# Dangerous Knowledge (2007) Movie Script

Beneath the surface of the world...

are the rules of science.

But beneath them, there is

a far deeper set of rules.

A matrix of pure mathematics,

which explains the nature

of the rules of science,

and how it is we can understand

them in the first place.

To see a world in a grain of sand,

And a heaven in a wild flower,

Hold infinity in the palm of your hand,

And eternity in an hour.

What is the system that...

that everything has to adhere to,

if there is no God?

You had these ideas, and...

and you had to be very careful

because at any moment,

they would bite you.

They sounded great but

they were very dangerous.

But then of course,

people get scared.

So they pull back from

the edge of the precipice.

Well, this is not a matter

of liking it or not...

You have here this proof and...

one has to live with it.

This film, is about how a small

group of the most brilliant minds,

unraveled our old cosey certainties

about maths and the universe.

It is also about how once they

had looked at these problems,

they could not look away...

and pursued the questions

to the brink of insanity,

and then over it,

to madness and suicide.

But for all their tragedies,

what they saw, is still true.

Their contempories largely rejected

the significance of their work,

and we have yet

to fully inhered it.

Today, we still stand

only on the threshold,

of the world they saw.

My name is David Malone.

And this is my hommage,

to former great thinkers,

who without most of us,

even having heard of them,

have profoundly influenced

the nature of our age,

and who's stories have, i think,

an important message for us today.

This is Halle.

A provincial town

in Eastern Germany,

where Martin Luther once

preached the reformation.

Our story starts here,

at the towns university

with a mathematics professor.

A man called: Georg Cantor,

who started a revolution he

never really meant to start.

But which eventually threatened

to shake the whole of mathematics

and science on it's foundations.

And he started this revolution by

asking himself a simple question:

how big is infinity?

Cantor is wonderful

because it's so crazy.

It's the equivalent

of being on drugs.

It's just an incredible

feat of imagination.

Georg Cantor is one of the greatest

mathematicians of the world.

Others before him, going back

to the Ancient Greeks at least,

had asked the question.

But it was Cantor, who made the

journey no one else ever had,

and found the answer.

But he paid a price

for his discovery.

This is the only bust

there is of Georg Cantor.

It was made just one

year before he died,

and he died utterly alone,

in an insane asylum.

The question is:

what could the greatest mathematician

of his century have seen,

that could drive him insane?

If all that Cantor had seen

was mathematics,

then his story would be

of limited interest.

But from the beginning,

Cantor realised his work

had far wider significance.

He believed,

it could take the human mind,

towards greater,

trancendent truth and certainty.

What he never suspected,

was that eventually his maths

would make that certainty

ever more elusive.

Perhaps even destroy the

possibility of ever reaching it.

If you want to understand Georg

Cantor you have to understand

he was a religious man.

Though not in a conventional sense.

He almost certainly

came to this church,

but that's not his God.

He wasn't interested in a God

who's mysteries were

redemption and resurrection.

Ever since he was just a boy,

he had heard what he called:

a secret voice,

calling him to mathematics.

That voice which he

heard all his life,

in his mind, was God.

So for Cantor,

his mathematics of infinity

had to be correct,

because God, the 'True Infinite',

had revealed it to him.

These things which are

now hidden from you,

will be brought into the light.

If you look at Cantor's

last major publication,

about Set Theory, in 1895.

It starts with three aphorisms,

and it's third motto

is from the bible,

and it's from Corinthians.

And it's, you know:

that which has been hidden

to you, will eventually

be brought into the light.

And Cantor i think, really

believed he was the messenger.

That this theory had been hidden.

He was God's means,

of bringing this Theory of

the Infinite to the world.

There is no contradiction for

Cantor between his religious

thinking and his mathematics.

He understood or

he was thinking that,

his ideas were a gift of God.

My view was that Cantor was

trying to understand God

and that this was really...

like a mathematical theology

that he was doing.

Cantor's God was the 'Creator God'.

The God who set the planets

spinning in their orbits.

Who's mysteries were the eternal

and perfect laws of motion.

Laws who's discovery had

launched the modern world,

and allowed us to see the world

as curves, trajectories and forces,

and which would one day

even put men on the moon.

The eternal certainties

layed down by God

which Newton and Leibniz

had discovered.

And it was infinity which

lay at the heart of it all.

But, there was a problem with it.

If you look at that beautiful

smooth curve of motion,

you notice it's not

actually smooth.

It's made of an infinite number of

infinitesimally small straight lines.

And each line is an instant

in which nothing moves.

But like frames of film, if you run

one after another, you get motion.

And it works. The whole thing

relies on infinity but it works.

And because it worked,

everyone said:

alright, we don't

understand infinity.

Just leave it alone.

Cantor comes along and says

no, if this whole thing

rests on infinity,

we have to understand it.

And now remember

Cantor is a religious man.

So for him, that symbol...

isn't just a scientific mystery.

It's a religious mystery as well.

Here's the maths that God uses

to keep creation in motion.

And at it's heart, lies the

deeper mystery of infinity.

But it was a mystery,

that had so far defied every

mind that had looked at it.

The first Modern Thinker to

confront the infinite was Galileo.

And he tried to do it using the circle.

This is how he did it.

He said: first of all,

draw a circle.

Now put a triangle inside it,

then a square and keep on going.

Keep adding sides.

Eventually he realised that

what the circle is...

is a shape with infinately many,

infinitesimally small sides.

Which seems great.

Now you can hold the infinite

and the infinitesimal

in your hands.

But as soon as he done that,

he realised it actually opened up

a horrible illogical paradox.

Because he said: alright, let's

draw a bigger circle outside.

And now with an infinitely sharp

pencil, draw from the center...

infinitely sharp lines. One for each

of the lines on the inner circle.

There's an infinate number of them,

that should be enough

for the inner circle.

But now extend those lines out

till they meet the outer circle.

Now, those lines are diverging...

which means when you

get to the outer circle,

if you look really carefully,

there will be gaps.

There won't be enough.

Galileo just said:

that makes no sense.

If there's an infinite number

it should be enough!

At which point he said:

we just can't understand the infinite!

Maybe God can, but with

our finite minds, we can't.

So, let's use the

concept if we must...

but let's not try

and understand infinity.

And that's exactly

how they left it...

until Georg Cantor came along.

At first it must have

seemed to Cantor,

that God really was on his side.

In the space of only a few years,

he married, began a family,

and published his

first ground-breaking paper

about infinity.

Where previously infinity had just

been a vague number without end,

Cantor saw a whole

new world opening up.

Cantor did a new step and he said:

i want to add one plus one.

And Cantor said:

ok, why can i not add

infinity plus infinity?

That's also possible!

And this was a starting

point of his theory.

Cantor found he could add

and subtract infinities...

and in fact discovered there

was a vast new mathematics

of the infinite.

You really finally feel

for the first time,

that the infinite is no longer

this amorphous concept:

well, it's infinite.

And that's all you

can say about it.

But Cantor says:

no!

There's a way you can

make this very precise

and i can make it

very definite as well.

By 1872, Cantor is a man inspired.

He's already grasped and understood,

the nature of real infinity,

which no one before him had done,

but in that same year,

he come's up here to the Alps...

to meet the only other man

who really understood his work:

a mathematician called,

Richard Dedekind.

And this time,

is probably the happiest and most

inspired period of Cantor's life.

Within a year of there meeting, he

announces an astonishing discovery:

that beyond infinity,

there's another larger infinity,

and possibly even a whole

hierarchy of different infinities.

Though it is contrary

to every intuition,

Cantor began to see that some

infinities are bigger than others.

He already knew that when

you looked at the number line,

it divided up,

into an infinite number

of whole numbers and fractions.

But Cantor found that as he

looked closer at this line,

that infinite though the

fractions are, each one...

is separated from the next by

a wilderness of other numbers.

Irrational numbers like pi.

Which require an infinite

number of decimal places

just to define them.

Against all logic,

the infinity of these numbers,

was unmeasurably, uncountably

larger than the first.

What had frightened Galileo,

Cantor had proved:

there was a larger infinity!

Today, Cantor's genius

continues to inspire the work

of some of the

greatest mathematicians.

Greg Chaiton, is recognised

as one of the most brilliant.

Well, infinity was

always there but it...

they tried to contain it.

They tried to...

to keep it in a cage.

And, people would talk

about potential infinity

as opposed to actual infinity.

But Cantor just goes all the way.

He just goes totally berserk.

And then you find that

you have infinities and

bigger infinities and

even bigger infinities

and for any infinite

series of infinities,

there are infinities that are

bigger than all of them.

And you get numbers so big

that you wonder

how you could even name them?

You know infinities so big that

you can't even give them names?

This is just...

It's just fantastic stuff!

So in a way what he's saying is,

giving any set of concepts,

i'm going to invent

something that's bigger.

