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00:01There 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. Many people have in fact described maths as
00:34the underlying language of the universe. But how did it get there?
00:41Even after thousands of years this question causes controversy. We still
00:48can't agree on what maths actually is or where it comes from. Is it something
00:53that's invented like a language or is it something that we have nearly discovered?
00:58I think discovered. Invented. It's both. I have no idea. Oh my god! Why does any of this matter?
01:08Well maths underpins just about everything in our modern world. From computers and
01:14mobile phones to our understanding of human biology and our place in the
01:20universe. My name is Hannah Fry and I'm a mathematician. In this series I will
01:28explore how the greatest thinkers in history have tried to explain the
01:33origins of maths extraordinary power.
01:36You've ruined his equation. I'm going to look at how in ancient times our ancestors thought
01:45maths was a gift from the gods. How in the 17th and 18th centuries we invented new mathematical
01:52systems and used them to create the scientific and industrial revolutions. And I'll reveal how
02:00in the 20th and 21st centuries radical new theories are forcing us to question once again everything
02:09we thought to be new about maths and the universe. The unexpected should be expected because why would
02:17reality down there bear any resemblance to reality up here? In this episode I discover how maths led
02:24Victorian scientists into a world of invisible forces and particles we cannot see. Now this couldn't be
02:33a coincidence and I reveal why the concept of infinity broke the rules about where maths comes from.
02:42I'm very tormented by infinity. Is infinity real? I do not know the answer to that question.
02:54Our world is governed by the rules of science. But science wouldn't work if it wasn't for a far
03:07deeper set of rules. Those of mathematics.
03:14It predicts the movement of the planets and the ebb and flow of the tides. If you look hard enough at
03:20anything you'll find mathematics hiding underneath. If maths is the language of the universe then where do
03:30numbers come from? Before we learn that one plus one equals two the idea of one and two still existed.
03:41The nature of oneness and twoness has always been there.
03:48The concept of numbers is something universal. All around the world and in every language we
03:55understand the idea of what one or two means. And this raises an intriguing question.
04:03Is maths all in our heads? Is it something that we have invented? A language that we use to describe
04:13the universe? Or is it an external physical reality? Something that exists completely independently
04:21of us humans? Something that's just out there waiting to be discovered?
04:26In ancient times we were in awe of the power of maths. Seen as a gift from the gods it was considered
04:38pure and complete. But through the centuries maths developed. It wasn't complete after all. New areas
04:46and techniques have been invented. And the more we explored science the more it became obvious that we
04:52couldn't just rely on simple experiments. We needed a theory and crucially a mathematical description to be able to
05:02understand the world around us. Things that seem obvious at first often have a habit of melting away when
05:12exposed to the rigor of experimentation. The problem for humans is overriding our instinct to trust our
05:22intuition. Our senses aren't always the best guide to the truth. The Greek philosopher Aristotle fell into this
05:32trap when he famously declared that something heavy will fall quicker than something that's light.
05:41To him it seemed blindingly obvious and for centuries nobody disagreed with it.
05:46On the face of fear you might think that suggesting that heavier objects
05:52fall faster than light objects was quite a sensible idea. After all if you drop them at the same height
05:59a hammer lands first.
06:06But a 16th century scientist and mathematician called Galileo Galilei had a different explanation.
06:13He believed Aristotle had failed to consider something crucial.
06:21The incredible fact is not that Aristotle was wrong
06:26but that his law of motion stood unchallenged for almost 2000 years.
06:34How could such a flawed idea survive for so long?
06:38Well to be fair there are a few reasons. You can see the hammer hitting the ground earlier than the feather.
06:45The reason for that of course is air resistance and Galileo argues that if you dropped them in a vacuum
06:52they would land at exactly the same time.
06:56To come up with this theory Galileo imagined the idea of a vacuum in which air resistance didn't exist
07:03and created a series of laws that described the motion of falling objects.
07:09They completely overturned Aristotle's ideas.
07:15Over 300 years after Galileo's prediction, Apollo 15 astronaut David Scott gave the theory its most dramatic test.
07:25Well in my left hand I have a feather. In my right hand a hammer.
07:32And I'll drop the two of them here and hopefully they'll hit the ground at the same time.
07:38How about that? Mr Galileo was correct.
07:42With no air resistance on the moon, the hammer and feather hit the lunar surface at the same time.
07:49A physical description of the world on its own isn't enough. It has to go hand in hand with mathematics
08:01before you can truly discover the nature of reality.
