Royal Society. Royal Society. OK. Thank you.
Anything in particular you're doing there tonight, or is it just...?
I've got to attend some presentation of, you know the scientist, Stephen Hawking?
Oh, yeah. Yeah, I know who he is.
Well, I'm doing this programme about equations
and he deals with the enormity of the universe... OK. ..and his tool
for scientific research is the equation, the mathematical equation.
Oh, OK.
My name's Matthew Collings.
I'm an artist and art critic.
That's what I know and understand,
but I'm about to enter an alien world.
To me, equations have always been incomprehensible hieroglyphs.
What do they describe?
Are they just a mathematical game?
In this film, I'll learn about some of the most important equations in science.
They're actually masterpieces that explain the universe we live in.
APPLAUSE
I would like to thank Dame Stephanie Shirley
for commissioning this magnificent portrait.
It will be an honour to have my picture
join the Royal Society's collection of the greats of British science.
I only wish I could remain looking as good as this picture!
LAUGHTER
With art, I think beauty is very important
and I'm always trying to define it, and work out what it is.
Now I want to apply that knowledge to mathematics and maybe understand why scientists talk
of "beautiful" equations.
Hello, Stephen. I'm Matthew Collings.
I'm doing this BBC programme about equations and beauty.
Hello. That sounds an interesting idea.
Thank you. I look forward to speaking...
I'm glad the most respected living scientist thinks I'm on to something.
It's a busy night for Stephen, so I've arranged to meet him again in a week's time.
Now, you yourself must work with equations?
I do, I do, I'm...
I mean, technically I'm an astrophysicist.
I'm really what we call a theorist. What I like to do is noodle around
with equations and work things out and make predictions.
Equations is what I do every day. I mean, my colleagues who do astronomy
like to show pretty pictures and beautiful pictures of the cosmos.
I like to show equations. Very much so.
Come and visit me in Oxford and I will tell you all about this. We can't do this here,
but in Oxford we've got blackboards and I can explain the beauty of equations.
Fantastic, thank you very much, Pedro.
I've come to the University of Oxford to take up Pedro's invitation.
And he's going to tell me about the most famous equation of all,
the one that everyone's heard of,
E = mc2.
This equation conjures up a whole load of thoughts in my mind,
but the main ones are that it's got something to do with the atomic bomb
and of course, it's by Einstein.
But there's cultural knowledge and then there's maths.
I don't know anything at all about how E = mc2 works.
When Einstein first published the equation in 1905,
it started a scientific revolution.
Hey! Hello. Hello, Pedro.
How are you? Very nice to see you again.
Thanks for coming. Well, it's a pleasure.
Thank you very much for having me. Now you've got this tall order
to explain to me so that I can totally understand it.
We'll give it a go, we'll give it a go. Let me just clear this up.
'Uh-oh, what am I doing?
'Pedro lives and breathes abstract numbers.
'I'm an art guy who left school when I was 13.'
So what maths do you know? Well, I must confess that I don't know any maths, any geometry,
or any algebra or anything in that realm of experience.
I'm completely ignorant about all that. OK.
I know about art and that's about it.
Ok, that's a good starting point. Let me get a pen.
It seems a very bad starting point to me, but...!
So, you know nothing about what an equation is?
Only, uh, I think it's a sort of...
code or some kind of metaphor for the natural world.
It's the natural world reduced to a formula.
That's pretty good. Let's start with a really famous one.
Have you ever seen this equation?
Well, I've certainly heard of it. I know it's E = mc2. Very good.
E...
E stands for energy. Do you know what energy is?
It's a difficult question, so...
You're having energy as you talk to me. Yes.
A certain amount of energy is keeping me alive so I don't die and decay.
Very good, very good.
That's the limit. That's what I think energy is.
Energy, I mean, energy is kind of a funny thing to try to define.
The best way I can think of it is, it's the capacity to do things.
It's the capacity to lift something up, to heat something up. All right.
Then you've got this thing here. Do you know what the m stands for?
I think it stands for mass.
Exactly. Mass, and mass is basically the amount of stuff in a thing.
So when you pick up a book, it's the amount of stuff that that book is made of. Mass is kind of interesting.
For example, suppose you've got a nail and you weigh it, all right?
And then you leave it out in the air and you weigh it three weeks later, it will have rusted and so...
Its mass has changed. More particle things. It's gone up, exactly.
Stuff has stuck onto it, there have been chemical reactions so the mass of it really does have to do
with what it's made of and how it changes. And then we've got this thing over here, the c.
Do you know what the c is? No. There's no reason for you to know, it's the speed of light. OK?
C is incredibly important because c is the speed at which light rays propagate through empty space.
I know what squared is, that means a thing multiplied by itself. Exactly.
