Magic Numbers: Hannah Fry's Mysterious World of Maths (2018–…): Season 1, Episode 3 - Weirder and Weirder - full transcript
(soft piano music)
- There is a mystery at
the heart of our universe.
A puzzle that, so far, no-one
has been able to solve.
I can't, this is too weird!
- Welcome to my world!
- If we can solve this mystery,
it will have profound
consequences for all of us.
That mystery is why
mathematical rules and patterns
seem to infiltrate pretty much everything
in the world around us.
Many people have, in fact, described maths
as the underlying
language of the universe.
But how did it get there?
Even after thousands of years,
this question causes controversy.
We still can't agree on
what maths actually is
or where it comes from.
Is it something that's
invented, like a language?
Or is it something that
we've merely discovered?
- I think discovered.
- Invented.
- It's both.
- I have no idea!
- Oh, my God!
Why does any of this matter?
Well, maths underpins
just about everything
in our modern world,
from computers and mobile phones
to our understanding of human biology
and our place in the universe.
My name is Hannah Fry
and I'm a mathematician.
In this series, I will explore
how the greatest thinkers in history
have tried to explain the origins
of maths' extraordinary power.
(Hannah laughing)
You've ruined his equation!
I'm going to look at
how, in ancient times,
our ancestors thought maths
was a gift from the gods.
How, in the 17th and 18th centuries,
we invented new mathematical
systems and used them
to create the scientific
and industrial revolutions.
And I'll reveal how, in the
20th and 21st centuries,
radical new theories are forcing
us to question, once again,
everything we thought we knew
about maths and the universe.
- The unexpected should be expected,
because why would reality down
there bear any resemblance
to reality up here.
- In this episode, I explore paradoxes
within modern mathematics.
Who shaves the barber?
And I discover the very weird worlds
that maths seems to be leading us into.
(Hannah screaming)
(dramatic music)
Maths is very much part
of our modern world.
Even the images you're watching now
are essentially numbers
processed by computers.
Sorry, guys. Would you
mind taking a photo of me?
- Oh, sure.
- Give me one second.
Today, maths is at the
heart of big business,
in the development of new software,
such as facial recognition technology.
All of which, fundamentally,
is based on mathematical algorithms.
And it matters because copyright
issues and legal ownership
can depend on where that maths comes from.
You can phrase the question like this,
is maths a genuine, fundamental
part of our universe,
something that we have discovered?
Or is it merely invented?
A language that we've created
just to describe the world around us?
(people screaming)
Mathematicians have argued
over this idea for centuries.
And even today, this question
is a thought-provoking
and challenging dilemma.
So far, I've explored
how, in ancient times,
maths was revered as a gift from the gods.
Perfect, complete and
gratefully discovered by humans.
But through the ages,
new areas of mathematics,
like algebra and the concept of zero,
have, quite simply, been invented.
But for most of us, we
normally think of maths
as just a series of objective facts
based in logic that someone,
somewhere has discovered.
Facts that we all start
to learn at school.
If you're anything like me,
you'll remember maths at school
being taught as a series of rules.
It was very logical, it was
very ordered, very complete.
Very black and white.
There were right and wrong answers,
which you didn't necessarily
get in other subjects
like art or like music,
which were much more about preferences,
about opinions and about
cultural differences.
It felt like the mathematical
rules were intrinsically true.
But why?
What are the fundamental
mathematical laws?
To answer that question, you
have to categorize everything.
You have to boil maths down
into distinct groups of objects
in something called set theory.
Set theory is a language
that talks about groups,
or sets, of items.
So, for example, the set of odd numbers
are all the whole numbers
that cannot be neatly divided by two.
And the set of even
numbers are those that can.
This reveals a basic rule.
Adding an odd number to an even
one produces an odd number.
From simple rules like these,
you can build up more
and more complex rules
and relations of maths.
But there's a problem with set theory.
A paradox at the heart
of mathematical rules
which caused a bit of a crisis
at the start of the 20th century.
You can discover this paradox yourself
by going to your local
hairdresser or gentleman's barber,
and trying to define what you find
in a concise and complete way.
- Hello.
- Hello.
- I was wondering if you could help me.
I am looking for the very
definition of a barber.
- [Barber] I think I can help with that.
- Mathematicians took the same approach
to precisely define the laws of maths.
So, if you were looking
it up in a dictionary,
that one sentence that
defined what a barber was,
what would you say it was?
- Cut men's hair.
- [Hannah] Cut men's hair.
But that could be a
hairdresser though, right?
A hairdresser?
It needs to be a unique
definition for barbers.
Barbers, and only barbers.
- Cos there's the shaving
element, as well, isn't there?
- [Hannah] Yeah, that's true.
- I've never had a shave
in a hairdresser's.
- No, that's true.
- I've had a chat.
- The chat?
(Hannah laughing)
- Yeah.
- It's a fair point.
- It's a very important part of it.
- You do hear some
stories, being a barber.
- So, actually, I suppose, the shave thing
is something that only barbers do.
So, someone who shaves men.
But a barber doesn't shave all men.
And I need a phrase that uniquely
and completely identifies
a barber and no one else.
OK, let's see where we are, then.
So, we've got, a barber shaves all men,
but only the men who shave
but don't shave themselves?
- Yes.
- Yes.
- All right, I think we've
settled on something now.
We've agreed on, "A barber shaves all men,
"and only those men, who shave
but do not shave themselves."
Sound about right?
I mean, it doesn't exactly
roll off the tongue!
But I think it's fairly accurate.
But, hang on a second!
There's a bit of a paradox here.
Who shaves the barber?
- [Barber] Well, can a
barber not shave himself?
- But if he does shave himself,
then our definition here says
that he doesn't shave himself.
Let me clarify that.
If he doesn't shave himself,
then according to the definition
he's one of the men shaved by the barber.
So, he does shave himself.
Attempting to create a
mathematically precise definition
creates a contradiction where the barber
both shaves himself and
doesn't shave himself.
- First, the bristles.
- [Hannah] This is known
as the barber's paradox.
- [Barber] You've got it!
- I want to do it perfectly!
- Perfect, OK.
- It is an illustration of the paradox
at the heart of mathematics,
which was discovered in 1901
by one of my favorite
troublemakers, Bertrand Russell.
The problem for maths was
that Russell's paradox
undermines the logic of defining things,
like odd or even numbers,
by putting them into categories or sets.
Over here, I have got a
set of clipper attachments.
And in there, I have got a set of things
that aren't clipper attachments.
Clipper attachment goes in there.
Not a clipper attachment, goes in there.
Clipper.
Not a clipper.
Now, the question is,
where does this bag belong?
It's clearly not a clipper attachment.
Is it going to attach to a clipper?
No, it's not, which means
it needs to go in there,
but we've got a problem,
because this sink is supposed
to only contain things
that are not clipper attachments.
Which means that the contents of the bag
can't go in the sink.
Since the bag, or set,
is not, in itself, a clipper attachment,
but, by its definition,
contains clipper attachments,
we can't easily categorize
where the set belongs.
Similarly, the barber can't,
in a logically consistent way
be contained in a set of
people that do shave themselves
or the set of people who don't.
Russell's paradox shows that
there is a logical problem
with trying to categorize
anything into coherent sets,
whether it's barbers, clipper
attachments, or even numbers,
and this logical puzzle
exposed a fault in the bedrock
on which all the rest of maths is built.
If the foundations are shaky,
how can we trust everything else?
- Bertrand Russell
realized that mathematics
was on much shakier ground
than people had originally thought.
It turned out to be much, much harder
to really lay a solid foundation for maths
that everybody agreed on,
and this is still wonderfully
controversial to this day.
- That's what you do in
science and mathematics.
You take a sledgehammer.
You smash at whatever structure,
whatever edifice you've built.
You try to find the weaknesses
and that allows you to figure out
what needs to be shored up.
And that's really, I think, the
legacy that Russell left us.
- I think of it as, in
some ways, the death knell,
or at least a major challenge,
the attempt to ground
mathematics in logic.
And that's the thing
that becomes really hard
in light of Russell's paradox.
- Russell's paradox caused a real crisis
amongst mathematicians.
Suddenly, maths was uncertain.
It was fallible.
And if it has these fundamental problems,
how can it possibly be discovered?
So, does that mean that
maths has to be invented?
Just a human language
and all of the flaws that come with it?
If maths is merely an
invention of the human mind,
it's perhaps not that surprising
that it's not perfect.
But I don't think that I'm ready to accept
the invention argument quite yet.
Maths just seems to be too
good at predicting the behavior
of the world in ways that we
never could have imagined.
Because, just as Bertrand Russell
was exposing the limitations
of maths in one way,
another titan of the 20th
century, Albert Einstein,
was pulling it back in a
completely different direction.
(baby giggling)
Take what is probably the most
famous equation in the world.
With just five symbols,
it looks so simple.
Almost childish.
Yet, it contains some incredibly powerful
mathematical and philosophical concepts.
I'm talking, of course,
about E equals MC squared.
So, E, that's energy.
That is equal to M, that's mass.
Times by a constant, C.
It's the speed of light.
Squared.
There is so much more to this
equation than meets the eye.
It is Einstein's discovery
that matter and energy are equivalent,
and that has profound consequences.
This equation gives us
one of the immutable laws in the universe,
that nothing can travel faster
than the speed of light.
Try this one.
The reasoning is this.
Making something move requires more energy
than keeping it at rest.
And because this C here is a constant,
if the energy goes up by
accelerating something,
accelerating something,
So, that means that you or I
actually weigh a tiny bit more
when we're moving in a car or a plane.
