Beautiful Equations


Beautiful Equations

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-Royal Society.

-Royal Society. OK.

-Thank you.

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Anything in particular you're doing there tonight, or is it just...?

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I've got to attend some presentation of, you know the scientist, Stephen Hawking?

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Oh, yeah. Yeah, I know who he is.

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Well, I'm doing this programme about equations

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-and he deals with the enormity of the universe...

-OK.

-..and his tool

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for scientific research is the equation, the mathematical equation.

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Oh, OK.

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My name's Matthew Collings.

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I'm an artist and art critic.

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That's what I know and understand,

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but I'm about to enter an alien world.

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To me, equations have always been incomprehensible hieroglyphs.

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What do they describe?

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Are they just a mathematical game?

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In this film, I'll learn about some of the most important equations in science.

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They're actually masterpieces that explain the universe we live in.

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APPLAUSE

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I would like to thank Dame Stephanie Shirley

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for commissioning this magnificent portrait.

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It will be an honour to have my picture

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join the Royal Society's collection of the greats of British science.

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I only wish I could remain looking as good as this picture!

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LAUGHTER

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With art, I think beauty is very important

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and I'm always trying to define it, and work out what it is.

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Now I want to apply that knowledge to mathematics and maybe understand why scientists talk

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of "beautiful" equations.

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Hello, Stephen. I'm Matthew Collings.

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I'm doing this BBC programme about equations and beauty.

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Hello. That sounds an interesting idea.

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Thank you. I look forward to speaking...

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I'm glad the most respected living scientist thinks I'm on to something.

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It's a busy night for Stephen, so I've arranged to meet him again in a week's time.

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Now, you yourself must work with equations?

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I do, I do, I'm...

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I mean, technically I'm an astrophysicist.

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I'm really what we call a theorist. What I like to do is noodle around

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with equations and work things out and make predictions.

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Equations is what I do every day. I mean, my colleagues who do astronomy

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like to show pretty pictures and beautiful pictures of the cosmos.

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I like to show equations. Very much so.

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Come and visit me in Oxford and I will tell you all about this. We can't do this here,

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but in Oxford we've got blackboards and I can explain the beauty of equations.

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Fantastic, thank you very much, Pedro.

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I've come to the University of Oxford to take up Pedro's invitation.

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And he's going to tell me about the most famous equation of all,

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the one that everyone's heard of,

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E = mc2.

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This equation conjures up a whole load of thoughts in my mind,

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but the main ones are that it's got something to do with the atomic bomb

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and of course, it's by Einstein.

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But there's cultural knowledge and then there's maths.

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I don't know anything at all about how E = mc2 works.

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When Einstein first published the equation in 1905,

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it started a scientific revolution.

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-Hey!

-Hello.

-Hello, Pedro.

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-How are you?

-Very nice to see you again.

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-Thanks for coming.

-Well, it's a pleasure.

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Thank you very much for having me. Now you've got this tall order

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to explain to me so that I can totally understand it.

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We'll give it a go, we'll give it a go. Let me just clear this up.

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'Uh-oh, what am I doing?

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'Pedro lives and breathes abstract numbers.

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-'I'm an art guy who left school when I was 13.'

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-So what maths do you know?

-Well, I must confess that I don't know any maths, any geometry,

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or any algebra or anything in that realm of experience.

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-I'm completely ignorant about all that.

-OK.

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I know about art and that's about it.

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Ok, that's a good starting point. Let me get a pen.

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It seems a very bad starting point to me, but...!

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So, you know nothing about what an equation is?

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Only, uh, I think it's a sort of...

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code or some kind of metaphor for the natural world.

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It's the natural world reduced to a formula.

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That's pretty good. Let's start with a really famous one.

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Have you ever seen this equation?

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-Well, I've certainly heard of it. I know it's E = mc2.

-Very good.

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E...

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E stands for energy. Do you know what energy is?

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It's a difficult question, so...

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-You're having energy as you talk to me.

-Yes.

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A certain amount of energy is keeping me alive so I don't die and decay.

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Very good, very good.

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That's the limit. That's what I think energy is.

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Energy, I mean, energy is kind of a funny thing to try to define.

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The best way I can think of it is, it's the capacity to do things.

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-It's the capacity to lift something up, to heat something up.

-All right.

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Then you've got this thing here. Do you know what the m stands for?

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I think it stands for mass.

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Exactly. Mass, and mass is basically the amount of stuff in a thing.

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So when you pick up a book, it's the amount of stuff that that book is made of. Mass is kind of interesting.

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For example, suppose you've got a nail and you weigh it, all right?

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And then you leave it out in the air and you weigh it three weeks later, it will have rusted and so...

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-Its mass has changed. More particle things.

-It's gone up, exactly.

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Stuff has stuck onto it, there have been chemical reactions so the mass of it really does have to do

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with what it's made of and how it changes. And then we've got this thing over here, the c.

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-Do you know what the c is?

-No.

-There's no reason for you to know, it's the speed of light. OK?

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C is incredibly important because c is the speed at which light rays propagate through empty space.

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-I know what squared is, that means a thing multiplied by itself.

-Exactly.

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So this is a kind of fascinating statement.

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This is saying suppose you have some mass, right,

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it's possible to convert that mass into a certain amount of energy.

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I can see that E = mc2, like all equations,

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is about balancing two sides.

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That's what the equals sign is all about.

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So this equation allows us to calculate how much energy

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is contained in any given mass.

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It's a surprise to me that it applies to everything.

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Toothpaste, a book, a nail, or uranium for that matter.

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This equation is universal. And since "c2" is such a big number

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a tiny lump of matter contains an enormous amount of energy.

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What this equation doesn't tell you is how to unlock that energy.

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The most dramatic proof that the equation was true

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came 40 years after Einstein first worked it out,

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when the atomic bomb was dropped on Hiroshima.

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Pedro walked me through the chilling sums.

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You have a mass which is something like half a gramme,

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I write it as a kilogramme.

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You need the speed of light and the speed of light looks like this -

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-it's about 300 million metres per second.

-OK.

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OK. And we can work out how much energy there is, OK?

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All we've got to do we say that energy is going to be that mass

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times the speed of light, squared, and what you get is this.

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Joules. This is the unit of energy.

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So you get a phenomenal amount of energy.

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Now, if I told you this was something like 15 kilotons of TNT from,

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something the size of a pill,

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-giving enough energy...

-Producing an explosion of 15 kilotons...