So this is a kind of fascinating statement.
This is saying suppose you have some mass, right,
it's possible to convert that mass into a certain amount of energy.
I can see that E = mc2, like all equations,
is about balancing two sides.
That's what the equals sign is all about.
So this equation allows us to calculate how much energy
is contained in any given mass.
It's a surprise to me that it applies to everything.
Toothpaste, a book, a nail, or uranium for that matter.
This equation is universal. And since "c2" is such a big number
a tiny lump of matter contains an enormous amount of energy.
What this equation doesn't tell you is how to unlock that energy.
The most dramatic proof that the equation was true
came 40 years after Einstein first worked it out,
when the atomic bomb was dropped on Hiroshima.
Pedro walked me through the chilling sums.
You have a mass which is something like half a gramme,
I write it as a kilogramme.
You need the speed of light and the speed of light looks like this -
it's about 300 million metres per second. OK.
OK. And we can work out how much energy there is, OK?
All we've got to do we say that energy is going to be that mass
times the speed of light, squared, and what you get is this.
Joules. This is the unit of energy.
So you get a phenomenal amount of energy.
Now, if I told you this was something like 15 kilotons of TNT from,
something the size of a pill,
giving enough energy... Producing an explosion of 15 kilotons...
Which is equivalent tons of TNT.
I want to throw you, I don't know if this is a stupid question and you might have nothing to say about it,
but supposing the sign for squared was changed to a three.
Would that just be nonsense, or...? It would be nonsense,
and the reason it would be nonsense is because we've tested it.
We've gone into a lab and tested this relationship. We've weighed something, done something to it,
weighed it again, worked out the amount of energy that came out and it was on balance
so the left side was balanced with the right side.
I'm impressed that E = mc2 was created before it was shown to be true.
The equation was a prophecy.
The five symbols explain the link between energy and all matter across the cosmos.
This universality is part of its power.
Einstein once claimed "the only physical theories that we're willing to accept are the beautiful ones".
But what do scientists mean by "beautiful"?
They talk about equations being testable, being universal.
Is that what they think beauty is?
I'm going to take you to the Rhodes building...
..because Einstein actually came here in, I think 1933,
to give the Herbert Spencer lecture. And it's an interesting lecture because it's a lecture
where he basically discusses his philosophy.
Right. Why he does science the way he does, and his craft, what he does as a theoretical physicist.
And he basically said two things.
The first is that the endgame of what he does is experience.
It's experiment. It's the natural world.
It's not theory for theory's sake. It's always relating to reality.
Exactly.
But the bulk of what he says is that what guides him
is mathematical beauty, or mathematical simplicity.
That's what guides his research. He says, "It is essential from our point of view that we can arrive
"at these constructions and the laws relating them, one with another, by adhering
"to the principle of searching for the mathematically simplest concepts and their connections".
So go for simplicity, go for the simplest relationships which are mathematically true,
and that underpins the way that he thought about what he did.
So he was telling people this in a lecture that was really about the philosophy of what he did. Exactly.
What his ultimate aims were, and what the use of what he did was to the world.
It's a practical philosophy, it's what he actually did on an everyday basis. That's how he worked.
So it gives us an insight into...
what science at that level is about.
Exactly. Yeah.
Einstein believed that the laws which govern the universe would have an elegant simplicity
and this would be shared by their equations.
I paint abstracts in collaboration with my partner, Emma.
It occurs to me that when we intuitively put shapes and colours together in a visual order,
we too, like people who come up with equations,
try to arrive at a convincing metaphor for nature.
For us, art tells you something important about the world.
This is a coloured engraving of Isaac Newton by William Blake.
It shows Newton studying a tiny corner of the world
with a pair of dividers.
Blake despised Newton, who he felt reduced
the magnificence of existence to cold and mechanistic equations.
So today I'm coming to a place...
..where I'm actually going to find out a bit more about what Newton actually did.
This is Newton's house, where he developed his ideas on gravity.
I'm going back to the 17th century because it's when scientists
first used equations to try to explain the natural order.
Hello. Hello, welcome to Woolsthorpe.
I'm Margaret Winn, the house steward. Pleased to meet you.
Likewise. Hi, I'm Ruth.
I'm professor of theoretical physics from Durham.
Thank you both very much for seeing me.
It's nice to meet you. Have you been here before?
No, never. This is my first time.
Right, well, this is the house Newton was born in, Christmas Day, 1642.
Yes. Ruth, the only thing I know about Newton is an image of him observing apples falling off a tree.
Falling from a tree! And he suddenly works out that that means gravity.
Well, we call him the father of modern science, and that is not an understatement.
We can date our modern way
of doing physics or science as trying to write down equations
as coming from Newton.