08:06Galileo was incredibly impressed, for good reason, at the power of mathematics to give us insights,
08:14to describe things that were happening, to articulate the patterns that the human brain is able to access.
08:22It almost seems miraculous that some symbols on a piece of paper can do that.
08:28And in that sense it might lead you to think that math is the language of reality.
08:33Galileo exclaimed that the world is a grand book written in the language of mathematics.
08:38I think the reason for this is that ultimately the world is completely mathematical and we're just
08:46discovering that bit by bit. He had this feeling that by using mathematics he could get into
08:57these things which he wanted to be inevitable. And mathematics gave him that certainty that things
09:04are inevitable. So he was the first to understand that in order to explain phenomena he needs to use mathematics.
09:15Galileo's theories, though ahead of their time, raised as many questions as they answered.
09:24There appeared to be some kind of a force that was pulling objects to the ground,
09:29but exactly what that force was, or how it worked, remained a mystery.
09:37Solving this mystery would take the genius of a 17th century Englishman. His name was Isaac Newton.
09:51I'm heading to North Wales to do something I'm not entirely happy about.
09:59So my director was looking for a clever way to illustrate gravity.
10:08And he came up with the bright idea to send me down the fastest zip wire in the world to head first as well.
10:18I wasn't in that meeting. I should have been in that meeting.
10:21The same force that brought Newton's apple to the ground is the thing that's going to be propelling me
10:31towards a quarry.
10:36What do I let myself in for?
10:38But all this is nothing compared to how Newton performed experiments on himself.
10:44Newton totally believed that the path to true knowledge lay in observation.
10:54So rather than just read a book on optics, say, he decided to experiment by poking a blunt needle
11:01into his own eye. Maybe don't try that one at home.
11:04He wasn't going to take someone else's word for it. He had to test these theories for himself.
11:12As he began wrestling with bigger ideas, such as gravity,
11:17only mathematics could help him find the answers.
11:22When you think about it, gravity is actually quite a strange beast.
11:26It creates this invisible force of attraction between me and everything around me,
11:31but one that's weak enough that I can easily overcome it just by moving my own muscles.
11:37Newton set out to find a way to describe this mysterious force.
11:43Originally described in words, his law of gravity was later written down in the form of an equation.
11:50Now, don't be fooled by its simplicity, because this guy packs a real punch.
12:02I'm using it to work out the force that will be acting on me as I head down the zip wire.
12:09To understand it, you need to know what all the letters stand for.
12:13So let's begin with if, the force.
12:16Newton says that between any two objects in the universe, there is an attractive force.
12:23And this force depends on the mass of those objects.
12:26There's capital M here, that's the mass of the Earth.
12:28And then slightly smaller, the little m there is me.
12:32That little m is my mass, and it's measured in kilograms.
12:38There's also g, the gravitational constant, which Newton knew had to exist,
12:42although he didn't know exactly the size of it at the time.
12:46And R there, which is the distance between me and the centre of the Earth.
12:51More generally, what this equation is saying is that the bigger the mass of your objects,
12:56like planets, for example, the bigger your force between them is going to be.
13:01And the greater the distance between objects, the bigger this R is, the weaker the force of gravity is going to be.
13:12So what does Newton say the force of gravity will be on me?
13:16So if you plug in all of the numbers into this equation, you can calculate the force on me
13:26as I travel down the wire, it works out to be 736.
13:33And the unit is Newton's.
13:37Newton arrived at his now famous formula after studying centuries worth of measurements
13:43from astronomers that had gone before him.
13:46His law of gravity not only explained why objects fall to the ground,
13:52it predicted the positions of every moon, planet or comet anywhere in the cosmos.
13:59That is one devastatingly powerful equation.
14:04This was Newton's genius.
14:06Once you've got a mathematical law, you can use it to apply it to anything.
14:10apples, planets and people.
14:13And if you can calculate exactly what that force will be,
14:16it means you can predict all kinds of other things,
14:19like my terminal velocity as I travel down to the bottom.
14:23So let's put it to the test.
14:29As the force of gravity pulls an object to the ground,
14:32it reaches a maximum speed.
14:35This is called its terminal velocity.
14:37Before you can calculate this figure, there is a bunch of things you need to consider,
14:42such as the gravity, drag and friction along the cable.
14:51Time to put my faith in Newton and the fastest zip line in the world.
14:56From my calculations, I reckon my terminal velocity is going to be about 90 miles an hour.
15:02I don't think I'm going to speak to this director again.
15:09What am I doing?
15:12Five, three, two, one.
15:23No!
15:32That was actually really fun.