The increase in mass
only becomes significant
when objects are moving at speeds
close to the speed of light.
As an object approaches
the speed of light,
its mass rises faster and faster,
which means it takes more
energy to accelerate it further.
It can't, therefore,
reach the speed of light,
because the mass becomes infinite,
and it would require an
infinite amount of energy
to get there.
You've ruined his equation!
As well as proving there's
a cosmological speed limit,
this single equation also explains
how all the stars in the
universe convert mass into energy
as they burn brightly in the night sky.
Einstein's famous
equation has proved itself
to be a remarkable match for reality
every time it's been put to the test.
Einstein had uncovered
one of the essential
mathematical rules underlying the cosmos.
It seems like clear
evidence that that maths,
at least, is discovered.
But Einstein didn't stop there.
Using the power of mathematics,
he brought about a fundamental shift
in our understanding of space and of time
and of how light travels through space.
To see that evidence for myself,
I've come to an observatory
to do some serious thinking
about what we actually see
when we look at stars in
the sky, such as our sun.
- If things were happening right now,
we wouldn't be able to see it
until eight and a half minutes later,
because that's how long it takes the light
to travel to the Earth.
- So, when you're looking at the sun,
you're seeing how it was
eight and a half minutes ago?
- Exactly.
And objects that are further away,
we see them as they were
further back in time.
So, for instance, there are
other stars in our galaxy
that are thousands of light years away,
so we see them as they were
thousands of years ago.
- So, when you look in a telescope,
and you're seeing them how they were
when people were building pyramids
and Pythagoras was discovering
his rules on Earth.
- Exactly, and we can even see things
that are even further away than that.
So, galaxies outside our own galaxy.
We see many of them as they were
a billion years ago or more.
- Gosh, goodness.
Does this work at smaller scales, then?
Is there, like, a limit
to how big something has
to be before this works?
I mean, I'm looking at you now, right?
Light, presumably, is taking
time to bounce off you
and for me to see you.
- Yes, it is.
But light travels at an
incredibly fast speed,
300,000 kilometers per second, roughly.
So, the time it takes
to travel from me to you
is a very, very tiny fraction of a second.
- But, in theory, I am
seeing you in the past.
- In theory, yes, you're
absolutely seeing me in the past.
- [Hannah] All of this
shows that we can never know
what the universe is like
at this very instant.
The universe is, remarkably,
not a thing that extends just in space,
but in time, as well.
This is fundamental to
Einstein's revolutionary insights
about our universe.
He realized that the very
concept of time is relative.
That is to say, it depends on the position
and movement of the observer.
He worked it out by thinking about events
that appear to be simultaneous.
- So, let's imagine that
you're in a hot-air balloon
floating above the observatory here,
and you're high enough that
you can see a flash of light
in London, say, and
another one in Portsmouth.
And let's assume that these
flashes of light go off
such that you see both of them happening
exactly simultaneously.
- So, from where I am, it looks like
they are both flashing their
lights at the same time?
- At exactly the same time.
But if I were in an aircraft
that was flying very fast towards London,
I would see the flash of light in London
before the flash of light from Portsmouth.
- Using the inescapable
logic of mathematics,
Einstein realized that if an observer
is moving towards one of the flashes,
they would see that flash
before the other one caught up with them.
So, for them, the flashes
are not simultaneous.
But who's?
OK, I mean, they did go off together.
Who's right?
Am I right in the hot-air balloon?
- In fact, there is no way
of saying that you are right
and I am wrong in how
we observe these events.
- This is called relativity.
So, our whole concept of time,
our whole concept of time
means what happens first,
what happens second,
comes down to where we
are and how we're moving.
- Exactly.
So, the concept of time
is now inextricably linked
to the positions in space and
your movement through space.
So, this is why we can't describe
space and time separately,
but we have to put them
together in space-time.
- You can't separate the two.
- You can't separate the two.
- And that all comes down to this idea
that Einstein managed to
prove via thought experiments.
- Yeah, that's the amazing thing about it.
Purely through thought experiments and--
- And a good bit of maths.
- And a good bit of maths.
A very good bit of maths, yes.
- Einstein was using the mathematics
to make sense of the universe,
and claiming that the universe
was nothing like what
anyone thought it was.
His concept of relativity flew
in the face of what people
had believed about space and
about time for centuries.
Whether that was the Greeks thinking
that the universe was
eternal and unchanging,
or Isaac Newton's more mobile
and mechanistic descriptions.
Einstein took his thoughts even further,
attempting to wrestle gravity
into a neat mathematical law.
He believed it was all down
to the strange behavior of space-time,
and if he was right,
as he laid out in the theory
of general relativity in 1916,
then gravity will even affect light.
If you've got a star shining
light from over here,
then you, the observer, over there,
will receive it in a straight line.
But, if there's a massive
object in the way,
you might think that you
won't be able to see the star.
However, Einstein predicted
that the mass of an object
will distort the space-time around it,
and anything moving through
that warped space-time
will have to follow the curves.
This warping of space-time, Einstein said,
is what we usually describe as gravity.
We think of gravity as keeping the planets
in orbit around our sun.
In fact, he said, it's the
result of the distortion
of space-time near massive objects.
And Einstein calculated the precise effect
it would have on light.
So, the starlight,
while still technically
traveling in a straight line,
will follow the curves of space
and appear around the object.
Einstein predicted that,
in exactly this way,
we should be able to observe
light from distant stars
getting bent as the stars
pass behind our sun.
But a theory is just a theory,
an invention of the mind.
It only becomes a discovery when proven
by practical measurement or experiment.
In the decade after Einstein's prediction,
solar eclipses around
the globe gave scientists
the chance to repeatedly test his theory.
The darkness of the eclipse allowed them
to actually see stars
passing close to the sun.
When scientists took the measurements,
they discovered that
light from a distant star
was bending around the sun
in exactly the way that
Einstein had predicted.
The mathematics of general
relativity was correct.
- With general relativity,
Einstein completely upended
our understanding of space,
time, matter, energy,
and kind of what else is there
to the nature of reality.
All of a sudden, we learn
that mass and energy
can warp the fabric of space and time
in this beautiful, interconnected dance
where the motion of matter
affects the warping of space and time,
which affects the motion of other matter.
- We used to think of space
as this boring static stage
upon which events unfolded.
Then Einstein told us that space is itself
an active player in this game,
like a stretchy rubber sheet.
And, yet, a substance perfectly described
by beautiful mathematical equations.
- I mean, how did he think of that?
How did he think of something like this?
- Einstein's description of gravity,
the warping of space-time,
accurately explains why
objects stay in orbit,
whether they're satellites
around the Earth
or galaxies around black holes.
His equations are being tested
and re-proven every day,
and without Einstein's
general theory of relativity,
modern communication, GPS
or satellite TV systems
couldn't even function.
Although this theory came from his mind,
from thinking about the problem,
rather than from real-world experiments,
it's still so good at predicting,
so perfectly capable of describing
what happens in the universe,
that it must be reflecting
some underlining mathematical truth.
And this lends quite a lot
of weight to the argument
that mathematics is discovered,
which is something that matches
up with my own experience,
because when you're toying
around with mathematics,
it really does feel as
though you're exploring
something that already exists.
But if we accept that
maths does already exist
and is an intrinsic part of nature,
then surely all the rules are out there
waiting to be discovered.
In some ways, mathematics is quite a lot
like a game of chess.
So, you have these very strict rules
that you're not allowed to break,
but within those rules,
there are all kinds of opportunities
to play around and be creative.
The only problem is that, in maths,
no-one tells you what those rules are.
We have to work them out for ourselves.
Most mathematicians like a challenge,
but this idea got blown apart
at a maths conference in 1930
in the Prussian city of Koenigsberg,
when two great mathematicians
and their conclusions collided.
On the one side, you
have got David Hilbert,
a mathematical king in every
possible sense of the word.
This is an enormously well-respected man
who laid down the gauntlet,
asking people to come up with
a fundamental set of rules
on which every mathematical
proof could be based.
On the other side was a young
academic called Kurt Godel.
In contrast to Hilbert, who
thought that mathematics
should be built from
the ground up by humans,
Godel thought that
mathematics was discovered.
He believed that mathematical
truths exist outside of us,
and that we have very little
say in what we can find.
That summit in Koenigsberg
can be seen as a clash
between those who thought that mathematics
is part of our fabric of
reality to be discovered,
and those who saw it as a
language under our control.
Hilbert was confident that humanity
would soon know all there
is to know in maths.
But Godel, who had also been trying
to find the rules of maths,
had come to the opposite conclusion.
In a side room at the summit,
Godel quietly announces that,
in fact, however hard you try,
there are always going to be some things
that are unknowable.
There are always going to be
parts of the mathematical game
that can't be fully explained.
And if you can't know all the rules,
how can you play the game?
According to Godel, any
rule-based maths system
is always going to have some things
that are either unknowable or unprovable,
and what's more, he could prove it,
which is kind of ironic,
if you think about it.
This was quickly accepted,
and became known as Godel's
Incompleteness Theorem.
And it puts an interesting
twist on our key question.
It shows that, even if mathematical rules
truly are part of the universe
and we're simply discovering them,
we are nevertheless
going to have to accept
some of those rules
without knowing how or why they are true.
- Normally, people think
that there's some intrinsic difference
between science and math on one hand,
and faith-based belief
systems on the other,
and yet what Godel's theorem
tells us is that's not true.
That there are things in mathematics
that you have to take on faith
or you can't do the mathematics.