That's very clear, that equations come from beginning of equations.
Yes, I think equations as a method, as a means of encapsulating,
of modelling, of saying what physics is and what the world around us is.
Why do you think the image of the apples falling
is needed in the mythology of Newton?
I think it's the link more than anything else. If we take an apple
and just look at what happens
as it goes up and down under gravity...
Gravity is something that I think we often take for granted.
And the apple falling, he realised that the same thing that made that apple fall down to the ground was
the same thing that kept the Moon going round the Earth, or indeed the Earth going round the Sun.
Begs a lot of questions. How does he go from that to realising something about that? A lot of hard work!
Is it assuming that perhaps there's some kind of force
connected to the moon that's similar to the thing that makes...?
That's right, so as soon as you start thinking about planetary bodies or things moving round,
round other objects... I think people at the time, they would have felt, "ell that's a mystery of God,
"and we're not supposed to understand that," but by using this apple
as this sort of metaphor for the moon or the sun,
Newton managed to say, "No, actually man can start plumbing those mysteries."
I'm absorbing everything you've told and as I'm starting to freeze a bit, maybe we could go in the house!
Shall we go up to the house? You talked about he didn't want
his theories to be thought of as the final word...
Newton realised that mathematics could provide a precise and universal language to describe
things as diverse as the fall of an apple and the orbit of the moon.
He put his ideas in a revolutionary book, Principia Mathematica.
I can barely tear my eyes away from this!
THEY ALL LAUGH
That is our very prized possession,
a third edition copy of the Principia Mathematica.
If we turn to the relevant page, I'm going to leave you to our book. Thank you very much. Wonderful.
The first things that we notice here is not a single equation,
having talked about Newton as being the father of modern science.
The other thing is of course it's in Latin, which the language of...
It's hard enough...!
The universal language at the time. Propositio eight, theorema eight.
Essentially, he is giving us his equation for gravity in words.
So he starts off, "Si globorum duroum in se mutuo gravitantium materia undique in regionibus..."
Newton's written version eventually formed the basis for the equation for gravity.
So I want Ruth to unpick the different elements,
a mathematical version of the words.
I've noticed you've got a book, so we could try and translate
what he said into one of these beautiful equations.
Absolutely, find a blank page.
Anywhere will do. Right.
So we write both of these objects as M1 and M2.
So these are just the masses. But then Newton talked about
the force between the two spheres,
these two bodies, is inversely proportional,
which means we divide, to the distance squared.
The two bodies or spheres could be any size -
the Earth, the Moon, or even an apple.
And this is G, for gravity.
It's actually called Newton's constant.
You've written out an equation for me there. Yes.
And earlier you threw an apple in the air and it fell to the ground.
Can you give me some numbers...
Certainly. ..that will show me what the apple is doing?
Let's talk about the apple.
OK, so one of the Ms is an apple? An apple. What does an apple weigh?
Half... ? You get a pound of apples so I guess that's about four apples, so that's a quarter of a pound.
We work in kilograms! SHE LAUGHS
Let's just say it's 200 grams.
But is the other M the Earth? You don't need explaining, do you? You know this already. I'm guessing!
But I don't know how much the Earth weighs, I very rarely buy one from the grocer!
Well, I can give you a rough idea.
Five times ten to the 24 kilograms. Got it.
So it's ten to the power of 24.
Which is a trillion trillion.
OK, good.
Now the radius of the Earth is 6,000 kilometres.
We've got a 200 gram apple,
several trillion kilograms Earth
and radius of Earth 6,000.
Now, what I'm going to do is cancel all those off to make it easy.
Then what I end up with is a simple ten on the top
times the 0.2 of the apple.
So what this tells me is the force the apple feels is its mass
times this ten and this is...
Mass times ten. Ten is... This is the sort of,
how fast gravity is going to cause the apple to start to fall.
So the force on the apple here is simply two.
And that unit is called the Newton.
Ah! N for Newton.
N for Newton. Fantastic.
The number of Newtons measures the force of gravity acting on the apple.
It's a complicated equation,
but I'm beginning to understand the key parts.
The force depends on the mass of the two objects
and the distance between them.
The bigger the objects, the bigger the force.
And the further apart they are, the weaker the force.
The two masses, M1 and M2, could be anything.
The earth and the apple. Or the earth and the moon.
Or the earth and the sun.
Ruth told me Newton's equation allowed us to understand why
the moons and planets move around the solar system.
His equation seemed to make sense of, well, the universe.
So the equation itself, F = G x M1 x M2/R squared,
that's Newton's equation of gravity, but how we use it,
this is a sort of process, you know doing science,
of calculating things, of making predictions.
You've now showed me how we use that equation.
