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00:01There is a mystery at the heart of our universe.
00:06A puzzle that so far no one has been able to solve.
00:10It's kind of too weird.
00:12Welcome to my world.
00:14If we can solve this mystery, it will have profound consequences for all of us.
00:20That mystery is why mathematical rules and patterns seem to infiltrate pretty much everything in the world around us.
00:30Many people have, in fact, described maths as the underlying language of the universe.
00:38But how did it get there?
00:41Even after thousands of years, this question causes controversy.
00:47We still can't agree on what maths actually is or where it comes from.
00:52Is it something that's invented like a language or is it something that we have merely discovered?
00:58I think discovered.
01:00Invented.
01:01It's both.
01:02I have no idea.
01:04Oh my god!
01:06Why does any of this matter?
01:08Well, maths underpins just about everything in our modern world.
01:13From computers and mobile phones to our understanding of human biology and our place in the universe.
01:21My name is Hannah Fry and I'm a mathematician.
01:27In this series, I will explore how the greatest thinkers in history have tried to explain the origins of maths extraordinary power.
01:36You've ruined his equation!
01:39I'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 the scientific and industrial revolutions.
01:58And I'll reveal how, in the 20th and 21st centuries, radical new theories are forcing us to question, once again, everything we thought to be new about maths and the universe.
02:13The unexpected should be expected, because why would reality down there bear any resemblance to reality up here?
02:20In this episode, I explore paradoxes within modern mathematics.
02:27Who shaves the barber?
02:29And I discover the very weird world that maths seems to be leading us into.
02:35maths is very much part of our modern world.
03:04Even the images you're watching now are essentially numbers processed by computers.
03:11Sorry guys, would you mind taking a photo of me?
03:15Oh sure!
03:16Give me one second.
03:17Today maths is at the heart of big business in the development of new software, such as facial recognition technology.
03:25All of which, fundamentally, is based on mathematical algorithms.
03:32And it matters because copyright issues and legal ownership can depend on where that maths comes from.
03:41You can phrase the question like this.
03:43Is maths a genuine, fundamental part of our universe?
03:47Something that we have discovered?
03:50Or is it merely invented?
03:53A language that we've created just to describe the world around us?
03:57Mathematicians have argued over this idea for centuries.
04:04And even today, this question is a thought-provoking and challenging dilemma.
04:09So far, I've explored how, in ancient times, maths was revered as a gift from the gods.
04:22Perfect, complete and gratefully discovered by humans.
04:28But through the ages, new areas of mathematics, like algebra and the concept of zero, have, quite simply, been invented.
04:39But for most of us, we normally think of maths as just a series of objective facts, based in logic that someone somewhere has discovered.
04:50Facts that we all start to learn at school.
04:54If you're anything like me, you'll remember maths at school being taught as a series of rules.
05:00It was very logical, it was very ordered, very complete, very black and white.
05:05There were right and wrong answers, which you didn't necessarily get in other subjects like art or like music,
05:13which were much more about preferences, about opinions and about cultural differences.
05:22It felt like the mathematical rules were intrinsically true.
05:26But why? What are the fundamental mathematical laws?
05:32To answer that question, you have to categorise everything.
05:35You have to boil maths down into distinct groups of objects in something called set theory.
05:42Set theory is a language that talks about groups or sets of items.
05:48So, for example, the set of odd numbers are all the whole numbers that cannot be uniquely divided by two.
06:00And the set of even numbers are those that can.
06:03This reveals a basic rule. Adding an odd number to an even one produces an odd number.
06:16From simple rules like these, you can build up more and more complex rules and relations of maths.
06:23But there's a problem with set theory, a paradox at the heart of mathematical rules which cause a bit of a crisis at the start of the 20th century.
06:36You can discover this paradox yourself by going to your local hairdresser or gentleman's barber and trying to define what you find in a concise and complete way.
06:48Hello. Hello.
06:49I was wondering if you could help me. I am looking for the very definition of a barber.
06:54I think I can help with that.
06:55Mathematicians took the same approach to precisely define the laws of maths.
07:00So, if you were looking it up in a dictionary, like one sentence that defined what a barber was, what would you say it was?
07:08Cut men's hair.
07:09Cut men's hair.
07:10But that could be a hairdresser though, right? Hairdresser.
07:14It needs to be a unique definition for barbers. Barbers and only barbers.
07:19There's a shaving element as well, isn't there?
07:21Yeah, that's true.
07:22I've never had a shave in a hairdresser.
07:23No, that's true.
07:24The chat.
07:25The chat.
07:26That's true.
07:27Yeah.
07:28That's true.
07:29It's a very important barber.
07:30You do hear some stories being a barber.
07:33So, actually, I suppose the shave thing is something that only barbers do.
07:38So, someone who shaves men.
07:42But a barber doesn't shave all men.
07:44And I need a phrase that uniquely and completely identifies a barber and no one else.