To me, this was an
astounding thing to realize.
- We're going to have to accept
that we can't give maths a
foundation in formal laws
or in logic in the way
that we thought we could.
- I think it's enormously
exciting that math,
in some sense, is open-ended.
So, in a sense, it puts an end
to one way of thinking about
mathematics, but I think,
it actually adds color and
richness to the subject
because it's just going to keep on going.
(soft piano music)
- So, what does Godel's
Incompleteness Theorem
mean for our view of the universe
and the parts that maths plays in it?
Well, it depends on what
you're trying to use maths for.
If your goal is to use it to
describe what's around you,
then it still offers a
very detailed picture,
enough to navigate your
way through the universe
and to explain its features.
Sure, the map is not going to
be the same as the terrain,
but even if maths is a bit
incomplete around the edges,
you could argue that it
doesn't really matter.
Although Godel proves it's not possible
to formalize all of maths,
it is possible to formalize
all the mathematics
we actually need to use.
Take flying as an example.
Now, I did my PhD in the
mathematics of aerodynamics,
and that means I spent four
years poring over equations
for wing sections and wind speeds.
It's stuff that I know
like the back of my hand.
But does that qualify me for going up
in one of these on my own?
Absolutely not!
And on the other hand, these
guys don't really need to know
any of this stuff to make them
graceful acrobats in the air.
Not having a complete understanding
doesn't always matter.
We've still flown successfully
for over 100 years.
And now, it's my turn.
- [Instructor] And then
this is your diagonal line.
The strap that comes across.
This will dig in a little bit on take-off
when you're leaning forwards
and running down the hill.
- [Hannah] I can handle it.
- It should be a little bit uncomfortable.
- [Hannah] I can handle it.
- [Instructor] Don't
worry too much about it.
- And do you have quite a good feel
for where the thermals are?
- You have to have the
right weather conditions.
So, if you imagine a hill that
faces totally into the wind,
that's well drained, maybe darker,
and it will create this
kind of pool of warm air,
and then it will,
once it kind of reaches a
decent temperature difference,
it bubbles up through the atmosphere.
- Yeah, it's almost like we've
got kind of opposing skills.
And, like, they're sort
of about the same thing,
but you don't need my
skills to do what you do,
and I couldn't do what you do.
- I guess the ground-speed element
has a bit of maths in there.
I always thought the lesson
bit of maths to begin with.
Where's the wind coming from?
How strong it is?
How fast am I going to go, if
I'm pointing into the wind?
- But you're not solving
Navier-Stokes equations, are you?
- [Instructor] I don't
even know what that means!
- Yeah, exactly.
(both laughing)
Before the theoretical analysis
of aviation came along,
the practical side of flying
was mere trial and error.
Now, we have a much more
reliable understanding
of what keeps us aloft,
and it doesn't really matter
if the maths behind it
is, ultimately, a bit
fuzzy around the edges.
In the real world, the best that we can do
is just accept Godel's
Incompleteness Theorem
and get on with life.
Hey!
That's amazing!
- [Instructor] There's a thermal!
- Woohoo!
- You're feeling much more alert.
A bit stronger?
- Yeah.
We have to put aside, for the moment,
the question of whether maths
is invented or discovered,
because it now looks like
we may have to determine
which part of maths we're asking about.
You see, for me, Godel's work
highlights the distinction
between pure theoretical maths
and practical applied maths.
So, here is how I see things.
With mathematics,
there's a split down the
middle of the subject,
because the story changes
depending on what world you start with,
whether it's the real one
or one that exists in our imaginations.
And right now, when we're flying,
this is very much in the
realm of applied mathematics,
where everything is tangible and practical
and a little bit imprecise.
But, alongside that,
is where the more theoretical
pure mathematics lives.
That's where you have your proofs,
your paradoxes, and
incompleteness theorems.
A realm which doesn't match
up with a physical reality.
A sort of imperfect perfection.
Even though I instinctively
feel that maths is discovered,
I like that there is this
pure theoretical part of maths
that isn't found in reality.
And, since the maths there
doesn't need to match reality,
it's a convenient place where we can leave
all the weird contradictory
bits that we come across.
Woo!
However, I might have
it the wrong way round.
Although pure theoretical maths
seems rather divorced from reality,
that might merely reflect
the fact that reality
is not quite what we think it is.
And it's a reality that we can uncover
through the strange
maths of quantum physics.
The weirdest worlds that
most of us have come across
are likely to be in fiction, such as this,
Alice's Adventures In Wonderland.
The author, Lewis Carroll,
real name Charles Dodgson,
was actually a mathematics don at Oxford,
and a staunch traditionalist.
It's generally believed that
much of this surreal story
is a thinly veiled satire
on the new avant-garde maths
that was flourishing when
he was writing in the 1860s.
Still feels relevant today,
and applies equally well to
the new weird kid on the block,
quantum physics.
Take a close look at the
physical world around us,
and you can reduce it all to maths.
The solid bricks of our houses
or the blood cells in our veins
can all be reduced down into chemicals,
which comprise elements,
which themselves are made up of atoms,
comprising a tiny nucleus
of protons and neutrons
and electrons buzzing around
in a cloud of mostly emptiness.
The protons and neutrons in turn
are built from smaller subatomic particles
that we can't directly observe.
We can only verify their existence
using experiments and mathematics.
As we delve deeper into this world,
scientists have discovered
something very strange indeed.
We can never actually
know the precise location
of most particles in this
subatomic or quantum realm.
All we can know is the likelihood
of them being somewhere,
a mathematical formula that describes
the probability of their position.
All of this means we are,
fundamentally, at a quantum level,
just a great fuzz of
energy and probabilities.
I'm not sure Lewis Carroll
would have liked that.
And the only way to explore
this ill-defined quantum world.
Oh, hello.
Is through mathematics,
perfectly equipped to handle
strange probabilities.
It seems like there's
quite a lot of uncertainty
in quantum physics.
Does that bother you?
- Um, no.
When I heard that things
were, you know, uncertain,
and also against our common
sense in quantum physics,
then I thought, "Oh, wow,
that sounds interesting.
"I want to know more about that."
- The pivotal maths
behind the quantum world
was first laid out
by Austrian physicist
Erwin Schrodinger in 1926.
His equations accurately describe
the unusual behavior
of subatomic particles.
OK, all right, I'll tell you what, then.
Quantum physics lesson
101, where do we start?
Give me the lesson.
- Um, OK.
I would say we would have
to start with superposition.
So, let's talk about electrons.
So, they're a very small particle,
and they can be in two states.
They have a state called the spin,
and the spin can be pointing up or down.
So, if we were in the classical world,
the spin could only be either up or down.
But in the quantum world, the
spin is in a superposition,
which means it can be up
and down at the same time.
- Let me see if I understand this, then.
So, superposition is where something
is and isn't something at the same time?
- Yes.
We can think about some examples.
So, let's say that we have a cup,
and the cup is full of
water, that's one state.
Another possible state
is that the cup is empty.
So, if we were to bring the quantum ideas
to the classical world,
we would say the state, one
possible state of the cup,
would be to be empty and
full at the same time.
- [Hannah] OK, which you never see
in the world that we're living in,
you never see a cup that's full and empty.
- Yes, we don't.
- [Hannah] But you see this
a lot in the quantum world?
- Yes, superpositions
are an essential part
of the quantum world.
- Like a light being on
and off at the same time.
- Exactly.
Or the cake being eaten or
not eaten at the same time.
- [Hannah] OK.
It's a very tough idea to get around.
- Yes, yes.
- Given two possible outcomes,
in the quantum world,
we now have to allow for a third one,
the combination of both outcomes.
At the quantum scale, you can
have your cake and eat it.
This is such a weird idea.
How do we know it's real?
- Well, because we've done
many experiments to prove it
that show exactly that behavior.
- What does that experiment look like?
- Well, if we put it, say,
in terms of things we
have here on the table.
We could think about,
let's say that I wanted
this piece of sugar
to come into my cup,
but there is this pot in the middle.
So then, if the sugar is going
to come from here to my cup,
it could either go this way
or that way in the classical world.
But in a quantum experiment,
it can take both routes at the same time,
and I would be able to
distinguish that it did that
if I did a quantum experiment.
- I can't, this is too weird!
- Welcome to my world.
So, you go through this
whole transition, from first,
the ideas and the mathematics,
and then up to showing
it in the experiment.
- [Hannah] What came out
of Schrodinger's mouth
was a prediction of
something even stranger
that can sometimes be
produced when particles
interact in the quantum world,
a phenomenon called entanglement.
All right, tell me about
entanglement, then.
- OK, so take two electrons.
If the electrons are entangled,
and I do something to
one of the electrons,
for example change the
direction of the spin,
that will instantaneously affect
the state of the other electron,
even if they are separated long distances.
- How far away are they from each other?
- Well, they can be a few centimeters,
but now the latest experiments,
they're using satellites,
show entanglement across 1,200 kilometers.
- What?
- Yes.
- You've got something
over here, and you do?
And something 1,200 kilometers away.
You do something to one, and it instantly.
The other one instantly
knows what's happened?
- Yes, you'll affect the state
of the other one instantly.
- [Hannah] Apparently,
there is no cause or link.
The only thing we can say
is that the two particles
are synchronized.
How does one know what
the other one's doing?
- Well, that we're still
trying to understand,
because that's what mathematics tells us,
and then we can show it in the experiment,
but we're still struggling to
understand what that means.
And one of the reasons why
we don't understand it.
And, you know, like you're asking,
is because we don't see
it in our everyday lives.