So this is...

this is paradoxical essentially.

So there's something inherently

ungraspable, that escapes you

from this conception.

So it's absolutely breathtaking.

It's great stuff!

Now, it may not have

anything to do with

partial differential equations,

building bridges,

designing airfoiles, but who cares?

The shear audacity

of Cantor's ideas,

had thrown open the doors,

and changed mathematics forever.

And he knew it!

We can't know

exactly how he felt...

but Greg Chaitin has also felt those

rare moments of profound insight.

You know, here we are

down in the forest and...

and we can't see very

far in any direction.

And you struggle up,

ignoring the fact that

you're tired and weary.

You struggle up a mountain,

and the higher you go

the more beautiful and

breathtaking the views are.

And then...

If you're lucky you get

to the top of the mountain.

and...that can be a transcendant

experience, you know...

A spiritual person would say

they feel closer to God.

You have this breathtaking view.

All of a sudden you can see...

in all directions,

and things make sense.

It's beautiful to

understand something

that you couldn't

understand before,

but the problem is,

the moment you understand one thing,

that raises more questions.

So in other words,

the moment you climb one mountain,

then you see off in the distance...

Behind the haze are

much higher mountains.

His theory is all about the fact

that the mountains get

higher and higher.

And no range is ever enough because

there are always mountain ranges

beyond any range that you can

understand or conceive of.

So this has a tremendously

liberating effect on mathematics,

or it ought to!

But then of course,

people get scared.

So they pull back from

the edge of the precipice.

What was inspiring for Cantor,

frightened his critics.

They saw mathematics as the

pursuit of clarity and certainty.

Everything Cantor was doing:

his irrational numbers

and his illogical infinities,

seemed to them to be

eating away at certainty.

He soon faced the deep

and implacable hostility.

This is the main lecture theatre

in the university

where Cantor spent

his entire professional life.

A life that he felt trapped in.

And i think there's

some justification.

Other mathematicians,

actually tried to prevent

Cantor publishing his papers.

Cantor always dreamed that

he'd receive an invitation

to one of the great universities

like Vienna or Berlin,

but they were invitations

which never came.

And he was also

attacked personally.

The great mathematician

Henri Poincar, said..

that Cantor's mathematics

was a sickness

from which one day

maths would recover.

And worse...

His one time friend

and teacher, Kronecker...

said that Cantor was

a corrupter of youth.

Cantor felt,

that he and his ideas

were being caged,

or quarantined here

as if they were,

some kind of sickness.

The genie...

got out of the bottle.

It was a very

dangerous genie because

you see, the concepts,

that Cantor played with

are intrinsically

inherently self-contradictory.

And people don't like

to face up to that.

They've emasculated Set Theory.

They have this..this version,

which is safe, called:

"Zermelo-Fraenkel Set Theory".

Which is a sort of

a watered-down...

But you see, that takes

all the fun out of it!

The...for me, the fun...

Cantor was...he was...

He was playing on the edge!

You know, the idea was,

you had these ideas, and...

and you had to be very careful

because at any moment

they would bite you.

They sounded great but they

were very dangerous.

You see, they were

almost self-contradictory.

The notion of the

'Set of Everything' for example,

is self-contradictory.

And...it's...

and people got frightened.

His critics feared Cantor

was going to dislodge

the certainty and clarity

vital, to mathematics and logic,

which might not be able

to be put back.

It seemed Cantor had opened maths

to the very thing it was

supposed to save us from:

irresolvable uncertainty.

Cantor knew the only way to

convince his critics,

was to make his theory complete.

Could he show there was

a logic to his infinities?

Some system,

that bound them all together?

What he absolutely

must decide now,

is, what's the relationship

between them.

If he can do that,

then his theory is perfect.

If he can't,

then all he has is bits.

So he has to decide

what's the relationship

between them.

And that question,

is the 'Continuum Hypothesis'.

No matter how isolated he became,

the more he was opposed,

the more he struggled.

Where another person might

have given up, Cantor didn't.

Clinical psychologist,

Dr. Louis Sass,

suggests it is precisely this

ability to be isolated,

which is key, to Cantor's genius.

I think, that willingness to

step into a realm...

you know, beyond the...

the taken for granted,

is abolutely essential.

But i think if you're a person

who takes that step,

in a way you're already doomed,

to living outside in some way.

So, you know...

It's not as if it's only the

intellectual project itself

that takes you out there.

There is something about you

as a person, that is just...

That unnaturalness, so to speak,

comes so naturally to you.

Cantor was trapped.

There were too many things that

went to the core of who he was,

for him to be able to give up.

When Cantor was just a boy,

his father sent him a letter...

which became his most

precious possession,

and which he carried

with him all his life.

In it, his father told him,

how the whole family

looked to him,

to achieve greatness.

How he would come to nothing,

if he did not have the courage to

overcome criticism and adversity.

How he must trust in God to

guide him, and never give up.

And he never did.

Well i think, here you come to

the root of the problem for Cantor

of a theory,

that he was certain, was correct,

in part because he believed that

it had come to him

as a message from God.

There's a very important

religious aspect to Cantor's...

struggle to deal

with the infinite,

and face the problems of...

not being able to resolve many

of the open questions

that he himself raised

for the first time.

By 1894, Cantor has

been working solidly

on the Continuum Hypothesis

for over two years.

At the same time,

the personal and professional

attacks on him...

have become more

and more extreme.

In fact he writes to a friend

saying he's not sure

he can take them anymore.

And indeed, he can't.

By May of that year, he has

a massive nervous breakdown.

His daughter describes how his

whole personality is transformed.

He will rant and rave,

and then fall completely and

uncommunicatively silent.

Eventually, he's brought here...

to the 'Nervenklinik' in Halle,

which is...an asylum.

Today, we would say Cantor suffered

from manic depressive illness.

From Cantor's time,

we have left,

the case notes of

most of his psychiatrists.

In the notes for example,

we see that he, at times,

was quite disturbed,

was screaming...

and see that he was

really suffering from...

severe bouts of mania.

Sometimes he would be angry...

and he would have

ideas of grandeur

and sometimes he had

also ideas of persecution.

After his breakdown, everything

about Cantor is transformed.

He tells a friend he's not sure he'll

ever be able to do mathematics again.

He asks the university if he

can stop teaching maths

and teach philosophy instead.

But interestingly,

during this whole time...

despite having claimed, him not

being able to do mathematics again,

he never stops working on

the Continuum Hypothesis.

It's as if...

he just can't put it down,

can't look away.

You can only think:

i must find the proof!

This i can understand because

when you are a mathematician,

then you are for all the

time a mathematician.

It's a form of...living for you.

You must think

about mathematics, and...

you can't think anything else...

the whole day.

You are thinking and

thinking and thinking.

And you say:

i must find it!

I must, i must, i must!

You can't think anything else.

In August of 1884,

he writes a letter,

to his friend and colleague,

the last man who still

publishes his work.

A man called, Mittag-Leffler.

And the letter is ecstatic.

He says: i've done it!

I've proved the Continuum

Hypothesis. It's true.

And he promises that he'll send

the proof in the following weeks.

But the proof never comes.

Instead, three months later,

a second letter arrives.

And in this one, you can

feel Cantor's embarrassment.

He says: i'm sorry i should never

have claimed that i proved it.

And he says: my beautiful

proof lies all in ruins.

And you can see the wreckage

of his work, in the letter.

But then, three weeks after that,

this letter arrives:

and in it he says:

i've proved that the Continuum

Hypothesis is not true.

And this pattern continues.

He proves that it is true...

and then he's convinced

that it's not true.

Back and forth.

And in fact,

what Cantor is doing...

is driving himself slowly insane.

One of the things that will happen

especially in the early stages and,

the stages just before

a schizophrenic break,

but also in the early stages,

will be that the patient is...

in a way,

looking too hard at the world

and too concentrated away.

As a kind of rigidity of

the perceptual stance.

When he could not solve

the Continuum Hypothesis,

Cantor came to describe

the infinite, as an abyss.

A chasm perhaps,

between what he had seen...

and what he knew must be there,

but could never reach.

What can happen, is that

some object in the world that...

that the rest of us would just...

consider just a sort of

random thing there,

seem somehow symbolic in some way.

There's a way in which in

order to understand something

you have to look very hard at it.

But you also have to be able to

sort of move away from it

and kind of see it

in a kind of context.

And the person who stares too hard

can often lose that sense of context.

Cantor never fully recovered.

For the rest of his life...

he would be drawn back to work

on the problem he could not solve.

And each time,

it would hurt him, profoundly.

In 1899, Cantor had

returned once again

to work on the

Continuum Hypothesys.

And again it made him ill

and he returned to the asylum.

He was just recovering

from this breakdown,

when his son Rudolf died, suddenly.

Four days short of

his thirteenth birthday.

