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There is a mystery at
the heart of our universe.
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A puzzle that, so far, no-one
has been able to solve.
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I can't... This is too weird!
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Welcome to my world!
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If we can solve this mystery,
it will have profound consequences
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for all of us.
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That mystery is why mathematical
rules and patterns seem
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to infiltrate pretty much everything
in the world around us.
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Many people have,
in fact, described maths as
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the underlying
language of the universe.
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But how did it get there?
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Even after thousands of years,
this question causes controversy.
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We still can't agree on what maths
actually is or where it comes from.
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Is it something that's
invented, like a language?
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Or is it something
that we've merely discovered?
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I think discovered.
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Invented. It's both.
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I have no idea!
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Oh, my God!
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Why does any of this matter?
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Well, maths underpins
just about everything
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in our modern world,
from computers and mobile phones,
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to our understanding of human
biology and our place
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in the universe.
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My name is Hannah Fry
and I'm a mathematician.
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In this series, I will explore
how the greatest thinkers in history
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have tried to explain the origins
of maths' extraordinary power.
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SHE LAUGHS
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You've ruined his equation!
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I'm going to look at how,
in ancient times, our ancestors
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thought maths was
a gift from the gods.
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How, in the 17th and 18th centuries,
we invented new mathematical systems
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and used them to create
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the scientific and industrial
revolutions.
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And I'll reveal how,
in the 20th and 21st centuries,
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radical new theories are forcing us
to question, once again, everything
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we thought we knew about
maths and the universe.
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The unexpected should be expected,
because why would reality
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down there bear any resemblance
to reality up here?
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In this episode, I explore paradoxes
within modern mathematics.
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Who shaves the barber?
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And I discover the very weird
worlds that maths seems
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to be leading us into.
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SHE SCREAMS
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Maths is very much part
of our modern world.
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Even the images you're watching now
are essentially numbers
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processed by computers.
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Sorry, guys.
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Would you mind taking a photo of me?
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Oh, sure. Give me one second.
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Today, maths is at the heart of big
business, in the development
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of new software, such as facial
recognition technology.
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All of which, fundamentally,
is based on mathematical algorithms.
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And it matters because copyright
issues and legal ownership
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can depend on where
that maths comes from.
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You can phrase
the question like this -
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is maths a genuine, fundamental
part of our universe,
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something that we have discovered?
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Or is it merely invented?
A language that we've created
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just to describe
the world around us?
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THEY SCREAM
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Mathematicians have argued
over this idea for centuries.
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And even today, this question
is a thought-provoking
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and challenging dilemma.
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So far, I've explored how,
in ancient times, maths was revered
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Perfect, complete and gratefully
discovered by humans.
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But through the ages,
new areas of mathematics,
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like algebra and the concept
of zero,
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have, quite simply, been invented.
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But for most of us, we normally
think of maths as just a series
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of objective facts based
in logic that someone,
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somewhere has discovered.
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Facts that we all start
to learn at school.
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If you're anything like me,
you'll remember maths at school
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being taught as a series of rules.
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It was very logical, it was very
ordered, very complete.
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Very black and white.
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There were right and wrong answers,
which you didn't necessarily get in
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other subjects like art or
like music,
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which were much more about
preferences, about opinions
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and about cultural differences.
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It felt like the mathematical rules
were intrinsically true.
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But why?
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What are the fundamental
mathematical laws?
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To answer that question,
you have to categorise everything.
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You have to boil maths
down into distinct groups of objects
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in something called set theory.
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Set theory is a language
that talks about groups,
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or sets, of items.
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So, for example, the set of odd
numbers are all the whole numbers
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that cannot be neatly
divided by two.
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And the set of even numbers
are those that can.
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This reveals a basic rule.
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Adding an odd number to an even
one produces an odd number.
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From simple rules like these, you
can build up more and more
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complex rules and relations of
maths.
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But there's a problem
with set theory.
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A paradox at the heart
of mathematical rules which caused
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a bit of a crisis at the start
of the 20th century.
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You can discover this paradox
yourself by going to your local
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hairdresser or gentleman's barber,
and trying to define what you find
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in a concise and complete way.
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Hello. Hello.
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I was wondering
if you could help me.
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I am looking for the very
definition of a barber.
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I think I can help with that.
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Mathematicians took the same
approach to precisely define
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the laws of maths.
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So, if you were looking
it up in a dictionary,
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that one sentence that defined
what a barber was,
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what would you say it was?
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Cut men's hair.
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But that could be a hairdresser
though, right? A hairdresser?
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It needs to be a unique
definition for barbers.
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Barbers, and only barbers.
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Cos there's the shaving element,
as well, isn't there?
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Yeah, that's true. I've never
had a shave in a hairdresser's.
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No. That's true. I've had a chat.
The chat?!
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SHE LAUGHS
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Yeah. It's a fair point.
It's a very important part of it.
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You do hear some stories,
being a barber.
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So, actually, I suppose,
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the shave thing is something that
only barbers do. Mm-hm.
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So, someone who shaves men.
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But a barber doesn't shave all men.
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And I need a phrase that uniquely
and completely identifies a barber
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OK, let's see where we are, then.
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So, we've got, a barber shaves all
men, but only the men who shave
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but don't shave themselves?
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Yes. Yes.
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All right, I think we've settled
on something now.
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We've agreed on...
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Sound about right?
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I mean, it doesn't exactly
roll off the tongue!
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But I think it's fairly accurate.
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But, hang on a second!
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There's a bit of a paradox here.
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Who shaves the barber?
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Well, can a barber not shave
himself?
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But if he does shave himself,
then our
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definition here says
that he doesn't shave himself.
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Let me clarify that.
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If he doesn't shave himself,
then according to the definition,
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he's one of the men
shaved by the barber.
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So, he does shave himself.
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Attempting to create
a mathematically precise definition
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creates a contradiction
where the barber both shaves
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himself and doesn't shave himself.
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First, the bristles...
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This is known as
the barber's paradox.
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You've got it!
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I want to do it perfectly!
Perfect, OK.
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It is an illustration of the paradox
at the heart of mathematics,
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which was discovered in 1901 by one
of my favourite troublemakers,
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Bertrand Russell.
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The problem for maths
was that Russell's paradox
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undermines the logic
of defining things,
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like odd or even numbers,
by putting them into
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categories or sets.
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Over here, I have got a set of
clipper attachments.
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And in there, I have got a set of
things
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that aren't clipper
attachments.
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Clipper attachment goes in there...
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Not a clipper attachment,
goes in there.
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Clipper.
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Not a clipper.
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Now, the question is -
where does this bag belong?
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It's clearly not
a clipper attachment.
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Is it going to attach to a clipper?
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No, it's not, which means
it needs to go in there,
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but we've got a problem,
because this sink is supposed
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to only contain things that are not
clipper attachments.
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Which means that the contents
of the bag can't go in the sink.
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Since the bag, or set, is not, in
itself, a clipper attachment,
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but, by its definition,
contains clipper attachments,
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we can't easily categorise
where the set belongs.
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Similarly, the barber can't,
in a logically consistent way
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be contained in a set
of people that do
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shave themselves or the set
of people who don't.
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Russell's paradox shows
that there is a logical problem
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with trying to categorise
anything into coherent sets,
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whether it's barbers,
clipper attachments,
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or even numbers, and this logical
puzzle exposed a fault in
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the bedrock on which all the rest
of maths is built.
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If the foundations are shaky,
how can we trust everything else?
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Bertrand Russell realised
that mathematics was on much shakier
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ground than people had originally
thought.
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It turned out to be much, much
harder to really lay
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a solid foundation for maths
that everybody agreed on,
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and this is still wonderfully
controversial to this day.
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That's what you do
in science and mathematics.
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You take a sledgehammer.
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You smash at whatever structure,
whatever edifice you've built.
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You try to find the weaknesses
and that allows you to figure out
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what needs to be shored up.
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And that's really, I think,
the legacy that Russell left us.
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I think of it as...in some ways,
the death knell,
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or at least a major challenge,
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00:11:57,800 --> 00:11:59,840
the attempt to ground
mathematics in logic.
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00:11:59,840 --> 00:12:01,880
And that's the thing that
becomes really hard
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in light of Russell's paradox.
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Russell's paradox caused a real
crisis amongst mathematicians.
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Suddenly, maths was uncertain.
It was fallible.
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And if it has these
fundamental problems,
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how can it possibly be discovered?
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00:12:26,240 --> 00:12:28,920
So, does that mean that maths
has to be invented?
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Just a human language
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and all of the flaws
that come with it?
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If maths is merely an invention
of the human mind,
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it's perhaps not that surprising
that it's not perfect.
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But I don't think that I'm ready
to accept the invention
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argument quite yet.
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Maths just seems to be too good
at predicting the behaviour
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00:12:52,360 --> 00:12:55,840
of the world in ways that we never
could have imagined.
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Because, just as Bertrand Russell
was exposing
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the limitations of maths in one way,
another titan of the 20th century,
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00:13:03,960 --> 00:13:08,960
Albert Einstein, was pulling it back
in a completely different direction.
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BABY GIGGLES
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00:13:13,960 --> 00:13:17,800
Take what is probably the most
famous equation in the world.
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With just five symbols,
it looks so simple.
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It's almost childish.
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00:13:23,240 --> 00:13:26,840
Yet, it contains some incredibly
powerful mathematical
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and philosophical concepts.
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00:13:29,440 --> 00:13:33,720
I'm talking, of course,
about E = MC squared.
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So, "E"...that's energy.
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00:13:38,720 --> 00:13:41,760
That is equal to...
