place. So can we see, um, maths please? Oh, hold on.
33:11But this whole bit here. And look at that. Yeah. And also down at the bottom there.
33:17This time, when I'm thinking about maths, there are yellow areas in the right hand side of my brain.
33:23This is very different to the lack of activity seen when I was thinking about language.
33:35These scans reveal there seems to be a place in our brains where maths lives.
33:41What we're definitely able to say is this is not just the meanings of the words that you were reading.
33:46We're not just looking at you thinking about the meaning of words. You're seeing something that does
33:49seem to be qualitatively different for the maths. Maths is real. Maths is real. At least in my head.
34:02I tell you what really struck me about that conversation with Sophie just then
34:07is that it doesn't matter whether you're doing two plus two equals four or whether you're answering
34:14these much higher level maths questions. It's the same bit of your brain that's doing the
34:19grunt work. It's not the same thing that does words or language. You're seeing these problems and
34:25you're manipulating them in your mind.
34:32Research with similar experiments shows it's broadly the same for all of us.
34:37In your brain and mine, there is a specific place where we do maths.
34:42maths. But this doesn't prove that maths is something we discover.
34:48It could still be an invention, just one that we learn at school.
34:56To get to the bottom of this question, I need some volunteers who've never had a maths lesson
35:02in their lives.
35:08Denise, just put one of those on there. Very nice. That worked very nicely, didn't it?
35:13Yeah, you liked that, did you?
35:15Dr Sam Wass is an experimental psychologist at the University of East London.
35:21Helping him with some experiments are six months old Ira and Leo, who's just under a year.
35:26To begin with, each child is placed in a room where they're shown a series of images.
35:35Sam uses a battery of tests to analyse how they react to different situations.
35:44The first experiment uses eye-tracking technology to see how the baby follows the movement
35:50of a piggy puppet. Is that good? So here we can see a feed-out of what the child is looking at,
35:56and those two red dots are where the baby is looking.
36:04What we're presenting is a puppet that jumps up and then disappears. And it jumps up and disappears
36:10two times in a row, and then it stops. We present this same sequence again and again.
36:15And as the baby watches it again and again, their looking times, the amount of attention
36:20that they're paying to the screen diminishes. And that tells us that the child has learnt this sequence.
36:25Now, instead of popping up twice as expected, the puppet appears three times.
36:32Does the child notice the difference between the two-ness of it popping up twice in a row,
36:36and the three-ness of it popping up three times? And if it does, then that tells us that the child
36:42understands the difference between two and three. These tests reveal that the child is surprised
36:49when the puppet appears more often. When larger scale experiments were carried out by researchers
36:56in the US, the results suggested that infants do have a sense of quantity.
37:02So this research is really important because it suggested that even infants as young as five months
37:09old can do the basics of addition and subtraction. They know the difference between one plus one equals
37:15two and one plus one equals one, which is an incorrect conclusion. And that though is a really,
37:21really strong, provocative finding. This idea that the concept of mathematics and the basics of
37:27mathematics rules might be hardwired, might be innate in our genetic code.
37:32This research isn't conclusive, but it does suggest we all come pre-programmed to do maths. Some argue
37:40that we evolved this maths part of our brains to discover the world of mathematical truths.
37:47The evidence for maths being discovered is compelling. We found patterns in nature. The latest technology has
37:58uncovered startling patterns in viruses and scans reveal there's a part of our brains where maths lives.
38:09But this question is too important to leave the evidence here and move on. If it is discovered,
38:17if it lives in this other world, can we trust what it's telling us? How do we know that our idea of
38:26numbers is right? How do we know someone isn't just going to come along at some point and say,
38:30well actually, you've got that completely wrong and one plus one doesn't equal two after all? How do we know
38:36we can rely on the maths that we take for granted?
38:40What you need to be sure of is your foundations. If they're shaky, then all of your carefully
38:49constructed ideas come crashing down. And there was one mathematician who understood this only too well.