Cantor wrote to a friend,

saying how his son had

had a great musical talent,

just as he had had

when he was a boy.

But he had set music aside,

in order to go into mathematics.

And now with the death

of his son,

he felt that, his own dream

of musical fulfillment

had died with him.

Cantor went on to say,

that he could no longer even

remember why he himself...

had left music,

in order to go into maths.

That secret voice, which had once

called him on to mathematics,

and given meaning

to his life and work,

the voice he identified with God...

That voice too, had left him.

Here we have to leave Georg Cantor,

because if we treat

Cantor's story in isolation,

it makes it into a tragic

but obscure footnote,

to the broader sweep of history.

Where as in fact, the fear that

Cantor had dislodged something,

was part of a much broader feeling,

that things once felt to

be solid, were slipping.

A feeling seen more clearly in the

story of his great contemporary:

a man called, Ludwig Boltzmann.

Just as Cantor had revolutionary

ideas in mathematics and was opposed,

so Boltzmann, his contemporary,

had revolutionary ideas in physics,

and was equally opposed.

This is Ludwig Boltzmann's grave.

And that...carved on it,

is the equation which killed him.

And it did so, because like Cantor...

Boltzmann's ideas were out

of step with his times.

Cantor had undermined the ideal of a

timeless and perfect logic in maths.

Boltzmann's formula and

his destiny...

was to undermine the ideal

of a timeless order in physics.

Together, their ideas were part of

a general undermining of certainty,

in the wider world

outside of maths and physics.

Boltzmann's and Cantor's times

craved certainty,

in politics, in art, as well as

in science and philosophy.

They were times that looked on

the surface, solid and certain...

but felt themselves to be

teetering and sliding.

The old order was dying.

And it was as if they could already

feel disaster's gravitational pull.

In Vienna,

which was called by Karl Kraus:

"Laboratory for Apocalypse",

there was this feeling that...

this political construct

of the Habsburg Empire

couldn't last for much longer.

They were very strange times.

On the one hand,

those in power spent 20 years

building the monuments

of imperial Vienna,

to declare that this order, firm on

it's foundations, would last forever!

The rich man would

always be in his castle;

the poor man always at his gate.

But on the other hand, the empire

was actually on it's last legs.

And the intellectual

tenor of the times,

was summed up by

the poet, Hofmannsthal...

who said that, what previous

generations believed to be firm,

was in fact, what he called:

"das Gleitende".

The slipping, or sliding away

of the world.

I thing that describes

the feeling in Vienna.

In other places too,

but particularly in Vienna in

this capital of an empire that...

hadn't crumbled yet but,

it looked like...set to break down.

This characterizes it very well.

And it was against this background,

that the scientific questions of

Boltzmann's times were understood.

This is the Great Courtyard

of the Univerity of Vienna...

and these are the busts

of all of the greats who

have ever thaugt here.

Boltzmann's is here too.

But many of his contemporaries,

men, more influential in their

day even than he was,

lined up to oppose

him and his ideas.

But their opposition was as much

ideological, as it was scientific.

The physics of Boltzmann's time,

were still the physics of certainty.

Of an ordered universe,

determined from above,

by predictable and timeless

God-given laws.

Boltzmann suggested,

that the order of the world was

not imposed from above by God,

but emerged from below.

From the random bumping of atoms.

A radical idea,

at odds with his times,

but the foundation of ours.

Professor Mussardo, lives and works

in Trieste, on the Adriatic Coast.

Not far from where

Boltzmann's live ended.

He is an expert on Boltzmann, and

works on the same kind of physics.

I think that there were two

reasons why he could not...

get fully accepted and recognized

by the German physicists.

One of them was,

that he based all his theories on

atoms, that people can't see.

And this was the reason of the very,

very strong criticism by Ernst Mach.

One of the most influential...

Philosophers of Science

at that time.

So the criticism of Mach

was simply:

i can't see an atom...

I don't need them,

they don't exist.

So why should we bring

them in the game?

Worse than insisting on the reality

of something people could not see,

to base physics on atoms,

meant to base it on things who's

behaviour was to complex to predict.

Which meant an entirely

new kind of physics.

One based on probabilities,

not certainties.

But then there was a second aspect;

it was revolutionary as well.

And this consists in...

putting forth and emphasizing

the role of probability,

in the physics world.

And people were used to the laws of

physics and science as being exact.

Once established,

they stay there forever.

There is no room for uncertainty.

So, introducing into the game,

two ingredients like invisible

atoms and probabilities,

means there is no certainty.

You can predict what is probably

going on, but not certainly.

Well, this really contrasted

very, very much with the...

the scientific spirit of the time,

and therefore this produced trouble.

Boltzmann's genius, was that

he could accept probability.

This meant he could begin to

understand complex phenomenon,

like fire and water and life.

Things which traditional physics;

the physics of mechanics,

never could.

But because his solution

relied on probability,

and probability

undermines certainty,

no one wanted to hear him.

And so just like Cantor,

he faced implacable opposition

which he too, found extremely

difficult to deal with.

It seems like Boltzmann was just

the wrong man in the wrong place.

Absolutely...absolutely.

Absolutely, it's true.

It's true.

He could just had his idea

twenty years later...

And in England he would have been

the most succesful physicist

of that time, it's true.

Somehow he met all his enemies.

So he met Ernst Mach, often.

Their careers even crossed,

in a very...in a very...

He meets all his enemies

but none of his friends.

Not his friends, it's true.

It is not hard to see

how Boltzmann's ideas

where so radically at

odds with his times.

Especially when applied not just

to physics but to the social world.

Classical science, classical physics...

gives you this image of a

God-ordered creation.

Where everything is set in stone,

according to perfect

and eternal rules.

Everything is predictable.

Everything has it's place and

everything is in it's place.

But when you come here, to the

Central Cemetery of Vienna,

you see the idealised

vision of that idea.

Because here,

everything is predictable.

Everybody does have a place

and everybody is in their place.

But the problem is...

while such certainty might

seem desirable politically,

the real world...the living world,

the world described by

thermodynamics, just isn't like that.

A timeless and perfect world

never changes, but it is dead.

The real world,

the thermodynamic world is alive

precisely because it

is full of change.

But of course...

that life giving change also

brings with it, disorder and decay.

But then the problem arises,

if you then say:

well, Newtonian Mechanics

on which you are depending

is reversible in time.

So how can you derive a law,

which is asymmetrical in time

from basic principals which

are symmetrical in time.

You run the clock backwards,

it's just as good

Newtonian physics as

you run it forwards.

Yet the entropy

increases in the future.

But it is precisely this

accumulation of disorder and decay,

that science calls 'entropy',

which Bolzmann had understood.

In short this is really

the 'arrow of time'.

I mean, you can

measure the arrow of time

just seeing how things become

more and more disordered.

So it's a natural tendency

in the world

that Bolzmann quantified precisely.

Indeed i call him

the genius of disorder.

Boltzmann's work on entropy,

showed why no system can be perfect.

Why there must always

be some disorder.

It also revolutionized

the idea of time in physics.

In classical physics,

everything, including time

can run equally well

forwards as backwards.

Yet in thermodynamics, while

everything else is reversible,

time moves inexorably forward,

like an arrow.

The idea of entropy,

had a profound philosophical

and political significance.

Entropy is what changes

the ticking of a clock

into the destroyer of all things.

It is wat underlies the inexorable

passage from youth to old age.

Entropy is decay, and with the

decay nothing lasts forever.

Boltzmann had in essence,

captured mortality in an equation.

Physics now declared...

that no order, not even a

God-given one, will last forever.

That there was no natural order

that God had set in stone,

had already been pointed out by the

scientist Boltzmann most admired:

Charles Darwin.

In place of timeless perfection,

was a dance of

evolution and extinction.

With his equation of entropy,

Boltzmann brought this

picture of constant change,

into the very heart

of physics itself.

Did Boltzmann understand

the similarity?

Almost certainly.

When Boltzmann was asked how

his century would be remembered,

he did not chose a physicist.

He said it would be

the century of Darwin.

So he likes in Darwin, the momentum,

the evolution of life,

that is not static,

the fact that he's progressive.

Progress sometimes

has a jump in it.

And the fact that he can adress a...

a "life aspect",

with ideas of science,

that before was kind of

an ideological ground.

Bolzmann's ideas, like Cantor's and

Darwin's were revolutionary,

even though he did not

think of them that way.

But his times were frightened times.

Times when people felt new ideas,

could upset societies

fragile structure...

and bring it down.

At the end of the 19th century,

Viennese society was searching for

some certainty, some principal...

wether it was in politics,

philosophy, the arts or science.

But there didn't appear

to be any philosophy,

capable of holding

everyone together.

Upon which everything

else could be based.