15:38Okay, I also need to check my speed prediction.
15:41Now, disclaimer, just before I came down, they added some flags to the back of me,
15:47just to slow me down, because the wind's picked up, as you can probably hear.
15:51So, I don't think I'm going to quite hit 90, but let's have a look here.
15:55There's a big spike there on the graph, and it says it's 49 seconds for one mile,
16:02which is about, what, 75 miles an hour, something like that?
16:10Not bad, not bad. For a back-of-the-envelope calculation, not bad.
16:13The power of Newton's equation was that it could explain and predict so much about the universe.
16:29It allowed us to think of nature as ordered, not just on Earth, but throughout the cosmos.
16:36The key breakthrough of Newton was that he had the audacity to shatter this idea that
16:46Earth rules are different from Heaven rules, and the Moon doesn't fall down because it's
16:50made of Heaven stuff. And say, wait a minute, maybe all things obey the same physical laws.
16:57His laws of force and of motion were not meant to merely apply in, say, the heavenly realms,
17:03or just on Earth. They were meant to apply everywhere, and the idea was the whole of nature
17:06would really be captured by this single set of laws.
17:09I mean, the fact that we can write equations and know how to power a rocket and have it land
17:14on the Moon and come back. Holy cow! I mean, we take these things for granted,
17:20but think about the power of equations to give us the trajectory and figure out how to accomplish
17:26this incredible feat. That is thrilling.
17:30If evidence is needed to prove maths is discovered, part of the fabric of reality, then surely this is
17:39it. How could something we invented in our brain have the power to reveal the workings of the universe?
17:47And the extraordinary power of mathematics wasn't just confined to the stars.
17:58By the end of the 18th century, scientists and engineers were using it to drive innovation on a
18:03grand scale, what became known as the Industrial Revolution.
18:12This changed everything. People didn't just live and work in the fields anymore.
18:17There was an explosion of growth in towns and cities,
18:20cities as employment switched to factories and driving this entire revolution with the invention of the steam engine.
18:33The impact of this new technology was profound. It opened up the country not just to people and goods,
18:40but to ideas. New ways of doing things were propelling us into the age of the machine.
18:48How fast does it go? 25 miles an hour maximum.
18:52What are we doing now? About 15.
18:54And yeah, your speedometer goes up to 100.
18:57Yeah, come over there.
19:00Behind all of this were the essential calculations of the machine age.
19:06How strong the materials were, how hot or cold something might get.
19:10It was mathematics that was used to design faster and more efficient machines.
19:17So how hot does it get in there?
19:19Fahrenheit, it goes to about two and a half thousand degrees.
19:22Two and a half, what's that himself there?
19:24I'm not sure.
19:28Hot, very hot.
19:29New skills were required in all of this.
19:35So whereas before you would have craftsmen using hand tools, now you had people in factories operating machinery.
19:44But there's also a sea change here in the way that we think.
19:47It's a belief that while the natural world might not be tamed, it can at least be bent to our will.
19:53The Industrial Revolution marked a major turning point in history.
20:00From textiles to iron production and the spread of the railways, almost every aspect of daily life
20:06was influenced in some way.
20:09And at the heart of this revolution was mathematics.
20:12Now this is a world that feels firmly rooted in reality.
20:18We can trust the numbers and we know that they're not going to let us down.
20:22So forget all of your airy-fairy philosophical stuff here.
20:25This is maths in action.
20:28It's big, it's bold and actually it's pretty amazing.
20:32Technological miracles were coming thick and fast.
20:39Mathematics had given us a description of how the world works that was driving our understanding forward.
20:46But also, it could hint at how seemingly separate things could be connected.
20:52By the 19th century, mathematicians and scientists began to wonder what else was out there just waiting to be discovered.
21:10They soon turned their attention to the invisible link between electricity and magnetism.
21:16Both had been known about for centuries, from the raw power of lightning to navigation by means of a ship's compass.
21:28But they'd always been thought of as two very different things.
21:34It was a working-class son of the Industrial Revolution, Michael Faraday,
21:40who was the first person to see a connection between the two.
21:43I've come to the Royal Institution, to the place where Faraday had his laboratory.
21:52To see if electricity and magnetism were linked, Faraday ran a series of experiments.
21:57He took a wire that had electricity passing through it,
22:02and he watched as it moved the needle of a compass.
22:08The electric wire and the magnetic needle weren't touching,
22:12and yet one was having an effect on the other.
22:15What was the connection?
22:17Faraday looked deeper.