38:56His name was Euclid. Around 300 BC in Alexandria, he wrote one of the most famous and important books of all time,
39:07the elements. He was trying to go right back to the beginning to find the smallest elements on which you
39:15can build the vast gigantic structure of mathematics. If you have a little flick through, you can see the kind
39:22of things that Euclid was considering. So here it says that you can draw a straight line between any
39:28two points, which seems blindingly obvious. And here it says that all right angles are the same.
39:36Now these are quite simple concepts, but I think that they really illustrate just how exhaustive
39:41Euclid had to be to build the foundations for what was to come.
39:45He took statements like these, which mathematicians assumed were true, and put them to the test.
39:54He then set out to prove a whole host of other theories based on these fundamental building blocks.
40:03This was really the first time that someone had written down formal proofs for mathematical assumptions.
40:10Now, mathematical proof isn't like scientific proof or proof in a court of law. There's no room for
40:16reasonable doubt here. Instead, if something is true mathematically once, then it is true forever. And
40:23that is why this book is so important.
40:30It's the reason why Euclid's Elements is still relevant today. Every page within it is as true now as it
40:39ever was.
40:43And from that point of view, it really does feel like we're tapping into a world that already exists.
40:49One that's just out there waiting to be discovered.
40:54Unless, of course, you throw a spanner in the works, change the language of maths and invent a better way
41:02of doing things. Suddenly, this rock-solid world of God-given truths might feel decidedly shaky.
41:09One thing we know about languages is that they never stand still. They're constantly evolving to meet the
41:25challenges of a changing world. Forty-seven. Forty-nine. Forty-nine.
41:32Forty-nine. Forty-nine.
41:34Muy bien. Let's go for another tricky one.
41:39For centuries, the language of maths was thought to be fixed and unchangeable.
41:44That is, until something was found to be missing.
41:49It is the number zero.
41:52What exactly is zero?
41:57A zero means nothing.
42:00If you've got zero flowers, you've got no flowers.
42:05And if you've got zero of something, you've got nothing.
42:10So you can't really do anything with the zero.
42:13I don't really use it when I'm counting in numbers.
42:17Before the 7th century, neither did anyone else.
42:21Though people have always understood the concept of having nothing,
42:25the concept of zero is relatively new.
42:30We had numbers and could count, but zero didn't exist.
42:37If you think about it for long enough, zero is actually quite a strange concept.
42:43It's almost as though the absence of anything becomes something.
42:49Is it just a number or an idea?
42:52And how can something with no value have quite so much power?
42:57It's not exactly clear who first thought of zero.
43:02It might have originated in China or India.
43:05What we do know is that zero arrived in Europe from the Middle East
43:10at about the same time as the Christian Crusades against Islam,
43:14when ideas coming out of the Arab world were often met with suspicion.
43:19The West already had a numerical system, Roman numerals.
43:24They did the job, but were a bit unwieldy.
43:28For example, the number 1958 is written as MCMLVIII.
43:34And no matter where you place, say, the letter C,
43:38it will always represent the number 100.
43:42It was good for its time, but times change and a better system was needed.
43:49Zero was different.
43:54Where you place zero could change the values of the numbers around it.
43:59Think of the difference between 11 and 101.
44:03Although the concept of zero might have been created elsewhere,
44:08it was in India that zero started to be accepted as a proper number.
44:14This is a page from the Indian Bakshali manuscript from around AD 225,
44:23which shows the dots above the characters representing zero.
44:27This is the earliest known use of the symbol zero that we know today.
44:33For almost a thousand years, Indian mathematicians worked happily with zero,
44:41while their Western counterparts ploughed on with the Roman numerals.
44:45That was until Italian mathematician Fibonacci recognised its potential.
44:50Now, he'd been educated in North Africa, so he'd seen this number system working firsthand.
44:57Zero is a placeholder, signifying the absence of a value.
45:02Zero is also a number in its own right.