So when the university

commissioned Gustav Klint,

to paint a ceiling to

celebrate philosophy,

this is what they got:

Such was the outrage,

that twenty professors petitioned,

to have the painting removed.

Now whatever else it is,

it's not a celebration of certainty.

The radicals of Bolzmann's times,

knew, the old order, with it's

worn-out certainties was doomed.

But Viennese culture, was not

ready to embrace the new.

And Boltzmann,

was caught in the middle.

As a scientist, his personality

entered deeply into the game,

because he was very stubborn.

Not self-ironic.

He could not take criticism.

He always took it personally,

and Boltzmann was definately

a passionate man.

He used to swing rapidly from

incredible joy to deep depression.

As Boltzmann got older, and more

exhausted from the struggle,

these mood swings became

more and more severe.

More and more of

Boltzmann's energy,

was aborbed in trying

to convince his opponents,

that his theory was correct.

He wrote:

no sacrifice is too high for

this goal, which represents

the whole meaning of my life.

In the last year of Boltzmann's life,

he didn't do any research at all.

I'm talking about

the last ten years.

He was fully immersed in dispute,

philosophical dispute...

Tried to make his point,

writing books,

which were most of

the time the same,

repeating the same

concept and so on.

So you can see he was in a loop...

that didn't go ahead.

But by the beginning of the 1900's,

the struggle was

getting too hard for him.

Boltzmann had discovered one

of the fundamental equations

which makes the universe work,

and he had dedicated his life to it.

The philosopher Bertrand Russel

said that for any great thinker,

this discovery that everything

flows from these fundamental laws,

comes, as he described it,

whith the overwhelming

force of a revelation.

Like a palace, emerging

from the autumn mist,

as the traveller ascends

an Italian hillside.

And so it was for Boltzmann.

But for him,

that palace was here,

at Duino in Italy,

where he hung himself.

In 1906,

Boltzmann came here to Duino,

with his wife and

daughter on holiday.

Exhausted and demoralised,

his ideas still not accepted.

While they were out walking,

he killed himself,

and left no note of explaination.

Of course we can never know

what Boltzmann was thinking,

but i think we have clues.

Boltzmann knew what it was,

to be in the grip of a

beatiful and powerful idea.

He once wrote that,

what the poet laments,

holds for the mathematician:

that he writes his works,

whith the blood of his heart.

So we know that he

was a passionate man.

But i think there is another clue.

At the start of one of Boltzmann's

major scientific papers,

he quotes three lines

from Goethe's Faust:

"Bring forth what is true."

"Write it so it's clear."

"Defend it to your last breath."

Which of course he does.

But i think there's

something deeper here.

Why quote Faust, at the

start of a scientific paper?

The pact, that Faust

makes with the devil,

is that the devil will give

him all of the knowledge

and all of the experience

that he wants,

so long as he never asks to stay,

in any one moment.

And i think when

Boltzmann came here,

to this beautiful place,

after thirty years of fighting

for what he believed in,

he simply said:

i want to stay here, in this

perfect, beautiful moment.

I don't want to have to leave.

I want time, for me, to stop.

The great and controversial thing

that Boltzmann had done,

was to introduce,

into the unchanging perfection

of classical physics,

the notion of real time.

Of irreversible change.

And yet it was this man,

who in his final moments,

wanted time to stop.

So ironically, Boltzmann was

vindicated just after his death.

If he would have

waited a little longer,

Boltzmann would have been

one of the fathers of the

revolution of the

twentieth century fysics.

Yet Boltzmann died as he had lived:

out of step with his times.

He had sawn the seeds of

uncertainty and fysics,

but no school of followers

took up his work.

Against all the odds,

it was Cantor,

who had uncovered the

uncertainty in mathematics,

around whom followers

where gathering.

A new generation of mathematicians

and philosphers were convinced:

if only they could solve the

problems and paradoxes

that had defeated Cantor,

maths could be made perfect again.

The most prominent amongst them,

Hilbert, declared:

the definitive clarification of the

nature of the infinite,

has become necessary for the honour

of human understanding itself.

They were so concerned to

find some kind of certainty,

they had come to believe

that the only kind of understanding

that was really worth anything,

was the logical and the provable.

And a measure of how desperate this

attempt to find the perfect system

of reasoning and logic

had become, is this:

three volumes of the Principia

Mathematica, published in 1910.

It takes a huge chunk

of this volume,

just to prove, that one

plus one equals two.

And a large part of that proof,

revolves around the problems

of the finite and the infinite,

and the paradoxes that

Cantor's work had trown up.

But despite the Principia,

there was now the feeling

that the logic of maths,

had undone itself,

and it was Cantor's fault.

As the Austrian writer, Musil

wrote at the time:

suddenly mathematicians, those

working in the innermost region,

discovered that something

in the foundations,

could absolutely not

be put in order.

Indeed, they took

a look at the bottom,

and found that the whole edifice,

was standing on air.

Cantor had stretched the limits of

maths and logic to breaking point,

and paid for it.

Much of the last

twenty years of his life,

was spent in and out

of the asylum.

The last time that Cantor came

here to the Nervenklinik in Halle,

was in 1917, and he truly

did not want to be here.

He wrote to his wife,

begging her to let him come home.

He was one of only

two civilians left here.

The rest of the place, was filled

with the casualties of World War I.

But of the 6th of January 1918,

the greatest mathematician

of his century,

died alone in his room,

his great project still unfinished.

Cantor had dislodged the pebble,

which would one day

start a landslide.

For him, it had all

been held together.

The paradoxes resolved, in God.

But what holds our ideas together,

when God is dead?

Without God,

the pebble is dislodged,

and the avalanche is unleashed,

and World War I, had killed God.

Here at last,

was the slippage.

Well, hasn't there always been a

desire in the history of the West

to find certainty or...maybe,

there wasn't so much a desire

in earlier era's because, the

assumption was that we had that.

You know, there was God!

And, you know even Descartes,

despite all of his scepticism,

assumes...

for him unproblematically,

that there is a God.

So what happens when that really,

really comes in to question?

After the death of God,

so to speak.

And along with the death of God

is a...is a loss of faith in some...

supernatural order,

of which we are a small part.

No one won the Great War.

Nothing was resolved at Versaille.

It was merely an armistice.

And none of the intellectual

crises that proceded it,

had been resolved either.

Things like the Principia, had

merely papered over the cracks.

In a way, the Principia was

like the Versaille Treaty,

only a lot more substantial.

This is basicly ten thousend

tonnes of intellectual concrete

poured over the

cracks in mathematics.

And for a while, it looked

like it really might hold.

But then a young man came here

to the university of Vienna,

to this library.

His name was Kurt Gdel.

And the work that he did here,

brought that dream of finding

the perfect system of reasoning

and logic, crashing down.

Gdel was born the year

Boltzmann died: 1906.

He was an insatiably

questioning boy,

growing up in unstable times.

His family,

called him: "Mister Why".

But by the time he

went to university,

World War I was over.

But Austria like the rest of Europe,

was in the grip of the depression,

and Hitler was forming

the National Socialist Party.

Gdel for his part,

became one of a brilliant group

of young philosophers,

political thinkers,

poets and scientists,

known as 'The Vienna Circle'.

Chaos was good because it ment

that there was no central authority

that was imposing ideas

so individuals could come

up with their own ideas.

The chaos around them,

on the one hand

had a liberating effect.

And on the other hand they were

desperately searching for ideas,

that they could believe in because

everything else around them

was crumbling in a heap.

So you'd want to

find some beautiful ideas

that you could believe in.

Though Gdel was surrounded by

radicals and revolutionary thinkers,

he was not one himself.

He was an unworldly and exact man,

who believed, like Hilbert,

that maths at least,

could be made whole again.

But it was not to be.

He certainly did not start out,

with trying to explode

Hilbert's program also.

In fact,

i think it came to Gdel...

ultimately as a surprise when

he showed that the next step,

to show the completeness of

arythmetic, was unachievable.

There was actually something

very mysterious happening

in pure mathematics.

In it's own way as mysterious as

black holes, the big bang,

as quantum uncertainty in the atom.

And this was Gdel's

Incompleteness Theorem.

And at that time,

there was a mystery there.

The one place where you don't

expect there to be mystery

is in pure reason!

Because pure reason should be black

and white. It should be really clear.

But, pure reason,

the clearest thing there is,

was revealing that there were

thing that were unclear.

This is one of the cafs

where the Vienna Circle

used to meet regularly.

Late summer of 1930,

Gdel came to the caf

with two eminent colleagues.

Towards the end

of their conversation,

he just mentioned an idea

he'd been working on,

which he called

the 'Incompleteness Theory'.

And what he told them,

was that he had just proved,

that all systems of

mathematical logic, were limited.

That there would always be

some things wich while true,

would never be able to

be proved to be true.