07:51Okay, let's see where we are then.
07:52So, we've got a barber shaves all men but only the men who shave but don't shave themselves.
07:59Yes.
08:00Yes.
08:01Alright, I think we've settled on something now.
08:03We've agreed on a barber shaves all men and only those men who shave but do not shave themselves.
08:14Sound about right?
08:15Well, it doesn't exactly run off the tongue.
08:17I think it's fairly accurate.
08:18But hang on a second.
08:20There's a bit of a paradox here.
08:23Who shaves the barber?
08:25Well, can a barber not shave himself?
08:28But if he does shave himself, then our definition here says that he doesn't shave himself.
08:35Let me clarify that.
08:36If he doesn't shave himself, then according to the definition he's one of the men shaved by the barber.
08:47So, he does shave himself.
08:50Attempting to create a mathematically precise definition creates a contradiction where the barber both shaves himself and doesn't shave himself.
09:00Push the bristles into the face.
09:03This is known as the barber's paradox.
09:06Let's get this bit.
09:07You got it?
09:08Perfect.
09:09Perfect.
09:10Okay.
09:11It is an illustration of the paradox at the heart of mathematics, which was discovered in 1901 by one of my favourite troublemakers, Bertrand Russell.
09:22The problem for maths was that Russell's paradox undermines the logic of defining things, like odd or even numbers, by putting them into categories or sets.
09:34Over here, I have got a set of clipper attachments.
09:38And in there, I have got a set of things that aren't clipper attachments.
09:42Clipper attachment goes in there.
09:45Not a clipper attachment goes in there.
09:49Clipper.
09:52Not a clipper.
09:55Now, the question is, where does this bag belong?
09:59It's clearly not a clipper attachment.
10:01Is it going to attach to a clipper?
10:03No, it's not.
10:04Which means it needs to go in there.
10:07But we've got a problem.
10:08Because this sink is supposed to only contain things that are not clipper attachments.
10:15Which means that the contents of the bag can't go in the sink.
10:19Since the bag, or set, is not in itself a clipper attachment, but by its definition contains clipper attachments, we can't easily categorise where the set belongs.
10:35Similarly, the barber can't, in a logically consistent way, be contained in the set of people that do shape themselves, or the set of people who don't.
10:45Russell's paradox shows that there is a logical problem with trying to categorise anything into coherent sets.
10:56Whether it's barbers, clipper attachments, or even numbers.
11:00And this logical puzzle exposed a fault in the bedrock on which all the rest of maths is built.
11:07If the foundations are shaky, how can we trust everything else?
11:15Bertrand Russell realised that mathematics was on much shakier ground than people had originally thought.
11:25It turned out to be much, much harder to really lay a solid foundation for math that everybody agreed on.
11:33And this is still wonderfully controversial to this day.
11:36That's what you do in science, in mathematics, you take a sledgehammer, you smash at whatever structure, whatever edifice you've built, you try to find the weaknesses, and that allows you to figure out what needs to be shored up.
11:49And that's really, I think, the legacy that Russell left us.
11:53I think of it as, in some ways, the death knell, or at least a major challenge, the attempt to ground mathematics in logic.
11:59And that's the thing that becomes really hard in light of Russell's paradox.
12:12Russell's paradox caused a real crisis amongst mathematicians.
12:15Suddenly, maths was uncertain, was fallible.
12:18And if it has these fundamental problems, how can it possibly be discovered?
12:26So does that mean that maths has to be invented?
12:29Just a human language and all of the flaws that come with it.
12:36If maths is merely an invention of the human mind, it's perhaps not that surprising that it's not perfect.
12:42But I don't think I'm ready to accept the invention argument quite yet.
12:49Maths just seems to be too good at predicting the behaviour of the world in ways we never could have imagined.
12:56Because just as Bertrand Russell was exposing the limitations of maths in one way,
13:02another titan of the 20th century, Albert Einstein, was pulling it back in a completely different direction.
13:08Take what is probably the most famous equation in the world.
13:18With just five symbols, it looks so simple, it's almost childish.
13:23Yet it contains some incredibly powerful mathematical and philosophical concepts.
13:30I'm talking, of course, about E equals MC squared.
13:34So E, that's energy, that is equal to M, that's mass, times by a constant C, that's the speed of light, squared.
13:51Look!
13:53There is so much more to this equation than meets the eye.
13:57It is Einstein's discovery that matter and energy are equivalent.
14:02And that has profound consequences.
14:05This equation gives us one of the immutable laws in the universe,
14:11that nothing can travel faster than the speed of light.
14:15Watch on this one.
14:16The reasoning is this.
14:19Making something move requires more energy than keeping it at rest.
14:25And because this C here is a constant, if the energy goes up by accelerating something,
14:32the mass also has to increase.
14:35So that means that you or I actually weigh a tiny bit more when we're moving in a car or a plane.