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00:13:41,760 --> 00:13:44,040
.."M", that's mass.
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Times by a constant, C.
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It's the speed of light...
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..squared.
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00:13:53,000 --> 00:13:56,840
There is so much more
to this equation than meets the eye.
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It is Einstein's discovery
that matter and energy
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00:14:00,440 --> 00:14:05,320
are equivalent, and that
has profound consequences.
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00:14:05,320 --> 00:14:09,320
This equation gives us one
of the immutable laws
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00:14:09,320 --> 00:14:12,800
in the universe -
that nothing can travel faster
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00:14:12,800 --> 00:14:14,800
than the speed of light.
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00:14:14,800 --> 00:14:16,360
Try this one.
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00:14:16,360 --> 00:14:18,760
The reasoning is this...
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00:14:18,760 --> 00:14:23,760
Making something move requires
more energy than keeping it at rest.
240
00:14:24,480 --> 00:14:28,120
And because this "C"
here is a constant,
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00:14:28,120 --> 00:14:31,600
if the energy goes up by
accelerating something,
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00:14:31,600 --> 00:14:34,200
the mass also has to increase.
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00:14:35,200 --> 00:14:40,240
So, that means that you or I
actually weigh a tiny bit more
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00:14:40,240 --> 00:14:43,360
when we're
moving in a car or a plane.
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00:14:45,200 --> 00:14:49,920
The increase in mass only becomes
significant when objects are moving
246
00:14:49,920 --> 00:14:52,560
at speeds close
to the speed of light.
247
00:14:54,760 --> 00:14:57,800
As an object approaches
the speed of light,
248
00:14:57,800 --> 00:15:02,840
its mass rises faster and faster,
which means it takes more energy...
249
00:15:04,560 --> 00:15:07,320
..to accelerate it further.
250
00:15:07,320 --> 00:15:11,360
It can't, therefore, reach the speed
of light, because the mass becomes
251
00:15:11,360 --> 00:15:14,960
infinite, and it would require
an infinite amount of energy
252
00:15:14,960 --> 00:15:16,280
to get there.
253
00:15:19,440 --> 00:15:21,320
You've ruined his equation!
254
00:15:23,160 --> 00:15:26,600
As well as proving there's
a cosmological speed limit,
255
00:15:26,600 --> 00:15:31,480
this single equation also explains
how all the stars in the universe
256
00:15:31,480 --> 00:15:35,040
convert mass into energy
as they burn brightly
257
00:15:35,040 --> 00:15:36,280
in the night sky.
258
00:15:37,280 --> 00:15:41,080
Einstein's famous equation
has proved itself to be a remarkable
259
00:15:41,080 --> 00:15:45,000
match for reality every time it's
been put to the test.
260
00:15:52,640 --> 00:15:57,000
Einstein had uncovered one
of the essential mathematical rules
261
00:15:57,000 --> 00:15:59,760
underlying the cosmos.
262
00:15:59,760 --> 00:16:02,640
It seems like clear
evidence that that maths,
263
00:16:02,640 --> 00:16:05,200
at least, is discovered.
264
00:16:05,200 --> 00:16:07,920
But Einstein didn't stop there.
265
00:16:07,920 --> 00:16:13,000
Using the power of mathematics,
he brought about a fundamental shift
266
00:16:17,440 --> 00:16:19,200
travels through space.
267
00:16:21,040 --> 00:16:24,480
To see that evidence for myself,
I've come to an observatory to do
268
00:16:24,480 --> 00:16:28,400
some serious thinking
about what we actually see
269
00:16:28,400 --> 00:16:32,400
when we look at stars
in the sky, such as our sun.
270
00:16:34,160 --> 00:16:37,520
If things were happening right now,
we wouldn't be able to see
271
00:16:37,520 --> 00:16:40,120
it until 8.5 minutes
later, because that's how long
272
00:16:40,120 --> 00:16:42,280
it takes the light
to travel to the Earth.
273
00:16:42,280 --> 00:16:44,960
So, when you're looking at the sun,
you're seeing how it was
274
00:16:44,960 --> 00:16:47,400
8.5 minutes ago? Exactly.
275
00:16:47,400 --> 00:16:51,120
And objects that are further away,
we see them as they were further
276
00:16:51,120 --> 00:16:52,360
back in time.
277
00:16:52,360 --> 00:16:54,640
So, for instance, there are other
stars in our galaxy
278
00:16:54,640 --> 00:16:57,520
that are thousands
of light years away,
279
00:16:57,520 --> 00:17:00,320
so we see them as they were
thousands of years ago.
280
00:17:00,320 --> 00:17:03,720
So, when you look in a telescope,
and you're seeing them
281
00:17:03,720 --> 00:17:06,800
how they were when people
were building pyramids
282
00:17:06,800 --> 00:17:09,800
and Pythagoras was discovering
his rules on Earth.
283
00:17:09,800 --> 00:17:11,880
Exactly. And we can even see
things that are
284
00:17:11,880 --> 00:17:13,440
even further away than that.
285
00:17:13,440 --> 00:17:16,600
So, galaxies outside our own galaxy.
286
00:17:16,600 --> 00:17:18,920
We see many of them as they were
287
00:17:18,920 --> 00:17:21,200
a billion years ago or more.
Gosh. Goodness.
288
00:17:21,200 --> 00:17:24,560
Does this work at smaller scales,
then? Is there, like, a limit
289
00:17:24,560 --> 00:17:26,440
to how big something has to be
before this works?
290
00:17:26,440 --> 00:17:27,840
If you...
291
00:17:27,840 --> 00:17:29,560
I mean, I'm looking
at you now, right?
292
00:17:29,560 --> 00:17:32,760
Light, presumably,
is taking time to bounce
293
00:17:32,760 --> 00:17:34,760
off you and for me to see you.
294
00:17:34,760 --> 00:17:36,000
Yes, it is.
295
00:17:36,000 --> 00:17:38,800
But light travels at
an incredibly fast speed,
296
00:17:38,800 --> 00:17:41,520
300,000 km per second, roughly.
297
00:17:41,520 --> 00:17:45,880
So, the time it takes to travel
from me to you is very, very tiny
298
00:17:45,880 --> 00:17:47,160
fraction of a second.
299
00:17:47,160 --> 00:17:49,000
But, in theory, I am
seeing you in the past.
300
00:17:49,000 --> 00:17:52,640
In theory, yes, you're absolutely
seeing me in the past.
301
00:17:52,640 --> 00:17:56,720
All of this shows that we can never
know what the universe
302
00:17:56,720 --> 00:17:59,760
is like at this very instant.
303
00:17:59,760 --> 00:18:04,760
The universe is, remarkably, not
a thing that extends just in space,
304
00:18:04,760 --> 00:18:07,280
but in time, as well.
305
00:18:09,560 --> 00:18:13,400
This is fundamental to Einstein's
revolutionary insights
306
00:18:13,400 --> 00:18:15,760
about our universe.
307
00:18:15,760 --> 00:18:20,200
He realised that the very concept
of time is relative.
308
00:18:20,200 --> 00:18:22,120
That is to say, it depends on
309
00:18:22,120 --> 00:18:26,360
the position and movement
of the observer.
310
00:18:26,360 --> 00:18:29,840
He worked it out by thinking
about events that appear
311
00:18:29,840 --> 00:18:32,160
to be simultaneous.
312
00:18:32,160 --> 00:18:36,560
So, let's imagine that you're
in a hot-air balloon floating
313
00:18:36,560 --> 00:18:39,400
above the observatory here,
and you're high enough
314
00:18:39,400 --> 00:18:44,160
that you can see a flash of light
in London, say, and another
315
00:18:44,160 --> 00:18:45,760
one in Portsmouth.
316
00:18:45,760 --> 00:18:50,800
And let's assume that these flashes
of light go off such that you see
317
00:18:54,040 --> 00:18:56,360
So, from where I am, it looks like
they are both flashing
318
00:18:56,360 --> 00:18:57,840
their lights at the same time?
319
00:18:57,840 --> 00:18:59,160
At exactly the same time.
320
00:18:59,160 --> 00:19:04,120
But if I were in an aircraft
that was flying very fast
321
00:19:04,120 --> 00:19:08,200
towards London, I would see
the flash of light in London
322
00:19:08,200 --> 00:19:11,120
before the flash of light
from Portsmouth.
323
00:19:13,720 --> 00:19:17,560
Using the inescapable
logic of mathematics,
324
00:19:17,560 --> 00:19:21,520
Einstein realised that if
an observer is moving towards one
325
00:19:21,520 --> 00:19:25,760
of the flashes, they would see
that flash before the other one
326
00:19:25,760 --> 00:19:27,920
caught up with them.
327
00:19:27,920 --> 00:19:32,200
So, for them, the flashes
are not simultaneous.
328
00:19:32,200 --> 00:19:35,040
But who's...? OK... I mean,
they did go off together.
329
00:19:35,040 --> 00:19:36,120
Who's right?
330
00:19:36,120 --> 00:19:38,080
Am I right in the hot-air balloon?
331
00:19:38,080 --> 00:19:42,160
In fact, there is no way
of saying that you are right
332
00:19:42,160 --> 00:19:46,520
and I am wrong in how
we observe these events.
333
00:19:46,520 --> 00:19:49,360
This is called relativity.
334
00:19:49,360 --> 00:19:52,000
So, our whole concept of time,
our whole concept of time
335
00:19:52,000 --> 00:19:55,080
means what happens first,
what happens second,
336
00:19:55,080 --> 00:19:57,920
comes down to where we are
and how we're moving.