22:22What he did was to take a magnet, like this one,
22:25and a roll of copper wire wrapped around a cylinder, like this,
22:30and then to pass one through the other very quickly, like this.
22:34The wire surrounds the outside of the cylinder, so the magnet can't come into contact with it.
22:43And that's really all there is to it. There's nothing more complicated than that.
22:46The wire never touches the magnet, and yet, as you can see from these LEDs,
22:51probably not the originals, that is enough to generate electricity.
22:55Faraday realised there had to be some kind of invisible force working behind the scenes.
23:09And he had a clever idea of how to make it visible.
23:12What you do is you take a permanent magnet and you place some paper on top of it,
23:22and then take some iron filings and sprinkle them on top.
23:27Now, this, I think, is one of the most memorable experiments that you do at school,
23:32and I can remember that moment where you see the invisible force fields that's created by the magnet.
23:44As the iron filings fall onto the paper, they line up with the magnet's field lines.
23:53Now, this is just two-dimensional here, but actually these lines are three-dimensional.
23:58They come out and they warp and curve and wrap around the entire magnet.
24:09That's pretty cool, isn't it? It's pretty cool.
24:18Faraday's iron filings experiment revealed the existence of an invisible field stretching out into space.
24:26He could see the lines of the force, but he was an experimentalist and lacked a complete mathematical description of what was going on.
24:37As a result, many of his contemporaries dismissed his ideas as fanciful.
24:43It was the Scottish scientist James Clerk Maxwell who took Faraday's ideas
24:48and came up with a mathematical way to link electricity and magnetism.
24:56Drawing from the observations of previous scientists, Maxwell distilled electricity and magnetism down
25:03into four equations that worked for nearly every situation.
25:08The symbols themselves aren't important to the story.
25:13The key point is that Maxwell spotted a gap.
25:17The mathematics was telling him there was something missing in this last equation.
25:23He realised there has to be another term in this equation, one that looks like this.
25:36And essentially what it's saying is that if an electric field is moving, then a magnetic field will wrap itself around it.
25:46And it's mirrored by this equation up here, which says that if a magnetic field is moving, an electric field will wrap itself around it.
25:54With this missing piece in place, suddenly everything fitted together.
25:59Mathematics had led Maxwell to see the bigger picture.
26:04These equations are linking the two things together.
26:07Electricity to magnetism, magnetism to electricity, back and forth from one to the other.
26:12Using only mathematical ideas, Maxwell had found the evidence to prove that electricity and magnetism were inextricably linked.
26:25Together, electricity and magnetism formed what he called an electromagnetic field.
26:34This helped explain so much.
26:36The equations perfectly described what Faraday had seen with his experiments.
26:44But Maxwell didn't stop there.
26:46He showed how these field lines could move in time with each other, creating electromagnetic waves.
26:56By playing around with these equations, Maxwell could calculate the speed of this wave.
27:03And it came out to be about 300,000 kilometers a second.
27:08And that wasn't a random number.
27:10That was a number that Maxwell knew very well.
27:11Because it was the same as the speed of light in a vacuum.
27:17Now, this couldn't be a coincidence.
27:19You don't really get coincidences like that in the universe.
27:22There was only one possible explanation.
27:27Light had to be an electromagnetic wave.
27:33Maxwell's discoveries were genuinely revolutionary.
27:41He'd given us a unified theory for electricity and magnetism.
27:45And as an added bonus, an explanation of light itself.
27:50For the first time, an electric field, a magnetic field, and light could all be explained using a single theory.
27:59The elegance and simplicity of this solution was breathtaking.
28:06Surely nothing the human mind could conceive of would ever be capable of thinking up something so sublime.
28:15Equations that reveal new truths about the universe.
28:19It feels very much as if this answer was always out there.
28:25It just needed someone who thought differently to discover it.
28:29It's quite uncanny how mathematics has again and again predicted new things in the physical world that we weren't even looking for.
28:40You come up with novel predictions.
28:42You come up with ideas that there should be structures in the world that you haven't yet discovered.
28:48And then on inquiry, you discover those to be real.
28:52That's really extraordinary.
28:54I can tell you from my personal experience, it is shocking, not just surprising, but shocking that mathematics makes predictions about the world around us.
29:05The ancient Greeks found intriguing patterns in nature which seemed to follow the rules of maths.
29:12Then Newton showed us how mathematical equations had the power to predict the movement of the planets revealing an ordered universe.
29:21By the 19th century, the formidable power of maths allowed Maxwell to unify electricity and magnetism.
29:31It seemed inconceivable that maths could be anything other than something we discover.