So, let's say it's not part
of our experience and common sense.
But that doesn't mean it doesn't happen.
I don't want you to go crazy.
- So, quantum mathematics
has made predictions,
which have been discovered to be true.
But, despite that, the
quantum world is so weird,
it suggests to me that the maths
behind it is just invented.
It feels like what we're seeing
is evidence of a man-made
system being pushed too far.
These are the absurdities that appear
when it's applied to situations
it wasn't designed for.
But my quest to find the truth about maths
takes me back to nature.
There is amazing new evidence
that quantum processes
might actually be crucial
to our own existence,
and much of life on Earth.
That would strengthen the argument
that mathematical processes
are intrinsic to our world.
That maths is discovered.
It all comes down to the photosynthesis,
the process that converts sunlight
into chemical energy used in life.
It takes place in the
molecules called chlorophyll,
which can be found in
plants, algae and bacteria.
- In bacteria, we have something
that's similar to what we have in plants.
- So, this is the stuff
that captures the sunlight?
- Exactly.
Each of these molecules,
each of these little blue
things here that I'm showing,
is bacterial chlorophyll,
and if we take it apart,
it will capture light.
- The chlorophyll captures light
by absorbing particles
of light, or photons.
- So, a photon is absorbed,
and it's absorbed by all of them,
so energy shared by all of
these bacterial chlorophylls,
and that sharing, we call it
using a quantum superposition.
- Because it's coming in
and hitting one of these,
but all of them are somehow.
- In a way, it's as if
each of the electrons
of the chlorophylls are
talking to each other
and sharing the energy around.
- The subatomic particles
in the chlorophyll
are synchronized in a way
that can only be described
by quantum mechanics.
Does it do a good job?
I mean, is it efficient?
- That is part of why
photosynthesis is efficient.
Because, by sharing the
energy among all of them,
it's easier to transfer the
energy to another molecule.
Imagine if you have to
share the energy one by one,
you have to explore each part separately,
but if you share the energy all together,
you explore all the
parts at the same time.
- Every leaf on every plant on the planet
has been following these quantum rules
for millions of years.
And we still don't fully
understand how they do it.
Without quantum physics,
despite all the mathematical
uncertainties and ambiguities,
plants wouldn't produce
oxygen so efficiently.
And without oxygen, we wouldn't exist.
- The systems are amazing,
because they are effectively the interface
between using a little
bit of classical mechanics
and a little bit of quantum mechanics
to operate in a wonderful way.
- Ultimately, quantum mechanics
is at the heart of photosynthesis
and, well, I guess, all of life on Earth.
- It is, it is.
We can say life is nothing
but quantum mechanics
giving us energy.
- So, what does all this
mean for our key question
about the origins of maths?
There is no shortage of evidence
that mathematical rules
are intrinsic to the world.
We keep discovering them everywhere.
However, we now know we have to take
some of the maths on faith,
and believing in the numbers
is taking us to a very strange world,
with crazy notions like superposition
and entanglement at the core of it.
Quantum mathematics is inextricably linked
to the world as we know it.
linked to the world as we know it.
Because the world is actually
a whole lot weirder than we thought.
What quantum mechanics does do
is force us to question what is real.
And what is reality, anyway?
Just how much light can
mathematics shed on reality?
With the world stripped bare,
exposing the nuts and bolts of existence,
what does maths tell us
about this realm of subatomic particles?
The maths that underlies it
isn't particularly pretty,
but it can all be written
out in just one equation.
This is the formula that describes
the constituents of the universe.
It has become well enough
accepted to be called
the standard model of subatomic physics.
I told you it wasn't pretty.
Now, you're just going to have
to take my word for it on this one.
This equation encapsulates all
of the fundamental properties
all of the fundamental properties
But there are a couple of sticking points.
For one thing, no-one has
ever satisfactorily explained
how our common-sense,
day-to-day version of the world
emerges from this kind
of subatomic reality.
All of that fuzziness,
all of that uncertainty
in the quantum world,
just how does it end up
giving us that comfortable,
familiar solidity of the normal world?
At the other end of the spectrum,
the solar system and beyond
is beautifully and accurately described
by a different equation.
Einstein's general relativity.
This remarkable equation
tells you about gravity,
about the warping of space-time,
about general relativity.
And when you take these two together,
these two single mathematical sentences,
they are enough to tell
you everything you need
about the fundamental
behavior of the universe
and everything in it.
There is nothing more
articulate than mathematics.
Maths seems to be written
into the physical universe.
So, on the one hand, at
the teeny-tiny scale,
the standard model of particle physics
does this amazing job.
And in the ginormous
scale, general relativity,
I mean, you couldn't
ask for anything more.
There's just one problem when you try
and put these two together.
They're incompatible.
The problem is that general relativity
breaks down in the quantum world.
Gravity simply doesn't apply to particles
at the subatomic scale.
Meanwhile, quantum effects
are virtually never seen
at the scale of humans and
planets, where gravity rules.
You and I are never in a
superposition of existing
and not existing at the same time.
So, what does this mean for us?
Are there two different worlds
each obeying their own
sets of mathematical laws?
Solving this conundrum is
one of the biggest problems
that puzzles scientists today.
We will ever reconcile the two?
- I think it's perfectly plausible
that, within our lifetime, somebody,
maybe somebody watching this program,
will discover the mathematical structure
which unifies Einstein's
theory of relativity
with quantum mechanics
and just provides the perfect
description of this world.
And that would be really exciting.
- Will we have one?
How do I know?
We would all like to have one.
But, you know, maybe
we are not smart enough
to formulate a theory
that combines everything.
It's hard.
- I do believe that there
are good ideas out there
and that eventually, it
might take a long time,
but, eventually, humans
will work this out.
I'm confident about that.
- So, will we make it
all the way to include
all possible forces at all possible scales
with all possible forms of matter?
It's a hope I have for our
species, that's all I can say.
- The incompatibility of
these two great theories,
general relativity and quantum mechanics,
creates a serious obstacle
for believing that maths
is really discovered.
And there's a bigger hurdle to come.
Many of the best proposals
to unify general relativity
and the quantum world
have consequences that are even weirder
than the problems they're trying to solve.
They predict the existence
of multiple universes.
This idea is rooted in the
mathematical explanations
of the quantum world,
and the work of its founding
father, Erwin Schrodinger.
The mathematics in Schrodinger's equation
insists that particles can exist
in multiple states at the same time.
And Schrodinger himself says
that these possibilities
aren't just alternatives, but
really happen simultaneously.
This can lead to multiple universes.
And the maths also suggests
there's an infinite number of them,
each slightly different from the others.
Thank you.
Mathematically speaking,
in an infinite universe,
everything that's possible
has to happen somewhere.
Yeah, that's right.
Everything possible happens, somewhere.
Even Schrodinger acknowledged
that the consequences of his equation
describing the quantum
world might seem lunatic.
But if there's one thing I've learned,
it's that you should trust the maths.
So, maybe our experience isn't special,
maybe our reality isn't unique after all.
- There are so many distinct
avenues of investigation
that lead to the
possibility of a multiverse.
From our studies of
unification and string theory,
from our studies of quantum mechanics,
even from the study of space
going on infinitely far.
Even that gives rise to a
version of the multiverse.
- If we're going to reject
everything that just seems weird,
we're almost guaranteed to reject
the true theories of the future
when they get discovered.
I think we should just chill out,
accept that the world is weird
and that's just part of its charm.
And trust the math.
(dramatic music)
- So, why does all of this matter?
Well, if maths really is discovered,
then there is an intrinsic truth
behind the maths we uncover
however weird that truth seems to be.
If maths is invented,
then how do we know what is true or false?
Is it true purely because we define it so?
And how does it relate to the real world
that we all experience?
In this series, we've seen that maths
can explain so much of our world,
from aerodynamics to planetary orbits,
from the subatomic world
to processes crucial to life on Earth.
And that is something I just
can't accept as a coincidence.
So, here's my take on things.
For me, it's almost as though you have
this alternate parallel mathematical world
that hides just beneath our own.
You can't see it, you can't touch it.
The only way that you can explore it
is by using the language
that we've invented.
All of those symbols and
equations and conventions
are our only tools of navigation,
and they are undoubtedly man-made.
But once you're inside that world,
once you're exploring the landscape
that mathematics has
laid out in front of you,
I am absolutely convinced
that you are on a voyage of discovery.
It is a world without a human designer.
So, ultimately, I think it's both.
Mathematics is a little bit of invention
and a lot of discovery.
Mathematicians will
probably never all agree,
and maybe we will never
find a definitive answer,
but the consequences of having that debate
is why it really matters.
- We have used mathematics
for a much deeper understanding of nature
and of the universe in general.
We know about the universe now,
things that, a few hundred years ago,
people didn't even know what to ask.
- Searching for the truth about maths
has, over 2,000 years of history,
transformed the human experience.
Discovering patterns everywhere in nature
has given us structure and
beauty and inspiration.
Inventing new areas of maths
has led to an explosion of technology
that, ultimately, underpins
modern trade and computing.
We have discovered powerful rules
that we continue to use to explore,
enhance and explain the world around us.
And we have had a tantalizing glimpse
of what could be to come.
- It's quite possible that
what we have been doing in
science for all the centuries
is, in some sense, looking for
our keys under the lamp post.
We have been able to use mathematics
to describe what happens out there,
but that could be the tip
of an iceberg of reality
that we as yet don't have
any understanding of,
haven't yet had any contact with.
- But most of all, I think
that asking questions
about the origins and truth of maths
has given us a purpose,
it's given us understanding.