337
00:19:57,920 --> 00:19:59,040
Exactly.
338
00:19:59,040 --> 00:20:04,120
So, the concept of time is now
inextricably linked to the positions
339
00:20:04,760 --> 00:20:07,760
in space and your movement
through space.
340
00:20:07,760 --> 00:20:12,240
So, this is why we can't describe
space and time separately,
341
00:20:12,240 --> 00:20:14,560
but we have to put them
together in space-time.
342
00:20:14,560 --> 00:20:17,800
You can't separate the two.
343
00:20:17,800 --> 00:20:22,840
And that all comes down to this idea
that Einstein managed to prove
344
00:20:23,480 --> 00:20:25,360
via thought experiments.
345
00:20:25,360 --> 00:20:27,320
Yeah, that's the amazing
thing about it.
346
00:20:27,320 --> 00:20:30,360
Purely through thought
experiments and...
347
00:20:30,360 --> 00:20:32,760
And a good bit of maths.
And a good bit of maths.
348
00:20:32,760 --> 00:20:34,920
A very good bit of maths, yes.
349
00:20:37,560 --> 00:20:42,520
Einstein was using the mathematics
to make sense of the universe,
350
00:20:42,680 --> 00:20:45,960
and claiming that the universe
was nothing like what anyone
351
00:20:45,960 --> 00:20:47,960
thought it was.
352
00:20:47,960 --> 00:20:51,080
His concept of relativity flew
in the face of what people
353
00:20:51,080 --> 00:20:55,960
had believed about space
and about time for centuries.
354
00:20:55,960 --> 00:20:59,320
Whether that was the Greeks thinking
that the universe was eternal
355
00:20:59,320 --> 00:21:02,040
and unchanging,
356
00:21:02,040 --> 00:21:06,680
or Isaac Newton's more mobile and
mechanistic descriptions.
357
00:21:08,600 --> 00:21:13,080
Einstein took his thoughts even
further, attempting to wrestle
358
00:21:13,080 --> 00:21:17,040
gravity into a neat
mathematical law.
359
00:21:17,040 --> 00:21:19,840
He believed it was all
down to the strange
360
00:21:19,840 --> 00:21:22,000
behaviour of space-time,
361
00:21:22,000 --> 00:21:25,800
and if he was right, as he laid out
in the theory of general relativity
362
00:21:25,800 --> 00:21:29,920
in 1916, then gravity
will even affect light.
363
00:21:31,920 --> 00:21:35,560
If you've got a star shining
light from over here,
364
00:21:35,560 --> 00:21:39,240
then you, the observer,
over there, will receive
365
00:21:39,240 --> 00:21:41,160
it in a straight line.
366
00:21:47,160 --> 00:21:52,200
But, if there's a massive
object in the way,
367
00:21:52,560 --> 00:21:56,360
you might think that you won't be
able to see the star.
368
00:21:56,360 --> 00:22:00,040
However, Einstein predicted
that the mass of an object
369
00:22:00,040 --> 00:22:04,720
will distort the space-time
around it, and anything moving
370
00:22:04,720 --> 00:22:06,840
through that warped space-time
371
00:22:06,840 --> 00:22:09,360
will have to follow the curves.
372
00:22:10,760 --> 00:22:14,680
This warping of space-time,
Einstein said, is what we usually
373
00:22:14,680 --> 00:22:16,560
describe as gravity.
374
00:22:17,680 --> 00:22:22,040
We think of gravity as keeping
the planets in orbit around our sun.
375
00:22:22,040 --> 00:22:26,360
In fact, he said, it's the result
of the distortion of space-time
376
00:22:26,360 --> 00:22:28,640
near massive objects.
377
00:22:28,640 --> 00:22:33,560
And Einstein calculated the precise
effect it would have on light.
378
00:22:33,560 --> 00:22:37,800
So, the starlight, while still
technically travelling in a straight
379
00:22:37,800 --> 00:22:42,720
line, will follow the curves of
space and appear around the object.
380
00:22:50,040 --> 00:22:53,520
Einstein predicted that,
in exactly this way,
381
00:22:53,520 --> 00:22:58,240
we should be able to observe light
from distant stars getting bent
382
00:22:58,240 --> 00:23:01,000
as the stars pass behind our sun.
383
00:23:05,280 --> 00:23:10,320
But a theory is just a theory,
an invention of the mind.
384
00:23:10,560 --> 00:23:14,120
It only becomes a discovery
when proven by practical
385
00:23:14,120 --> 00:23:16,000
measurement or experiment.
386
00:23:17,480 --> 00:23:20,200
In the decade after
Einstein's prediction,
387
00:23:20,200 --> 00:23:24,880
solar eclipses around the globe gave
scientists the chance to repeatedly
388
00:23:24,880 --> 00:23:26,200
test his theory.
389
00:23:27,400 --> 00:23:32,480
The darkness of the eclipse allowed
them to actually see stars passing
390
00:23:35,080 --> 00:23:36,760
When scientists took the
measurements,
391
00:23:36,760 --> 00:23:40,200
they discovered that light from a
distant star
392
00:23:40,200 --> 00:23:43,240
was bending around the sun in
exactly the way
393
00:23:43,240 --> 00:23:46,480
that Einstein had predicted.
394
00:23:46,480 --> 00:23:50,320
The mathematics of general
relativity was correct.
395
00:23:53,840 --> 00:23:56,520
With general relativity,
Einstein completely upended our
396
00:23:56,520 --> 00:23:59,080
understanding of space,
time, matter, energy,
397
00:23:59,080 --> 00:24:02,520
and kind of what else
is there to the nature of reality.
398
00:24:02,520 --> 00:24:05,000
All of a sudden, we learn
that mass and energy can warp
399
00:24:05,000 --> 00:24:06,640
the fabric of space and time
400
00:24:06,640 --> 00:24:09,080
in this beautiful,
interconnected dance
401
00:24:09,080 --> 00:24:12,240
where the motion of matter affects
the warping of space and time,
402
00:24:12,240 --> 00:24:14,720
which affects the motion
of other matter.
403
00:24:14,720 --> 00:24:17,440
We used to think of
space as this boring
404
00:24:17,440 --> 00:24:20,840
static stage upon
which events unfolded.
405
00:24:22,800 --> 00:24:27,160
Then Einstein told us that space is
itself an active player
406
00:24:27,160 --> 00:24:30,120
in this game, like
a stretchy rubber sheet.
407
00:24:30,120 --> 00:24:32,880
And, yet, a substance
perfectly described
408
00:24:32,880 --> 00:24:35,720
by beautiful mathematical equations.
409
00:24:35,720 --> 00:24:37,520
I mean, how did he think of that?
410
00:24:37,520 --> 00:24:41,160
How did he think of
something like this?
411
00:24:41,160 --> 00:24:45,080
Einstein's description of gravity,
the warping of space-time,
412
00:24:45,080 --> 00:24:48,600
accurately explains why
objects stay in orbit,
413
00:24:48,600 --> 00:24:51,320
whether they're satellites around
the Earth or galaxies
414
00:24:51,320 --> 00:24:53,520
around black holes.
415
00:24:53,520 --> 00:24:57,960
His equations are being tested
and reproven every day,
416
00:24:57,960 --> 00:25:00,880
and without Einstein's general
theory of relativity,
417
00:25:00,880 --> 00:25:04,680
modern communication, GPS or
satellite TV systems
418
00:25:04,680 --> 00:25:06,960
couldn't even function.
419
00:25:08,000 --> 00:25:10,640
Although this theory
came from his mind,
420
00:25:10,640 --> 00:25:13,840
from thinking about the problem,
rather than from real-world
421
00:25:13,840 --> 00:25:18,840
experiments, it's still
so good at predicting,
422
00:25:18,840 --> 00:25:23,240
so perfectly capable of describing
what happens in the universe,
423
00:25:23,240 --> 00:25:28,280
that it must be reflecting some
underlining mathematical truth.
424
00:25:28,680 --> 00:25:31,680
And this lends quite a lot of weight
to the argument that mathematics
425
00:25:31,680 --> 00:25:36,040
is discovered, which is something
that matches up with my own
426
00:25:36,040 --> 00:25:40,680
experience, because when you're
toying around with mathematics,
427
00:25:40,680 --> 00:25:44,720
it really does feel
as though you're exploring something
428
00:25:44,720 --> 00:25:46,160
that already exists.
429
00:25:51,400 --> 00:25:56,520
But if we accept that maths does
already exist and is an intrinsic
430
00:25:56,920 --> 00:26:02,000
part of nature, then surely
all the rules are out there waiting
431
00:26:02,440 --> 00:26:03,640
to be discovered.
432
00:26:06,520 --> 00:26:09,000
In some ways, mathematics
is quite a lot like
433
00:26:09,000 --> 00:26:10,640
a game of chess.
434
00:26:10,640 --> 00:26:14,440
So, you have these very strict rules
that you're not allowed to break,
435
00:26:14,440 --> 00:26:17,840
but within those rules,
there are all kinds of opportunities
436
00:26:17,840 --> 00:26:20,760
to play around and be creative.
437
00:26:20,760 --> 00:26:23,520
The only problem is that, in maths,
438
00:26:23,520 --> 00:26:27,160
no-one tells you
what those rules are.
439
00:26:27,160 --> 00:26:29,960
We have to work them
out for ourselves.
440
00:26:33,280 --> 00:26:36,680
Most mathematicians
like a challenge,
441
00:26:36,680 --> 00:26:41,600
but this idea got blown apart
at a maths conference in 1930
442
00:26:42,000 --> 00:26:47,080
in the Prussian city of Koenigsberg,
when two great mathematicians
443
00:26:47,640 --> 00:26:50,120
and their conclusions collided.