29:39But then something happened that turned this world view on its head.
29:44There was a new way to look at maths.
29:47Someone had invented a different way of doing things.
29:52Since the days of the Greek mathematician Euclid more than 2000 years ago,
29:58right angles and parallel lines, the kind we learned at school,
30:03have been the bedrock upon which all of geometry and our understanding of space is built.
30:08But in the 19th century, mathematicians started to wonder whether everything really was as it seemed,
30:19or whether there was the possibility of something a bit weird going on behind the scenes.
30:24You can see it with games like Pac-Man.
30:31What kind of a shape is the Pac-Man universe?
30:35Your instinctive answer might be a square, and you'd be right, sort of.
30:39You'd be right, sort of.
30:41For instance, if this little pink character exits to the left, it will re-enter on the right.
30:50Which actually makes this universe
30:53more of a cylinder.
30:57What's more, in other similar games, you can exit out of the top and re-enter at the bottom.
31:05Which means that these two loose ends have to bend around and connect up to one another.
31:11It's a bit of a strange idea to get your head around,
31:13but these kind of computer games are not played on a square.
31:17They are played on a doughnut.
31:22Once you move from a flat square to another shape,
31:26you can't take it for granted that geometry will follow the rules you've always expected it to.
31:32Behind the scenes, there can be something else going on entirely.
31:37But hold on to your hats, because this is all about to get much weirder.
31:43What would happen when I leave one of the courtyards?
31:56If the world was as Euclid says it is, and everything works normally,
32:01if I turned left four times, I would eventually get back to where I started.
32:08I've left the yellow courtyard. I've gone through orange, red and blue,
32:13and I'm back in yellow again. Nothing controversial here.
32:17But who says there has to be four courtyards next to each other?
32:22What if you got back to where you started after turning left only three times?
32:28But hang on, I hear you cry. That's impossible.
32:33Except it's not if you're living on a cube.
32:37Begin on this side, turn once, turn twice, turn three times, and you're back where you started.
32:45No longer was there only one description of space.
32:51By changing the rules, you could now choose a different type of geometry.
32:56It turns out there's many different ways to think about space.
33:00It would be very much like if somebody discovered Piccadilly Circus by taking a left turn where
33:08they had always taken a right turn before. People hadn't even thought that there could be a
33:11distinction between the physical space and the mathematical space that Euclid had studied with
33:17his axioms.
33:18Because all of a sudden, Euclidean geometry just looks like one way of describing a space.
33:24And in fact, you know, it happens to be a good one for describing the space we're sitting in right now.
33:28Not such a good one for describing space on astronomical scales, it turns out.
33:32So it's a little bit like a game.
33:34Namely, I teach you the rule of chess and we play chess.
33:38I change the rules and we play a different game, but we still can play a game.
33:43So that was the feeling that maybe it is all, you know, depending on which set of axioms you choose,
33:51you can get a new type of mathematics.
33:57But hang on a minute. If we can just make up a new type of geometry, then perhaps I've got this wrong.
34:03Maybe math is something we invent after all.
34:08So that's a good idea.
34:10With this newfound freedom, mathematicians began exploring ever more abstract ideas,
34:18the most intriguing of which was the notion of infinity.
34:31Can everybody show me this sign that we are going to be using to solve this problem?
34:38Off you go.
34:40From an early age, we all have an idea of what infinity is, but it's hard to pin down.
34:46Our minds aren't built to wrap themselves around the concept of something that is completely endless and boundless.
34:52And that makes describing exactly what infinity is pretty tricky.
35:01It's a number that keeps on going and never stops.
35:05The biggest number I could think of is 99 billion.
35:10400.
35:11Googleplex.
35:13There's nothing bigger than infinity because that's the biggest number that you could possibly need.
35:21I'm very tormented by infinity. I have a love-hate relationship with infinity.
35:29I love using it when I teach courses at MIT because it makes things so easy to derive and prove.
35:35But in my gut, I know there is no actual infinity. It's just a convenient approximation.
35:42Is infinity real? It's about as real as the number one or the number zero.
35:49It's a concept. It's a useful concept in describing a certain set of elements.
35:57And in that sense, yes, it's real.
36:02I think it's fair to say that nobody in the laboratory is ever going to have a dial that registers infinity,
36:10that measures infinity. We're never going to literally count to infinity. We can approach it,
36:16but from that point of view, I don't think we're ever going to embrace it the way that we embrace
36:20tables and chairs and finite objects.
36:23It's only by definition we can't go there. You can't get there.