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Which is equivalent tons of TNT.

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I want to throw you, I don't know if this is a stupid question and you might have nothing to say about it,

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but supposing the sign for squared was changed to a three.

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-Would that just be nonsense, or...?

-It would be nonsense,

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and the reason it would be nonsense is because we've tested it.

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We've gone into a lab and tested this relationship. We've weighed something, done something to it,

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weighed it again, worked out the amount of energy that came out and it was on balance

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so the left side was balanced with the right side.

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I'm impressed that E = mc2 was created before it was shown to be true.

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The equation was a prophecy.

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The five symbols explain the link between energy and all matter across the cosmos.

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This universality is part of its power.

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Einstein once claimed "the only physical theories that we're willing to accept are the beautiful ones".

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But what do scientists mean by "beautiful"?

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They talk about equations being testable, being universal.

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Is that what they think beauty is?

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I'm going to take you to the Rhodes building...

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..because Einstein actually came here in, I think 1933,

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to give the Herbert Spencer lecture. And it's an interesting lecture because it's a lecture

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where he basically discusses his philosophy.

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-Right.

-Why he does science the way he does, and his craft, what he does as a theoretical physicist.

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And he basically said two things.

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The first is that the endgame of what he does is experience.

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It's experiment. It's the natural world.

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It's not theory for theory's sake. It's always relating to reality.

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Exactly.

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But the bulk of what he says is that what guides him

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is mathematical beauty, or mathematical simplicity.

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That's what guides his research. He says, "It is essential from our point of view that we can arrive

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"at these constructions and the laws relating them, one with another, by adhering

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"to the principle of searching for the mathematically simplest concepts and their connections".

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So go for simplicity, go for the simplest relationships which are mathematically true,

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and that underpins the way that he thought about what he did.

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-So he was telling people this in a lecture that was really about the philosophy of what he did.

-Exactly.

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What his ultimate aims were, and what the use of what he did was to the world.

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It's a practical philosophy, it's what he actually did on an everyday basis. That's how he worked.

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So it gives us an insight into...

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what science at that level is about.

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-Exactly.

-Yeah.

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Einstein believed that the laws which govern the universe would have an elegant simplicity

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and this would be shared by their equations.

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I paint abstracts in collaboration with my partner, Emma.

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It occurs to me that when we intuitively put shapes and colours together in a visual order,

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we too, like people who come up with equations,

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try to arrive at a convincing metaphor for nature.

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For us, art tells you something important about the world.

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This is a coloured engraving of Isaac Newton by William Blake.

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It shows Newton studying a tiny corner of the world

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with a pair of dividers.

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Blake despised Newton, who he felt reduced

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the magnificence of existence to cold and mechanistic equations.

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So today I'm coming to a place...

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..where I'm actually going to find out a bit more about what Newton actually did.

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This is Newton's house, where he developed his ideas on gravity.

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I'm going back to the 17th century because it's when scientists

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first used equations to try to explain the natural order.

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-Hello.

-Hello, welcome to Woolsthorpe.

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I'm Margaret Winn, the house steward. Pleased to meet you.

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-Likewise.

-Hi, I'm Ruth.

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I'm professor of theoretical physics from Durham.

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Thank you both very much for seeing me.

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It's nice to meet you. Have you been here before?

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No, never. This is my first time.

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Right, well, this is the house Newton was born in, Christmas Day, 1642.

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Yes. Ruth, the only thing I know about Newton is an image of him observing apples falling off a tree.

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-Falling from a tree!

-And he suddenly works out that that means gravity.

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Well, we call him the father of modern science, and that is not an understatement.

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We can date our modern way

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of doing physics or science as trying to write down equations

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as coming from Newton.

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That's very clear, that equations come from beginning of equations.

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Yes, I think equations as a method, as a means of encapsulating,

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of modelling, of saying what physics is and what the world around us is.

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Why do you think the image of the apples falling

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is needed in the mythology of Newton?

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I think it's the link more than anything else. If we take an apple

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and just look at what happens

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as it goes up and down under gravity...

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Gravity is something that I think we often take for granted.

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And the apple falling, he realised that the same thing that made that apple fall down to the ground was

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the same thing that kept the Moon going round the Earth, or indeed the Earth going round the Sun.

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-Begs a lot of questions. How does he go from that to realising something about that?

-A lot of hard work!

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Is it assuming that perhaps there's some kind of force

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connected to the moon that's similar to the thing that makes...?

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That's right, so as soon as you start thinking about planetary bodies or things moving round,

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round other objects... I think people at the time, they would have felt, "ell that's a mystery of God,

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"and we're not supposed to understand that," but by using this apple

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as this sort of metaphor for the moon or the sun,

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Newton managed to say, "No, actually man can start plumbing those mysteries."

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I'm absorbing everything you've told and as I'm starting to freeze a bit, maybe we could go in the house!

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-Shall we go up to the house?

-You talked about he didn't want

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his theories to be thought of as the final word...

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Newton realised that mathematics could provide a precise and universal language to describe

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things as diverse as the fall of an apple and the orbit of the moon.

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He put his ideas in a revolutionary book, Principia Mathematica.

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I can barely tear my eyes away from this!

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THEY ALL LAUGH

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That is our very prized possession,

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a third edition copy of the Principia Mathematica.

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-If we turn to the relevant page, I'm going to leave you to our book.

-Thank you very much. Wonderful.

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The first things that we notice here is not a single equation,

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having talked about Newton as being the father of modern science.

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The other thing is of course it's in Latin, which the language of...

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It's hard enough...!

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-The universal language at the time.

-Propositio eight, theorema eight.

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Essentially, he is giving us his equation for gravity in words.

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So he starts off, "Si globorum duroum in se mutuo gravitantium materia undique in regionibus..."

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Newton's written version eventually formed the basis for the equation for gravity.

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So I want Ruth to unpick the different elements,

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a mathematical version of the words.

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I've noticed you've got a book, so we could try and translate

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what he said into one of these beautiful equations.

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Absolutely, find a blank page.

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-Anywhere will do.

-Right.

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So we write both of these objects as M1 and M2.

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So these are just the masses. But then Newton talked about

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the force between the two spheres,

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these two bodies, is inversely proportional,

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which means we divide, to the distance squared.

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The two bodies or spheres could be any size -

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the Earth, the Moon, or even an apple.

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And this is G, for gravity.

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It's actually called Newton's constant.

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-You've written out an equation for me there.

-Yes.

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And earlier you threw an apple in the air and it fell to the ground.