How we would use it, yes.
This is our paint, how we paint the world.
We paint it in equations.
In fact we use that a lot, we say, "I'm painting,"
you know, we tend to use this word, "painting".
If you, if you can use that metaphor of paint and colours etcetera
is there a place also for beauty in this world of calculating things?
I don't expect everyone to find this beautiful,
but it certainly is for us and for me.
Great!
'A few decades after Newton came up with his law, it was used
'to successfully predict the return of a comet, Halley's comet.
'His law of gravity had been confirmed.'
'With his equation, Newton had transformed
'the way mathematics modelled the world,
'and his work went unchallenged for over 200 years.'
'Everything changed at the beginning of the 20th century
'with the arrival of Einstein and his Theory of Relativity.
'In that same decade, something else entered, and that was modern art.
'In the world of art many believe that Picasso was involved
'in the same revolution as Einstein.'
Weirdly, the one place in which I had heard about relativity
before embarking on this programme was art school when I was young.
As art students we all had to absorb the idea that
relativity had something to do with cubist paintings.
I'm about to look at a cubist painting by Picasso from about 1909-1910.
It's of a woman in an armchair.
I think cubism was really seen as something quite terrifying and shocking when it first came out.
It's not like a Renaissance painting
where you feel you're looking through a kind of window onto the world.
With cubism the artist is deliberately confusing you
as to where thing are, and indeed what things are.
So that the space in the room
seems to be eating into the side of the woman.
And the textures of the room seem to be no different from the textures
of the woman. So there's all this moving around of objects and space
in a way that is deliberately confusing if you were thinking,
"Well, where is the thing that looks like ordinary reality?"
'I think it's right to say that cubism was a new kind of beauty
'that looked a bit like science.
'But I'm not convinced that cubism is science.
'I've arranged to meet historian of science, Arthur Miller,
'who's going to attempt to change my mind.'
I've got to tell you, Arthur, that at art school, and subsequently,
I felt oppressed by the idea that I had to think of a connection between
Einstein and relativity and cubism.
Einstein and relativity and Picasso.
But there is one in a sense that I'll say they both worked on
the same problem, the nature of space and time.
OK. The connection is that time and space are important to them both.
That's right.
Where I find the proposal difficult is that, just because
he's doing something with time and space that he's therefore
something like Einstein, or that cubism is something like science.
Cubism was very much of a scientific research programme, as I've said.
It had, you know, an explicit intent to reduce forms
to geometry. Why is that? Picasso...
Why is that scientific and not artistic?
I mean, medieval artists reduced forms to geometry,
and African artists reduce it to geometry, archaic art reduces it to geometry.
Well, that's because Picasso had in mind scientific texts as a way to do it.
For example, we know that he looked at a text written by a mathematician
and the text discussed how you represent in four dimensions
complex polyhedra and Picasso took a look at these.
Of course, he didn't know what the equations meant,
but when the author of the books specialised the equations
of the two dimensions and then could generate illustrations,
Picasso was interested in the illustrations.
It's correct to call Picasso a revolutionary artist,
it's not hyperbole, but for me, I don't know enough about Einstein
to see the way in which Einstein is a revolutionary too, or how
Einstein's ideas and Picasso's are the same level of revolution
and also going in the same direction.
Well, Einstein was a revolutionary scientist because what he did
was to go take the next step beyond Newton.
Newtonian science is based on our sense perceptions that all time,
your time is the same as my time.
What Einstein was able to do was to raise himself to
heights of abstraction so he could glimpse a world beyond appearances.
The real objective world out there where there is scientific truth.
I still think the connections between Einstein and Picasso are
more superficial than substantial,
but I am very interested to hear more about Einstein.
Arthur will attempt to explain to me
one of the key equations of the Special Theory of Relativity.
When Einstein came up with this equation,
he wasn't even officially a scientist.
The days when he wrote the relativity theory, he worked as a patent clerk
in the Swiss federal patent office in Bern.
In fact he worked there from 1902 to 1909.
He was also a conscientious daydreamer.
And in his dreams and visions he soared over the landscape
of physics and realised what the fundamental problem was.
The nature of space and time.
People were beginning to think that maybe there was something wrong with
classical, intuitive notions of space and time,
but they couldn't put their finger on it.
What they especially wanted to do was to leave alone the notion of time.
Why was time sacrosanct, because it was obvious what it was,
it didn't need any more inquiry, or they were afraid that they couldn't find out anything more?
It seemed that your time is the same as my time.
No matter how fast we're moving with respect to one another.
There's no mystery there. We know what time is.
That's right. It's like Superman said, "Leave time alone."
Don't mess with time. Don't mess with time, yeah.
OK I've got a book, if you've got a pen?