14:43The increase in mass only becomes significant when objects are moving at speeds close to the speed of light.
14:54As an object approaches the speed of light, its mass rises faster and faster,
15:02which means it takes more energy to accelerate it further.
15:06It can't therefore reach the speed of light because the mass becomes infinite and it would require an infinite amount of energy to get there.
15:19You've ruined this equation!
15:20As well as proving there's a cosmological speed limit, this single equation also explains how all the stars in the universe convert mass into energy as they burn brightly in the night sky.
15:36Einstein's famous equation has proved itself to be a remarkable match for reality every time it's been put to the test.
15:45Einstein had uncovered one of the essential mathematical rules underlying the cosmos.
15:59It seems like clear evidence that, that maths at least, is discovered.
16:04But Einstein didn't stop there.
16:07Using the power of mathematics, he brought about a fundamental shift in our understanding of space and of time and of how light travels through space.
16:21To see that evidence for myself, I've come to an observatory to do some serious thinking about what we actually see when we look at stars in the sky, such as our sun.
16:32If things were happening right now, we wouldn't be able to see it until eight and a half minutes later, because that's how long it takes the light to travel to the earth.
16:43So when you're looking at the sun, you're seeing how it was eight and a half minutes ago?
16:47Exactly. And objects that are further away, we see them as they were further back in time.
16:53So for instance, there are other stars in our galaxy that are thousands of light years away, so we see them as they were thousands of years ago.
17:00So when you look in a telescope and you're seeing them how they were when people were building pyramids and Pythagoras was discovering his rules on Earth.
17:10Exactly. And we can see things that are even further away than that.
17:14So galaxies outside our own galaxy, we see many of them as they were a billion years ago or more.
17:20Gosh, goodness. Does this work at smaller scales then? Is there like a limit to how big something has to be before this works?
17:27If you, I mean, I'm looking at you now, right? Light presumably is taking time to bounce off you and for me to see you.
17:34Yes, it is. But light travels at an incredibly fast speed, 300,000 kilometres per second, roughly. So the time it takes to travel from me to you is very, very tiny fraction of a second.
17:46But in theory, I am seeing you in the past. In theory, yes, you're absolutely seeing me in the past.
17:53All of this shows that we can never know what the universe is like at this very instant.
17:59The universe is, remarkably, not a thing that extends just in space, but in time as well.
18:09This is fundamental to Einstein's revolutionary insights about our universe.
18:16He realised that the very concept of time is relative.
18:20That is to say, it depends on the position and movement of the observer.
18:25He worked it out by thinking about events that appear to be simultaneous.
18:32So let's imagine that you're in a hot air balloon, jumping above the observatory here.
18:38And you're high enough that you can see a flash of light in London, say, and another one in Portsmouth.
18:45And let's assume that these flashes of light go off such that you see both of them happening exactly simultaneously.
18:54So from where I am, it looks like they're both flashing the lights at the same time?
18:58At exactly the same time. But if I were in an aircraft that was flying very fast towards London, I would see the flash of light in London before the flash of light from Portsmouth.
19:11Using the inescapable logic of mathematics, Einstein realised that if an observer is moving towards one of the flashes, they would see that flash before the other one caught up with them.
19:26So for them, the flashes are not simultaneous.
19:31But who's, okay, but I mean, they did go off together. Who's wrong? Am I right in the hot air balloon?
19:38In fact, there is no way of saying that you are right and I am wrong in how we observe these events.
19:45How we observe these events.
19:46This is called relativity.
19:49So our whole concept of time, our whole concept of timing, what happens first, what happens second, comes down to where we are and how we're moving.
19:57Exactly. So the concept of time is now inextricably linked to the positions in space and your movement through space.
20:08So this is why we can't describe space and time separately, but we have to put them together in space time.
20:15You can't separate the two.
20:16You can't separate the two.
20:18And that all comes down to this idea that Einstein managed to prove via thought experiments.
20:25Yeah, that's the amazing thing about it. Purely through thought experiments.
20:30And a good bit of maths.
20:32And a good bit of maths. A very good bit of maths.
20:38Einstein was using the mathematics to make sense of the universe and claiming that the universe was nothing like what anyone thought it was.
20:47His concept of relativity flew in the face of what people had believed about space and about time for centuries.
20:56Whether that was the Greeks thinking that the universe was eternal and unchanging.
21:02Or Isaac Newton's more mobile and mechanistic descriptions.
21:06Einstein took his thoughts even further, attempting to wrestle gravity into a neat mathematical law.
21:17He believed it was all down to the strange behaviour of space time.
21:22And if he was right, as he laid out in the theory of general relativity in 1916, then gravity will even affect light.
21:30If you've got a star shining light from over here, then you, the observer, over there, will receive it in a straight line.
21:42But if there's a massive object in the way, you might think you won't be able to see the star.
21:56However, Einstein predicted that the mass of an object will distort the space time around it.