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Can you give me some numbers...

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-Certainly.

-..that will show me what the apple is doing?

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Let's talk about the apple.

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-OK, so one of the Ms is an apple?

-An apple. What does an apple weigh?

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-Half... ?

-You get a pound of apples so I guess that's about four apples, so that's a quarter of a pound.

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-We work in kilograms!

-SHE LAUGHS

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Let's just say it's 200 grams.

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-But is the other M the Earth?

-You don't need explaining, do you? You know this already.

-I'm guessing!

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But I don't know how much the Earth weighs, I very rarely buy one from the grocer!

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Well, I can give you a rough idea.

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-Five times ten to the 24 kilograms.

-Got it.

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So it's ten to the power of 24.

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Which is a trillion trillion.

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OK, good.

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Now the radius of the Earth is 6,000 kilometres.

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We've got a 200 gram apple,

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several trillion kilograms Earth

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and radius of Earth 6,000.

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Now, what I'm going to do is cancel all those off to make it easy.

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Then what I end up with is a simple ten on the top

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times the 0.2 of the apple.

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So what this tells me is the force the apple feels is its mass

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times this ten and this is...

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-Mass times ten.

-Ten is... This is the sort of,

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how fast gravity is going to cause the apple to start to fall.

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So the force on the apple here is simply two.

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And that unit is called the Newton.

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Ah! N for Newton.

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-N for Newton.

-Fantastic.

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The number of Newtons measures the force of gravity acting on the apple.

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It's a complicated equation,

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but I'm beginning to understand the key parts.

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The force depends on the mass of the two objects

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and the distance between them.

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The bigger the objects, the bigger the force.

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And the further apart they are, the weaker the force.

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The two masses, M1 and M2, could be anything.

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The earth and the apple. Or the earth and the moon.

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Or the earth and the sun.

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Ruth told me Newton's equation allowed us to understand why

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the moons and planets move around the solar system.

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His equation seemed to make sense of, well, the universe.

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So the equation itself, F = G x M1 x M2/R squared,

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that's Newton's equation of gravity, but how we use it,

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this is a sort of process, you know doing science,

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of calculating things, of making predictions.

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You've now showed me how we use that equation.

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How we would use it, yes.

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This is our paint, how we paint the world.

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We paint it in equations.

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In fact we use that a lot, we say, "I'm painting,"

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you know, we tend to use this word, "painting".

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If you, if you can use that metaphor of paint and colours etcetera

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is there a place also for beauty in this world of calculating things?

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I don't expect everyone to find this beautiful,

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but it certainly is for us and for me.

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Great!

0:22:200:22:22

'A few decades after Newton came up with his law, it was used

0:22:370:22:40

'to successfully predict the return of a comet, Halley's comet.

0:22:400:22:45

'His law of gravity had been confirmed.'

0:22:450:22:48

'With his equation, Newton had transformed

0:22:520:22:55

'the way mathematics modelled the world,

0:22:550:22:58

'and his work went unchallenged for over 200 years.'

0:22:580:23:02

'Everything changed at the beginning of the 20th century

0:23:150:23:18

'with the arrival of Einstein and his Theory of Relativity.

0:23:180:23:22

'In that same decade, something else entered, and that was modern art.

0:23:220:23:27

'In the world of art many believe that Picasso was involved

0:23:270:23:30

'in the same revolution as Einstein.'

0:23:300:23:33

Weirdly, the one place in which I had heard about relativity

0:23:350:23:38

before embarking on this programme was art school when I was young.

0:23:380:23:43

As art students we all had to absorb the idea that

0:23:430:23:48

relativity had something to do with cubist paintings.

0:23:480:23:53

I'm about to look at a cubist painting by Picasso from about 1909-1910.

0:24:000:24:07

It's of a woman in an armchair.

0:24:070:24:09

I think cubism was really seen as something quite terrifying and shocking when it first came out.

0:24:090:24:17

It's not like a Renaissance painting

0:24:220:24:24

where you feel you're looking through a kind of window onto the world.

0:24:240:24:29

With cubism the artist is deliberately confusing you

0:24:290:24:33

as to where thing are, and indeed what things are.

0:24:330:24:37

So that the space in the room

0:24:370:24:41

seems to be eating into the side of the woman.

0:24:410:24:44

And the textures of the room seem to be no different from the textures

0:24:440:24:48

of the woman. So there's all this moving around of objects and space

0:24:480:24:55

in a way that is deliberately confusing if you were thinking,

0:24:550:24:59

"Well, where is the thing that looks like ordinary reality?"

0:24:590:25:02

'I think it's right to say that cubism was a new kind of beauty

0:25:080:25:11

'that looked a bit like science.

0:25:110:25:14

'But I'm not convinced that cubism is science.

0:25:140:25:17

'I've arranged to meet historian of science, Arthur Miller,

0:25:170:25:21

'who's going to attempt to change my mind.'

0:25:210:25:25

I've got to tell you, Arthur, that at art school, and subsequently,

0:25:250:25:29

I felt oppressed by the idea that I had to think of a connection between

0:25:290:25:35

Einstein and relativity and cubism.

0:25:350:25:39

Einstein and relativity and Picasso.

0:25:390:25:41

But there is one in a sense that I'll say they both worked on

0:25:410:25:44

the same problem, the nature of space and time.

0:25:440:25:47

OK. The connection is that time and space are important to them both.

0:25:470:25:51

That's right.

0:25:510:25:52

Where I find the proposal difficult is that, just because

0:25:520:25:59

he's doing something with time and space that he's therefore

0:25:590:26:03

something like Einstein, or that cubism is something like science.

0:26:030:26:09

Cubism was very much of a scientific research programme, as I've said.

0:26:090:26:12

It had, you know, an explicit intent to reduce forms

0:26:120:26:16

-to geometry.

-Why is that?

-Picasso...

0:26:160:26:20

Why is that scientific and not artistic?

0:26:200:26:22

I mean, medieval artists reduced forms to geometry,

0:26:220:26:25

and African artists reduce it to geometry, archaic art reduces it to geometry.

0:26:250:26:29

Well, that's because Picasso had in mind scientific texts as a way to do it.

0:26:290:26:35

For example, we know that he looked at a text written by a mathematician

0:26:350:26:40

and the text discussed how you represent in four dimensions

0:26:400:26:44

complex polyhedra and Picasso took a look at these.