What Gdel showed in

his Incompleteness Theorem,

is that, no matter how large

you make your basis of reasoning,

your axioms, your set of axioms,

in arythmetic, there would

always be statements

that are true

but can not be proven.

No matter how much data

you have, to build on,

you will never...

prove all true statements!

What this meant, was that the

great Renaissance dream,

that one day, maths and logic

would be able to prove all things

and give us a godlike knowledge.

That dream was over!

But this idea was so far away,

from what anyone else

was working on,

what anyone else even suspected,

that neither of these colleagues

understood what he had just told them.

It was as if...

there was an explosion, but the

blast wave hadn't hit them yet.

Unaware of what had happened

in the caf, the very next day,

Hilbert, now the grand old

man of mathematics,

stood up and gave a

lecture in Knigsberg,

in which he said:

"We must know!"

"We will know!"

The irony was,

that the very day before,

Gdel had proved, that there were

some things, we would never know.

Some, didn't like it.

Some...

In particular for instance, Hilbert.

It seems that at the beginning, he

was quite annoyed and even angry.

This is not a matter

of liking it or not...

You have here this proof and...

one has to live with it.

Are there any holes

in Gdels argument?

No, there are not.

This was a perfect argument.

This argument was so

crystal clear and obvious.

Gdel had joined Hilbert,

in trying to solve the paradoxes,

uncovered by Cantor.

Instead, he had just proved,

that would never happen!

His work, springing directly

from Cantor's work on infinity,

proved, the paradoxes

were unsolvable,

and there would be more of them.

But being right,

didn't make him popular.

So here we are again in the Great

Courtyard of Vienna University

with the busts of all

of the great thinkers...

except for Kurt Gdel.

There's no bust to Gdel here.

And i can't help but

feel that at least

part of the reason

that he's not here,

is simply due to the

nature of his ideas.

Ah! Well you see,

nobody wants to face him.

In my opinion nobody wants to

face the consequences of Gdel.

You see, basically people want to

go ahead with formal systems anyway,

as if Hilbert had it all right.

You see?

And in my opinion, Gdel explodes

that formalist view of mathematics.

that you can just mechanically grind

away on a fixed set of concepts.

So even though i believe

Gdel pulled out the rug

out from under it intellectually,

nobody wants to face that fact.

So there's a very ambivalent

attitude to Gdel.

Even now, a century after his birth.

A very ambivalent attitude.

On the one hand, he's the

greatest logician of all time

so logicians will claim him,

but on the other hand,

they don't want,

people who are not logicians

to talk about the consequences

of Gdels work, because the obvious

conclusion from Gdels work

is that logic is a failure.

Let's move on to something else.

And this would destroy the field.

Gdel too, felt the effects

of his conclusion.

As he worked out the true

extent of what he had done,

Incompleteness began to

eat away at his own beliefs

about the nature of mathematics.

His health began to deteriorate,

and he began to worry about

the state of his mind.

In 1934,

he had his first breakdown.

But is was after he

recovered however,

that his real troubles began,

when he made a fateful decision.

Almost as soon as Gdel has

finished the Incompleteness Theorem,

he decides to work on the great

unsolved problem of modern mathematics:

Cantor's 'Continuum Hypothesis'.

And this is the effect

that it has on him.

These are some pages from one

of Gdel's workbooks

and they all, look like this.

Beautifully neat,

beautifully logical.

Except for this one.

This is the workbook,

where he's working on

the Continuum Hypothesis.

Gdel, like Cantor before him,

could neither solve the

problem, nor put it down.

Even as it made him unwell.

There could be a danger...

a danger in it.

And perhaps there's also a danger

in it at the more existential

or personal, psychological level.

If you're a person,

who is already prone to

the kind of exaggerated...

intellectual, self-reflection,

self-conscienceness...

you may find that your,

intellectual work is

exaggerating, exacerbating

that tendency, which...

which of course can make

life more difficult to live.

He calls this the

worst year of his life.

He has a massive

nervous breakdown,

and ends up in a sanitorium,

just like Cantor.

We're talking about people

here who, of course are...

are capable of, and

maybe afflicted with,

the capacity to care

very, very much

about things that are

very, very abstract.

To really lose themselves in

these intellectual problems.

One of the sanatoria that Gdel

spent some time in, is here:

the Purkersdorf Sanatorium,

just outside of Vienna.

The Purkersdorf itself, was

build to embody the philosophy

that the calm,

smooth lines of rationalism,

are the cure for madness.

Ironic then,

that Gdel, driven mad by

pushing the limits of rationalism,

should come here to recover.

But while the man who had

proved, there was a limit

to rational certainty,

was in the sanatorium,

outside, a greater

madness was unfolding...

as a nation threw itself into

the arms of a demagogue

who promised, there was certainty.

Gdel's madness passed.

Austria's didn't.

In 1939, Gdel himself was

attacked by a group of Nazi thugs.

That same year, he reluctantly

left Austria, for America.

It was during these pre-war years,

that another brilliant young man,

Alan Turing, enters our story.

Turing is most famous, for his

wartime work at Bletchley Park,

breaking the German Enigma code.

But he is also the man,

who made Gdel's already

devastating Incompleteness Theorem,

even worse.

Turing was a much more

practical man than Gdel.

And simply wanted to make Gdel's

theorem clearer, and simpler.

How to do it, came to him,

as he said later...in a vision.

That vision...was the computer.

The invention that has

shaped the modern world,

was first imagined

simply as the means,

to make Gdel's Incompleteness

Theorem, more concrete.

Because for many, Gdel's proof

had simply been too abstract.

It's an absolutely

devastating result,

from a philosophical

point of view,

we still haven't absorbed.

But the proof was too superficial.

It didn't get at the real heart

of what was going on.

It was more tantalizing

than anything else.

It was not a good

reason for something so...

devastating and fundamental.

It was too clever by half.

It was too superficial.

It said: i'm unprovable.

You know, so what?

This doesn't give you any insight

into how serious the problem is.

But Turing, five years later...

his approach to Incompleteness...

that, I felt...

was getting more

in the right direction.

Turing recast Incompleteness,

in terms of computers.

and showed, that since

they are logic machines,

Incompleteness meant,

there would always some problems

they would never solve.

A machine,

fed one of these problems,

would never stop.

And worse...

Turing proved,

there was no way

of telling beforehand,

which these problems were.

Gdel had proved,

that in all systems of logic,

there would be some

unsolvable problems.

Which is bad enough.

Then Turing comes along,

and makes matters much worse.

At least with Gdel,

there was the hope,

that you could distinguish

between the provable

and the unprovable,

and simply leave the

unprovable to one side.

What Turing does,

is prove that in fact

there is no way of telling

which will be

the unprovable problems.

So how do you know,

when to stop?

You'll never know whether

the problem you're working on

is simply

extraordinarily difficult,

or if it's

fundamentally unprovable.

And that...

is Turing's 'halting problem'.

But Turing makes it

very down to earth,

because he talks about machines,

and he talks about whether

a machine will halt or not.

It's there in his paper.

He didn't call it...

didn't speak of it in those terms

but the ideas are really

there in his original paper.

That's where i learned them.

And this sounds so

concrete and down to earth.

You know, computers are

physical devices and you just...

You started running, and...

there are two possibilities:

if you start a program running,

a self-contained program running,

you know, with no input-output.

It's just there!

It's running on a computer.

And one possibility is

it's going to stop, eventually,

saying, i finished the work.

Come up with

an answer and stop...

Done!...Finished!

The other possibility is,

it's going to be searching forever

and never find what it's looking

for, never finish the calculation.

Just go on forever.

It's one or the other.

The problem is...

How can we tell that a

program is never going to stop?

And the answer is: there's no

systematic, general way to do it.

And this is Turing's

version of Incompleteness.

Turing get's Incompleteness;

Gdel's profound discovery,

he get's it as a corollary of

something more basic

which is uncomputability.

Things which, can not be calculated.

Things which no

computer can calculate.

In certain domains, most things

can not be calculated.

But that's your work isn't it?

You come along and make it worse, again!

I do my best.

As if the news wasn't bad enough!

Yeah, i do my best.

Some of it is already

contained there in...

in Gdel's...in Turing's paper

although he doesn't emphasize it.

Startling as the

halting problem was,

the really profound part of

Incompleteness for Turing,

was not what it said

about logic or computers,

but what it said about us,

and our minds.

Were we,

or weren't we, computers?

It was the question that went

to the heart of who Turing was.

Turing was a man

of two great loves.

The first, was for a young man:

Christopher Morcom.

The second,

was for the computer which

he felt he had

brought into this world.

His love for Christopher,

had a unique place in his life,

because Christopher had died,

tragically young.

Turing never recaptured

that first pure love,

but never let go of the memory,

of what it had been.

But when Turing developed

the idea of the computer,

he began to fall in love

in a very different way,

with the sheer power,

of what he had imagined.