22:04And anything moving through that warped space time will have to follow the curves.
22:09This warping of space time, Einstein said, is what we usually describe as gravity.
22:17We think of gravity as keeping the planet in orbit around our sun.
22:22In fact, he said, it's the result of the distortion of space time near massive objects.
22:29And Einstein calculated the precise effect it would have on light.
22:33So, the starlight, while still technically travelling in a straight line, will follow the curves of space and appear around the object.
22:45Einstein predicted that, in exactly this way, we should be able to observe light from distant stars getting bent, as the stars pass behind our sun.
23:01But a theory is just a theory, an invention of the mind.
23:11It only becomes a discovery when proven by practical measurement or experiment.
23:16In the decade after Einstein's prediction, solar eclipses around the globe gave scientists the chance to repeatedly test his theory.
23:27The darkness of the eclipse allowed them to actually see stars passing close to the sun.
23:34When scientists took the measurements, they discovered that light from a distant star was bending around the sun in exactly the way that Einstein had predicted.
23:47The mathematics of general relativity was correct.
23:54With general relativity, Einstein completely upended our understanding of space, time, matter, energy, and kind of what else is there to the nature of reality.
24:02All of a sudden, we learned that mass and energy can warp the fabric of space and time in this beautiful interconnected dance,
24:09where the motion of matter affects the warping of space and time, which affects the motion of other matter.
24:15We used to think of space as this boring static stage upon which events unfolded.
24:22Then Einstein told us that space is itself an active player in this game, like a stretchy rubber sheet.
24:29And yet, a substance perfectly described by beautiful mathematical equations.
24:36I mean, how did he think of that? How did he think of something like this?
24:40Einstein's description of gravity, the warping of space-time, accurately explains why objects stay in orbit,
24:48whether they're satellites around the Earth or galaxies around black holes.
24:52His equations are being tested and reproven every day, and without Einstein's general theory of relativity,
25:01modern communication, GPS or satellite TV systems couldn't even function.
25:06Although this theory came from his mind, from thinking about the problem rather than from real-world experiments,
25:15it's still so good at predicting, so perfectly capable of describing what happens in the universe,
25:23that it must be reflecting some underlying mathematical truth.
25:28And this lends quite a lot of weight to the argument that mathematics is discovered,
25:34which is something that matches up with my own experience.
25:37Because when you're toying around with mathematics,
25:41it really does feel as though you're exploring something that already exists.
25:46But if we accept that maths does already exist, and is an intrinsic part of nature,
25:59then surely all the rules are out there waiting to be discovered.
26:07In some ways, mathematics is quite a lot like a game of chess.
26:10So you have these very strict rules that you're not allowed to break.
26:15But within those rules, there are all kinds of opportunities to play around and be creative.
26:21The only problem is that in maths, no one tells you what those rules are.
26:27We have to work them out for ourselves.
26:33Most mathematicians like a challenge.
26:36But this idea got blown apart at a maths conference in 1930,
26:42in the Prussian city of Königsberg,
26:45when two great mathematicians and their conclusions collided.
26:51On the one side, you have got David Hilbert,
26:55a mathematical king in every possible sense of the word.
26:58This is an enormously well-respected man who laid down the gauntlet,
27:02asking people to come up with a fundamental set of rules
27:07on which every mathematical proof could be based.
27:10On the other side was a young academic called Kurt Gödel.
27:15In contrast to Hilbert, who thought that mathematics should be built from the ground up by humans,
27:21Gödel thought that mathematics was discovered.
27:24He believed that mathematical truths exist outside of us and that we have very little say in what we can find.
27:34That summit in Königsberg can be seen as a clash between those who thought that mathematics is part of our fabric of reality to be discovered,
27:44and those who saw it as a language under our control.
27:50Hilbert was confident that humanity would soon know all there is to know in maths.
27:56But Gödel, who had also been trying to find the rules of maths, had come to the opposite conclusion.
28:03In a side room at the summit, Gödel quietly announces that in fact,
28:09however hard you try, there are always going to be some things that are unknowable.
28:17There are always going to be parts of the mathematical game that can't be fully explained.
28:24And if you can't know all the rules, how can you play the game?
28:28According to Gödel, any rule-based maths system is always going to have some things that are either unknowable or unprovable.
28:36And what's more, he could prove it.
28:41It's kind of ironic, if you think about it.
28:44This was quickly accepted and became known as Gödel's incompleteness theorem.
28:50And it puts an interesting twist on our key question.
28:53It shows that even if mathematical rules truly are part of the universe and we're simply discovering them,
29:00we are nevertheless going to have to accept some of those rules without knowing how or why they are true.
29:11Normally people think that there's some intrinsic difference between science and math on one hand,
29:17and faith-based belief systems on the other.
29:21And yet what Gödel's theorem tells us is that's not true.
29:25That there are things in mathematics that you have to take on faith or you can't do the mathematics.