0:26:440:26:47

Of course, he didn't know what the equations meant,

0:26:470:26:49

but when the author of the books specialised the equations

0:26:490:26:53

of the two dimensions and then could generate illustrations,

0:26:530:26:57

Picasso was interested in the illustrations.

0:26:570:27:00

It's correct to call Picasso a revolutionary artist,

0:27:000:27:04

it's not hyperbole, but for me, I don't know enough about Einstein

0:27:040:27:07

to see the way in which Einstein is a revolutionary too, or how

0:27:070:27:11

Einstein's ideas and Picasso's are the same level of revolution

0:27:110:27:16

and also going in the same direction.

0:27:160:27:18

Well, Einstein was a revolutionary scientist because what he did

0:27:180:27:21

was to go take the next step beyond Newton.

0:27:210:27:26

Newtonian science is based on our sense perceptions that all time,

0:27:260:27:31

your time is the same as my time.

0:27:310:27:33

What Einstein was able to do was to raise himself to

0:27:330:27:36

heights of abstraction so he could glimpse a world beyond appearances.

0:27:360:27:40

The real objective world out there where there is scientific truth.

0:27:400:27:45

I still think the connections between Einstein and Picasso are

0:27:470:27:51

more superficial than substantial,

0:27:510:27:53

but I am very interested to hear more about Einstein.

0:27:530:27:57

Arthur will attempt to explain to me

0:27:570:27:59

one of the key equations of the Special Theory of Relativity.

0:27:590:28:02

When Einstein came up with this equation,

0:28:050:28:07

he wasn't even officially a scientist.

0:28:070:28:11

The days when he wrote the relativity theory, he worked as a patent clerk

0:28:110:28:14

in the Swiss federal patent office in Bern.

0:28:140:28:17

In fact he worked there from 1902 to 1909.

0:28:170:28:20

He was also a conscientious daydreamer.

0:28:200:28:22

And in his dreams and visions he soared over the landscape

0:28:220:28:25

of physics and realised what the fundamental problem was.

0:28:250:28:30

The nature of space and time.

0:28:300:28:32

People were beginning to think that maybe there was something wrong with

0:28:320:28:36

classical, intuitive notions of space and time,

0:28:360:28:39

but they couldn't put their finger on it.

0:28:390:28:41

What they especially wanted to do was to leave alone the notion of time.

0:28:410:28:45

Why was time sacrosanct, because it was obvious what it was,

0:28:450:28:48

it didn't need any more inquiry, or they were afraid that they couldn't find out anything more?

0:28:480:28:52

It seemed that your time is the same as my time.

0:28:520:28:54

No matter how fast we're moving with respect to one another.

0:28:540:28:57

There's no mystery there. We know what time is.

0:28:570:28:59

That's right. It's like Superman said, "Leave time alone."

0:28:590:29:03

-Don't mess with time.

-Don't mess with time, yeah.

0:29:030:29:06

OK I've got a book, if you've got a pen?

0:29:060:29:09

Absolutely, let me show you one of the spectacular results

0:29:090:29:14

of relativity theory.

0:29:140:29:15

Let's do a little thought experiment.

0:29:150:29:17

Suppose here is Matt one standing on a train platform

0:29:170:29:24

and here is Matt two, just call him Matt, standing on a train

0:29:240:29:29

and he's moving along with some velocity, call it V,

0:29:290:29:33

relative to the Matt standing on the platform.

0:29:330:29:35

The Matt on the moving train is wearing a wristwatch

0:29:350:29:39

and his time, call it t prime,

0:29:390:29:42

and call the times of all the clocks on the platform t.

0:29:420:29:45

And what we want to do is to compare the time

0:29:450:29:48

on Matt's wristwatch with clocks that remain at rest on the platform.

0:29:480:29:55

They all read the same time.

0:29:550:29:57

I'm going to assume that, even though the clocks are at rest

0:29:570:30:00

and my clock is moving that they're all the same,

0:30:000:30:03

because clocks always tell the same time, assuming they're all synchronised.

0:30:030:30:06

One would think so, yeah. Now let's call the Matt on the train...

0:30:060:30:09

But you're going to show me that they don't.

0:30:090:30:11

I'm going show you that they don't, convince you that they don't.

0:30:110:30:15

t prime and t.

0:30:150:30:16

Now it turns out Matt on the train's time t prime is

0:30:160:30:18

equal to t times the square root of 1 - V squared over C squared.

0:30:180:30:24

So the time here is equal to something complicated.

0:30:240:30:26

It's not just the same as that time.

0:30:260:30:28

No, it's not the same as that time. Your time is not the same as my time.

0:30:280:30:31

These two times are different...

0:30:310:30:33

'If I understand the equation correctly, it says something unbelievable -

0:30:330:30:38

'that time runs at different rates depending on how fast you're moving.

0:30:380:30:43

'Take a train zooming through a station.

0:30:430:30:46

'This equation predicts that a clock on the train, reading time t-,

0:30:460:30:50

'would run slower than clocks reading time t

0:30:500:30:54

'on the station platform.

0:30:540:30:57

'I've never noticed it and here's why.

0:30:570:31:00

'This bit of the equation is what makes the two clock times different,

0:31:000:31:04

'but it only has a significant effect if the velocity, V,

0:31:040:31:07

'of the train is very fast, close to the speed of light.

0:31:070:31:12

'If the train could reach the speed of light, you get 1 - 1,

0:31:120:31:16

'which equals zero.

0:31:160:31:18

'And then t- equals zero.

0:31:180:31:21

'Relative to the platform, time on the train completely stops.

0:31:210:31:26

'This stretching of time seems impossible

0:31:280:31:31

'but according to Arthur it's been proven by practical experiment.'

0:31:310:31:35

Now that's really something. That's wild.

0:31:400:31:42

And he realised that's because time is a relative quantity.

0:31:420:31:46

Just as I discussed with you.

0:31:460:31:50

Your time is only the same as my time if we're standing still

0:31:500:31:53

next to each other, but if you go away and come back, your clock,

0:31:530:31:57

although it'd be very difficult to perceive it,

0:31:570:31:59

will read a slower time than mine.

0:31:590:32:01

Well, I'm taking in a lot of what you're saying so that

0:32:010:32:04

-I'm far more informed than I was before you spoke.

-Good.

0:32:040:32:07

But the thing that's really big for me is this idea

0:32:070:32:10

of the physical nature of time and that seems a marvellous idea.

0:32:100:32:14

Oh, it turns out that there's not space and time. There's space-time.