36:26Try and get closer to infinity. It always stays just as far away.
36:32You might imagine that something as abstract as infinity is not very useful.
36:36But in reality, infinity offers a way to solve problems that previously would have seemed impossible.
36:46If you wanted to know the distance between the UK and New York,
36:52you could try and use a ruler on a globe like this.
36:57You'd have some trouble because of course the world is round and curves, unlike straight lines,
37:03are quite tricky to measure.
37:05Good luck in geography class with a globe and a measuring stick.
37:10But what if rather than just using one ruler, you use too much smaller rulers and use how they overlap
37:20to wrap around the curve of the Earth? Now, by doing that, you're not going to get the exact distance
37:26between London and New York, but you're going to get a much better approximation for it.
37:31And you can imagine the more and more rulers that you use, the better they'll wrap around
37:36the curve of the globe and the better an approximation you'll end up with.
37:42So here's the key idea. If you zoom in enough on any curve, it will start to look straight.
37:50And if you have an infinite number of teeny tiny rulers, you can perfectly measure the length
37:56of any curve just by adding up all of those straight lines.
38:03It's only by harnessing the power of infinity that any of this is possible.
38:10OK, so why should you care? Well, it's not just the Earth that's got curves.
38:15Because everything from the movement of satellites in the sky to the rise and fall of the stock market,
38:21to understanding how our human behavior changes over time, all of them rely on this idea of infinity.
38:33Relying on an idea we don't really understand isn't something that sits comfortably with mathematicians.
38:39In 1924, the renowned German mathematician David Hilbert created a famous thought experiment
38:51to try and help explain infinity.
38:58He did it by imagining a large hotel.
39:01But this was no ordinary hotel. It had an infinite number of rooms.
39:15Hi. Hiya.
39:16Can I have a room for tonight, please?
39:17Sorry, ma'am, we're fully booked tonight.
39:19Oh, you haven't got any rooms at all?
39:21Unfortunately not, sorry.
39:23Oh.
39:23Hilbert wondered what would happen if all the rooms were full and a guest like me turned up.
39:36Would there be room for one more in the infinite hotel?
39:42So today I've turned up and the place is fully booked.
39:46They're saying I haven't got any rooms at all whatsoever.
39:50I've tried to ask them if they know who I am, but apparently they're not familiar with my
39:55back catalogue of extremely niche online maths videos, if you can believe it.
40:01Even in a hotel with an infinite number of rooms, there's a problem.
40:05The manager can't just put me in the last room, because in an infinite hotel there is no last room.
40:12So if the hotel is full, how do I find a bed for the night?
40:19All we have to do is politely ask the person staying in room one to move along into room two,
40:26the person in room two to move to room three, three to four, four to five, and so on, and so on, and so on.
40:36As there's no last room, if you move everyone along by one room number, every guest has somewhere to sleep.
40:47And that leaves room one for me.
40:52Even if the hotel is full, a room can always be found.
40:56That's because infinity plus one is still infinity, so there's always room at the infinity hotel,
41:05because you can always add on an extra room at the beginning to make infinity just that little bit bigger.
41:12And if my friend wants to come and stay too, well, infinity plus two is still infinity,
41:18which is perfect for a girls weekend away.
41:21I told you it was weird.
41:29That's the thing about infinity. It's a very slippery beast.
41:38There was one mathematician who set out to tame the infinite beast.
41:43His name was Georg Cantor, and the question he wanted to answer sounded deceptively simple.
41:52How big is infinity?
41:56With that one simple question, Cantor would start a revolution,
42:01one that would have a profound effect on the foundations of mathematics.
42:06I've come to Halle in Germany. It was here that Cantor taught in the city's university.
42:20For him, infinity was the key that opened the door to a new mathematical landscape.
42:29I don't know about you, but I find it quite hard to picture in my head
42:34the size of something like our solar system or our galaxy, the Milky Way.
42:40These distances are so big that they defy our imagination.
42:45But each of these things scales into insignificance.
42:49They are infinitesimally small when compared to the vastness of infinity.
42:57While the idea of infinity was known to the ancient Greeks,
43:01some of Cantor's contemporaries saw it as an offshoot of maths,
43:04rather than anything worth understanding in its own right.
43:10This wasn't good enough for Cantor.
43:12If our knowledge of the world is built on infinity, he said,
43:17we can't just accept it. We have to understand it.
43:20To get a handle on infinity, take a look at these two sets of numbers.
43:30Let's imagine that along here, you've got all of the natural numbers, the counting numbers.