29:30To me, this was an astounding thing to realize.
29:33We're going to have to accept that we can't give math a foundation in formal rules or in logic in the way that we thought we could.
29:41I think it's enormously exciting that math in some sense is open-ended.
29:48So in a sense, it puts an end to one way of thinking about mathematics,
29:54but I think it actually adds color and richness to the subject because it's just going to keep on going.
30:00So what does Gödel's incompleteness theorem mean for our view of the universe and the parts that maths plays in it?
30:16Well, it depends on what you're trying to use maths for.
30:20If your goal is to use it to describe what's around you, then it still offers a very detailed picture,
30:26enough to navigate your way through the universe and to explain its features.
30:31Sure, the map is not going to be the same as the terrain,
30:34but even if maths is a bit incomplete around the edges, you could argue that it doesn't really matter.
30:43Although Gödel proves it's not possible to formalize all of maths,
30:48it is possible to formalize all the mathematics we actually need to use.
30:52Take flying as an example. Now, I did my PhD in the mathematics of aerodynamics,
30:59and that means I spent four years poring over equations for wing sections and wind speed.
31:05It's stuff that I know like the back of my hand.
31:08But does that qualify me for going up in one of these on my own?
31:13Absolutely not.
31:14And on the other hand, these guys don't really need to know any of this stuff to make them graceful acrobats in the air.
31:24Not having a complete understanding doesn't always matter.
31:28We've still flown successfully for over a hundred years.
31:31And now, it's my turn.
31:33And then this is your strap that comes across.
31:37This will dig in a little bit on take-off when you're leaning forward and running down the hill.
31:43I can handle it. It should be a little bit uncomfortable.
31:45I can handle it. Don't worry too much about it.
31:47And do you have quite a good feel for where the thermals are?
31:50You have to have the right weather conditions.
31:51So, if you imagine a hill that faces totally into the wind, that's well drained, maybe darker,
31:58and it will create this kind of pool of warm air.
32:00And then it will, once it kind of reaches a decent temperature difference, it bobbles up through the atmosphere.
32:06It's almost like we've got kind of opposing skills.
32:09And like, they're sort of about the same thing, but they, you don't need my skills to do what you do,
32:13and I couldn't do what you do.
32:15I guess the ground speed element, there's a bit of mass in there.
32:17Like, we start the lesson with a bit of mass to begin with.
32:19Where's the wind coming from? How strong it is? How fast am I going to go if I point into wind?
32:24But you're not solving Navier-Stokes equations, are you?
32:26I don't even know what that means.
32:27Yeah, exactly.
32:31Before the theoretical analysis of aviation came along,
32:35the practical side of flying was mere trial and error.
32:39Now we have a much more reliable understanding of what keeps us aloft.
32:43And it doesn't really matter if the mass behind it is, ultimately, a bit fuzzy around the edges.
32:52In the real world, the best that we can do is just accept Gödel's Incompleteness Theorem and get on with life!
33:03Ho-hey!
33:04It's amazing!
33:05There's a thermal!
33:14Yeah!
33:16It's got a bit stronger.
33:18Yeah.
33:21We have to put aside for the moment the question of whether maths is invented or discovered.
33:27Because it now looks like we may have to determine which part of maths we are asking about.
33:33You see, for me, Gödel's work highlights the distinction between pure theoretical maths and practical applied maths.
33:44So here is how I see things.
33:46With mathematics, there's a split down the middle of the subject,
33:49because the story changes depending on what world you start with,
33:53whether it's the real one or one that exists in our imaginations.
33:57And right now, when we're flying, this is very much in the realm of applied mathematics,
34:02where everything is tangible and practical and a little bit imprecise.
34:08But alongside that is where the more theoretical, pure mathematics lives.
34:17That's where you have your proofs, your paradoxes and incompleteness theorems.
34:22A realm which doesn't match up with a physical reality, a sort of imperfect perfection.
34:27Even though I instinctively feel that maths is discovered, I like that there is this pure theoretical part of maths that isn't found in reality.
34:39And since the maths there doesn't need to match reality, it's a convenient place where we can leave all the weird contradictory bits that we come across.
34:51However, I might have it the wrong way round.
35:05Although pure theoretical maths seems rather divorced from reality,
35:10that might merely reflect the fact that reality is not quite what we think it is.
35:15And it's a reality that we can uncover through the strange maths of quantum physics.
35:25The weirdest worlds that most of us have come across are likely to be in fiction, such as this, Alice's Adventures in Wonderland.
35:34Now, the author Lewis Carroll, real name Charles Dodgson, was actually a mathematics don at Oxford and a staunch traditionalist.
35:42It's generally believed that much of this surreal story is a thinly veiled satire on the new avant-garde maths that was flourishing when he was writing in the 1860s.
35:54Still feels relevant today and applies equally well to the new weird kid on the block, quantum physics.
36:02Take a close look at the physical world around us and you can reduce it all to maths.