45:05It allowed you to write down numbers and manipulate them much more quickly and easily than Roman numerals.
45:12Realising all this, Fibonacci champions the new number and brought it to the attention of Western Europe.
45:19Zero wasn't something that we discovered so much as something that was created as part of a new language to describe numbers.
45:29And that's not to say it isn't useful.
45:32The whole of modern technology is literally built on ones and zeros.
45:36But suddenly maths feels like something we've come up with.
45:40Something we've invented.
45:43We needed a more user-friendly numerical system,
45:47so someone came up with the clever idea of zero.
45:51Not a gift from the gods, but a smart way to make counting more convenient.
45:58This is intriguing evidence that maths might be invented after all.
46:04A product of our intellect and imagination.
46:08Once the idea of zero had been widely accepted, mathematicians could relax.
46:14All conceivable numbers lay out on a single line with no holes and no gaps to speak of.
46:21Over here you have the positive numbers.
46:23One sheep, two donkeys, the kind of stuff you find in real life.
46:27And in the other direction, all the negative numbers.
46:31It's a bit trickier to imagine what negative one sheep looks like.
46:38The number line stretches out in both directions all the way to infinity.
46:43And zero sits proudly in the middle.
46:46Everything was well in number land.
46:49Or was it?
46:51This is where it all starts to get a bit strange.
46:55Because there are some numbers that are simply weird.
46:59There are some fundamental rules of maths that you learned at school.
47:03Two times two equals four.
47:06Three times three equals nine.
47:10A positive number multiplied by itself equals another positive number.
47:16Nothing controversial so far.
47:18Curiously, a negative number multiplied by itself also gives a positive number.
47:25Why is that?
47:28Well, this is not a maths lecture.
47:30So let's just accept it as a fact and move on.
47:34In fact, if you take any number and multiply it by itself, or square it,
47:40then the answer is always going to be positive.
47:44If plus two squared gives me plus four,
47:48and minus two squared gives me plus four,
47:53what do I have to square to get minus four?
47:56But it's a question without an answer.
48:00There is no number that when multiplied by itself gives a negative answer.
48:05That is, unless you invent your own.
48:09Meet I, a number we simply made up.
48:14Not everyone was keen.
48:20It became known as an imaginary number.
48:23A deliberately chosen derogatory term to scoff at its existence.
48:29It turns out I is really useful.
48:34Especially when it comes to simplifying problems with things like electricity or wireless technologies.
48:40Things that otherwise would seem impossible to solve.
48:43Essentially, if you're working with waves, you will use I.
48:48This imaginary number broke all the rules.
48:54It didn't come from this world of ethereal numbers.
48:57It wasn't God-given.
48:59It was very definitely invented.
49:02If you can have one imaginary number, why can't you have two?
49:06Or three?
49:07Or infinitely many of them?
49:10Why can't you have negative imaginary numbers as well?
49:13Why can't, in fact, imaginary numbers have their very own number line?
49:18Exactly the same as the real one, just on a different axis.
49:23The number line isn't a single line at all.
49:27Numbers are two-dimensional.
49:30You might think this all sounds a bit airy-fairy.
49:40Imaginary numbers that we just made up.
49:43But if you've ever flown in an aircraft, you've already trusted your life to these strange numbers.
49:50At Gatwick Airport, air traffic controllers here rely on radar to keep everything moving safely and quickly.
50:07Once you get busy, you definitely need radar.
50:11So the busier the tower, the busier the operation, you need radar.
50:15Radar works by sending out radio waves and examining that part of the signal that's reflected back.
50:22The complex equations that allow us to filter out the correct signal from other conflicting frequencies is heavily dependent on imaginary numbers.
50:33In this case, separating out moving objects like planes from flocks of birds or stationary objects.
50:40Imaginary numbers are a very efficient tool to be able to manipulate radio waves.
50:47Imaginary numbers are fundamental to the operation.
50:50Imaginary numbers allow us to track planes in real time.