444
00:26:51,160 --> 00:26:54,520
On the one side,
you have got David Hilbert,
445
00:26:54,520 --> 00:26:57,960
a mathematical king in every
possible sense of the word.
446
00:26:57,960 --> 00:27:02,400
This is an enormously
well-respected man who laid down
the gauntlet,
447
00:27:02,400 --> 00:27:06,880
asking people to come up
with a fundamental set of rules
448
00:27:06,880 --> 00:27:10,800
on which every mathematical
proof could be based.
449
00:27:10,800 --> 00:27:15,760
On the other side was a young
academic called Kurt Godel.
450
00:27:15,840 --> 00:27:18,720
In contrast to Hilbert,
who thought that mathematics
451
00:27:18,720 --> 00:27:21,520
should be built from
the ground up by humans,
452
00:27:21,520 --> 00:27:25,560
Godel thought that
mathematics was discovered.
453
00:27:25,560 --> 00:27:29,360
He believed that mathematical
truths exist outside of us,
454
00:27:29,360 --> 00:27:34,000
and that we have very little say
in what we can find.
455
00:27:34,000 --> 00:27:37,600
That summit in Koenigsberg
can be seen as a clash
456
00:27:37,600 --> 00:27:41,920
between those who thought that
mathematics is part of our fabric
457
00:27:41,920 --> 00:27:46,920
of reality to be discovered,
and those who saw it as a language
458
00:27:47,400 --> 00:27:49,920
under our control.
459
00:27:49,920 --> 00:27:54,600
Hilbert was confident that humanity
would soon know all there is to know
460
00:27:54,600 --> 00:27:55,880
in maths.
461
00:27:55,880 --> 00:27:58,680
But Godel, who had also been trying
to find
462
00:27:58,680 --> 00:28:02,680
the rules of maths,
had come to the opposite conclusion.
463
00:28:03,760 --> 00:28:08,800
In a side room at the summit,
Godel quietly announces that,
464
00:28:09,040 --> 00:28:14,000
in fact, however hard you try, there
are always going to be some
465
00:28:14,000 --> 00:28:17,000
things that are unknowable.
466
00:28:17,000 --> 00:28:21,040
There are always going to be parts
of the mathematical game
467
00:28:21,040 --> 00:28:23,920
that can't be fully explained.
468
00:28:23,920 --> 00:28:27,600
And if you can't know all the rules,
how can you play the game?
469
00:28:27,600 --> 00:28:31,960
According to Godel, any rule-based
maths system is always
470
00:28:31,960 --> 00:28:35,720
going to have some things
that are either unknowable
471
00:28:35,720 --> 00:28:40,640
or unprovable, and what's more,
he could prove it -
472
00:28:40,640 --> 00:28:43,320
which is kind of ironic, if you
think about it.
473
00:28:44,400 --> 00:28:46,840
This was quickly accepted,
and became known as
474
00:28:46,840 --> 00:28:49,480
Godel's Incompleteness Theorem.
475
00:28:49,480 --> 00:28:53,080
And it puts an interesting twist
on our key question.
476
00:28:53,080 --> 00:28:58,080
It shows that, even if mathematical
rules truly are part of the universe
477
00:28:58,520 --> 00:29:02,360
and we're simply discovering them,
we are nevertheless going
478
00:29:02,360 --> 00:29:06,280
to have to accept some
of those rules without knowing
479
00:29:06,280 --> 00:29:09,080
how or why they are true.
480
00:29:11,120 --> 00:29:14,120
Normally, people think
that there's some intrinsic
481
00:29:14,120 --> 00:29:19,120
difference between science and math
on one hand, and faith-based belief
482
00:29:19,520 --> 00:29:22,080
systems on the other,
and what Godel's theorem
483
00:29:22,080 --> 00:29:25,160
tells us is that's not true...
484
00:29:25,160 --> 00:29:27,840
That there are things in mathematics
that you have to take on faith
485
00:29:27,840 --> 00:29:30,320
or you can't do the mathematics.
486
00:29:30,320 --> 00:29:33,040
To me, this was an astounding
thing to realise.
487
00:29:33,040 --> 00:29:36,840
We're going to have to accept
that we can't give maths
488
00:29:36,840 --> 00:29:39,760
a foundation in formal laws
or in logic
489
00:29:39,760 --> 00:29:41,440
in the way that we thought we could.
490
00:29:41,440 --> 00:29:44,840
I think it's enormously exciting
491
00:29:44,840 --> 00:29:48,480
that math, in some sense,
is open-ended.
492
00:29:48,480 --> 00:29:52,680
So, in a sense, it puts an end
to one way of thinking
493
00:29:52,680 --> 00:29:57,400
about mathematics, but I think, it
actually adds colour and richness
494
00:29:57,400 --> 00:30:00,960
to the subject because it's just
going to keep on going.
495
00:30:07,920 --> 00:30:11,840
So, what does Godel's Incompleteness
Theorem mean for our view
496
00:30:11,840 --> 00:30:15,880
of the universe and the parts
that maths plays in it?
497
00:30:15,880 --> 00:30:20,600
Well, it depends on what
you're trying to use maths for.
498
00:30:20,600 --> 00:30:24,520
If your goal is to use
it to describe what's around you,
499
00:30:24,520 --> 00:30:26,760
then it still offers
a very detailed picture -
500
00:30:26,760 --> 00:30:28,800
enough to navigate your way
through the universe
501
00:30:28,800 --> 00:30:31,120
and to explain its features.
502
00:30:31,120 --> 00:30:34,200
Sure, the map is not going
to be the same as the terrain,
503
00:30:34,200 --> 00:30:37,640
but even if maths is a bit
incomplete around the edges,
504
00:30:37,640 --> 00:30:40,360
you could argue that it
doesn't really matter.
505
00:30:42,960 --> 00:30:47,800
Although Godel proves it's not
possible to formalise all of maths,
506
00:30:47,800 --> 00:30:52,800
it is possible to formalise all the
mathematics we actually need to use.
507
00:30:53,560 --> 00:30:55,560
Take flying as an example.
508
00:30:55,560 --> 00:30:59,120
Now, I did my PhD in the mathematics
of aerodynamics,
509
00:30:59,120 --> 00:31:02,560
and that means I spent four years
poring over equations
510
00:31:02,560 --> 00:31:05,400
for wing sections and wind speeds.
511
00:31:05,400 --> 00:31:08,240
It's stuff that I know
like the back of my hand.
512
00:31:08,240 --> 00:31:12,360
But does that qualify me for going
up in one of these on my own?
513
00:31:12,360 --> 00:31:14,560
Absolutely not!
514
00:31:14,560 --> 00:31:17,840
And on the other hand,
these guys don't really need to know
515
00:31:17,840 --> 00:31:22,120
any of this stuff to make them
graceful acrobats in the air.
516
00:31:23,520 --> 00:31:27,840
Not having a complete understanding
doesn't always matter.
517
00:31:27,840 --> 00:31:31,400
We've still flown successfully
for over 100 years.
518
00:31:31,400 --> 00:31:33,280
And now, it's my turn.
519
00:31:34,920 --> 00:31:37,160
And then this is your...
Diagonal line.
520
00:31:37,160 --> 00:31:38,560
The strap that comes across.
521
00:31:38,560 --> 00:31:41,000
This will dig in a little bit
on take-off when you're leaning
522
00:31:41,000 --> 00:31:42,640
forwards and running down the hill.
523
00:31:42,640 --> 00:31:45,040
I can handle it. It should be
a little bit uncomfortable.
524
00:31:45,040 --> 00:31:46,960
I can handle it. Don't worry
too much about it.
525
00:31:46,960 --> 00:31:49,360
And do you have quite a good feel
for where the thermals are?
526
00:31:49,360 --> 00:31:51,360
You have to have the
right weather conditions.
527
00:31:51,360 --> 00:31:54,680
So, if you imagine a hill that faces
totally into the wind,
528
00:31:54,680 --> 00:31:57,640
that's well drained, maybe darker,
and it will create this
529
00:31:57,640 --> 00:32:01,760
kind of pool of warm air, and then
it will... Once it kind of reaches
530
00:32:01,760 --> 00:32:04,720
a decent temperature difference, it
bubbles up through the atmosphere.
531
00:32:04,720 --> 00:32:07,960
Yeah, it's almost like we've got
kind of opposing skills. Yeah.
532
00:32:07,960 --> 00:32:10,480
And, like...they're sort of about
the same thing, but they...
533
00:32:10,480 --> 00:32:12,320
You don't need my skills
to do what you do,
534
00:32:12,320 --> 00:32:14,240
and I couldn't do what you do.
Mm-hm.
535
00:32:14,240 --> 00:32:17,000
I guess the ground-speed element
has a bit of maths in there. Yeah.
536
00:32:17,000 --> 00:32:19,200
I always thought the lesson bit of
maths to begin with.
537
00:32:19,200 --> 00:32:21,120
Where's the wind coming from?
How strong it is.
538
00:32:21,120 --> 00:32:23,400
How fast am I going to go,
if I'm pointing into the wind?
539
00:32:23,400 --> 00:32:25,880
But you're not solving
Navier-Stokes equations, are you?
540
00:32:25,880 --> 00:32:27,960
I don't even know what that means!
Yeah, exactly.
541
00:32:27,960 --> 00:32:29,240
THEY LAUGH
542
00:32:30,480 --> 00:32:34,160
Before the theoretical analysis
of aviation came along,
543
00:32:34,160 --> 00:32:38,520
the practical side of flying was
mere trial and error.