Ultimately, maths has given us meaning.
(dramatic music)
- There is a mystery at
the heart of our universe.
A puzzle that, so far, no-one
has been able to solve.
I can't, this is too weird!
- Welcome to my world!
- If we can solve this mystery,
it will have profound
consequences for all of us.
That mystery is why
mathematical rules and patterns
seem to infiltrate pretty much everything
in the world around us.
Many people have, in fact, described maths
as the underlying
language of the universe.
But how did it get there?
Even after thousands of years,
this question causes controversy.
We still can't agree on
what maths actually is
or where it comes from.
Is it something that's
invented, like a language?
Or is it something that
we've merely discovered?
- I think discovered.
- Invented.
- It's both.
- I have no idea!
- Oh, my God!
Why does any of this matter?
Well, maths underpins
just about everything
in our modern world,
from computers and mobile phones
to our understanding of human biology
and our place in the universe.
My name is Hannah Fry
and I'm a mathematician.
In this series, I will explore
how the greatest thinkers in history
have tried to explain the origins
of maths' extraordinary power.
(Hannah laughing)
You've ruined his equation!
I'm going to look at
how, in ancient times,
our ancestors thought maths
was a gift from the gods.
How, in the 17th and 18th centuries,
we invented new mathematical
systems and used them
to create the scientific
and industrial revolutions.
And I'll reveal how, in the
20th and 21st centuries,
radical new theories are forcing
us to question, once again,
everything we thought we knew
about maths and the universe.
- The unexpected should be expected,
because why would reality down
there bear any resemblance
to reality up here.
- In this episode, I explore paradoxes
within modern mathematics.
Who shaves the barber?
And I discover the very weird worlds
that maths seems to be leading us into.
(Hannah screaming)
(dramatic music)
Maths is very much part
of our modern world.
Even the images you're watching now
are essentially numbers
processed by computers.
Sorry, guys. Would you
mind taking a photo of me?
- Oh, sure.
- Give me one second.
Today, maths is at the
heart of big business,
in the development of new software,
such as facial recognition technology.
All of which, fundamentally,
is based on mathematical algorithms.
And it matters because copyright
issues and legal ownership
can depend on where that maths comes from.
You can phrase the question like this,
is maths a genuine, fundamental
part of our universe,
something that we have discovered?
Or is it merely invented?
A language that we've created
just to describe the world around us?
(people screaming)
Mathematicians have argued
over this idea for centuries.
And even today, this question
is a thought-provoking
and challenging dilemma.
So far, I've explored
how, in ancient times,
maths was revered as a gift from the gods.
Perfect, complete and
gratefully discovered by humans.
But through the ages,
new areas of mathematics,
like algebra and the concept of zero,
have, quite simply, been invented.
But for most of us, we
normally think of maths
as just a series of objective facts
based in logic that someone,
somewhere has discovered.
Facts that we all start
to learn at school.
If you're anything like me,
you'll remember maths at school
being taught as a series of rules.
It was very logical, it was
very ordered, very complete.
Very black and white.
There were right and wrong answers,
which you didn't necessarily
get in other subjects
like art or like music,
which were much more about preferences,
about opinions and about
cultural differences.
It felt like the mathematical
rules were intrinsically true.
But why?
What are the fundamental
mathematical laws?
To answer that question, you
have to categorize everything.
You have to boil maths down
into distinct groups of objects
in something called set theory.
Set theory is a language
that talks about groups,
or sets, of items.
So, for example, the set of odd numbers
are all the whole numbers
that cannot be neatly divided by two.
And the set of even
numbers are those that can.
This reveals a basic rule.
Adding an odd number to an even
one produces an odd number.
From simple rules like these,
you can build up more
and more complex rules
and relations of maths.
But there's a problem with set theory.
A paradox at the heart
of mathematical rules
which caused a bit of a crisis
at the start of the 20th century.
You can discover this paradox yourself
by going to your local
hairdresser or gentleman's barber,
and trying to define what you find
in a concise and complete way.
- Hello.
- Hello.
- I was wondering if you could help me.
I am looking for the very
definition of a barber.
- [Barber] I think I can help with that.
- Mathematicians took the same approach
to precisely define the laws of maths.
So, if you were looking
it up in a dictionary,
that one sentence that
defined what a barber was,
what would you say it was?
- Cut men's hair.
- [Hannah] Cut men's hair.
But that could be a
hairdresser though, right?
A hairdresser?
It needs to be a unique
definition for barbers.
Barbers, and only barbers.
- Cos there's the shaving
element, as well, isn't there?
- [Hannah] Yeah, that's true.
- I've never had a shave
in a hairdresser's.
- No, that's true.
- I've had a chat.
- The chat?
(Hannah laughing)
- Yeah.
- It's a fair point.
- It's a very important part of it.
- You do hear some
stories, being a barber.
- So, actually, I suppose, the shave thing
is something that only barbers do.
So, someone who shaves men.
But a barber doesn't shave all men.
And I need a phrase that uniquely
and completely identifies
a barber and no one else.
OK, let's see where we are, then.
So, we've got, a barber shaves all men,
but only the men who shave
but don't shave themselves?
- Yes.
- Yes.
- All right, I think we've
settled on something now.
We've agreed on, "A barber shaves all men,
"and only those men, who shave
but do not shave themselves."
Sound about right?
I mean, it doesn't exactly
roll off the tongue!
But I think it's fairly accurate.
But, hang on a second!
There's a bit of a paradox here.
Who shaves the barber?
- [Barber] Well, can a
barber not shave himself?
- But if he does shave himself,
then our definition here says
that he doesn't shave himself.
Let me clarify that.
If he doesn't shave himself,
then according to the definition
he's one of the men shaved by the barber.
So, he does shave himself.
Attempting to create a
mathematically precise definition
creates a contradiction where the barber
both shaves himself and
doesn't shave himself.
- First, the bristles.
- [Hannah] This is known
as the barber's paradox.
- [Barber] You've got it!
- I want to do it perfectly!
- Perfect, OK.
- It is an illustration of the paradox
at the heart of mathematics,
which was discovered in 1901
by one of my favorite
troublemakers, Bertrand Russell.
The problem for maths was
that Russell's paradox
undermines the logic of defining things,
like odd or even numbers,
by putting them into categories or sets.
Over here, I have got a
set of clipper attachments.
And in there, I have got a set of things
that aren't clipper attachments.
Clipper attachment goes in there.
Not a clipper attachment, goes in there.
Clipper.
Not a clipper.
Now, the question is,
where does this bag belong?
It's clearly not a clipper attachment.
Is it going to attach to a clipper?
No, it's not, which means
it needs to go in there,
but we've got a problem,
because this sink is supposed
to only contain things
that are not clipper attachments.
Which means that the contents of the bag
can't go in the sink.
Since the bag, or set,
is not, in itself, a clipper attachment,
but, by its definition,
contains clipper attachments,
we can't easily categorize
where the set belongs.
Similarly, the barber can't,
in a logically consistent way
be contained in a set of
people that do shave themselves
or the set of people who don't.
Russell's paradox shows that
there is a logical problem
with trying to categorize
anything into coherent sets,
whether it's barbers, clipper
attachments, or even numbers,
and this logical puzzle
exposed a fault in the bedrock
on which all the rest of maths is built.
If the foundations are shaky,
how can we trust everything else?
- Bertrand Russell
realized that mathematics
was on much shakier ground
than people had originally thought.
It turned out to be much, much harder
to really lay a solid foundation for maths
that everybody agreed on,
and this is still wonderfully
controversial to this day.
- That's what you do in
science and mathematics.
You take a sledgehammer.
You smash at whatever structure,
whatever edifice you've built.
You try to find the weaknesses
and that allows you to figure out
what needs to be shored up.
And that's really, I think, the
legacy that Russell left us.
- I think of it as, in
some ways, the death knell,
or at least a major challenge,
the attempt to ground
mathematics in logic.
And that's the thing
that becomes really hard
in light of Russell's paradox.
- Russell's paradox caused a real crisis
amongst mathematicians.
Suddenly, maths was uncertain.
It was fallible.
And if it has these fundamental problems,
how can it possibly be discovered?
So, does that mean that
maths has to be invented?
Just a human language
and all of the flaws that come with it?
If maths is merely an
invention of the human mind,
it's perhaps not that surprising
that it's not perfect.
But I don't think that I'm ready to accept
the invention argument quite yet.
Maths just seems to be too
good at predicting the behavior
of the world in ways that we
never could have imagined.
Because, just as Bertrand Russell
was exposing the limitations
of maths in one way,
another titan of the 20th
century, Albert Einstein,
was pulling it back in a
completely different direction.
(baby giggling)
Take what is probably the most
famous equation in the world.
With just five symbols,
it looks so simple.
Almost childish.
Yet, it contains some incredibly powerful
mathematical and philosophical concepts.
I'm talking, of course,
about E equals MC squared.
So, E, that's energy.
That is equal to M, that's mass.
Times by a constant, C.
It's the speed of light.
Squared.
There is so much more to this
equation than meets the eye.
It is Einstein's discovery
that matter and energy are equivalent,
and that has profound consequences.
This equation gives us
one of the immutable laws in the universe,
that nothing can travel faster
than the speed of light.
Try this one.
The reasoning is this.
Making something move requires more energy
than keeping it at rest.
And because this C here is a constant,
if the energy goes up by
accelerating something,
accelerating something,
So, that means that you or I
actually weigh a tiny bit more
when we're moving in a car or a plane.