50:55Without them, we never would have been able to use radar in our skies.
50:59When I started this investigation going back to the time of the ancient Greeks, it did seem like maths could only be discovered.
51:19There were too many coincidences, too many mathematical patterns popping up all over the place.
51:25But if we can invent the rules, create new numbers, and they work, then perhaps I've got it wrong.
51:34Maybe maths is invented after all.
51:38The concept of zero or negative numbers or complex numbers or imaginary numbers, they cause great consternation to the cultures that first invented or encountered them.
51:51There 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.
52:03Zero kind of and maybe came from that observation.
52:07The power of mathematics lies in the way its language and symbols have allowed us to manipulate the world.
52:16But this was a world that followed the rules of God and the church.
52:21By the 17th century, a new breed of intellectual was emerging, not afraid to challenge authority.
52:28There was one man who dared to question all of the philosophical and scientific assumptions that had gone before.
52:37This was someone who was trying to promote a new way of thinking, using reason, experimentation and observation.
52:43This was the young Frenchman called René Descartes.
52:49It was while in a restless sleep in 1619 that Descartes experienced a series of dreams that would change his life and mathematics.
53:00The first two could be better described as nightmares.
53:06But the third dream, the third dream was intriguing.
53:15As his eyes scanned the room, he saw books on the bedroom table that appeared and then disappeared.
53:24He opened one book of poems and at random caught the opening line of one, which read,
53:31What road shall I pursue in life?
53:36Then someone appeared out of thin air and recited another verse, saying simply,
53:42What is and is not.
53:48As with dreams, it's all about the interpretation you place upon them.
53:53In Descartes' case, the effect of these dreams was profound.
53:57He was convinced that the dreams were pointing him in a single direction, bringing together the whole of human knowledge by the means of reason.
54:14He was nothing if not ambitious, but his genius led to perhaps one of the greatest advances ever in the field of mathematics.
54:26As with so many brilliant ideas, it was deceptively simple.
54:30Let's say that I'm meeting a friend for a coffee.
54:35Now, I'm standing at the end of Ensley Gardens and they are somewhere over on Gordon Street.
54:42It'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.
54:48In this case, three streets down and one along.
54:51It sounds like an incredibly simple idea, but actually, it revolutionised mathematics.
55:02He showed that a pair of numbers can uniquely determine the position of a point in space.
55:14It sounds trivial, but this was just the start.
55:17It gets more interesting when you apply this idea to curves.
55:25As this point moves around a circle, its coordinates change.
55:29And we can write down an equation that precisely and uniquely characterises this circle.
55:37For the first time, shapes could be described by a formula.
55:43By uniting the language of numbers and equations and symbols with shapes,
55:50Descartes was able to expand the horizons of mathematics,
55:54thus laying the foundations for the modern scientific world.
56:00What Descartes and the other trailblazers like him did was to question the accepted wisdom of the time.
56:08They thought differently and the result was that they delivered monumental breakthroughs
56:14for our understanding of the universe.
56:18Descartes lived in a time when many philosophers backed up their arguments with appeals to God.
56:24But Descartes preferred to place his trust in the power of human logic and maths.
56:31He believed all ideas should have their foundations in experience and reason,
56:37rather than tradition and authority.
56:40It still feels like maths belongs to a discovered world.
56:46But after Descartes, it's a world that is increasingly devoid of a divine influence.
56:53And we started this episode with just one question.
56:56Is mathematics invented or discovered?
56:59And based on the evidence so far, I'm leaning quite heavily towards discovered.
57:04Because it doesn't seem to me to be possible that something so all-encompassing
57:09could be the product of the human mind alone.
57:13But than thought, what I'd love for him…
57:15…knowingly.
57:17My Lily was, are my Ellen.
57:19prophets talking about justice.
57:20When dogs, are, are they!
57:21What are they?
57:23Who are they?
57:25Who are they!
57:27What should I declare, something I am.