He fell in love,

with the fantastic idea,

that one day, computers

would be more than machines.

They would be like children,

capable of learning,

thinking and communicating.

And the scientist in him,

could also see, that if our

minds were like computers,

then here, in our hands, was the

means to understand ourselves.

What started with Cantor,

as a question from

pure mathematics,

about the nature of infinity,

in Gdels hands,

became a question

about the limits of logic.

And now with Turing,

it comes into focus

as a queston about us,

and the nature of our minds.

There is this sort of standard view

that Turing was a computationalist.

And certainly, in a

certain stage of his life,

he did take that point of view.

He said: well, maybe you can

make one of these machines,

imitate the human mind.

But he was of course well aware

of these limitations of computers

and that was one of his

important results of his own.

I think he may have

shifted his view...

he may have vacillated a bit,

and had one view and then another

but then, when he really developed

the computers as actual machines,

he sort of took of and thought,

maybe these really are,

going to...

It's a kind of...

When you get into a scientific

thing, you get...totally...

You think, you know,

maybe this is solving all problems

but without realising the

limitations that are there,

and which are part of

his own...his own theories.

Turing understood,

that Gdel's and his own work,

said that if our

minds were computers,

then Incompleteness

would apply to us,

and the limitations of logic,

would be our limitations.

We would not be capable of leaps

of imagination, beyond logic.

Turing's personality is one thing.

His mathematics doesn't have to

be consistent with his personality.

There is his work on

artificial intelligence,

where i think he...

he does believe that...

machines could become

intelligent...just like people,

or better or different

but intelligent.

But if you look at his first paper,

when he points out

that machines have limits,

because there are numbers...

In fact most numbers,

can not be calculated

by any machine.

He's showing the power of the human

mind to imagine things that...

escape what any machine

could ever do...you see?

So that may go against

his own philosophy,

he may think of

himself as a machine,

but...his very first paper is...

is smashing machines.

It's creating machines and then

it's pointing out

their devastating limitations.

Turing was well aware

of these problems,

but desperately wanted to prove,

he could get the fullness

of the human mind

from mere computation.

And it wasn't just the scientist

in him, that wanted to do this.

Turing's personal philosophy,

which he stuck to all his life,

was to be free from hypocrisy,

compromise and deceit.

Turing was a homosexual,

when it was both illegal

and even dangerous.

Yet he never hid it,

nor made it an issue.

With computers, there are

no lies or hypocrisy.

If we were computers,

then we were the kind of creature,

Turing wanted us to be.

People could vacillate here.

They can have one view and

then wonder about this.

Is this really right?

And then have another view,

and play around.

If they're good scientists

they will do that.

They won't just doggedly

follow one point of view.

So i suspect Turing,

vacillated rather.

But, i think...

in a lot of his analysis

on criticisms of other people

who criticize his view,

he would show the flaws

in their arguments and say:

well look, you see:

it may still be...

despite all these theorems we

know about non-computability,

it still might be, that we

are computational entities,

and then point out:

well, because of this and

this loophole and so on.

And maybe he...

came to believe those loopholes

were sufficient to get him out.

But yet, he did do these things

like looking at oracle machines

which were sort of super

Turing machines; went beyond them.

They're not machines that you

could see any way of constructing

out of ordinary stuff.

But nevertheless,

as a theoretical entity,

these devices were...

theoretical things which would

go beyond, standard computers.

This tension, between the

human and the computational,

was central to Turing's life.

And he lived with it,

until the events

which led to his death.

After the war, Turing

increasingly found himself

drawing the attention

of the security services.

In the Cold War,

homosexuality was seen,

as not only illegal and immoral,

but also a security risk.

So when in March 1952,

he was arrested,

charged and found guilty

of engaging in a homosexual act,

the authorities decided, he was

a problem that needed to be fixed.

They would chemically castrate him

by injecting him with the

female hormone estrogen.

Turing was being treated

as no more than a machine,

chemically reprogrammed,

to eliminate the uncertainty

of his sexuality,

and the risk they felt it posed,

to security and order.

To his horror,

he found the treatment

affected his mind and his body.

He grew breasts,

his moods altered, and he

worried about his mind.

For a man who had always been

authentic, and at one with himself,

it was as if he had been

injected, with hypocrisy.

On the 7th of June 1954,

Turing was found dead.

At his bedside, an apple...

from which he had

taken several bites.

Turing had poisoned

the apple, with cyanide.

Turing was dead,

but his question was not.

Whether the mind was a computer,

and so limited by logic,

or somehow able

to transcend logic,

was now the question that came

to trouble the mind of Kurt Gdel.

Gdel was now working in America,

at the institute for advanced study,

where he continued to work,

as obsessively as he ever had.

Of course, Gdel recovered

from his time in the sanatorium,

but by the time he got here

to the Institute for Advanced

Study in America,

he was a very peculiar man.

One of the stories

they tell about him,

is if he was caught in the Commons,

with a crowd of other people,

he so hated physical contact,

that he would stand very still

so as to plot the

perfect course out,

so as not to have to

actually touch anyone.

He also felt he was being poisoned

by what he called "bad air",

from heating systems

and air conditioners.

And most of all, he thought

his food was being poisoned.

He insisted his wife,

taste all his food for him.

He would sometimes,

order oranges,

and then send them straight back

claiming they were poisoned.

Peculiar as Gdel was,

his genius was undimmed.

Unlike Turing,

Gdel could not believe

we were like computers.

He wanted to show

how the mind had a way

of reaching truth outside logic,

and what it would

mean, if it couldn't.

In principal you can

have a machine grinding away,

deducing all the consequences

of a fixed set of principles

and mathematics would

be static and dead.

I mean, it would just be

a question of mecanically...

deducing all the consequences.

And so...

and so mathematicians in a

sense would just be...machines.

I mean, Turing did think

that he was a machine.

I think he did.

And i think...

that paper on

the imitation game...

shows that.

And Gdel, clearly did not

think that he was a machine.

He thought that he was divine.

You know, that human beings

have a...devine spark in them

that enables them to create

new mathematics i think.

Why was Gdel, so convinced

humans had this spark of creativity?

The key to his believe,

comes from a deep conviction

he shared with one of the

few close friends, he ever had.

That other, Austrian genius,

who had settled at the institute:

Albert Einstein.

Einstein used to say

that he came here,

to the Institute for Advanced Study,

simply for the privilege of

walking home with Kurt Gdel.

But what was it that held this

most unlikely of couples together?

Because on the one hand,

you've got the warm

and avuncular Einstein

and on the other,

the rather cold, wizened,

and withdrawn Kurt Gdel.

And the answer i think,

comes from something

else that Einstein said.

He said that,

God may be subtle

but he's not malicious.

What does that mean?

What it means for Einstein,

is that however complicated

the universe might be,

there will always be beautiful

rules, by which it works.

Gdel believed the same idea

from his point of view to mean,

that, God would never

have put us into a creation,

that we could then not understand.

The question is,

how is it that Kurt Gdel can

believe that God isn't malicous?

That it's all understandable?

Because Gdel is

the man who has proved,

that some things can not be

proven logically and rationally.

So surely, God must be malicious.

The way he gets out of it,

is that Gdel, like Einstein,

believes deeply in intuition.

That we can know things,

outside of logic,

because we just...intuit them.

And they believe it

because they have both felt it.

They've both had

their moments of intuition.

Just like Cantor had had his.

He talks about new principals...

that the mathematician...

closing your eyes,

tuning out the real world,

you can try to perceive,

directly by your

mathematical intuition,

the platonic world of ideas,

and come up with new principles,

which you can then

use to extend the...

the current set of

principles in mathematics.

And he viewed this as a way

of getting around, i think,

the limitations of his own theorem.

I don't think he thought

there was any limit

to the mathematics that

human beings were capable of.

But, how do you prove this?

The interpretation that

Gdel himself drew,

was that...

computers are limited.

He certainly tried again and

again, to work out that...

the human mind

transcends the computer.

In the sense that he can

understand things to be true,

that can not be proven,

by a computer program.

Gdel also was

wrestling with some...

finding means of knowledge,

which are not based on experience

and on mathematical reasoning,

but on some sort of intuition.

The frustration for Gdel,

was getting anyone to understand him.

I think people very often, for

some reason, misunderstand Gdel.

Certainly his intention.

Gdel was deliberatly

trying to show,

that, what one might call

"mathematical intuition".

He referred to, what he called,

"mathematical intuiton",

and he was...

demonstrating, clearly in

my mind demonstrated,

that this is outside

just following formal rules.

And, i don't know...

Some people...

picked up on what he did and said,

well, he's showing there are

unprovable results and

therefore beyond the mind.

What he really showed, was that

for any system that you adopt,

which, in the sense the mind has

been removed from it, because you...

The mind is used to

lay down the system.