0:32:140:32:19

Right, they are a single entity. Is entity the right word?

0:32:190:32:23

Time and space are connected by the velocity of light.

0:32:230:32:26

'That was definitely the hardest equation so far,

0:32:290:32:32

'not just the maths but because of the ideas it contained.

0:32:320:32:37

'You might be worrying about time on a tube train,

0:32:370:32:40

'but you wouldn't think time was actually changing shape.'

0:32:400:32:44

'Einstein worked out that time and space are inextricably linked

0:32:450:32:50

'through the speed of light.'

0:32:500:32:52

'It was a thought that it was simply impossible to have before,

0:32:550:33:00

'reality had changed,

0:33:000:33:01

'and Einstein did it with equations.'

0:33:010:33:04

'I'm beginning to get a crush on science.'

0:33:080:33:11

'Before, I literally didn't know what an equation was.

0:33:110:33:15

'Now, in some ways I know the basics of what an equation is,

0:33:150:33:21

'but I also know the implications of what an equation is,

0:33:210:33:25

'so there's a sort of excitement

0:33:250:33:26

'about the philosophy of an equation,'

0:33:260:33:29

or the use of an equation in some kind of profound way

0:33:290:33:32

as opposed to something like a railway timetable that tells you very detailed information.

0:33:320:33:36

You know the process of learning is a mixture of pain and pleasure.

0:33:360:33:42

It's quite hard to dislodge the pattern of the world that

0:33:420:33:45

you've already got in place, and bring in a whole load of new stuff.

0:33:450:33:49

You can appreciate it on mythological levels.

0:33:490:33:52

Someone's telling you the myth of equations, or the myth of science,

0:33:520:33:56

or the myth of Newton, or the myth of Einstein,

0:33:560:33:59

but they all do sound like myths to me.

0:33:590:34:01

But as the days go by they acquire more and more reality as each

0:34:010:34:06

scientist adds to the stories that the other scientists have told me.

0:34:060:34:11

There's one scientist who stands out in the story of equations,

0:34:200:34:25

because he took the idea of beauty in science further than anyone else.

0:34:250:34:30

His name is Paul Dirac.

0:34:300:34:33

He too revolutionised our view of the universe,

0:34:340:34:38

yet virtually no-one outside scientific circles has heard of Dirac.

0:34:380:34:43

So, I've arranged to meet the biographer of this mysterious genius.

0:34:430:34:49

This is a particularly favourite part of Cambridge for Paul Dirac.

0:34:510:34:56

Dirac was the greatest English theoretician since

0:34:560:34:59

Isaac Newton and that's how... That's his reputation in 1927,

0:34:590:35:03

when he was looking for what became his greatest achievement -

0:35:030:35:07

his equation.

0:35:070:35:08

Why is he... Being so great,

0:35:080:35:10

why is he totally unknown to the general public?

0:35:100:35:16

He actually wanted anonymity, he really had no interest at all in celebrity.

0:35:160:35:20

He simply wanted to get on with his work and be unknown

0:35:200:35:24

to the outside world.

0:35:240:35:27

I love the idea that for Dirac, beauty is important.

0:35:270:35:32

Is there a sense in which it is more important for him

0:35:320:35:37

than I've been hearing so far about other scientists?

0:35:370:35:39

Oh, yeah, Dirac was the first scientist actually to elevate this idea of beauty to a principle.

0:35:390:35:45

He called it the principle of mathematical beauty.

0:35:450:35:47

And what he meant by that was that as we advance in fundamental,

0:35:470:35:52

theoretical physics, the theories as they get closer and closer to nature,

0:35:520:35:56

become more and more beautiful.

0:35:560:35:58

So, for him, it was a method of sifting out theories,

0:35:580:36:03

right from wrong because if it wasn't beautiful, if it was ugly

0:36:030:36:07

in his opinion, it just wouldn't cut pass muster with nature.

0:36:070:36:11

So for him, a theory had to be beautiful for it to stand a chance of describing nature.

0:36:110:36:17

Incredible.

0:36:170:36:19

Here's a scientist who insisted science went through a "filter"

0:36:190:36:23

of beauty. And by pursuing beauty, you end up with truth.

0:36:230:36:27

It's an idea that's often used metaphorically, but Dirac meant it literally.

0:36:270:36:33

This is the Bridge of Sighs, which he walked across as a Fellow.

0:36:330:36:36

He walked back to his rooms here and this is where he did his great work on the Dirac Equation.

0:36:360:36:43

In fact, he was staying in a room just here.

0:36:430:36:47

That's where he was working in the late months of 1927

0:36:470:36:52

on what came to be known as the Dirac Equation, one of the greatest achievements in modern science.

0:36:520:36:57

Here we are, Room A4. Newcourt. Where Dirac discovered his great equation.

0:37:090:37:14

Completely free of distraction. The only noise you get is a bit of noise from the punters outside.

0:37:140:37:19

Apart from that, no radio, just nothing.

0:37:190:37:22

Dirac was not given to luxury. In late 1927, all he did, apparently, was to work on that equation.

0:37:220:37:29

Tell me about that equation, what was he trying to accomplish with it?

0:37:290:37:34

Well, what he was trying to do was come up with an equation for the electron,

0:37:340:37:38

the first material fundamental particle to have been discovered.

0:37:380:37:44

-What does that mean, "the first fundamental material particle"?

-OK.

0:37:440:37:49

A fundamental particle has no constituents.

0:37:490:37:52

It's a completely basic particle, you can't subdivide it.

0:37:520:37:56

The point of the tiny, tiny thing, this electron,

0:37:560:37:59

-is that nothing else is more basic than it.

-That's right.

0:37:590:38:03

So you had a chance of giving a fundamental description in nature.

0:38:030:38:08

I've got a notebook in my bag.

0:38:080:38:12

-If I give that to you and you find a blank page...

-Yep.

0:38:120:38:15

And I then give you my pen, could you write out for me

0:38:150:38:19

-the equation...

-I will.

0:38:190:38:21

-..that Dirac came up with.

-I will.

0:38:210:38:23

It's called the Dirac Equation?

0:38:230:38:25

That's right. This is the Dirac Equation.

0:38:250:38:28

And this equation applies to every electron that's ever existed, or ever will exist,

0:38:280:38:35

in the entire universe, so this is the ultimate compact equation that

0:38:350:38:40

has this universal significance.

0:38:400:38:43

This is a miracle, one of the miracles of 20th century science.