544
00:32:38,520 --> 00:32:41,120
Now, we have a much
more reliable understanding
545
00:32:41,120 --> 00:32:44,680
of what keeps us aloft, and it
doesn't really matter
546
00:32:44,680 --> 00:32:47,640
if the maths behind it is,
ultimately,
547
00:32:47,640 --> 00:32:49,840
a bit fuzzy around the edges.
548
00:32:51,920 --> 00:32:55,320
In the real world, the best that we
can do is just accept
549
00:32:55,320 --> 00:32:58,440
Godel's Incompleteness Theorem
and get on with life.
550
00:33:02,720 --> 00:33:03,960
Hey!
551
00:33:03,960 --> 00:33:05,160
It's amazing.
552
00:33:12,280 --> 00:33:14,360
There's a thermal! Yeah!
553
00:33:14,360 --> 00:33:16,080
Woo! Woohoo!
554
00:33:16,080 --> 00:33:18,760
You're feeling much more alert.
A bit stronger? Yeah.
555
00:33:21,160 --> 00:33:23,200
We have to put aside,
for the moment,
556
00:33:23,200 --> 00:33:26,640
the question of whether maths
is invented or discovered,
557
00:33:26,640 --> 00:33:28,200
because it now looks like
558
00:33:28,200 --> 00:33:32,920
we may have to determine which part
of maths we're asking about.
559
00:33:34,320 --> 00:33:37,800
You see, for me, Godel's work
highlights the distinction
560
00:33:37,800 --> 00:33:42,760
between pure theoretical maths and
practical applied maths.
561
00:33:43,600 --> 00:33:45,760
So, here is how I see things...
562
00:33:45,760 --> 00:33:48,600
With mathematics, there's a split
down the middle of the subject,
563
00:33:48,600 --> 00:33:53,080
because the story changes depending
on what world you start with,
564
00:33:53,080 --> 00:33:57,840
whether it's the real one or one
that exists in our imaginations.
565
00:33:59,200 --> 00:34:02,080
And right now, when we're flying,
this is very much in the realm
566
00:34:02,080 --> 00:34:06,280
of applied mathematics, where
everything is tangible and practical
567
00:34:06,280 --> 00:34:11,240
and a little bit imprecise.
But, alongside that,
568
00:34:12,560 --> 00:34:16,800
is where the more theoretical pure
mathematics lives.
569
00:34:16,800 --> 00:34:19,520
That's where you have your proofs,
your paradoxes,
570
00:34:19,520 --> 00:34:21,720
and incompleteness theorems.
571
00:34:21,720 --> 00:34:24,640
A realm which doesn't match
up with a physical reality.
572
00:34:24,640 --> 00:34:27,360
A sort of imperfect perfection.
573
00:34:29,840 --> 00:34:34,280
Even though I instinctively feel
that maths is discovered,
574
00:34:34,280 --> 00:34:37,600
I like that there is this pure
theoretical part of maths
575
00:34:37,600 --> 00:34:39,640
that isn't found in reality.
576
00:34:42,240 --> 00:34:45,480
And, since the maths there
doesn't need to match reality,
577
00:34:45,480 --> 00:34:47,520
it's a convenient place where we can
578
00:34:47,520 --> 00:34:51,560
leave all the weird contradictory
bits that we come across.
579
00:34:56,160 --> 00:34:57,400
Yeah! Woo!
580
00:35:00,640 --> 00:35:04,240
However, I might have it
the wrong way round.
581
00:35:05,320 --> 00:35:09,840
Although pure theoretical maths
seems rather divorced from reality,
582
00:35:09,840 --> 00:35:13,240
that might merely reflect
the fact that reality is not
583
00:35:13,240 --> 00:35:15,120
quite what we think it is.
584
00:35:16,560 --> 00:35:21,040
And it's a reality that we can
uncover through the strange maths
585
00:35:21,040 --> 00:35:23,000
of quantum physics.
586
00:35:25,080 --> 00:35:28,520
The weirdest worlds that most of us
have come across are likely
587
00:35:28,520 --> 00:35:33,520
to be in fiction, such as this,
Alice's Adventures In Wonderland.
588
00:35:33,520 --> 00:35:36,560
The author, Lewis Carroll -
real name Charles Dodgson -
589
00:35:36,560 --> 00:35:39,440
was actually
a mathematics don at Oxford,
590
00:35:39,440 --> 00:35:41,880
and a staunch traditionalist.
591
00:35:43,320 --> 00:35:47,040
It's generally believed that much
of this surreal story is a thinly
592
00:35:47,040 --> 00:35:51,160
veiled satire on the new avant-garde
maths that was flourishing
593
00:35:51,160 --> 00:35:54,200
when he was writing in the 1860s.
594
00:35:54,200 --> 00:35:58,320
Still feels relevant today,
and applies equally well
595
00:35:58,320 --> 00:36:02,240
to the new weird kid on the block,
quantum physics.
596
00:36:04,280 --> 00:36:07,000
Take a close look at
the physical world around us,
597
00:36:07,000 --> 00:36:10,360
and you can reduce it all to maths.
598
00:36:10,360 --> 00:36:14,040
The solid bricks of our houses
or the blood cells in our veins
599
00:36:14,040 --> 00:36:16,880
can all be reduced
down into chemicals,
600
00:36:16,880 --> 00:36:19,280
which comprise elements,
601
00:36:19,280 --> 00:36:24,320
which themselves are made
up of atoms, comprising a tiny
602
00:36:24,360 --> 00:36:27,800
nucleus of protons and neutrons
and electrons buzzing
603
00:36:27,800 --> 00:36:30,800
around in a cloud
of mostly emptiness.
604
00:36:34,560 --> 00:36:38,640
The protons and neutrons in turn
are built from smaller subatomic
605
00:36:38,640 --> 00:36:42,120
particles that we can't
directly observe.
606
00:36:42,120 --> 00:36:47,160
We can only verify their existence
using experiments and mathematics.
607
00:36:49,280 --> 00:36:51,400
As we delve deeper into this world,
608
00:36:51,400 --> 00:36:56,440
scientists have discovered something
very strange indeed.
609
00:36:56,440 --> 00:36:59,920
We can never actually know
the precise location
610
00:36:59,920 --> 00:37:04,960
of most particles in this
subatomic or quantum realm.
611
00:37:05,400 --> 00:37:09,080
All we can know is the likelihood
of them being somewhere,
612
00:37:09,080 --> 00:37:12,160
a mathematical formula that
describes the probability
613
00:37:12,160 --> 00:37:13,920
of their position.
614
00:37:15,200 --> 00:37:19,760
All of this means we are,
fundamentally, at a quantum level,
615
00:37:19,760 --> 00:37:24,040
just a great fuzz of
energy and probabilities.
616
00:37:24,040 --> 00:37:26,520
I'm not sure Lewis Carroll
would have liked that.
617
00:37:30,240 --> 00:37:34,520
And the only way to explore
this ill-defined quantum world...
618
00:37:35,600 --> 00:37:37,600
Oh! Hey. Hello.
619
00:37:37,600 --> 00:37:39,800
..is through mathematics -
620
00:37:39,800 --> 00:37:43,640
perfectly equipped to handle
strange probabilities.
621
00:37:46,240 --> 00:37:49,720
It seems like there's quite a lot
of uncertainty in quantum physics.
622
00:37:49,720 --> 00:37:51,960
Does that bother you?
623
00:37:51,960 --> 00:37:53,160
Um, no.
624
00:37:53,160 --> 00:37:56,520
When I heard that things were, you
know, uncertain,
625
00:37:56,520 --> 00:37:59,920
and also against our common sense
in quantum physics,
626
00:37:59,920 --> 00:38:02,320
then I thought, like, "Oh, wow!
That sounds interesting.
627
00:38:02,320 --> 00:38:04,520
"I want to know more about that."
628
00:38:04,520 --> 00:38:08,960
The pivotal maths behind the quantum
world was first laid out by Austrian
629
00:38:08,960 --> 00:38:13,080
physicist Erwin Schrodinger in 1926.
630
00:38:13,080 --> 00:38:16,960
His equations accurately describe
the unusual behaviour
631
00:38:16,960 --> 00:38:19,040
of subatomic particles.
632
00:38:20,480 --> 00:38:22,400
OK, all right,
I'll tell you what, then...
633
00:38:22,400 --> 00:38:26,120
Quantum physics lesson 101,
where do we start?
634
00:38:26,120 --> 00:38:27,320
Give me the lesson.
635
00:38:27,320 --> 00:38:30,480
Um, OK, I would say we would
have to start with superposition.
636
00:38:30,480 --> 00:38:32,960
So, let's talk about electrons.
637
00:38:32,960 --> 00:38:37,520
So, they're a very small particle,
and they can be in two states.
638
00:38:37,520 --> 00:38:39,480
They have a state called the spin,
639
00:38:39,480 --> 00:38:42,520
and the spin can be pointing
up or down.
640
00:38:42,520 --> 00:38:46,640
So, if we were in the classical
world, the spin could only be either
641
00:38:46,640 --> 00:38:48,480
up or down.
642
00:38:48,480 --> 00:38:51,920
But in the quantum world,
the spin is in a superposition,
643
00:38:51,920 --> 00:38:56,240
which means it can be up and down
at the same time.
644
00:38:56,240 --> 00:38:58,080
Let me see if I understand this,
then.
645
00:38:58,080 --> 00:39:03,000
So, superposition is where something
is and isn't something
646
00:39:03,120 --> 00:39:05,440
at the same time? Yes.