The increase in mass
only becomes significant
when objects are moving at speeds
close to the speed of light.
As an object approaches
the speed of light,
its mass rises faster and faster,
which means it takes more
energy to accelerate it further.
It can't, therefore,
reach the speed of light,
because the mass becomes infinite,
and it would require an
infinite amount of energy
to get there.
You've ruined his equation!
As well as proving there's
a cosmological speed limit,
this single equation also explains
how all the stars in the
universe convert mass into energy
as they burn brightly in the night sky.
Einstein's famous
equation has proved itself
to be a remarkable match for reality
every time it's been put to the test.
Einstein had uncovered
one of the essential
mathematical rules underlying the cosmos.
It seems like clear
evidence that that maths,
at least, is discovered.
But Einstein didn't stop there.
Using the power of mathematics,
he brought about a fundamental shift
in our understanding of space and of time
and of how light travels through space.
To see that evidence for myself,
I've come to an observatory
to do some serious thinking
about what we actually see
when we look at stars in
the sky, such as our sun.
- If things were happening right now,
we wouldn't be able to see it
until eight and a half minutes later,
because that's how long it takes the light
to travel to the Earth.
- So, when you're looking at the sun,
you're seeing how it was
eight and a half minutes ago?
- Exactly.
And objects that are further away,
we see them as they were
further back in time.
So, for instance, there are
other stars in our galaxy
that are thousands of light years away,
so we see them as they were
thousands of years ago.
- So, when you look in a telescope,
and you're seeing them how they were
when people were building pyramids
and Pythagoras was discovering
his rules on Earth.
- Exactly, and we can even see things
that are even further away than that.
So, galaxies outside our own galaxy.
We see many of them as they were
a billion years ago or more.
- Gosh, goodness.
Does this work at smaller scales, then?
Is there, like, a limit
to how big something has
to be before this works?
I mean, I'm looking at you now, right?
Light, presumably, is taking
time to bounce off you
and for me to see you.
- Yes, it is.
But light travels at an
incredibly fast speed,
300,000 kilometers per second, roughly.
So, the time it takes
to travel from me to you
is a very, very tiny fraction of a second.
- But, in theory, I am
seeing you in the past.
- In theory, yes, you're
absolutely seeing me in the past.
- [Hannah] All of this
shows that we can never know
what the universe is like
at this very instant.
The universe is, remarkably,
not a thing that extends just in space,
but in time, as well.
This is fundamental to
Einstein's revolutionary insights
about our universe.
He realized that the very
concept of time is relative.
That is to say, it depends on the position
and movement of the observer.
He worked it out by thinking about events
that appear to be simultaneous.
- So, let's imagine that
you're in a hot-air balloon
floating above the observatory here,
and you're high enough that
you can see a flash of light
in London, say, and
another one in Portsmouth.
And let's assume that these
flashes of light go off
such that you see both of them happening
exactly simultaneously.
- So, from where I am, it looks like
they are both flashing their
lights at the same time?
- At exactly the same time.
But if I were in an aircraft
that was flying very fast towards London,
I would see the flash of light in London
before the flash of light from Portsmouth.
- Using the inescapable
logic of mathematics,
Einstein realized that if an observer
is moving towards one of the flashes,
they would see that flash
before the other one caught up with them.
So, for them, the flashes
are not simultaneous.
But who's?
OK, I mean, they did go off together.
Who's right?
Am I right in the hot-air balloon?
- In fact, there is no way
of saying that you are right
and I am wrong in how
we observe these events.
- This is called relativity.
So, our whole concept of time,
our whole concept of time
means what happens first,
what happens second,
comes down to where we
are and how we're moving.
- Exactly.
So, the concept of time
is now inextricably linked
to the positions in space and
your movement through space.
So, this is why we can't describe
space and time separately,
but we have to put them
together in space-time.
- You can't separate the two.
- You can't separate the two.
- And that all comes down to this idea
that Einstein managed to
prove via thought experiments.
- Yeah, that's the amazing thing about it.
Purely through thought experiments and--
- And a good bit of maths.
- And a good bit of maths.
A very good bit of maths, yes.
- Einstein was using the mathematics
to make sense of the universe,
and claiming that the universe
was nothing like what
anyone thought it was.
His concept of relativity flew
in the face of what people
had believed about space and
about time for centuries.
Whether that was the Greeks thinking
that the universe was
eternal and unchanging,
or Isaac Newton's more mobile
and mechanistic descriptions.
Einstein took his thoughts even further,
attempting to wrestle gravity
into a neat mathematical law.
He believed it was all down
to the strange behavior of space-time,
and if he was right,
as he laid out in the theory
of general relativity in 1916,
then gravity will even affect light.
If you've got a star shining
light from over here,
then you, the observer, over there,
will receive it in a straight line.
But, if there's a massive
object in the way,
you might think that you
won't be able to see the star.
However, Einstein predicted
that the mass of an object
will distort the space-time around it,
and anything moving through
that warped space-time
will have to follow the curves.
This warping of space-time, Einstein said,
is what we usually describe as gravity.
We think of gravity as keeping the planets
in orbit around our sun.
In fact, he said, it's the
result of the distortion
of space-time near massive objects.
And Einstein calculated the precise effect
it would have on light.
So, the starlight,
while still technically
traveling in a straight line,
will follow the curves of space
and appear around the object.
Einstein predicted that,
in exactly this way,
we should be able to observe
light from distant stars
getting bent as the stars
pass behind our sun.
But a theory is just a theory,
an invention of the mind.
It only becomes a discovery when proven
by practical measurement or experiment.
In the decade after Einstein's prediction,
solar eclipses around
the globe gave scientists
the chance to repeatedly test his theory.
The darkness of the eclipse allowed them
to actually see stars
passing close to the sun.
When scientists took the measurements,
they discovered that
light from a distant star
was bending around the sun
in exactly the way that
Einstein had predicted.
The mathematics of general
relativity was correct.
- With general relativity,
Einstein completely upended
our understanding of space,
time, matter, energy,
and kind of what else is there
to the nature of reality.
All of a sudden, we learn
that mass and energy
can warp the fabric of space and time
in this beautiful, interconnected dance
where the motion of matter
affects the warping of space and time,
which affects the motion of other matter.
- We used to think of space
as this boring static stage
upon which events unfolded.
Then Einstein told us that space is itself
an active player in this game,
like a stretchy rubber sheet.
And, yet, a substance perfectly described
by beautiful mathematical equations.
- I mean, how did he think of that?
How did he think of something like this?
- Einstein's description of gravity,
the warping of space-time,
accurately explains why
objects stay in orbit,
whether they're satellites
around the Earth
or galaxies around black holes.
His equations are being tested
and re-proven every day,
and without Einstein's
general theory of relativity,
modern communication, GPS
or satellite TV systems
couldn't even function.
Although this theory came from his mind,
from thinking about the problem,
rather than from real-world experiments,
it's still so good at predicting,
so perfectly capable of describing
what happens in the universe,
that it must be reflecting
some underlining mathematical truth.
And this lends quite a lot
of weight to the argument
that mathematics is discovered,
which is something that matches
up with my own experience,
because when you're toying
around with mathematics,
it really does feel as
though you're exploring
something that already exists.
But if we accept that
maths does already exist
and is an intrinsic part of nature,
then surely all the rules are out there
waiting to be discovered.
In some ways, mathematics is quite a lot
like a game of chess.
So, you have these very strict rules
that you're not allowed to break,
but within those rules,
there are all kinds of opportunities
to play around and be creative.
The only problem is that, in maths,
no-one tells you what those rules are.
We have to work them out for ourselves.
Most mathematicians like a challenge,
but this idea got blown apart
at a maths conference in 1930
in the Prussian city of Koenigsberg,
when two great mathematicians
and their conclusions collided.
On the one side, you
have got David Hilbert,
a mathematical king in every
possible sense of the word.
This is an enormously well-respected man
who laid down the gauntlet,
asking people to come up with
a fundamental set of rules
on which every mathematical
proof could be based.
On the other side was a young
academic called Kurt Godel.
In contrast to Hilbert, who
thought that mathematics
should be built from
the ground up by humans,
Godel thought that
mathematics was discovered.
He believed that mathematical
truths exist outside of us,
and that we have very little
say in what we can find.
That summit in Koenigsberg
can be seen as a clash
between those who thought that mathematics
is part of our fabric of
reality to be discovered,
and those who saw it as a
language under our control.
Hilbert was confident that humanity
would soon know all there
is to know in maths.
But Godel, who had also been trying
to find the rules of maths,
had come to the opposite conclusion.
In a side room at the summit,
Godel quietly announces that,
in fact, however hard you try,
there are always going to be some things
that are unknowable.
There are always going to be
parts of the mathematical game
that can't be fully explained.
And if you can't know all the rules,
how can you play the game?
According to Godel, any
rule-based maths system
is always going to have some things
that are either unknowable or unprovable,
and what's more, he could prove it,
which is kind of ironic,
if you think about it.
This was quickly accepted,
and became known as Godel's
Incompleteness Theorem.
And it puts an interesting
twist on our key question.
It shows that, even if mathematical rules
truly are part of the universe
and we're simply discovering them,
we are nevertheless
going to have to accept
some of those rules
without knowing how or why they are true.
- Normally, people think
that there's some intrinsic difference
between science and math on one hand,
and faith-based belief
systems on the other,
and yet what Godel's theorem
tells us is that's not true.
That there are things in mathematics
that you have to take on faith
or you can't do the mathematics.
To me, this was an
astounding thing to realize.