0:38:430:38:46

You've shown me the miracle, now tell me what it is.

0:38:460:38:50

I see something like "I followed by squiggle, followed by P followed by

0:38:500:38:55

"a squiggle, followed by equals, followed by m,

0:38:550:38:57

"followed by squiggle."

0:38:570:38:58

OK, you say, "I, gamma, P, psi = M psi".

0:38:580:39:04

OK, so it's like E = M C squared, only you say these new things that

0:39:040:39:09

he thought up himself, a bit like the Lord of the Rings language.

0:39:090:39:12

-That's right.

-And what is the most important symbol there?

0:39:120:39:15

Right, OK.

0:39:150:39:16

This is called a spinner, all right?

0:39:160:39:19

This is a thing that encodes the information about the behaviour of the electron.

0:39:190:39:24

So, you tell the equation what situation the electron is in and out

0:39:240:39:30

of the equation comes the prediction for how the electron will behave.

0:39:300:39:35

What's the thing in the ordinary world that is the closest that

0:39:350:39:40

-I could visualise, to tell me what a spinner really means?

-There is none.

0:39:400:39:44

-OK, so I've got to accept that.

-Exactly.

-Fine.

0:39:440:39:46

This was a complete Dirac concoction, right?

0:39:460:39:49

So spinners didn't exist before him?

0:39:490:39:51

-No, they didn't.

-Do you have to learn his new language before you can say that equation?

-Yeah.

0:39:510:39:55

Seriously, people for six months a year were struggling. Brilliant,

0:39:550:39:59

world-leading physicists had no clue about what this equation meant.

0:39:590:40:03

This is why he was so far ahead of his time, they were having to say,

0:40:030:40:07

"What the hell do these symbols mean?"

0:40:070:40:08

It was on extremely good ground and moreover...

0:40:080:40:11

'If it stumps the world's top scientists then I think it's OK for it to be beyond me.

0:40:110:40:17

'This really is a foreign language.

0:40:170:40:19

'But I was getting a broader sense of how equations have advanced knowledge.'

0:40:190:40:25

I do feel from your talk that I'm starting to get a picture filled in

0:40:270:40:32

for me of science, the big points.

0:40:320:40:34

Newton, Einstein and now Dirac.

0:40:340:40:37

-That's right.

-And a sort of journey that the spheres, the planets,

0:40:370:40:44

the stars, this earth, everything on it, all the objects

0:40:440:40:48

can be somehow described and understood in mechanical terms.

0:40:480:40:53

That's right. Einstein said that the most incomprehensible thing

0:40:530:40:57

about the universe is that it is comprehensible.

0:40:570:41:00

And Dirac, Newton, Einstein, they all had faith that they could,

0:41:000:41:06

if they thought hard enough, they could come up with these laws that

0:41:060:41:09

describe nature at a fundamental level.

0:41:090:41:11

But faith doesn't produce more faith, it actually produces equations.

0:41:110:41:14

Oh, absolutely.

0:41:140:41:16

-It's not like a faith that you can't verify.

-Faith oils the works.

-Yeah.

0:41:160:41:20

Dirac actually said that the principle of mathematical beauty was a kind of religion to him.

0:41:200:41:27

He actually used those words because he really did believe

0:41:270:41:30

with all his heart and soul that a mathematically beautiful theory

0:41:300:41:36

was going to be the kind of theory that nature backed

0:41:360:41:38

and that that was the direction in which you should travel, so he really did believe that.

0:41:380:41:43

It was an article of faith.

0:41:430:41:45

Why is the spinner beautiful?

0:41:450:41:48

This is beautiful because Dirac used this equation

0:41:480:41:52

to predict the first example of anti-matter.

0:41:520:41:56

This was perhaps the greatest triumph of 20th century physics.

0:41:560:42:00

Now just to give you a sense of how monumental that is,

0:42:000:42:04

now cosmologists believe that the very beginning of the universe,

0:42:040:42:08

half the universe was anti-matter.

0:42:080:42:11

So by that token, Dirac conceived, using this equation,

0:42:110:42:15

half the universe in his head.

0:42:150:42:17

'Scientists now stand in awe of Dirac's Equation.

0:42:320:42:37

'But at the time, things were very different.

0:42:370:42:39

'In the late 1920's, anti-matter was totally unknown.

0:42:390:42:44

'The idea that every electron, proton, and neutron

0:42:440:42:48

'had an opposite partner was preposterous.

0:42:480:42:51

'If his equation predicted this make-believe stuff

0:42:510:42:56

'then it must be wrong.'

0:42:560:42:58

OK, so what we can do now is go into the teaching lab.

0:43:000:43:04

What we have is an experiment set up where we can

0:43:040:43:07

actually see tracks of particles that have been produced by anti-matter.

0:43:070:43:11

So you'll be showing me some anti-matter in action.

0:43:110:43:14

'Five years after Dirac came up with his prediction,

0:43:140:43:17

'anti-matter was discovered.'

0:43:170:43:19

'The equation had turned out to be true.

0:43:210:43:24

'Now, I too want to see the proof.'

0:43:240:43:26

This is the first practical place I've been to.

0:43:260:43:30

I'm surprised at how quaint everything looks.

0:43:300:43:33

This is a very simple experiment. This is very low tech.

0:43:330:43:36

You could do this in your kitchen.

0:43:360:43:38

-Really?

-OK, so, this is a magnet.

0:43:380:43:41

It's a fairly powerful magnet and we're going to put dry ice on here, so that will be very cold.

0:43:410:43:47

-A sort of cookery element at the moment.

-It is yeah.

0:43:470:43:50

Cooking fish in salt.

0:43:500:43:52

Now the Perspex box is going to go on top.

0:43:570:44:00

And there's alcohol that we put in the upper layer.

0:44:000:44:03

In order to see the tracks, they're actually quite faint,

0:44:030:44:07

-we have to illuminate it with a very bright lamp.

-OK.

0:44:070:44:10

And then, one of the other ingredients that we should

0:44:100:44:13

explain here is the radioactive sources that we're going to use.

0:44:130:44:16

So we have two radioactive sources.

0:44:160:44:18

One emits electrons and the other emits positrons.

0:44:180:44:22

And so what we have here is the isotope of strontium called strontium 90.

0:44:220:44:26

'Glen told me these radioactive materials would let us see the tracks of electrons.

0:44:260:44:32

'And more importantly, the anti-matter partner to the electron.

0:44:320:44:36

'Known as the positron, this is the particle predicted by Dirac's Equation.'