647
00:39:05,440 --> 00:39:08,600
We can think about some examples.
648
00:39:08,600 --> 00:39:13,600
So, let's say that we have a cup,
and the cup is full of water
649
00:39:13,680 --> 00:39:15,640
or... That's one state.
650
00:39:15,640 --> 00:39:19,000
Another possible state
is that the cup is empty.
651
00:39:19,000 --> 00:39:23,160
So, if we were to bring the quantum
ideas to the classical world,
652
00:39:23,160 --> 00:39:26,920
we would say the state,
one possible state of the cup,
653
00:39:26,920 --> 00:39:29,840
would be to be empty
and full at the same time.
654
00:39:29,840 --> 00:39:34,080
OK, which you never see
in the world that we're living
655
00:39:34,080 --> 00:39:36,320
in, you never see a cup
that's full and empty.
656
00:39:36,320 --> 00:39:38,720
Yes, we don't.
657
00:39:38,720 --> 00:39:40,600
But you see this a lot
in the quantum world?
658
00:39:40,600 --> 00:39:45,680
Yes, superpositions are an essential
part of the quantum world.
659
00:39:45,840 --> 00:39:48,280
Like a light being on
and off at the same time.
660
00:39:48,280 --> 00:39:52,600
Exactly. Or the cake being eaten or
not eaten at the same time.
661
00:39:52,600 --> 00:39:56,080
OK. It's a very tough idea to get
around. Yes, yes.
662
00:39:56,080 --> 00:39:59,480
Given two possible outcomes,
in the quantum world,
663
00:39:59,480 --> 00:40:04,520
we now have to allow for a third one
- the combination of both outcomes.
664
00:40:05,360 --> 00:40:08,960
At the quantum scale, you can have
your cake and eat it.
665
00:40:10,040 --> 00:40:12,280
This is such a weird idea.
666
00:40:12,280 --> 00:40:15,120
How do we know it's real?
667
00:40:15,120 --> 00:40:18,240
Well, because we've done many
experiments to prove it
668
00:40:18,240 --> 00:40:20,720
that show exactly that behaviour.
669
00:40:20,720 --> 00:40:22,680
What does that experiment look like?
670
00:40:22,680 --> 00:40:27,600
Well, if we put it, say, in terms
of things we have here on the table.
671
00:40:27,600 --> 00:40:31,360
We could think about... Let's say
that I wanted this piece of sugar
672
00:40:31,360 --> 00:40:36,320
to come into my cup,
but there is this pot in the middle.
673
00:40:37,080 --> 00:40:41,160
So then, if the sugar is going to
come from here to my cup,
674
00:40:41,160 --> 00:40:46,120
it could either go this way or that
way in the classical world.
675
00:40:47,480 --> 00:40:51,880
But in a quantum experiment, it can
take both routes at the same time,
676
00:40:51,880 --> 00:40:55,000
and I would be able to distinguish
that it did that
677
00:40:55,000 --> 00:40:56,560
if I did a quantum experiment.
678
00:40:56,560 --> 00:40:59,200
I can't.... This is too weird!
679
00:40:59,200 --> 00:41:01,800
Welcome to my world.
680
00:41:01,800 --> 00:41:04,480
So, you go through this whole
transition, from first,
681
00:41:04,480 --> 00:41:08,560
the ideas and the mathematics,
and then up to showing
682
00:41:08,560 --> 00:41:10,000
it in the experiment.
683
00:41:11,480 --> 00:41:14,760
What came out of Schrodinger's mouth
was a prediction of something
684
00:41:14,760 --> 00:41:19,520
even stranger that can sometimes be
produced when particles interact
685
00:41:19,520 --> 00:41:24,520
in the quantum world -
a phenomenon called entanglement.
686
00:41:24,520 --> 00:41:26,880
All right, tell me
about entanglement, then.
687
00:41:26,880 --> 00:41:29,040
OK, so take two electrons.
688
00:41:29,040 --> 00:41:32,880
If the electrons are entangled,
and I do something to one
689
00:41:32,880 --> 00:41:37,920
of the electrons, for example change
the direction of the spin,
690
00:41:38,240 --> 00:41:42,520
that will instantaneously affect
the state of the other electron,
691
00:41:42,520 --> 00:41:45,520
even if they are separated long
distances.
692
00:41:45,520 --> 00:41:48,360
How far away are
they from each other?
693
00:41:48,360 --> 00:41:50,600
Well, they can be
a few centimetres...
694
00:41:50,600 --> 00:41:53,640
but now the latest experiments,
695
00:41:53,640 --> 00:41:58,680
they're using satellites, show
entanglement across 1,200 km.
696
00:42:00,200 --> 00:42:01,640
What? Yes.
697
00:42:01,640 --> 00:42:04,040
You've got something over here,
698
00:42:04,040 --> 00:42:08,960
and you do...? And something
1,200 km away.
699
00:42:08,960 --> 00:42:11,720
You do something to one,
and it instantly...
700
00:42:11,720 --> 00:42:14,160
The other one instantly
knows what's happened?
701
00:42:14,160 --> 00:42:18,320
Yes, you'll affect the state
of the other one instantly.
702
00:42:18,320 --> 00:42:21,720
Apparently, there is
no cause or link.
703
00:42:21,720 --> 00:42:26,360
The only thing we can say is that
the two particles are synchronised.
704
00:42:26,360 --> 00:42:29,640
How does one know
what the other one's doing?
705
00:42:29,640 --> 00:42:33,280
Well, that we're still trying to
understand, because that's
706
00:42:33,280 --> 00:42:36,840
what mathematics tells us,
and then we can show
707
00:42:36,840 --> 00:42:40,240
it in the experiment,
but we're still are struggling
708
00:42:40,240 --> 00:42:42,680
to understand what that means.
709
00:42:42,680 --> 00:42:45,400
And one of the reasons
why we don't understand it...
710
00:42:45,400 --> 00:42:48,160
And, you know, like you're asking,
is because we don't see
711
00:42:48,160 --> 00:42:49,920
it in our everyday lives.
712
00:42:49,920 --> 00:42:54,400
So, let's say it's not part of our
experience and common sense.
713
00:42:54,400 --> 00:42:57,120
But that doesn't mean it doesn't
happen.
714
00:42:58,280 --> 00:43:00,080
I don't want you to go crazy...
715
00:43:00,080 --> 00:43:02,720
So, quantum mathematics
has made predictions,
716
00:43:02,720 --> 00:43:05,440
which have been
discovered to be true.
717
00:43:07,280 --> 00:43:12,080
But, despite that,
the quantum world is so weird,
718
00:43:12,080 --> 00:43:16,760
it suggests to me that the maths
behind it is just invented.
719
00:43:18,280 --> 00:43:21,800
It feels like what we're seeing
is evidence of a man-made system
720
00:43:21,800 --> 00:43:23,560
being pushed too far.
721
00:43:24,880 --> 00:43:27,880
These are the absurdities that
appear when it's applied
722
00:43:27,880 --> 00:43:31,240
to situations it
wasn't designed for.
723
00:43:37,440 --> 00:43:42,440
But my quest to find the truth
about maths takes me back to nature.
724
00:43:43,760 --> 00:43:46,840
There is amazing new evidence
that quantum processes
725
00:43:46,840 --> 00:43:50,240
might actually be crucial to our own
existence,
726
00:43:50,240 --> 00:43:52,520
and much of life on Earth.
727
00:43:54,760 --> 00:43:58,760
That would strengthen the argument
that mathematical processes
728
00:43:58,760 --> 00:44:01,400
are intrinsic to our world.
729
00:44:01,400 --> 00:44:03,240
That maths is discovered.
730
00:44:05,480 --> 00:44:08,320
It all comes down
to the photosynthesis,
731
00:44:08,320 --> 00:44:13,400
the process that converts sunlight
into chemical energy used in life.
732
00:44:13,400 --> 00:44:16,040
It takes place in the molecules
called chlorophyll,
733
00:44:16,040 --> 00:44:19,920
which can be found in plants,
algae and bacteria.
734
00:44:21,520 --> 00:44:23,760
In bacteria, we have something
that's similar
735
00:44:23,760 --> 00:44:25,160
to what we have in plants.
736
00:44:25,160 --> 00:44:28,240
So, this is the stuff
that captures the sunlight?
737
00:44:28,240 --> 00:44:32,000
Exactly. Each of these molecules,
each of these little blue things
738
00:44:32,000 --> 00:44:34,640
here that I'm showing, is bacterial
chlorophyll, and if we take
739
00:44:34,640 --> 00:44:37,240
it apart, it will capture light.
740
00:44:37,240 --> 00:44:40,800
The chlorophyll captures light
by absorbing particles
741
00:44:40,800 --> 00:44:43,640
of light, or photons.
742
00:44:43,640 --> 00:44:48,320
So, a photon is absorbed,
and it's absorbed by all of them,
743
00:44:48,320 --> 00:44:51,600
so energy shared by all of these
bacterial chlorophylls,
744
00:44:51,600 --> 00:44:54,960
and that sharing, we call it
using a quantum superposition.
745
00:44:54,960 --> 00:44:57,280
Because it's coming in and hitting
one of these,
746
00:44:57,280 --> 00:44:58,960
but all of them are somehow...
747
00:44:58,960 --> 00:45:02,520
In a way, it's as if each of the
electrons of the chlorophylls
748
00:45:02,520 --> 00:45:07,520
are talking to each other
and sharing the energy around.