- We're going to have to accept
that we can't give maths a
foundation in formal laws
or in logic in the way
that we thought we could.
- I think it's enormously
exciting that math,
in some sense, is open-ended.
So, in a sense, it puts an end
to one way of thinking about
mathematics, but I think,
it actually adds color and
richness to the subject
because it's just going to keep on going.
(soft piano music)
- So, what does Godel's
Incompleteness Theorem
mean for our view of the universe
and the parts that maths plays in it?
Well, it depends on what
you're trying to use maths for.
If your goal is to use it to
describe what's around you,
then it still offers a
very detailed picture,
enough to navigate your
way through the universe
and to explain its features.
Sure, the map is not going to
be the same as the terrain,
but even if maths is a bit
incomplete around the edges,
you could argue that it
doesn't really matter.
Although Godel proves it's not possible
to formalize all of maths,
it is possible to formalize
all the mathematics
we actually need to use.
Take flying as an example.
Now, I did my PhD in the
mathematics of aerodynamics,
and that means I spent four
years poring over equations
for wing sections and wind speeds.
It's stuff that I know
like the back of my hand.
But does that qualify me for going up
in one of these on my own?
Absolutely not!
And on the other hand, these
guys don't really need to know
any of this stuff to make them
graceful acrobats in the air.
Not having a complete understanding
doesn't always matter.
We've still flown successfully
for over 100 years.
And now, it's my turn.
- [Instructor] And then
this is your diagonal line.
The strap that comes across.
This will dig in a little bit on take-off
when you're leaning forwards
and running down the hill.
- [Hannah] I can handle it.
- It should be a little bit uncomfortable.
- [Hannah] I can handle it.
- [Instructor] Don't
worry too much about it.
- And do you have quite a good feel
for where the thermals are?
- You have to have the
right weather conditions.
So, if you imagine a hill that
faces totally into the wind,
that's well drained, maybe darker,
and it will create this
kind of pool of warm air,
and then it will,
once it kind of reaches a
decent temperature difference,
it bubbles up through the atmosphere.
- Yeah, it's almost like we've
got kind of opposing skills.
And, like, they're sort
of about the same thing,
but you don't need my
skills to do what you do,
and I couldn't do what you do.
- I guess the ground-speed element
has a bit of maths in there.
I always thought the lesson
bit of maths to begin with.
Where's the wind coming from?
How strong it is?
How fast am I going to go, if
I'm pointing into the wind?
- But you're not solving
Navier-Stokes equations, are you?
- [Instructor] I don't
even know what that means!
- Yeah, exactly.
(both laughing)
Before the theoretical analysis
of aviation came along,
the practical side of flying
was mere trial and error.
Now, we have a much more
reliable understanding
of what keeps us aloft,
and it doesn't really matter
if the maths behind it
is, ultimately, a bit
fuzzy around the edges.
In the real world, the best that we can do
is just accept Godel's
Incompleteness Theorem
and get on with life.
Hey!
That's amazing!
- [Instructor] There's a thermal!
- Woohoo!
- You're feeling much more alert.
A bit stronger?
- Yeah.
We have to put aside, for the moment,
the question of whether maths
is invented or discovered,
because it now looks like
we may have to determine
which part of maths we're asking about.
You see, for me, Godel's work
highlights the distinction
between pure theoretical maths
and practical applied maths.
So, here is how I see things.
With mathematics,
there's a split down the
middle of the subject,
because the story changes
depending on what world you start with,
whether it's the real one
or one that exists in our imaginations.
And right now, when we're flying,
this is very much in the
realm of applied mathematics,
where everything is tangible and practical
and a little bit imprecise.
But, alongside that,
is where the more theoretical
pure mathematics lives.
That's where you have your proofs,
your paradoxes, and
incompleteness theorems.
A realm which doesn't match
up with a physical reality.
A sort of imperfect perfection.
Even though I instinctively
feel that maths is discovered,
I like that there is this
pure theoretical part of maths
that isn't found in reality.
And, since the maths there
doesn't need to match reality,
it's a convenient place where we can leave
all the weird contradictory
bits that we come across.
Woo!
However, I might have
it the wrong way round.
Although pure theoretical maths
seems rather divorced from reality,
that might merely reflect
the fact that reality
is not quite what we think it is.
And it's a reality that we can uncover
through the strange
maths of quantum physics.
The weirdest worlds that
most of us have come across
are likely to be in fiction, such as this,
Alice's Adventures In Wonderland.
The author, Lewis Carroll,
real name Charles Dodgson,
was actually a mathematics don at Oxford,
and a staunch traditionalist.
It's generally believed that
much of this surreal story
is a thinly veiled satire
on the new avant-garde maths
that was flourishing when
he was writing in the 1860s.
Still feels relevant today,
and applies equally well to
the new weird kid on the block,
quantum physics.
Take a close look at the
physical world around us,
and you can reduce it all to maths.
The solid bricks of our houses
or the blood cells in our veins
can all be reduced down into chemicals,
which comprise elements,
which themselves are made up of atoms,
comprising a tiny nucleus
of protons and neutrons
and electrons buzzing around
in a cloud of mostly emptiness.
The protons and neutrons in turn
are built from smaller subatomic particles
that we can't directly observe.
We can only verify their existence
using experiments and mathematics.
As we delve deeper into this world,
scientists have discovered
something very strange indeed.
We can never actually
know the precise location
of most particles in this
subatomic or quantum realm.
All we can know is the likelihood
of them being somewhere,
a mathematical formula that describes
the probability of their position.
All of this means we are,
fundamentally, at a quantum level,
just a great fuzz of
energy and probabilities.
I'm not sure Lewis Carroll
would have liked that.
And the only way to explore
this ill-defined quantum world.
Oh, hello.
Is through mathematics,
perfectly equipped to handle
strange probabilities.
It seems like there's
quite a lot of uncertainty
in quantum physics.
Does that bother you?
- Um, no.
When I heard that things
were, you know, uncertain,
and also against our common
sense in quantum physics,
then I thought, "Oh, wow,
that sounds interesting.
"I want to know more about that."
- The pivotal maths
behind the quantum world
was first laid out
by Austrian physicist
Erwin Schrodinger in 1926.
His equations accurately describe
the unusual behavior
of subatomic particles.
OK, all right, I'll tell you what, then.
Quantum physics lesson
101, where do we start?
Give me the lesson.
- Um, OK.
I would say we would have
to start with superposition.
So, let's talk about electrons.
So, they're a very small particle,
and they can be in two states.
They have a state called the spin,
and the spin can be pointing up or down.
So, if we were in the classical world,
the spin could only be either up or down.
But in the quantum world, the
spin is in a superposition,
which means it can be up
and down at the same time.
- Let me see if I understand this, then.
So, superposition is where something
is and isn't something at the same time?
- Yes.
We can think about some examples.
So, let's say that we have a cup,
and the cup is full of
water, that's one state.
Another possible state
is that the cup is empty.
So, if we were to bring the quantum ideas
to the classical world,
we would say the state, one
possible state of the cup,
would be to be empty and
full at the same time.
- [Hannah] OK, which you never see
in the world that we're living in,
you never see a cup that's full and empty.
- Yes, we don't.
- [Hannah] But you see this
a lot in the quantum world?
- Yes, superpositions
are an essential part
of the quantum world.
- Like a light being on
and off at the same time.
- Exactly.
Or the cake being eaten or
not eaten at the same time.
- [Hannah] OK.
It's a very tough idea to get around.
- Yes, yes.
- Given two possible outcomes,
in the quantum world,
we now have to allow for a third one,
the combination of both outcomes.
At the quantum scale, you can
have your cake and eat it.
This is such a weird idea.
How do we know it's real?
- Well, because we've done
many experiments to prove it
that show exactly that behavior.
- What does that experiment look like?
- Well, if we put it, say,
in terms of things we
have here on the table.
We could think about,
let's say that I wanted
this piece of sugar
to come into my cup,
but there is this pot in the middle.
So then, if the sugar is going
to come from here to my cup,
it could either go this way
or that way in the classical world.
But in a quantum experiment,
it can take both routes at the same time,
and I would be able to
distinguish that it did that
if I did a quantum experiment.
- I can't, this is too weird!
- Welcome to my world.
So, you go through this
whole transition, from first,
the ideas and the mathematics,
and then up to showing
it in the experiment.
- [Hannah] What came out
of Schrodinger's mouth
was a prediction of
something even stranger
that can sometimes be
produced when particles
interact in the quantum world,
a phenomenon called entanglement.
All right, tell me about
entanglement, then.
- OK, so take two electrons.
If the electrons are entangled,
and I do something to
one of the electrons,
for example change the
direction of the spin,
that will instantaneously affect
the state of the other electron,
even if they are separated long distances.
- How far away are they from each other?
- Well, they can be a few centimeters,
but now the latest experiments,
they're using satellites,
show entanglement across 1,200 kilometers.
- What?
- Yes.
- You've got something
over here, and you do?
And something 1,200 kilometers away.
You do something to one, and it instantly.
The other one instantly
knows what's happened?
- Yes, you'll affect the state
of the other one instantly.
- [Hannah] Apparently,
there is no cause or link.
The only thing we can say
is that the two particles
are synchronized.
How does one know what
the other one's doing?
- Well, that we're still
trying to understand,
because that's what mathematics tells us,
and then we can show it in the experiment,
but we're still struggling to
understand what that means.
And one of the reasons why
we don't understand it.
And, you know, like you're asking,
is because we don't see
it in our everyday lives.
So, let's say it's not part
of our experience and common sense.
But that doesn't mean it doesn't happen.