0:44:360:44:41

It emits positrons and we'll see tracks that are very similar.

0:44:410:44:44

Maybe slightly lower energy actually and they will be bending to the left.

0:44:440:44:49

So that really is the demonstration, that we have two types of particles that really look very similar

0:44:490:44:55

in terms of the tracks that they make,

0:44:550:44:56

except that one is positively charged and the other is negatively charged.

0:44:560:45:01

Yeah, yeah, I saw one going that way.

0:45:020:45:04

Furthermore, they should be bending to the right and they are.

0:45:040:45:08

Yeah, they're thin and irregular. It's like a string of beads almost.

0:45:080:45:12

OK, so all I've really convinced you that you can see so far are

0:45:120:45:15

bog standard electrons.

0:45:150:45:17

Even at the bog standard level, it's pretty impressive.

0:45:170:45:20

We're all made of plenty of those.

0:45:200:45:22

And so, maybe what we can try now, is to put in the positron source.

0:45:220:45:26

What we should see, is that they will bend in the opposite direction.

0:45:260:45:30

-The other one slotted in scientifically.

-That's right.

0:45:300:45:33

We're just going to hold it on to the entrance way.

0:45:330:45:37

Now I should expect to see things going to the left.

0:45:370:45:40

I'm seeing activity but not necessarily lines going to the left.

0:45:430:45:49

'We'd seen the electrons bend to the right.

0:45:500:45:53

'Now Glen hoped that we might spot the rarer anti-matter tracks

0:45:530:45:58

'as they curve towards the other side.'

0:45:580:46:01

One there! Very, very clear.

0:46:050:46:08

-There you go.

-Fantastic!

0:46:080:46:10

So that's the first time in this experiment that I've seen the anti-matter.

0:46:100:46:13

That was definitely coming from the source.

0:46:130:46:16

The amazing thing is to have something from...

0:46:180:46:21

a sort of comic world of science fiction, anti-matter,

0:46:210:46:26

-to have it presented to us in reality.

-There we go.

0:46:260:46:33

Except I wasn't looking at that one.

0:46:330:46:35

Every 30, 40 seconds a little blip occurs

0:46:350:46:39

within a sort of 10p size radius of the source.

0:46:390:46:44

It shoots out, curls around, doesn't go very far.

0:46:470:46:51

-One there, very curly one, shot right round!

-Very good.

0:46:510:46:56

-Yeah, yeah.

-So, we're really seeing a physical thing,

0:46:560:47:01

which connects to the very complicated mind-world of Paul Dirac.

0:47:010:47:06

That's right. Somehow the existence of anti-matter

0:47:060:47:09

emerges as a necessary consequence of the theory that he wrote down

0:47:090:47:13

and that's pretty difficult to see. To just look at his equation and say

0:47:130:47:16

that should give us anti-matter, but really if you analyse it carefully

0:47:160:47:20

it's clear that that is one of its necessary predictions and that's what you're seeing.

0:47:200:47:24

So, those curves and blips in that sort of molten sea,

0:47:240:47:27

is the Dirac Equation being shown to us in physical form.

0:47:270:47:31

'These elusive symbols point to a beautiful idea.

0:47:430:47:47

'There is something magical about them.

0:47:470:47:50

'The existence of anti-matter proved his theory true.

0:47:500:47:55

'Keats' romantic poem goes "Beauty is truth, truth is beauty,"

0:47:550:48:00

'as if one leads to the other.

0:48:000:48:02

'And that's exactly what Dirac, the scientist, believed.

0:48:020:48:06

'That the search for beauty powers the advance of science.'

0:48:060:48:11

I'm reading a paper by Dirac, which he delivered in February 1939.

0:48:200:48:26

He says, "What makes the theory of relativity so acceptable to

0:48:260:48:29

"physicists in spite of its going against the principle of simplicity,

0:48:290:48:34

"is its great mathematical beauty.

0:48:340:48:37

"This is a quality which cannot be defined any more than beauty in art

0:48:370:48:41

"can be defined, but which people who study mathematics usually

0:48:410:48:44

"have no difficulty in appreciating."

0:48:440:48:47

So, he's saying that beauty in art can't be ultimately defined

0:48:470:48:51

any more than beauty in anything can be ultimately defined.

0:48:510:48:54

But what he is saying is that people in the world of very, very high

0:48:540:48:59

and complex mathematics agree that beauty is something that

0:48:590:49:05

they all appreciate and follow.

0:49:050:49:08

And it may be that what Dirac is saying is that

0:49:080:49:11

there's a sort of high or true or pure beauty that

0:49:110:49:14

mathematicians are interested in,

0:49:140:49:17

which sounds to me a bit like the inner, true, deep beauty of art.

0:49:170:49:22

But you have to go on a bit of a journey to find,

0:49:220:49:25

you can't expect it to come leaping out and waving at you

0:49:250:49:28

straightaway when you haven't really bothered to get involved with art

0:49:280:49:33

and try and find out what it is.

0:49:330:49:35

I like these buildings very much.

0:49:570:50:00

But I think they have a sort of comic element.

0:50:000:50:03

They seem like a Hollywood mock up of some kind of scientific base

0:50:030:50:09

where something sinister is being worked out behind the scenes.

0:50:090:50:15

You wouldn't even really think you were in England.

0:50:150:50:17

You could be anywhere in the world.

0:50:170:50:19

I'm ending my foray into science with an equation about black holes.

0:50:220:50:29

I'd always thought they were the stuff of science fiction,

0:50:290:50:32

but the inner workings of black holes are explained

0:50:320:50:35

by the fifth of my great equations.

0:50:350:50:39

All the previous equations have come from historical figures - Newton, Einstein and Dirac.

0:50:400:50:47

This will be my chance to hear about the entropy equation direct from its creator, Stephen Hawking.

0:50:470:50:54

And find out if he agrees with Paul Dirac about beauty and the truth of science.

0:50:540:50:59

Thanks very much for allowing me into your department, Stephen.

0:51:090:51:13

Can I ask you straight away,

0:51:130:51:15

is beauty important for you in your scientific work?

0:51:150:51:18

I don't know about beauty,

0:51:180:51:20

but the fundamental laws of the universe should be elegant.

0:51:200:51:25

What do you mean by elegant?

0:51:250:51:28

An equation is elegant if it is short, simple and explains

0:51:280:51:34

properties of the universe that were previously not accounted for.

0:51:340:51:39

My most elegant equation is very simple.