749
00:45:07,760 --> 00:45:11,640
The subatomic particles
in the chlorophyll are synchronised
750
00:45:11,640 --> 00:45:15,680
in a way that can only be described
by quantum mechanics.
751
00:45:15,680 --> 00:45:16,960
Does it do a good job?
752
00:45:16,960 --> 00:45:18,320
I mean, is it efficient?
753
00:45:18,320 --> 00:45:21,040
That is part of why photosynthesis
is efficient.
754
00:45:21,040 --> 00:45:24,840
Because, by sharing the energy
among all of them,
755
00:45:24,840 --> 00:45:27,800
it's easier to transfer the energy
to another molecule.
756
00:45:27,800 --> 00:45:30,360
Imagine if you have to share
the energy one by one,
757
00:45:30,360 --> 00:45:32,640
you have to explore
each part separately,
758
00:45:32,640 --> 00:45:35,200
but if you share the energy
all together, you explore
759
00:45:35,200 --> 00:45:37,080
all the parts at the same time.
760
00:45:38,800 --> 00:45:42,600
Every leaf on every
plant on the planet
761
00:45:42,600 --> 00:45:46,320
has been following these quantum
rules for millions of years.
762
00:45:46,320 --> 00:45:49,920
And we still don't fully
understand how they do it.
763
00:45:51,520 --> 00:45:55,640
Without quantum physics,
despite all the mathematical
764
00:45:55,640 --> 00:45:59,640
uncertainties and ambiguities,
plants wouldn't produce
765
00:45:59,640 --> 00:46:02,280
oxygen so efficiently.
766
00:46:02,280 --> 00:46:06,080
And without oxygen,
we wouldn't exist.
767
00:46:06,080 --> 00:46:09,320
The systems are amazing,
because they are effectively
768
00:46:09,320 --> 00:46:12,360
the interface between using a little
bit of classical mechanics
769
00:46:12,360 --> 00:46:14,680
and a little bit of quantum
mechanics to operate
770
00:46:14,680 --> 00:46:15,880
in a wonderful way.
771
00:46:15,880 --> 00:46:20,880
Ultimately, quantum mechanics
is at the heart of photosynthesis
772
00:46:21,320 --> 00:46:23,800
and, well, I guess,
all of life on Earth.
773
00:46:23,800 --> 00:46:25,840
It is. It is.
774
00:46:25,840 --> 00:46:30,520
We can say life is nothing but
quantum mechanics giving us energy.
775
00:46:33,760 --> 00:46:36,960
So, what does all this mean
for our key question
776
00:46:36,960 --> 00:46:39,240
about the origins of maths?
777
00:46:39,240 --> 00:46:42,760
There is no shortage of evidence
that mathematical rules
778
00:46:42,760 --> 00:46:45,160
are intrinsic to the world.
779
00:46:45,160 --> 00:46:48,040
We keep discovering them everywhere.
780
00:46:48,040 --> 00:46:52,760
However, we now know we have to take
some of the maths on faith,
781
00:46:52,760 --> 00:46:56,800
and believing in the numbers is
taking us to a very strange world,
782
00:46:56,800 --> 00:47:00,760
with crazy notions like
superposition and entanglement
783
00:47:00,760 --> 00:47:02,560
at the core of it.
784
00:47:03,640 --> 00:47:08,480
Quantum mathematics is inextricably
linked to the world as we know it.
785
00:47:08,480 --> 00:47:10,720
Or, as we knew it.
786
00:47:10,720 --> 00:47:14,000
Because the world is actually
a whole lot weirder than we thought.
787
00:47:14,000 --> 00:47:19,000
What quantum mechanics does do is
force us to question what is real.
788
00:47:21,280 --> 00:47:23,520
And what is reality, anyway?
789
00:47:29,200 --> 00:47:34,280
Just how much light
can mathematics shed on reality?
790
00:47:34,720 --> 00:47:38,440
With the world stripped bare,
exposing the nuts and bolts
791
00:47:38,440 --> 00:47:41,160
of existence,
what does maths tell us
792
00:47:41,160 --> 00:47:44,720
about this realm of subatomic
particles?
793
00:47:46,240 --> 00:47:49,680
The maths that underlies it
isn't particularly pretty,
794
00:47:49,680 --> 00:47:53,800
but it can all be written out in
just one equation.
795
00:47:55,640 --> 00:48:00,640
This is the formula that describes
the constituents of the universe.
796
00:48:01,040 --> 00:48:03,600
It has become well enough accepted
797
00:48:03,600 --> 00:48:08,200
to be called the standard model
of subatomic physics.
798
00:48:08,200 --> 00:48:10,200
I told you it wasn't pretty.
799
00:48:10,200 --> 00:48:13,840
Now, you're just going to have to
take my word for it on this one.
800
00:48:13,840 --> 00:48:18,640
This equation encapsulates
all of the fundamental properties
801
00:48:18,640 --> 00:48:20,440
of the subatomic world.
802
00:48:21,520 --> 00:48:25,200
But there are a couple of sticking
points.
803
00:48:25,200 --> 00:48:28,520
For one thing, no-one has ever
satisfactorily explained
804
00:48:28,520 --> 00:48:33,560
how our common-sense, day-to-day
version of the world emerges
805
00:48:34,720 --> 00:48:38,640
from this kind of subatomic reality.
806
00:48:38,640 --> 00:48:42,640
All of that fuzziness, all of that
uncertainty in the quantum world,
807
00:48:42,640 --> 00:48:46,920
just how does it end up giving us
that comfortable,
808
00:48:46,920 --> 00:48:50,440
familiar solidity of the normal
world?
809
00:48:54,000 --> 00:48:58,760
At the other end of the spectrum,
the solar system and beyond
810
00:48:58,760 --> 00:49:03,760
is beautifully and accurately
described by a different equation.
811
00:49:05,200 --> 00:49:08,960
Einstein's general relativity.
812
00:49:08,960 --> 00:49:13,360
This remarkable equation
tells you about gravity,
813
00:49:13,360 --> 00:49:17,880
about the warping of space-time,
about general relativity.
814
00:49:17,880 --> 00:49:19,760
And when you take these two
together,
815
00:49:19,760 --> 00:49:22,480
these two single mathematical
sentences,
816
00:49:22,480 --> 00:49:25,880
they are enough to tell you
everything you need
817
00:49:25,880 --> 00:49:30,880
about the fundamental behaviour of
the universe and everything in it.
818
00:49:32,320 --> 00:49:35,880
There is nothing more
articulate than mathematics.
819
00:49:37,480 --> 00:49:42,000
Maths seems to be written
into the physical universe.
820
00:49:43,240 --> 00:49:47,360
So, on the one hand,
at the teeny-tiny scale,
821
00:49:47,360 --> 00:49:51,560
the standard model of particle
physics does this amazing job.
822
00:49:51,560 --> 00:49:55,040
And in the ginormous scale,
general relativity,
823
00:49:55,040 --> 00:49:58,720
I mean, you couldn't ask for
anything more.
824
00:49:58,720 --> 00:50:03,680
There's just one problem when you
try and put these two together.
825
00:50:08,880 --> 00:50:11,760
The problem is that general
relativity breaks down
826
00:50:11,760 --> 00:50:13,520
in the quantum world.
827
00:50:13,520 --> 00:50:18,480
Gravity simply doesn't apply
to particles at the subatomic scale.
828
00:50:18,480 --> 00:50:21,840
Meanwhile, quantum effects
are virtually never seen
829
00:50:21,840 --> 00:50:25,960
at the scale of humans and planets,
where gravity rules.
830
00:50:25,960 --> 00:50:30,920
You and I are never
in a superposition of existing
831
00:50:30,920 --> 00:50:34,080
and not existing at the same time.
832
00:50:34,080 --> 00:50:36,120
So, what does this mean for us?
833
00:50:36,120 --> 00:50:40,800
Are there two different worlds,
each obeying their own sets
834
00:50:40,800 --> 00:50:43,600
of mathematical laws?
835
00:50:43,600 --> 00:50:46,520
Solving this conundrum is one
of the biggest problems
836
00:50:46,520 --> 00:50:49,280
that puzzles scientists today.
837
00:50:49,280 --> 00:50:52,080
Will we ever reconcile the two?
838
00:50:52,080 --> 00:50:57,040
I think it's perfectly plausible
that, within our lifetime, somebody,
839
00:51:00,920 --> 00:51:05,520
the mathematical structure
which unifies Einstein's theory
840
00:51:05,520 --> 00:51:09,080
of relativity with quantum mechanics
and just provides the perfect
841
00:51:09,080 --> 00:51:11,120
description of this world.
842
00:51:11,120 --> 00:51:14,480
And that would be really exciting.
843
00:51:14,480 --> 00:51:16,400
Will we have one?
844
00:51:16,400 --> 00:51:18,240
How do I know?
845
00:51:18,240 --> 00:51:20,600
We would all like to have one.
846
00:51:20,600 --> 00:51:24,440
But, you know, maybe we are not
smart enough
847
00:51:24,440 --> 00:51:27,920
to formulate a theory that combines
everything.
848
00:51:27,920 --> 00:51:30,840
It's hard.
849
00:51:30,840 --> 00:51:35,760
I do believe that there are good
ideas out there and that eventually,
850
00:51:35,760 --> 00:51:37,560
it might take a long time,
but, eventually,
851
00:51:37,560 --> 00:51:39,880
humans will work this out.
852
00:51:39,880 --> 00:51:41,600
I'm confident about that.
853
00:51:41,600 --> 00:51:44,560
So, will we make it all the way
to include all possible forces
854
00:51:44,560 --> 00:51:49,600
at all possible scales with all
possible forms of matter?