Absolutely, let me show you one of the spectacular results
of relativity theory.
Let's do a little thought experiment.
Suppose here is Matt one standing on a train platform
and here is Matt two, just call him Matt, standing on a train
and he's moving along with some velocity, call it V,
relative to the Matt standing on the platform.
The Matt on the moving train is wearing a wristwatch
and his time, call it t prime,
and call the times of all the clocks on the platform t.
And what we want to do is to compare the time
on Matt's wristwatch with clocks that remain at rest on the platform.
They all read the same time.
I'm going to assume that, even though the clocks are at rest
and my clock is moving that they're all the same,
because clocks always tell the same time, assuming they're all synchronised.
One would think so, yeah. Now let's call the Matt on the train...
But you're going to show me that they don't.
I'm going show you that they don't, convince you that they don't.
t prime and t.
Now it turns out Matt on the train's time t prime is
equal to t times the square root of 1 - V squared over C squared.
So the time here is equal to something complicated.
It's not just the same as that time.
No, it's not the same as that time. Your time is not the same as my time.
These two times are different...
'If I understand the equation correctly, it says something unbelievable -
'that time runs at different rates depending on how fast you're moving.
'Take a train zooming through a station.
'This equation predicts that a clock on the train, reading time t-,
'would run slower than clocks reading time t
'on the station platform.
'I've never noticed it and here's why.
'This bit of the equation is what makes the two clock times different,
'but it only has a significant effect if the velocity, V,
'of the train is very fast, close to the speed of light.
'If the train could reach the speed of light, you get 1 - 1,
'which equals zero.
'And then t- equals zero.
'Relative to the platform, time on the train completely stops.
'This stretching of time seems impossible
'but according to Arthur it's been proven by practical experiment.'
Now that's really something. That's wild.
And he realised that's because time is a relative quantity.
Just as I discussed with you.
Your time is only the same as my time if we're standing still
next to each other, but if you go away and come back, your clock,
although it'd be very difficult to perceive it,
will read a slower time than mine.
Well, I'm taking in a lot of what you're saying so that
I'm far more informed than I was before you spoke. Good.
But the thing that's really big for me is this idea
of the physical nature of time and that seems a marvellous idea.
Oh, it turns out that there's not space and time. There's space-time.
Right, they are a single entity. Is entity the right word?
Time and space are connected by the velocity of light.
'That was definitely the hardest equation so far,
'not just the maths but because of the ideas it contained.
'You might be worrying about time on a tube train,
'but you wouldn't think time was actually changing shape.'
'Einstein worked out that time and space are inextricably linked
'through the speed of light.'
'It was a thought that it was simply impossible to have before,
'reality had changed,
'and Einstein did it with equations.'
'I'm beginning to get a crush on science.'
'Before, I literally didn't know what an equation was.
'Now, in some ways I know the basics of what an equation is,
'but I also know the implications of what an equation is,
'so there's a sort of excitement
'about the philosophy of an equation,'
or the use of an equation in some kind of profound way
as opposed to something like a railway timetable that tells you very detailed information.
You know the process of learning is a mixture of pain and pleasure.
It's quite hard to dislodge the pattern of the world that
you've already got in place, and bring in a whole load of new stuff.
You can appreciate it on mythological levels.
Someone's telling you the myth of equations, or the myth of science,
or the myth of Newton, or the myth of Einstein,
but they all do sound like myths to me.
But as the days go by they acquire more and more reality as each
scientist adds to the stories that the other scientists have told me.
There's one scientist who stands out in the story of equations,
because he took the idea of beauty in science further than anyone else.
His name is Paul Dirac.
He too revolutionised our view of the universe,
yet virtually no-one outside scientific circles has heard of Dirac.
So, I've arranged to meet the biographer of this mysterious genius.
This is a particularly favourite part of Cambridge for Paul Dirac.
Dirac was the greatest English theoretician since
Isaac Newton and that's how... That's his reputation in 1927,
when he was looking for what became his greatest achievement -
his equation.
Why is he... Being so great,
why is he totally unknown to the general public?
He actually wanted anonymity, he really had no interest at all in celebrity.
He simply wanted to get on with his work and be unknown
to the outside world.
I love the idea that for Dirac, beauty is important.
Is there a sense in which it is more important for him
than I've been hearing so far about other scientists?
Oh, yeah, Dirac was the first scientist actually to elevate this idea of beauty to a principle.
He called it the principle of mathematical beauty.
And what he meant by that was that as we advance in fundamental,
theoretical physics, the theories as they get closer and closer to nature,
become more and more beautiful.
So, for him, it was a method of sifting out theories,
right from wrong because if it wasn't beautiful, if it was ugly
in his opinion, it just wouldn't cut pass muster with nature.