43:34So one, two, three, four, five, six, seven, eight, and so on.
43:40Now there's going to be an infinite number of these.
43:43Now next to it, let's put the even numbers.
43:46So two, four, six, eight, and so on.
43:52On the surface of it, it looks like this infinity will be bigger than that one.
43:59As both of these lines will go on forever, it seems obvious that the infinity of one,
44:06two, three, four will be bigger than the infinity of the even numbers, two, four, six, eight.
44:13After all, there's only half as many of those.
44:17But actually, if you shuffle all of these along, they actually match up rather nicely.
44:21So one goes with two, two goes with four, three goes with six, and so on, and so on.
44:28Neither of these lists are ever going to run out.
44:33As each list of numbers never stops, every counting number can always find an even number to pair up with.
44:43As a result, both infinite lists of numbers have to be the same size.
44:50We know this is true because we can count them.
44:54I know that seems like a bit of a strange idea, but just go with me on this for a second.
44:58Because you can start at the beginning and work your way up counting as you go.
45:03The first number, the second number, the third number, and so on, and so on.
45:08Now it's true that you would have to carry on counting forever,
45:11but you could be sure that you wouldn't miss any of the numbers as you went.
45:16Even though the infinity of the counting numbers looks bigger than the infinity of the even numbers,
45:23they're actually the same size.
45:27Next, Cantor tried something different.
45:29He set out to count all the numbers between zero and one.
45:35Where is the most sensible place to begin?
45:37Is it 0.1?
45:39Well, no, because 0.01 is smaller.
45:43And it can't be 0.01 either, because 0.001 is smaller still, and 0.001 is smaller still.
45:51Wherever you try and start, I can always find another number to squish in.
45:58And that means there is no sensible place to start.
46:04However hard you try, you can't count up the number of numbers between 0 and 1.
46:12This infinity is uncountable.
46:16Cantor's disturbing conclusion was that some infinities are bigger than others.
46:23The sheer audacity of his work set off a quiet revolution in the world of mathematics.
46:30If Cantor thought that his work was going to be welcomed with open arms,
46:34then he was to be sorely disappointed.
46:37He was attacked on all sides by his academic colleagues.
46:41They called him a scientific charlatan and a corrupter of the youth.
46:46And some even tried to sabotage the publication of his works.
46:52Could it be that Cantor's ideas on infinity were merely a product of his own imagination,
46:59something he invented?
47:03His work on infinity consumed every waking minute.
47:08In May of 1884, he suffered a nervous breakdown.
47:13Eventually, he was brought here to the Nerf and Centre in Halle, a psychiatric hospital.
47:22How did Cantor's desire to tame the infinite impact on his illness?
47:28I'm meeting the hospital's director, Dr. Frank Pillman.
47:31This, for example, is a case note from 1907.
47:36Mania, an acute episode of periodic circular psychosis.
47:43This is what we would today call bipolar disorder.
47:48There are some people who have suggested that the struggle that he was having with his mental health
47:56was exacerbated by his fight to try and find these answers around infinity.
48:01What's your opinion on that?
48:02I would feel that the intellectual occupation with mathematical theories is
48:09nothing that makes you prone to get a psychiatric illness.
48:13As far as we know about his personality, he's always been described as a very ambitious
48:22person, certainly creative.
48:25Of course, he tried to solve some unsolvable problems, but I think that's the life of every mathematician.
48:33It's probably true, the struggle, the struggle with very difficult problems.
48:47This is a memorial to Cantor.
48:50He was feared by his critics because he dared to question their assumptions of conventional mathematics.
48:57His work on infinity was crucial for building more complex mathematical ideas that we rely on today.
49:09This is where mathematics starts to stray much more into the realms of the abstract.
49:14Infinities, bigger infinities, countable and uncountable infinities.
49:19These are not things that you tend to find in the physical world.
49:23So is it all just a product of our intellect and imagination?
49:28Is this mathematics invented?
49:32Certainly, when you just take the basic concept of infinity, it's meant to be the biggest possible
49:36thing, right? And then someone tells you that there's lots of infinities.
49:39So it's certainly a very puzzling concept, but it's an essential one. It's an essential feature of
49:45huge numbers of mathematical systems.
49:47Insofar that mathematics arises as an interaction between reality and conscious rational minds,
49:57and that's what creates mathematics, I would say infinity is real in that sense.
50:03If you ask me, is it real in actual reality? I do not know the answer to that question,
50:10nor do I know how to find the answer to that question.
50:14Some people find it emotionally disturbing, this idea that reality is bigger than we thought.