36:09The solid bricks of our houses or the blood cells in our veins can all be reduced down into chemicals which comprise elements,
36:19which themselves are made up of atoms, comprising a tiny nucleus of protons and neutrons and electrons buzzing around in a cloud of mostly emptiness.
36:30The protons and neutrons in turn are built from smaller subatomic particles that we can't directly observe.
36:41We can only verify their existence using experiments and mathematics.
36:46As we delve deeper into this world, scientists have discovered something very strange indeed.
36:57We can never actually know the precise location of most particles in this subatomic or quantum realm.
37:04All we can know is the likelihood of them being somewhere, a mathematical formula that describes the probability of their position.
37:15All of this means we are fundamentally, at the quantum level, just a great fuzz of energy and probabilities.
37:24I'm not sure Lewis Carroll would have liked that.
37:26And the only way to explore this ill-defined quantum world
37:37is through mathematics, perfectly equipped to handle strange probabilities.
37:43It seems like there's quite a lot of uncertainty in quantum physics. Does that bother you?
37:52Um, no. When I heard that things were, you know, uncertain and also against our common sense in quantum physics,
38:00then I thought like, oh, wow, that sounds interesting. I want to know more about that.
38:03The pivotal maths behind the quantum world was first laid out by Austrian physicist Erwin Schrödinger in 1926.
38:13His equations accurately describe the unusual behaviour of subatomic particles.
38:20OK, all right, tell you what then. Quantum physics lesson 101, where do we start? Give me a little second.
38:27OK, I would say we have to start with superposition.
38:29So let's talk about electrons. So there's a very small particle and they can be in two states.
38:38They have a state with spin and the spin can be pointing up or down.
38:43So if we were in the classical world, the spin could only be either up or down.
38:49But in the quantum world, the spin is in a superposition, which means it can be up and down at the same time.
38:56Let me see if I understand this then. So superposition is where something is and isn't something at the same time.
39:05Yes, we can think about some examples. So let's say that we have a cup and the cup is full of water or that's one state.
39:15Another possible state is that the cup is empty. So if we were bringing the quantum ideas to the classical world, we would say that the state, one possible state of the cup would be to be empty and full at the same time.
39:29OK, which you never see in the world that we're living in. You never see a cup that's full and empty.
39:36Yes, we don't.
39:38But you see this a lot in the quantum world.
39:41Yes, superpositions are an essential part of the quantum world.
39:46Like a light being on and off at the same time.
39:48Exactly, or the cake being eaten or not eaten at the same time.
39:52OK, it's a very tough idea to get around.
39:54Yes, yes.
39:55Given two possible outcomes, in the quantum world, we now have to allow for a third one, the combination of both outcomes.
40:05At the quantum scale, you can have your cake and eat it.
40:09This is such a weird idea. How do we know it's real?
40:15Well, because we've done many experiments to prove it that show exactly that behaviour.
40:20What does that experiment look like?
40:23Well, if we put it, say, in terms of things we have here in the table, we could think about, let's say that I wanted this piece of sugar to come into my cup.
40:34But there's this pot in the middle.
40:36So then if the sugar is going to come from here to my cup, it could either go this way or that way in the classical world.
40:48But in a quantum experiment, it can take both roots at the same time.
40:53And I would be able to distinguish that it did that if I did a quantum experiment.
40:57It's kind of too weird.
40:58Welcome to my world.
41:00So you go through this whole transition from first the ideas and the mathematics and then up to showing it in the experiment.
41:11What came out of Schrodinger's maths was a prediction of something even stranger that can sometimes be produced when particles interact in the quantum world.
41:21A phenomenon called entanglement.
41:23All right, tell me about entanglement then.
41:27OK, so take two electrons.
41:29If the electrons are entangled and I do something to one of the electrons, I, for example, change the direction of the spin, that will instantaneously affect the state of the other electron, even if they're separated long distances.
41:45How far away are they from each other?
41:48Well, they can be a few centimeters, but now the latest experiments using satellites show entanglement across 1,200 kilometers.
42:00What?
42:01Yes.
42:02You've got something over here and you do, and something at 1,200 kilometers away.
42:09You do something to one and it instantly, the other one instantly knows what's happened.
42:14Yes, you affect the state of the other one instantly.
42:18Apparently there is no causal link.
42:22The only thing we can say is that the two particles are synchronized.
42:25How does one know what the other one's doing?
42:29Well, that we're still trying to understand because that's what mathematics tells us, and then we can show it in the experiment, but we're still struggling to understand what that means.
42:41And one of the reasons why we don't understand it in, you know, like you're asking, is because we don't see it in our everyday life.
42:50So, let's say it's not part of our experience and common sense, but that doesn't mean it doesn't happen.
43:00So quantum mathematics has made predictions which have been discovered to be true.
43:05But despite that, the quantum world is so weird, it suggests to me that the math behind it is just invented.