855
00:51:49,600 --> 00:51:53,640
It's a hope I have for our species,
that's all I can say.
856
00:51:55,880 --> 00:51:59,440
The incompatibility
of these two great theories,
857
00:51:59,440 --> 00:52:02,440
general relativity
and quantum mechanics,
858
00:52:02,440 --> 00:52:05,680
creates a serious obstacle
for believing that maths
859
00:52:05,680 --> 00:52:08,120
is really discovered.
860
00:52:08,120 --> 00:52:11,320
And there's a bigger hurdle to come.
861
00:52:11,320 --> 00:52:16,000
Many of the best proposals to unify
general relativity and the quantum
862
00:52:16,000 --> 00:52:19,800
world have consequences
that are even weirder
863
00:52:19,800 --> 00:52:22,920
than the problems
they're trying to solve.
864
00:52:22,920 --> 00:52:27,600
They predict the existence
of multiple universes.
865
00:52:27,600 --> 00:52:31,360
This idea is rooted
in the mathematical explanations
866
00:52:31,360 --> 00:52:34,520
of the quantum world, and the work
of its founding father,
867
00:52:34,520 --> 00:52:36,160
Erwin Schrodinger.
868
00:52:36,160 --> 00:52:40,360
The mathematics in Schrodinger's
equation insists that particles
869
00:52:40,360 --> 00:52:44,880
can exist in multiple
states at the same time.
870
00:52:44,880 --> 00:52:48,720
And Schrodinger himself says
that these possibilities aren't
871
00:52:48,720 --> 00:52:53,400
just alternatives,
but really happen simultaneously.
872
00:52:58,480 --> 00:53:02,360
This can lead to multiple universes.
873
00:53:02,360 --> 00:53:06,600
And the maths also suggests
there's an infinite number of them,
874
00:53:06,600 --> 00:53:09,640
each slightly different
from the others.
875
00:53:09,640 --> 00:53:12,440
Thank you.
876
00:53:14,880 --> 00:53:18,560
ALL: Mathematically speaking,
in an infinite universe,
877
00:53:18,560 --> 00:53:23,400
everything that's
possible has to happen somewhere.
878
00:53:23,400 --> 00:53:25,280
Yeah, that's right.
879
00:53:25,280 --> 00:53:27,960
Everything possible happens.
880
00:53:27,960 --> 00:53:29,400
Somewhere.
881
00:53:29,400 --> 00:53:32,040
Even Schrodinger acknowledged
that the consequences
882
00:53:32,040 --> 00:53:37,080
of his equation describing the
quantum world might seem lunatic.
883
00:53:37,320 --> 00:53:39,720
But if there's one
thing I've learned,
884
00:53:39,720 --> 00:53:42,080
it's that you should
trust the maths.
885
00:53:42,080 --> 00:53:44,520
So, maybe our experience
isn't special,
886
00:53:44,520 --> 00:53:48,240
maybe our reality isn't unique
after all.
887
00:53:53,640 --> 00:53:58,440
There are so many distinct avenues
of investigation that lead
888
00:53:58,440 --> 00:54:01,320
to the possibility of a multiverse.
889
00:54:01,320 --> 00:54:03,840
From our studies of unification
and string theory,
890
00:54:03,840 --> 00:54:06,360
from our studies of quantum
mechanics, even from the study
891
00:54:06,360 --> 00:54:09,080
of space going on infinitely far.
892
00:54:09,080 --> 00:54:11,640
Even that gives rise
to a version of the multiverse.
893
00:54:11,640 --> 00:54:15,240
If we're going to reject everything
that just seems weird, we're almost
894
00:54:15,240 --> 00:54:18,800
guaranteed to reject the true
theories of the future
895
00:54:18,800 --> 00:54:20,560
when they get discovered.
896
00:54:20,560 --> 00:54:25,320
I think we should just chill out,
accept that the world is weird
897
00:54:25,320 --> 00:54:27,920
and that's just part of its charm.
898
00:54:27,920 --> 00:54:29,400
And trust the math.
899
00:54:37,080 --> 00:54:40,160
So, why does all of this matter?
900
00:54:40,160 --> 00:54:43,240
Well, if maths
really is discovered,
901
00:54:43,240 --> 00:54:47,560
then there is an intrinsic truth
behind the maths we uncover -
902
00:54:47,560 --> 00:54:51,160
however weird that truth
seems to be.
903
00:54:51,160 --> 00:54:53,640
If maths is invented,
then how do we know
904
00:54:53,640 --> 00:54:55,680
what is true or false?
905
00:54:55,680 --> 00:54:58,920
Is it true purely
because we define it so?
906
00:54:58,920 --> 00:55:00,960
And how does it relate
to the real world
907
00:55:00,960 --> 00:55:02,720
that we all experience?
908
00:55:02,720 --> 00:55:04,080
Clearance...
909
00:55:05,360 --> 00:55:08,480
In this series, we've seen that
maths can explain
910
00:55:08,480 --> 00:55:09,920
so much of our world,
911
00:55:09,920 --> 00:55:13,720
from aerodynamics to planetary
orbits, from the subatomic world
912
00:55:13,720 --> 00:55:17,680
to processes crucial to life
on Earth.
913
00:55:17,680 --> 00:55:22,680
And that is something I just can't
accept as a coincidence.
914
00:55:23,120 --> 00:55:25,320
So, here's my take on things.
915
00:55:25,320 --> 00:55:28,960
For me, it's almost
as though you have this alternate
916
00:55:28,960 --> 00:55:33,960
parallel mathematical world
that hides just beneath our own.
917
00:55:34,120 --> 00:55:36,320
You can't see it,
you can't touch it.
918
00:55:36,320 --> 00:55:40,960
The only way that you can explore it
is by using the language
919
00:55:40,960 --> 00:55:42,680
that we've invented.
920
00:55:42,680 --> 00:55:47,720
All of those symbols and equations
and conventions are our only tools
921
00:55:48,280 --> 00:55:52,800
of navigation, and they are
undoubtedly man-made.
922
00:55:52,800 --> 00:55:57,480
But once you're inside that world,
once you're exploring the landscape
923
00:55:57,480 --> 00:56:00,880
that mathematics has laid
out in front of you,
924
00:56:00,880 --> 00:56:05,840
I am absolutely convinced that you
are on a voyage of discovery.
925
00:56:06,560 --> 00:56:10,560
It is a world without a
human designer.
926
00:56:10,560 --> 00:56:13,600
So, ultimately, I think it's both.
927
00:56:13,600 --> 00:56:18,400
Mathematics is a little bit of
invention and a lot of discovery.
928
00:56:21,840 --> 00:56:26,600
Mathematicians will probably never
all agree, and maybe we will never
929
00:56:26,600 --> 00:56:30,560
find a definitive answer,
but the consequences
930
00:56:30,560 --> 00:56:35,200
of having that debate
is why it really matters.
931
00:56:35,200 --> 00:56:40,240
We have used mathematics for a much
deeper understanding of nature
932
00:56:40,400 --> 00:56:42,760
and of the universe in general.
933
00:56:42,760 --> 00:56:45,920
We know about the universe now,
things that,
934
00:56:45,920 --> 00:56:47,720
a few hundred years ago,
935
00:56:47,720 --> 00:56:51,280
people didn't even know what to ask.
936
00:56:51,280 --> 00:56:56,360
Searching for the truth about maths
has, over 2,000 years of history,
937
00:56:56,520 --> 00:57:00,040
transformed the human experience.
938
00:57:00,040 --> 00:57:04,400
Discovering patterns everywhere
in nature has given us structure
939
00:57:04,400 --> 00:57:07,200
and beauty and inspiration.
940
00:57:08,800 --> 00:57:13,800
Inventing new areas of maths has led
to an explosion of technology
941
00:57:13,880 --> 00:57:18,560
that, ultimately, underpins modern
trade and computing.
942
00:57:18,560 --> 00:57:23,520
We have discovered powerful rules
that we continue to use to explore,
943
00:57:23,520 --> 00:57:26,400
enhance and explain
the world around us.
944
00:57:27,520 --> 00:57:32,480
And we have had a tantalising
glimpse of what could be to come.
945
00:57:32,920 --> 00:57:35,680
It's quite possible
that what we have been doing
946
00:57:35,680 --> 00:57:39,600
in science for all the centuries is,
in some sense,
947
00:57:39,600 --> 00:57:43,040
looking for our keys
under the lamp post.
948
00:57:43,040 --> 00:57:47,960
We have been able to use mathematics
to describe what happens out there,
949
00:57:47,960 --> 00:57:51,600
but that could be the tip
of an iceberg of reality
950
00:57:51,600 --> 00:57:54,640
that we as yet don't have any
understanding of,
951
00:57:54,640 --> 00:57:57,680
haven't yet had any contact with.
952
00:57:57,680 --> 00:58:01,960
But most of all, I think that asking
questions about the origins
953
00:58:01,960 --> 00:58:05,040
and truth of maths has given us a
purpose,
954
00:58:05,040 --> 00:58:08,000
it's given us understanding.
955
00:58:08,000 --> 00:58:11,240
Ultimately, maths has given us
meaning.
956
00:58:18,400 --> 00:58:22,320
What is it that makes our world work
the way that it does?
957
00:58:22,320 --> 00:58:25,960
Explore more about the magic
and mystery of mathematics
958
00:58:25,960 --> 00:58:29,600
and how it impacts our everyday
lives by going to the web address
959
00:58:29,600 --> 00:58:32,720
below and following links
to the Open University.
82404
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