So for him, a theory had to be beautiful for it to stand a chance of describing nature.
Incredible.
Here's a scientist who insisted science went through a "filter"
of beauty. And by pursuing beauty, you end up with truth.
It's an idea that's often used metaphorically, but Dirac meant it literally.
This is the Bridge of Sighs, which he walked across as a Fellow.
He walked back to his rooms here and this is where he did his great work on the Dirac Equation.
In fact, he was staying in a room just here.
That's where he was working in the late months of 1927
on what came to be known as the Dirac Equation, one of the greatest achievements in modern science.
Here we are, Room A4. Newcourt. Where Dirac discovered his great equation.
Completely free of distraction. The only noise you get is a bit of noise from the punters outside.
Apart from that, no radio, just nothing.
Dirac was not given to luxury. In late 1927, all he did, apparently, was to work on that equation.
Tell me about that equation, what was he trying to accomplish with it?
Well, what he was trying to do was come up with an equation for the electron,
the first material fundamental particle to have been discovered.
What does that mean, "the first fundamental material particle"? OK.
A fundamental particle has no constituents.
It's a completely basic particle, you can't subdivide it.
The point of the tiny, tiny thing, this electron,
is that nothing else is more basic than it. That's right.
So you had a chance of giving a fundamental description in nature.
I've got a notebook in my bag.
If I give that to you and you find a blank page... Yep.
And I then give you my pen, could you write out for me
the equation... I will.
..that Dirac came up with. I will.
It's called the Dirac Equation?
That's right. This is the Dirac Equation.
And this equation applies to every electron that's ever existed, or ever will exist,
in the entire universe, so this is the ultimate compact equation that
has this universal significance.
This is a miracle, one of the miracles of 20th century science.
You've shown me the miracle, now tell me what it is.
I see something like "I followed by squiggle, followed by P followed by
"a squiggle, followed by equals, followed by m,
"followed by squiggle."
OK, you say, "I, gamma, P, psi = M psi".
OK, so it's like E = M C squared, only you say these new things that
he thought up himself, a bit like the Lord of the Rings language.
That's right. And what is the most important symbol there?
Right, OK.
This is called a spinner, all right?
This is a thing that encodes the information about the behaviour of the electron.
So, you tell the equation what situation the electron is in and out
of the equation comes the prediction for how the electron will behave.
What's the thing in the ordinary world that is the closest that
I could visualise, to tell me what a spinner really means? There is none.
OK, so I've got to accept that. Exactly. Fine.
This was a complete Dirac concoction, right?
So spinners didn't exist before him?
No, they didn't. Do you have to learn his new language before you can say that equation? Yeah.
Seriously, people for six months a year were struggling. Brilliant,
world-leading physicists had no clue about what this equation meant.
This is why he was so far ahead of his time, they were having to say,
"What the hell do these symbols mean?"
It was on extremely good ground and moreover...
'If it stumps the world's top scientists then I think it's OK for it to be beyond me.
'This really is a foreign language.
'But I was getting a broader sense of how equations have advanced knowledge.'
I do feel from your talk that I'm starting to get a picture filled in
for me of science, the big points.
Newton, Einstein and now Dirac.
That's right. And a sort of journey that the spheres, the planets,
the stars, this earth, everything on it, all the objects
can be somehow described and understood in mechanical terms.
That's right. Einstein said that the most incomprehensible thing
about the universe is that it is comprehensible.
And Dirac, Newton, Einstein, they all had faith that they could,
if they thought hard enough, they could come up with these laws that
describe nature at a fundamental level.
But faith doesn't produce more faith, it actually produces equations.
Oh, absolutely.
It's not like a faith that you can't verify. Faith oils the works. Yeah.
Dirac actually said that the principle of mathematical beauty was a kind of religion to him.
He actually used those words because he really did believe
with all his heart and soul that a mathematically beautiful theory
was going to be the kind of theory that nature backed
and that that was the direction in which you should travel, so he really did believe that.
It was an article of faith.
Why is the spinner beautiful?
This is beautiful because Dirac used this equation
to predict the first example of anti-matter.
This was perhaps the greatest triumph of 20th century physics.
Now just to give you a sense of how monumental that is,
now cosmologists believe that the very beginning of the universe,
half the universe was anti-matter.
So by that token, Dirac conceived, using this equation,
half the universe in his head.
'Scientists now stand in awe of Dirac's Equation.
'But at the time, things were very different.
'In the late 1920's, anti-matter was totally unknown.
'The idea that every electron, proton, and neutron
'had an opposite partner was preposterous.
'If his equation predicted this make-believe stuff
'then it must be wrong.'
OK, so what we can do now is go into the teaching lab.