43:18It feels like what we're seeing is evidence of a man-made system being pushed too far.
43:25These are the absurdities that appear when it's applied to situations it wasn't designed for.
43:35But my quest to find the truth about maths takes me back to nature.
43:43There is amazing new evidence that quantum processes might actually be crucial to our own existence and much of life on Earth.
43:55That would strengthen the argument that mathematical processes are intrinsic to our world, that maths is discovered.
44:03It all comes down to photosynthesis, the process that converts sunlight into chemical energy used in life.
44:13It takes place in molecules called chlorophyll, which can be found in plants, algae and bacteria.
44:19In bacteria, we have something that's similar to what we have in plants.
44:25So this is the stuff that captures the sunlight?
44:28Exactly. Each of these molecules, each of these little blue things here that I'm showing, is a bacterochlorophyll.
44:34And if we take it apart, it will capture light.
44:36The chlorophyll captures light by absorbing particles of light, or photons.
44:44So a photon is absorbed, and it's absorbed by all of them.
44:49So energy is shared by all of these bacterochlorophylls.
44:52And that sharing, we call it, is in a quantum superposition.
44:54Because it's coming in and hitting one of these, but all of them are somehow...
44:59In a way, it's as if each of the electrons of the chlorophylls are talking to each other, and sharing the energy around.
45:07The subatomic particles in the chlorophyll are synchronised in a way that can only be described by quantum mechanics.
45:15Does it do a good job? I mean, is it efficient?
45:18That is part of why photosynthesis is efficient.
45:21Because by sharing the energy among all of them, it's easier to transfer the energy to another molecule.
45:28Imagine if you have to share the energy one by one, you have to explore each path separately.
45:33But if you share the energy all together, you explore all the paths at the same time.
45:36Every leaf on every plant on the planet has been following these quantum rules for millions of years.
45:47And we still don't fully understand how they do it.
45:51Without quantum physics, despite all the mathematical uncertainties and ambiguities,
45:58plants wouldn't produce oxygen so efficiently.
46:02And without oxygen, we wouldn't exist.
46:06These systems are amazing because they are effectively at the interface between using a little bit of classical mechanics
46:12and a little bit of quantum mechanics to operate in a wonderful way.
46:16Ultimately, quantum mechanics is at the heart of photosynthesis and, well, I guess all of life on Earth.
46:24It is. It is. We can say life is nothing but quantum mechanics giving us energy.
46:30So what does all this mean for our key question about the origins of maths?
46:39There is no shortage of evidence that mathematical rules are intrinsic to the world.
46:44We keep discovering them everywhere.
46:48However, we now know we have to take some of that maths on faith.
46:53And believing in the numbers is taking us to a very strange world, with crazy notions like superposition and entanglement at the core of it.
47:02Quantum mathematics is inextricably linked to the world as we know it, or as we knew it, because the world is actually a whole lot weirder than we thought.
47:14What quantum mechanics does do is force us to question what is real.
47:20And what is reality, anyway?
47:29Just how much light can mathematics shed on reality?
47:35With the world stripped bare, exposing the nuts and bolts of existence,
47:40what does maths tell us about this realm of subatomic particles?
47:44The maths that underlies it isn't particularly pretty, but it can all be written out in just one equation.
47:55This is the formula that describes the constituents of the universe.
48:01It has become well enough accepted to be called the standard model of subatomic physics.
48:07I told you it wasn't pretty.
48:10Now, you're just going to have to take my word for it on this one.
48:14This equation encapsulates all of the fundamental properties of the subatomic world.
48:21But there are a couple of sticking points.
48:24For one thing, no one has ever satisfactorily explained how our common sense, day-to-day version of the world emerges from this kind of subatomic reality.
48:39All of that fuzziness, all of that uncertainty in the quantum world, just how does it end up giving us that comfortable, familiar solidity of the normal world?
48:50At the other end of the spectrum, the solar system and beyond is beautifully and accurately described by a different equation.
49:05Einstein's general relativity.
49:07And this remarkable equation tells you about gravity, about the warping of space-time, about general relativity.
49:17And when you take these two together, these two single mathematical sentences, they're enough to tell you everything you need about the fundamental behaviour of the universe and everything in it.
49:31There is nothing more articulate than mathematics.
49:36Maths seems to be written into the physical universe.
49:42So on the one hand, at the teeny tiny scale, the standard model of particle physics does this amazing job.
49:51And in the ginormous scale, general relativity, I mean, you couldn't ask for anything more.
49:57There's just one problem when you try and put these two together.
50:04They're incompatible.
50:08The problem is that general relativity breaks down in the quantum world.
50:14Gravity simply doesn't apply to particles at the subatomic scale.
50:17Meanwhile, quantum effects are virtually never seen at the scale of humans and planets where gravity rules.
50:26You and I are never in a superposition of existing and not existing at the same time.