I don't want you to go crazy.
- So, quantum mathematics
has made predictions,
which have been discovered to be true.
But, despite that, the
quantum world is so weird,
it suggests to me that the maths
behind it is just invented.
It feels like what we're seeing
is evidence of a man-made
system being pushed too far.
These are the absurdities that appear
when it's applied to situations
it wasn't designed for.
But my quest to find the truth about maths
takes me back to nature.
There is amazing new evidence
that quantum processes
might actually be crucial
to our own existence,
and much of life on Earth.
That would strengthen the argument
that mathematical processes
are intrinsic to our world.
That maths is discovered.
It all comes down to the photosynthesis,
the process that converts sunlight
into chemical energy used in life.
It takes place in the
molecules called chlorophyll,
which can be found in
plants, algae and bacteria.
- In bacteria, we have something
that's similar to what we have in plants.
- So, this is the stuff
that captures the sunlight?
- Exactly.
Each of these molecules,
each of these little blue
things here that I'm showing,
is bacterial chlorophyll,
and if we take it apart,
it will capture light.
- The chlorophyll captures light
by absorbing particles
of light, or photons.
- So, a photon is absorbed,
and it's absorbed by all of them,
so energy shared by all of
these bacterial chlorophylls,
and that sharing, we call it
using a quantum superposition.
- Because it's coming in
and hitting one of these,
but all of them are somehow.
- In a way, it's as if
each of the electrons
of the chlorophylls are
talking to each other
and sharing the energy around.
- The subatomic particles
in the chlorophyll
are synchronized in a way
that can only be described
by quantum mechanics.
Does it do a good job?
I mean, is it efficient?
- That is part of why
photosynthesis is efficient.
Because, by sharing the
energy among all of them,
it's easier to transfer the
energy to another molecule.
Imagine if you have to
share the energy one by one,
you have to explore each part separately,
but if you share the energy all together,
you explore all the
parts at the same time.
- Every leaf on every plant on the planet
has been following these quantum rules
for millions of years.
And we still don't fully
understand how they do it.
Without quantum physics,
despite all the mathematical
uncertainties and ambiguities,
plants wouldn't produce
oxygen so efficiently.
And without oxygen, we wouldn't exist.
- The systems are amazing,
because they are effectively the interface
between using a little
bit of classical mechanics
and a little bit of quantum mechanics
to operate in a wonderful way.
- Ultimately, quantum mechanics
is at the heart of photosynthesis
and, well, I guess, all of life on Earth.
- It is, it is.
We can say life is nothing
but quantum mechanics
giving us energy.
- So, what does all this
mean for our key question
about the origins of maths?
There is no shortage of evidence
that mathematical rules
are intrinsic to the world.
We keep discovering them everywhere.
However, we now know we have to take
some of the maths on faith,
and believing in the numbers
is taking us to a very strange world,
with crazy notions like superposition
and entanglement at the core of it.
Quantum mathematics is inextricably linked
to the world as we know it.
linked to the world as we know it.
Because the world is actually
a whole lot weirder than we thought.
What quantum mechanics does do
is force us to question what is real.
And what is reality, anyway?
Just how much light can
mathematics shed on reality?
With the world stripped bare,
exposing the nuts and bolts of existence,
what does maths tell us
about this realm of subatomic particles?
The maths that underlies it
isn't particularly pretty,
but it can all be written
out in just one equation.
This is the formula that describes
the constituents of the universe.
It has become well enough
accepted to be called
the standard model of subatomic physics.
I told you it wasn't pretty.
Now, you're just going to have
to take my word for it on this one.
This equation encapsulates all
of the fundamental properties
all of the fundamental properties
But there are a couple of sticking points.
For one thing, no-one has
ever satisfactorily explained
how our common-sense,
day-to-day version of the world
emerges from this kind
of subatomic reality.
All of that fuzziness,
all of that uncertainty
in the quantum world,
just how does it end up
giving us that comfortable,
familiar solidity of the normal world?
At the other end of the spectrum,
the solar system and beyond
is beautifully and accurately described
by a different equation.
Einstein's general relativity.
This remarkable equation
tells you about gravity,
about the warping of space-time,
about general relativity.
And when you take these two together,
these two single mathematical sentences,
they are enough to tell
you everything you need
about the fundamental
behavior of the universe
and everything in it.
There is nothing more
articulate than mathematics.
Maths seems to be written
into the physical universe.
So, on the one hand, at
the teeny-tiny scale,
the standard model of particle physics
does this amazing job.
And in the ginormous
scale, general relativity,
I mean, you couldn't
ask for anything more.
There's just one problem when you try
and put these two together.
They're incompatible.
The problem is that general relativity
breaks down in the quantum world.
Gravity simply doesn't apply to particles
at the subatomic scale.
Meanwhile, quantum effects
are virtually never seen
at the scale of humans and
planets, where gravity rules.
You and I are never in a
superposition of existing
and not existing at the same time.
So, what does this mean for us?
Are there two different worlds
each obeying their own
sets of mathematical laws?
Solving this conundrum is
one of the biggest problems
that puzzles scientists today.
We will ever reconcile the two?
- I think it's perfectly plausible
that, within our lifetime, somebody,
maybe somebody watching this program,
will discover the mathematical structure
which unifies Einstein's
theory of relativity
with quantum mechanics
and just provides the perfect
description of this world.
And that would be really exciting.
- Will we have one?
How do I know?
We would all like to have one.
But, you know, maybe
we are not smart enough
to formulate a theory
that combines everything.
It's hard.
- I do believe that there
are good ideas out there
and that eventually, it
might take a long time,
but, eventually, humans
will work this out.
I'm confident about that.
- So, will we make it
all the way to include
all possible forces at all possible scales
with all possible forms of matter?
It's a hope I have for our
species, that's all I can say.
- The incompatibility of
these two great theories,
general relativity and quantum mechanics,
creates a serious obstacle
for believing that maths
is really discovered.
And there's a bigger hurdle to come.
Many of the best proposals
to unify general relativity
and the quantum world
have consequences that are even weirder
than the problems they're trying to solve.
They predict the existence
of multiple universes.
This idea is rooted in the
mathematical explanations
of the quantum world,
and the work of its founding
father, Erwin Schrodinger.
The mathematics in Schrodinger's equation
insists that particles can exist
in multiple states at the same time.
And Schrodinger himself says
that these possibilities
aren't just alternatives, but
really happen simultaneously.
This can lead to multiple universes.
And the maths also suggests
there's an infinite number of them,
each slightly different from the others.
Thank you.
Mathematically speaking,
in an infinite universe,
everything that's possible
has to happen somewhere.
Yeah, that's right.
Everything possible happens, somewhere.
Even Schrodinger acknowledged
that the consequences of his equation
describing the quantum
world might seem lunatic.
But if there's one thing I've learned,
it's that you should trust the maths.
So, maybe our experience isn't special,
maybe our reality isn't unique after all.
- There are so many distinct
avenues of investigation
that lead to the
possibility of a multiverse.
From our studies of
unification and string theory,
from our studies of quantum mechanics,
even from the study of space
going on infinitely far.
Even that gives rise to a
version of the multiverse.
- If we're going to reject
everything that just seems weird,
we're almost guaranteed to reject
the true theories of the future
when they get discovered.
I think we should just chill out,
accept that the world is weird
and that's just part of its charm.
And trust the math.
(dramatic music)
- So, why does all of this matter?
Well, if maths really is discovered,
then there is an intrinsic truth
behind the maths we uncover
however weird that truth seems to be.
If maths is invented,
then how do we know what is true or false?
Is it true purely because we define it so?
And how does it relate to the real world
that we all experience?
In this series, we've seen that maths
can explain so much of our world,
from aerodynamics to planetary orbits,
from the subatomic world
to processes crucial to life on Earth.
And that is something I just
can't accept as a coincidence.
So, here's my take on things.
For me, it's almost as though you have
this alternate parallel mathematical world
that hides just beneath our own.
You can't see it, you can't touch it.
The only way that you can explore it
is by using the language
that we've invented.
All of those symbols and
equations and conventions
are our only tools of navigation,
and they are undoubtedly man-made.
But once you're inside that world,
once you're exploring the landscape
that mathematics has
laid out in front of you,
I am absolutely convinced
that you are on a voyage of discovery.
It is a world without a human designer.
So, ultimately, I think it's both.
Mathematics is a little bit of invention
and a lot of discovery.
Mathematicians will
probably never all agree,
and maybe we will never
find a definitive answer,
but the consequences of having that debate
is why it really matters.
- We have used mathematics
for a much deeper understanding of nature
and of the universe in general.
We know about the universe now,
things that, a few hundred years ago,
people didn't even know what to ask.
- Searching for the truth about maths
has, over 2,000 years of history,
transformed the human experience.
Discovering patterns everywhere in nature
has given us structure and
beauty and inspiration.
Inventing new areas of maths
has led to an explosion of technology
that, ultimately, underpins
modern trade and computing.
We have discovered powerful rules
that we continue to use to explore,
enhance and explain the world around us.
And we have had a tantalizing glimpse
of what could be to come.
- It's quite possible that
what we have been doing in
science for all the centuries
is, in some sense, looking for
our keys under the lamp post.
We have been able to use mathematics
to describe what happens out there,
but that could be the tip
of an iceberg of reality
that we as yet don't have
any understanding of,
haven't yet had any contact with.
- But most of all, I think
that asking questions
about the origins and truth of maths
has given us a purpose,
it's given us understanding.
Ultimately, maths has given us meaning.
(dramatic music)