What we have is an experiment set up where we can
actually see tracks of particles that have been produced by anti-matter.
So you'll be showing me some anti-matter in action.
'Five years after Dirac came up with his prediction,
'anti-matter was discovered.'
'The equation had turned out to be true.
'Now, I too want to see the proof.'
This is the first practical place I've been to.
I'm surprised at how quaint everything looks.
This is a very simple experiment. This is very low tech.
You could do this in your kitchen.
Really? OK, so, this is a magnet.
It's a fairly powerful magnet and we're going to put dry ice on here, so that will be very cold.
A sort of cookery element at the moment. It is yeah.
Cooking fish in salt.
Now the Perspex box is going to go on top.
And there's alcohol that we put in the upper layer.
In order to see the tracks, they're actually quite faint,
we have to illuminate it with a very bright lamp. OK.
And then, one of the other ingredients that we should
explain here is the radioactive sources that we're going to use.
So we have two radioactive sources.
One emits electrons and the other emits positrons.
And so what we have here is the isotope of strontium called strontium 90.
'Glen told me these radioactive materials would let us see the tracks of electrons.
'And more importantly, the anti-matter partner to the electron.
'Known as the positron, this is the particle predicted by Dirac's Equation.'
It emits positrons and we'll see tracks that are very similar.
Maybe slightly lower energy actually and they will be bending to the left.
So that really is the demonstration, that we have two types of particles that really look very similar
in terms of the tracks that they make,
except that one is positively charged and the other is negatively charged.
Yeah, yeah, I saw one going that way.
Furthermore, they should be bending to the right and they are.
Yeah, they're thin and irregular. It's like a string of beads almost.
OK, so all I've really convinced you that you can see so far are
bog standard electrons.
Even at the bog standard level, it's pretty impressive.
We're all made of plenty of those.
And so, maybe what we can try now, is to put in the positron source.
What we should see, is that they will bend in the opposite direction.
The other one slotted in scientifically. That's right.
We're just going to hold it on to the entrance way.
Now I should expect to see things going to the left.
I'm seeing activity but not necessarily lines going to the left.
'We'd seen the electrons bend to the right.
'Now Glen hoped that we might spot the rarer anti-matter tracks
'as they curve towards the other side.'
One there! Very, very clear.
There you go. Fantastic!
So that's the first time in this experiment that I've seen the anti-matter.
That was definitely coming from the source.
The amazing thing is to have something from...
a sort of comic world of science fiction, anti-matter,
to have it presented to us in reality. There we go.
Except I wasn't looking at that one.
Every 30, 40 seconds a little blip occurs
within a sort of 10p size radius of the source.
It shoots out, curls around, doesn't go very far.
One there, very curly one, shot right round! Very good.
Yeah, yeah. So, we're really seeing a physical thing,
which connects to the very complicated mind-world of Paul Dirac.
That's right. Somehow the existence of anti-matter
emerges as a necessary consequence of the theory that he wrote down
and that's pretty difficult to see. To just look at his equation and say
that should give us anti-matter, but really if you analyse it carefully
it's clear that that is one of its necessary predictions and that's what you're seeing.
So, those curves and blips in that sort of molten sea,
is the Dirac Equation being shown to us in physical form.
'These elusive symbols point to a beautiful idea.
'There is something magical about them.
'The existence of anti-matter proved his theory true.
'Keats' romantic poem goes "Beauty is truth, truth is beauty,"
'as if one leads to the other.
'And that's exactly what Dirac, the scientist, believed.
'That the search for beauty powers the advance of science.'
I'm reading a paper by Dirac, which he delivered in February 1939.
He says, "What makes the theory of relativity so acceptable to
"physicists in spite of its going against the principle of simplicity,
"is its great mathematical beauty.
"This is a quality which cannot be defined any more than beauty in art
"can be defined, but which people who study mathematics usually
"have no difficulty in appreciating."
So, he's saying that beauty in art can't be ultimately defined
any more than beauty in anything can be ultimately defined.
But what he is saying is that people in the world of very, very high
and complex mathematics agree that beauty is something that
they all appreciate and follow.
And it may be that what Dirac is saying is that
there's a sort of high or true or pure beauty that
mathematicians are interested in,
which sounds to me a bit like the inner, true, deep beauty of art.
But you have to go on a bit of a journey to find,
you can't expect it to come leaping out and waving at you
straightaway when you haven't really bothered to get involved with art
and try and find out what it is.
I like these buildings very much.
But I think they have a sort of comic element.
They seem like a Hollywood mock up of some kind of scientific base
where something sinister is being worked out behind the scenes.
You wouldn't even really think you were in England.
You could be anywhere in the world.