0:51:390:51:43

It is S = a quarter A.

0:51:430:51:47

Here, A is the area of the boundary of a black hole.

0:51:470:51:51

And S is its entropy, a measure of how much heat it contains.

0:51:510:51:55

What does that mean?

0:51:570:51:58

This equation shows that black holes aren't completely black.

0:52:000:52:06

They glow like hot bodies and lose energy and mass.

0:52:060:52:11

Eventually they will disappear in a tremendous explosion.

0:52:110:52:17

Why is that an elegant equation?

0:52:170:52:21

The equation came from a rather messy calculation.

0:52:220:52:27

It seemed a miracle that such a concise equation should result.

0:52:270:52:32

This equation unravels the physics of black holes,

0:52:340:52:38

one of the most mysterious objects in the universe.

0:52:380:52:41

As I understand it, the equation says that as stuff

0:52:410:52:44

falls into the black hole, the surface area of the black hole

0:52:440:52:48

gets bigger, and the entropy does too.

0:52:480:52:50

In 1975, when Stephen Hawking came up with his equation,

0:52:520:52:56

there was still some doubt as to whether black holes existed.

0:52:560:52:59

35 years on, all scientists agree they do.

0:52:590:53:03

Black holes have entered the realm of science fact.

0:53:030:53:07

While making this film, I found out that Paul Dirac

0:53:090:53:12

believed that it was more important to have beauty in one's equation

0:53:120:53:16

than to have the equation backed up by actual experiment.

0:53:160:53:21

Is this too extreme a view for you?

0:53:210:53:25

I think what Dirac meant was that although a beautiful equation

0:53:250:53:30

might not agree with experiment at a particular time,

0:53:300:53:33

it will eventually turn out to be true in the long run.

0:53:330:53:38

I think elegance is a good guide for equations but not an infallible one.

0:53:380:53:44

In art, an artist like Picasso say,

0:53:440:53:47

will just be working from hour to hour,

0:53:470:53:50

from work to work, pushing his ideas along with his work.

0:53:500:53:54

He doesn't necessarily think, "Now, I've discovered cubism."

0:53:540:53:58

That accolade will be bestowed upon his work

0:53:580:54:03

a bit later by other people.

0:54:030:54:04

But he probably will, at some point, think,

0:54:040:54:07

"I have made some kind of breakthrough here."

0:54:070:54:11

And I wonder if that breakthrough feeling,

0:54:110:54:14

if there's an equivalent for you in your type of enquiry.

0:54:140:54:18

There's nothing like the Eureka moment of discovering

0:54:200:54:24

something that no-one knew before.

0:54:240:54:27

I won't compare it to sex, but it lasts longer.

0:54:270:54:33

Thank you very much, Stephen.

0:54:330:54:35

Thank you.

0:54:350:54:37

The very fact that Stephen agreed to be interviewed by me,

0:54:410:54:45

when it's not an easy task for him,

0:54:450:54:49

it's not something that he does a lot,

0:54:490:54:54

proves to me that he believes in the thesis that beauty

0:54:540:54:58

is a significant element in the work of a theoretical scientist.

0:54:580:55:02

That making an equation calls for some kind of,

0:55:020:55:08

not just a sense of beauty, but almost a pursuit of it.

0:55:080:55:12

The pursuit of beauty really is a sort of driving force

0:55:120:55:17

in evolving an equation.

0:55:170:55:20

I've got to let all that sink in now.

0:55:200:55:25

I've been very happy to have my head crammed full of unfamiliar ideas,

0:56:020:56:06

but now there's one more thing I need to do.

0:56:060:56:10

Hello, Cary, how nice to see you!

0:56:100:56:13

I'm at the opening of my own exhibition, the work I do with my partner, Emma.

0:56:130:56:19

I've invited the scientists and most of them have turned up.

0:56:190:56:23

Throughout this film, the word "beauty" has often cropped up.

0:56:240:56:28

But it's hard to define and I can't help but feel that

0:56:280:56:30

while there are similarities, there are differences too

0:56:300:56:33

in what artists and scientists mean by beauty.

0:56:330:56:36

-Sorry, which are your paintings?

-All these paintings.

0:56:360:56:40

We do all these.

0:56:400:56:42

I'm trying to imagine what Paul Dirac would make of this painting.

0:56:420:56:46

My guess is he would ask you what you are representing here,

0:56:460:56:50

because he had a very literal mind.

0:56:500:56:52

I know what you're saying.

0:56:520:56:54

Are you conscious of representing anything?

0:56:540:56:56

There's no representation in the room at all,

0:56:560:56:59

but I think there is the idea of a model of the visual world.

0:56:590:57:03

There's a lot going on. There's a lot of them.

0:57:050:57:07

You come back in a couple of minutes from looking at something else. You can't find that order again.

0:57:070:57:12

Right.

0:57:120:57:13

That's really the point of them.

0:57:130:57:15

They should have a restlessly changing sense of order.

0:57:150:57:19

It's like looking at a fire where the fire always looks the same, but it's never exactly the same.

0:57:190:57:24

Exactly. Anything in nature that is permanent and changeable.

0:57:240:57:27

Are all the panels the same, or are they different?

0:57:280:57:31

I think they're all pretty different.

0:57:310:57:32

It's interesting because you'd look at it and think there's an algorithm

0:57:320:57:35

that tells you how you would paint that in terms of the things around.

0:57:350:57:39

But he says no, there's also a global point of view.

0:57:390:57:43

-A non-repeating pattern of some sort.

-Exactly.

0:57:450:57:48

Ultimately, it is highly mathematical,

0:57:480:57:50

but actually there is no... We didn't sit down and work it out.

0:57:500:57:54

Well, randomness is also mathematical.

0:57:540:57:57

It's interesting to see mathematical symmetries come out of aesthetic pursuits.

0:57:590:58:05

Well, that's arrived at...

0:58:050:58:06

'All my life, science has been totally out of my orbit.

0:58:060:58:10

'What was so illuminating for me in this programme was to

0:58:100:58:13

'find out that equations are the most important tool in science,

0:58:130:58:18

'forever pushing the boundaries of knowledge.

0:58:180:58:21

'And that the greatest and most beautiful equations

0:58:210:58:24

'have a life of their own.

0:58:240:58:26

'They've given us ideas beyond the human imagination.'

0:58:260:58:32

Subtitles by Red Bee Media Ltd

0:58:480:58:50

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0:58:500:58:52

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