50:33So what does this mean for us? Are there two different worlds, each obeying their own set of mathematical laws?
50:43Solving this conundrum is one of the biggest problems that puzzles scientists today.
50:49Will we ever reconcile the two?
50:51I think it's perfectly plausible that within our lifetime, somebody, maybe somebody watching this program, will discover the mathematical structure which unifies Einstein's theory of relativity with quantum mechanics and just provides a perfect description of this world.
51:11That would be really exciting.
51:14Will we have one? How do I know?
51:18We would all like to have one.
51:20But, you know, maybe we are not smart enough to formulate a theory that combines everything.
51:28It's hard.
51:29I do believe that there are good ideas out there and that eventually, it might take a long time, but eventually humans will work this out.
51:40I'm confident about that.
51:42So will we make it all the way to include all possible forces at all possible scales with all possible forms of matter?
51:48It's a hope I have for our species and that's all I can say.
51:52The incompatibility of these two great theories, general relativity and quantum mechanics, creates a serious obstacle for believing that maths is really discovered.
52:08And there's a bigger hurdle to come.
52:10Many of the best proposals to unify general relativity and the quantum world have consequences that are even weirder than the problems they are trying to solve.
52:23They predict the existence of multiple universes.
52:27This idea is rooted in the mathematical explanations of the quantum world and the work of its founding father, Erwin Schrödinger.
52:35The mathematics in Schrödinger's equation insists that particles can exist in multiple states at the same time.
52:45And Schrödinger himself says that these possibilities aren't just alternatives, but really happen simultaneously.
52:53This can lead to multiple universes, and the maths also suggests there's an infinite number of them, each slightly different from the others.
53:13Mathematically speaking, in the infinite universe, everything that's possible has to happen somewhere.
53:23Yep, that's right. Everything possible happens somewhere.
53:29Even Schrödinger acknowledged that the consequences of his equation describing the quantum world might seem lunatic.
53:36But if there's one thing I've learned is that you should trust the maths.
53:42So maybe our experience isn't special.
53:45Maybe our reality isn't unique after all.
53:48There are so many distinct avenues of investigation that lead to the possibility of a multiverse.
54:01From our studies of unification and string theory, from our studies of quantum mechanics, even from the study of space going on infinitely far.
54:09Even that gives rise to a version of the multiverse.
54:11If we're going to reject everything that just seems weird, we're almost guaranteed to reject the true theories of the future when they get discovered.
54:21I think we should just chill out, accept that the world is weird, and that's just part of its charm.
54:28And trust the math.
54:29So why does all of this matter?
54:40Well, if maths really is discovered, then there is an intrinsic truth behind the maths we uncover, however weird that truth seems to be.
54:51If maths is invented, then how do we know what is true or false?
54:55Is it true purely because we define it so?
54:59And how does it relate to the real world that we all experience?
55:05In this series, we've seen that maths can explain so much of our world.
55:10From aerodynamics to planetary orbits, from the subatomic world to processes crucial to life on Earth.
55:17And that is something I just can't accept as a coincidence.
55:22So here's my take on things.
55:25For me, it's almost as though you have this alternate, parallel, mathematical world that hides just beneath our own.
55:34You can't see it. You can't touch it.
55:37The only way that you can explore it is by using the language that we've invented.
55:42All of those symbols and equations and conventions are our only tools of navigation.
55:50And they are undoubtedly man-made.
55:53But once you're inside that world, once you're exploring the landscape that mathematics has laid out in front of you,
56:00I am absolutely convinced that you are on a voyage of discovery.
56:07It is a world without a human designer.
56:11So ultimately, I think it's both.
56:14Mathematics is a little bit of invention and a lot of discovery.
56:18Mathematicians will probably never all agree, and maybe we will never find a definitive answer.
56:29But the consequences of having that debate is why it really matters.
56:34We have used mathematics for a much deeper understanding of nature and of the universe in general.
56:43We know about the universe now things that a few hundred years ago people didn't even know what to ask.
56:51Searching for the truth about maths has, over 2,000 years of history, transformed the human experience.
56:59Discovering patterns everywhere in nature has given us structure, beauty and inspiration.
57:08Inventing new areas of maths has led to an explosion of technology that ultimately underpins modern trade and computing.
57:18We have discovered powerful rules that we continue to use to explore, enhance and explain the world around us.
57:26And we have had a tantalizing glimpse of what could be to come.
57:33It's quite possible that what we have been doing in science for all these centuries is, in some sense, looking for our keys under the lamppost.
57:43We have been able to use mathematics to describe what happens out there.
57:47But that could be the tip of an iceberg of reality that we, as yet, don't have any understanding of, haven't yet had any contact with.
57:57But most of all, I think that asking questions about the origins and truth of maths has given us a purpose.
58:05It's given us understanding.
58:08Ultimately, maths has given us meaning.
58:11The inference.
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58:23Transcription by CastingWords