50:20I actually find it kind of liberating. I think it would be rather claustrophobic
50:25if our reality were really small.
50:31Maths has taken us from a time when we could spot patterns in nature,
50:36to being able to describe the invisible forces that form the structure of the cosmos.
50:42To probe this hidden world, we've invented mathematical tools and equations.
50:49Maths has quietly, almost invisibly, revolutionized the way we understand our place in the universe.
50:57Today, the argument about whether maths is invented or discovered is much more than a philosophical debate.
51:07This is where it gets real. This jumble of pipes and wires looks chaotic,
51:13but it's at the cutting edge of science. If the researchers here succeed in their goal,
51:19they'll have found the answer to the world's energy needs. A form of power that's clean, renewable and free.
51:28I've come here today to the Cullum Center for Fusion Energy, where a group of people are trying to do
51:34something rather remarkable. They're taking a mathematical description of reality and trying to bend it to
51:40their will. Harnessing the power of a star and using it to change humanity's future.
51:49Controlling the power of a star such as our sun is, as you might imagine, incredibly difficult.
51:58The sun is one giant hot ball of gas called a plasma. Its heat is generated when atoms of hydrogen
52:05inside this plasma collide with each other very quickly, releasing vast amounts of energy.
52:13The challenge is to recreate that reaction down here on Earth, and the first step is to form the plasma.
52:23Within this shape, they're trying to recreate the conditions that you find in the inside of the sun
52:28and hold that plasma in place while it reaches temperatures of up to 200 million degrees Celsius.
52:39This doughnut-shaped space is called a tokamak. The most difficult part of this whole process is ensuring
52:48the plasma remains stable. If part of it touches the walls, the plasma cools and the reaction stops,
52:57trying to prevent that from happening is the job of Dr Anthony Shaw.
53:03The difficulty is that at 200 million degrees, you get quite a lot of extra effects coming in.
53:09It gets turbulent like the churning sea. There are various currents and turbulences and tides and all
53:16these things that make the behaviour of it very tricky to understand. And if you don't account for the
53:21right things at the right time, it'll do what it wants instead of what we want.
53:27Driving this behaviour are lots of subatomic reactions that no one has ever seen. The only
53:33reason we believe they exist is down to maths. Anthony and his colleagues are using maths to try and
53:42predict how these invisible particles will behave inside the plasma.
53:49So here we have a photograph that was taken inside the tokamak.
53:54You can see the hydrogen plasma here just glowing around the edges. And they've overlaid a photograph
54:01of the structure just so you can see roughly where it's sitting. For comparison, there is also a
54:07simulation of this, a mathematical simulation. And on this one you can see very clearly these little
54:14lines, they're called filaments. This is where wisps of plasma go out and touch the side. Now this one
54:20is purely mathematical, but what the physicists do is make comparisons between the two to see how well
54:28their mathematical version matches up to what really happened. And if you put these two side by side,
54:35you can see how well the mathematical version matches up with what really happened.
54:44By comparing the simulation of how the plasma is predicted to behave to what actually happened,
54:51it becomes clear that the mathematical model accurately predicted where the plasma would break down.
55:05Now the reason why this is important is because there's no limit really to the number of
55:10mathematical simulations you can run. But once you get them matching up to reality, once you know
55:16that your mathematical version is an accurate reflection of what's happening inside, that is the first
55:24step to being able to control your plasma. Nuclear fusion holds out the promise of almost unlimited supplies
55:33of clean energy. If they can ever solve this problem, the answer will lie in mathematics and its ability to
55:42describe an invisible world of subatomic particles and forces. The only way you know what's happening
55:50inside that plasma is by using mathematics. It's the maths that tells you how all of this works.
55:58In trying to replicate what's happening inside a star, we're pushing the boundaries of what science and maths is
56:05capable of. But we've been doing research in this area for decades, we've had the equations for even longer,
56:12and yet we're still not quite getting perfectly and neatly to the answer.
56:17If there are these gaps around the edges, if there are limits to how far the maths can take us,
56:24then how can it be discovered? Maybe it is just an invention after all.
56:30So where have we got to with our investigation of mathematics so far?
56:34Well, Newton came along with his fundamental laws of gravity that led to these incredibly powerful
56:43equations that can precisely predict the movement of planets in the universe. But they're not quite
56:50perfect. But then you have Cantor and his amazing ideas about different sizes of infinities. And
56:59mainly maths starts to go down a slightly different path. And the more you go down that road,
57:05the more it starts to feel like mathematics is invented.
57:29So
57:48you