But from thereon, it takes over.

And you ask what's it's scope?

And what Gdel showed,

is that it's scope

is always limited.

And that the mind

can go beyond it.

Here's the man who has said

certain things can not be proved,

within any rational

and logical system.

But he says, that doesn't matter,

because the human mind

isn't limited that way.

We have intuition!

But then of course the one thing

he really must prove to other people,

is the existence of intuition.

The one thing you'll

never be able to prove.

He has these drafts of papers where

he expresses himself very strongly.

But he didn't...

He wasn't satisfied with them.

Because he couldn't prove a theorem

about creativity or intuition.

It was just...

a gut feeling that he had.

And he wasn't satisfied with that.

And so Gdel,

like Cantor before him,

had finally found a problem,

he desperatly wanted to solve,

but could not.

He was now caught in a loop.

A logical paradox, from which

his mind could not escape.

And at the same time,

he slowly starved himself to death.

Using mathematics, to show

the limits of mathematics, is...

is psychologically

very contradictory.

It's clear in Gdel's case,

that he appreciated this.

His own life has this paradox.

What Gdel is,

is the mind thinking about itself,

and what it can achieve

at the deepest level.

Someone used the phrase:

"the Vertigo of the Modern".

You can be led into that particular

reflexive whirlpool where you're

beginning to think about

thinking about thinking...

about thinking about thinking...

and you find yourself entangled

in your own...in your own thoughts.

Well that seems to me, almost the

quintessence of the Modern moment

because there you have a...

what you could call

a paradox of self-reflection.

The kind of madness that you find

associated with Modernism,

is the kind of madness

that's bound up with,

not only rationality,

but with all the paradoxes that

arise from self-consciousness.

From the consciousness contemplating

it's own being as consciousness

or from logic contemplating

it's own being as logic.

Even though he's shown

that logic has certain limitations,

he's still, so drawn to that,

to the significance of the

rational and the logical,

that he desperately want's to

prove whatever is most important,

logically.

Even if it's an

alternative to logic.

How strange.

And what a testimony to his..

his inability to separate himself,

to detach himself from

the need for logical proof.

Gdel of all people...

At the beginning of our story,

Cantor had hoped,

that at it's deepest level,

mathematics would

rest on certainties.

Which for him,

were the mind of God.

But instead, he had

uncovered uncertainties.

Which Turing and Gdel then

proved, would never go away.

They were an inescapable part,

of the very foundations

of maths and logic.

The almost religious belief,

that there was a perfect logic,

which governed a

world of certainties,

had unraveled itself.

Logic, had revealed

the limitations of logic.

The search for certainty,

had revealed uncertainty.

I mean, there's a fashionable

solution to the problem,

which is basically,

in my opinion,

- people are going to

hate me for this -

is sweeping it under the carpet.

But you see, the problem is:

i don't think you want

to solve the problem.

I think it's much more fun

to live with the problem.

It's much more creative!

This notion of absolute certainty...

There is no absolute

certainty in human life.

But our knowledge, our possible

knowledge of this world of ideas,

can only be incomplete and finite,

because we are incomplete and finite.

The problem is that today,

some knowledge,

still feels too dangerous...

because our times

are not so different,

to Cantor, or Boltzmann,

or Gdel's time.

We too, feel things

we thought were solid,

being challenged...

feel our certainties slipping away.

And so, as then...

we still desperately want to

kling to a believe in certainty,

that makes us feel safe.

At the end of this journey,

the question i think

we are left with...

is actually the same as it was

in Cantor and Bolzmann's time:

are we grown up enough,

to live with uncertainties?

Or will we repeat the

mistakes of the 20th century,

and pledge blind allegiance,

to yet another certainty?

are the rules of science.

But beneath them, there is

a far deeper set of rules.

A matrix of pure mathematics,

which explains the nature

of the rules of science,

and how it is we can understand

them in the first place.

To see a world in a grain of sand,

And a heaven in a wild flower,

Hold infinity in the palm of your hand,

And eternity in an hour.

What is the system that...

that everything has to adhere to,

if there is no God?

You had these ideas, and...

and you had to be very careful

because at any moment,

they would bite you.

They sounded great but

they were very dangerous.

But then of course,

people get scared.

So they pull back from

the edge of the precipice.

Well, this is not a matter

of liking it or not...

You have here this proof and...

one has to live with it.

This film, is about how a small

group of the most brilliant minds,

unraveled our old cosey certainties

about maths and the universe.

It is also about how once they

had looked at these problems,

they could not look away...

and pursued the questions

to the brink of insanity,

and then over it,

to madness and suicide.

But for all their tragedies,

what they saw, is still true.

Their contempories largely rejected

the significance of their work,

and we have yet

to fully inhered it.

Today, we still stand

only on the threshold,

of the world they saw.

My name is David Malone.

And this is my hommage,

to former great thinkers,

who without most of us,

even having heard of them,

have profoundly influenced

the nature of our age,

and who's stories have, i think,

an important message for us today.

This is Halle.

A provincial town

in Eastern Germany,

where Martin Luther once

preached the reformation.

Our story starts here,

at the towns university

with a mathematics professor.

A man called: Georg Cantor,

who started a revolution he

never really meant to start.

But which eventually threatened

to shake the whole of mathematics

and science on it's foundations.

And he started this revolution by

asking himself a simple question:

how big is infinity?

Cantor is wonderful

because it's so crazy.

It's the equivalent

of being on drugs.

It's just an incredible

feat of imagination.

Georg Cantor is one of the greatest

mathematicians of the world.

Others before him, going back

to the Ancient Greeks at least,

had asked the question.

But it was Cantor, who made the

journey no one else ever had,

and found the answer.

But he paid a price

for his discovery.

This is the only bust

there is of Georg Cantor.

It was made just one

year before he died,

and he died utterly alone,

in an insane asylum.

The question is:

what could the greatest mathematician

of his century have seen,

that could drive him insane?

If all that Cantor had seen

was mathematics,

then his story would be

of limited interest.

But from the beginning,

Cantor realised his work

had far wider significance.

He believed,

it could take the human mind,

towards greater,

trancendent truth and certainty.

What he never suspected,

was that eventually his maths

would make that certainty

ever more elusive.

Perhaps even destroy the

possibility of ever reaching it.

If you want to understand Georg

Cantor you have to understand

he was a religious man.

Though not in a conventional sense.

He almost certainly

came to this church,

but that's not his God.

He wasn't interested in a God

who's mysteries were

redemption and resurrection.

Ever since he was just a boy,

he had heard what he called:

a secret voice,

calling him to mathematics.

That voice which he

heard all his life,

in his mind, was God.

So for Cantor,

his mathematics of infinity

had to be correct,

because God, the 'True Infinite',

had revealed it to him.

These things which are

now hidden from you,

will be brought into the light.

If you look at Cantor's

last major publication,

about Set Theory, in 1895.

It starts with three aphorisms,

and it's third motto

is from the bible,

and it's from Corinthians.

And it's, you know:

that which has been hidden

to you, will eventually

be brought into the light.

And Cantor i think, really

believed he was the messenger.

That this theory had been hidden.

He was God's means,

of bringing this Theory of

the Infinite to the world.

There is no contradiction for

Cantor between his religious

thinking and his mathematics.

He understood or

he was thinking that,

his ideas were a gift of God.

My view was that Cantor was

trying to understand God

and that this was really...

like a mathematical theology

that he was doing.

Cantor's God was the 'Creator God'.

The God who set the planets

spinning in their orbits.

Who's mysteries were the eternal

and perfect laws of motion.

Laws who's discovery had

launched the modern world,

and allowed us to see the world

as curves, trajectories and forces,

and which would one day

even put men on the moon.

The eternal certainties

layed down by God

which Newton and Leibniz

had discovered.

And it was infinity which

lay at the heart of it all.

But, there was a problem with it.

If you look at that beautiful

smooth curve of motion,

you notice it's not

actually smooth.

It's made of an infinite number of

infinitesimally small straight lines.

And each line is an instant

in which nothing moves.

But like frames of film, if you run

one after another, you get motion.

And it works. The whole thing

relies on infinity but it works.

And because it worked,

everyone said:

alright, we don't

understand infinity.

Just leave it alone.

Cantor comes along and says

no, if this whole thing

rests on infinity,

we have to understand it.

And now remember

Cantor is a religious man.

So for him, that symbol...

isn't just a scientific mystery.

It's a religious mystery as well.

Here's the maths that God uses

to keep creation in motion.

And at it's heart, lies the

deeper mystery of infinity.

But it was a mystery,

that had so far defied every

mind that had looked at it.

The first Modern Thinker to

confront the infinite was Galileo.

And he tried to do it using the circle.

This is how he did it.

He said: first of all,

draw a circle.

Now put a triangle inside it,

then a square and keep on going.

Keep adding sides.