I'm ending my foray into science with an equation about black holes.
I'd always thought they were the stuff of science fiction,
but the inner workings of black holes are explained
by the fifth of my great equations.
All the previous equations have come from historical figures - Newton, Einstein and Dirac.
This will be my chance to hear about the entropy equation direct from its creator, Stephen Hawking.
And find out if he agrees with Paul Dirac about beauty and the truth of science.
Thanks very much for allowing me into your department, Stephen.
Can I ask you straight away,
is beauty important for you in your scientific work?
I don't know about beauty,
but the fundamental laws of the universe should be elegant.
What do you mean by elegant?
An equation is elegant if it is short, simple and explains
properties of the universe that were previously not accounted for.
My most elegant equation is very simple.
It is S = a quarter A.
Here, A is the area of the boundary of a black hole.
And S is its entropy, a measure of how much heat it contains.
What does that mean?
This equation shows that black holes aren't completely black.
They glow like hot bodies and lose energy and mass.
Eventually they will disappear in a tremendous explosion.
Why is that an elegant equation?
The equation came from a rather messy calculation.
It seemed a miracle that such a concise equation should result.
This equation unravels the physics of black holes,
one of the most mysterious objects in the universe.
As I understand it, the equation says that as stuff
falls into the black hole, the surface area of the black hole
gets bigger, and the entropy does too.
In 1975, when Stephen Hawking came up with his equation,
there was still some doubt as to whether black holes existed.
35 years on, all scientists agree they do.
Black holes have entered the realm of science fact.
While making this film, I found out that Paul Dirac
believed that it was more important to have beauty in one's equation
than to have the equation backed up by actual experiment.
Is this too extreme a view for you?
I think what Dirac meant was that although a beautiful equation
might not agree with experiment at a particular time,
it will eventually turn out to be true in the long run.
I think elegance is a good guide for equations but not an infallible one.
In art, an artist like Picasso say,
will just be working from hour to hour,
from work to work, pushing his ideas along with his work.
He doesn't necessarily think, "Now, I've discovered cubism."
That accolade will be bestowed upon his work
a bit later by other people.
But he probably will, at some point, think,
"I have made some kind of breakthrough here."
And I wonder if that breakthrough feeling,
if there's an equivalent for you in your type of enquiry.
There's nothing like the Eureka moment of discovering
something that no-one knew before.
I won't compare it to sex, but it lasts longer.
Thank you very much, Stephen.
Thank you.
The very fact that Stephen agreed to be interviewed by me,
when it's not an easy task for him,
it's not something that he does a lot,
proves to me that he believes in the thesis that beauty
is a significant element in the work of a theoretical scientist.
That making an equation calls for some kind of,
not just a sense of beauty, but almost a pursuit of it.
The pursuit of beauty really is a sort of driving force
in evolving an equation.
I've got to let all that sink in now.
I've been very happy to have my head crammed full of unfamiliar ideas,
but now there's one more thing I need to do.
Hello, Cary, how nice to see you!
I'm at the opening of my own exhibition, the work I do with my partner, Emma.
I've invited the scientists and most of them have turned up.
Throughout this film, the word "beauty" has often cropped up.
But it's hard to define and I can't help but feel that
while there are similarities, there are differences too
in what artists and scientists mean by beauty.
Sorry, which are your paintings? All these paintings.
We do all these.
I'm trying to imagine what Paul Dirac would make of this painting.
My guess is he would ask you what you are representing here,
because he had a very literal mind.
I know what you're saying.
Are you conscious of representing anything?
There's no representation in the room at all,
but I think there is the idea of a model of the visual world.
There's a lot going on. There's a lot of them.
You come back in a couple of minutes from looking at something else. You can't find that order again.
Right.
That's really the point of them.
They should have a restlessly changing sense of order.
It's like looking at a fire where the fire always looks the same, but it's never exactly the same.
Exactly. Anything in nature that is permanent and changeable.
Are all the panels the same, or are they different?
I think they're all pretty different.
It's interesting because you'd look at it and think there's an algorithm
that tells you how you would paint that in terms of the things around.
But he says no, there's also a global point of view.
A non-repeating pattern of some sort. Exactly.
Ultimately, it is highly mathematical,
but actually there is no... We didn't sit down and work it out.
Well, randomness is also mathematical.
It's interesting to see mathematical symmetries come out of aesthetic pursuits.
Well, that's arrived at...
'All my life, science has been totally out of my orbit.
'What was so illuminating for me in this programme was to
'find out that equations are the most important tool in science,
'forever pushing the boundaries of knowledge.
'And that the greatest and most beautiful equations
'have a life of their own.
'They've given us ideas beyond the human imagination.'
Subtitles by Red Bee Media Ltd
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