Eventually he realised that

what the circle is...

is a shape with infinately many,

infinitesimally small sides.

Which seems great.

Now you can hold the infinite

and the infinitesimal

in your hands.

But as soon as he done that,

he realised it actually opened up

a horrible illogical paradox.

Because he said: alright, let's

draw a bigger circle outside.

And now with an infinitely sharp

pencil, draw from the center...

infinitely sharp lines. One for each

of the lines on the inner circle.

There's an infinate number of them,

that should be enough

for the inner circle.

But now extend those lines out

till they meet the outer circle.

Now, those lines are diverging...

which means when you

get to the outer circle,

if you look really carefully,

there will be gaps.

There won't be enough.

Galileo just said:

that makes no sense.

If there's an infinite number

it should be enough!

At which point he said:

we just can't understand the infinite!

Maybe God can, but with

our finite minds, we can't.

So, let's use the

concept if we must...

but let's not try

and understand infinity.

And that's exactly

how they left it...

until Georg Cantor came along.

At first it must have

seemed to Cantor,

that God really was on his side.

In the space of only a few years,

he married, began a family,

and published his

first ground-breaking paper

about infinity.

Where previously infinity had just

been a vague number without end,

Cantor saw a whole

new world opening up.

Cantor did a new step and he said:

i want to add one plus one.

And Cantor said:

ok, why can i not add

infinity plus infinity?

That's also possible!

And this was a starting

point of his theory.

Cantor found he could add

and subtract infinities...

and in fact discovered there

was a vast new mathematics

of the infinite.

You really finally feel

for the first time,

that the infinite is no longer

this amorphous concept:

well, it's infinite.

And that's all you

can say about it.

But Cantor says:

no!

There's a way you can

make this very precise

and i can make it

very definite as well.

By 1872, Cantor is a man inspired.

He's already grasped and understood,

the nature of real infinity,

which no one before him had done,

but in that same year,

he come's up here to the Alps...

to meet the only other man

who really understood his work:

a mathematician called,

Richard Dedekind.

And this time,

is probably the happiest and most

inspired period of Cantor's life.

Within a year of there meeting, he

announces an astonishing discovery:

that beyond infinity,

there's another larger infinity,

and possibly even a whole

hierarchy of different infinities.

Though it is contrary

to every intuition,

Cantor began to see that some

infinities are bigger than others.

He already knew that when

you looked at the number line,

it divided up,

into an infinite number

of whole numbers and fractions.

But Cantor found that as he

looked closer at this line,

that infinite though the

fractions are, each one...

is separated from the next by

a wilderness of other numbers.

Irrational numbers like pi.

Which require an infinite

number of decimal places

just to define them.

Against all logic,

the infinity of these numbers,

was unmeasurably, uncountably

larger than the first.

What had frightened Galileo,

Cantor had proved:

there was a larger infinity!

Today, Cantor's genius

continues to inspire the work

of some of the

greatest mathematicians.

Greg Chaiton, is recognised

as one of the most brilliant.

Well, infinity was

always there but it...

they tried to contain it.

They tried to...

to keep it in a cage.

And, people would talk

about potential infinity

as opposed to actual infinity.

But Cantor just goes all the way.

He just goes totally berserk.

And then you find that

you have infinities and

bigger infinities and

even bigger infinities

and for any infinite

series of infinities,

there are infinities that are

bigger than all of them.

And you get numbers so big

that you wonder

how you could even name them?

You know infinities so big that

you can't even give them names?

This is just...

It's just fantastic stuff!

So in a way what he's saying is,

giving any set of concepts,

i'm going to invent

something that's bigger.

So this is...

this is paradoxical essentially.

So there's something inherently

ungraspable, that escapes you

from this conception.

So it's absolutely breathtaking.

It's great stuff!

Now, it may not have

anything to do with

partial differential equations,

building bridges,

designing airfoiles, but who cares?

The shear audacity

of Cantor's ideas,

had thrown open the doors,

and changed mathematics forever.

And he knew it!

We can't know

exactly how he felt...

but Greg Chaitin has also felt those

rare moments of profound insight.

You know, here we are

down in the forest and...

and we can't see very

far in any direction.

And you struggle up,

ignoring the fact that

you're tired and weary.

You struggle up a mountain,

and the higher you go

the more beautiful and

breathtaking the views are.

And then...

If you're lucky you get

to the top of the mountain.

and...that can be a transcendant

experience, you know...

A spiritual person would say

they feel closer to God.

You have this breathtaking view.

All of a sudden you can see...

in all directions,

and things make sense.

It's beautiful to

understand something

that you couldn't

understand before,

but the problem is,

the moment you understand one thing,

that raises more questions.

So in other words,

the moment you climb one mountain,

then you see off in the distance...

Behind the haze are

much higher mountains.

His theory is all about the fact

that the mountains get

higher and higher.

And no range is ever enough because

there are always mountain ranges

beyond any range that you can

understand or conceive of.

So this has a tremendously

liberating effect on mathematics,

or it ought to!

But then of course,

people get scared.

So they pull back from

the edge of the precipice.

What was inspiring for Cantor,

frightened his critics.

They saw mathematics as the

pursuit of clarity and certainty.

Everything Cantor was doing:

his irrational numbers

and his illogical infinities,

seemed to them to be

eating away at certainty.

He soon faced the deep

and implacable hostility.

This is the main lecture theatre

in the university

where Cantor spent

his entire professional life.

A life that he felt trapped in.

And i think there's

some justification.

Other mathematicians,

actually tried to prevent

Cantor publishing his papers.

Cantor always dreamed that

he'd receive an invitation

to one of the great universities

like Vienna or Berlin,

but they were invitations

which never came.

And he was also

attacked personally.

The great mathematician

Henri Poincar, said..

that Cantor's mathematics

was a sickness

from which one day

maths would recover.

And worse...

His one time friend

and teacher, Kronecker...

said that Cantor was

a corrupter of youth.

Cantor felt,

that he and his ideas

were being caged,

or quarantined here

as if they were,

some kind of sickness.

The genie...

got out of the bottle.

It was a very

dangerous genie because

you see, the concepts,

that Cantor played with

are intrinsically

inherently self-contradictory.

And people don't like

to face up to that.

They've emasculated Set Theory.

They have this..this version,

which is safe, called:

"Zermelo-Fraenkel Set Theory".

Which is a sort of

a watered-down...

But you see, that takes

all the fun out of it!

The...for me, the fun...

Cantor was...he was...

He was playing on the edge!

You know, the idea was,

you had these ideas, and...

and you had to be very careful

because at any moment

they would bite you.

They sounded great but they

were very dangerous.

You see, they were

almost self-contradictory.

The notion of the

'Set of Everything' for example,

is self-contradictory.

And...it's...

and people got frightened.

His critics feared Cantor

was going to dislodge

the certainty and clarity

vital, to mathematics and logic,

which might not be able

to be put back.

It seemed Cantor had opened maths

to the very thing it was

supposed to save us from:

irresolvable uncertainty.

Cantor knew the only way to

convince his critics,

was to make his theory complete.

Could he show there was

a logic to his infinities?

Some system,

that bound them all together?

What he absolutely

must decide now,

is, what's the relationship

between them.

If he can do that,

then his theory is perfect.

If he can't,

then all he has is bits.

So he has to decide

what's the relationship

between them.

And that question,

is the 'Continuum Hypothesis'.

No matter how isolated he became,

the more he was opposed,

the more he struggled.

Where another person might

have given up, Cantor didn't.

Clinical psychologist,

Dr. Louis Sass,

suggests it is precisely this

ability to be isolated,

which is key, to Cantor's genius.

I think, that willingness to

step into a realm...

you know, beyond the...

the taken for granted,

is abolutely essential.

But i think if you're a person

who takes that step,

in a way you're already doomed,

to living outside in some way.

So, you know...

It's not as if it's only the

intellectual project itself

that takes you out there.

There is something about you

as a person, that is just...

That unnaturalness, so to speak,

comes so naturally to you.

Cantor was trapped.

There were too many things that

went to the core of who he was,

for him to be able to give up.

When Cantor was just a boy,

his father sent him a letter...

which became his most

precious possession,

and which he carried

with him all his life.

In it, his father told him,

how the whole family

looked to him,

to achieve greatness.

How he would come to nothing,

if he did not have the courage to

overcome criticism and adversity.

How he must trust in God to

guide him, and never give up.

And he never did.

Well i think, here you come to

the root of the problem for Cantor

of a theory,

that he was certain, was correct,

in part because he believed that

it had come to him

as a message from God.

There's a very important

religious aspect to Cantor's...

struggle to deal

with the infinite,

and face the problems of...

not being able to resolve many

of the open questions

that he himself raised

for the first time.

By 1894, Cantor has

been working solidly

on the Continuum Hypothesis

for over two years.

At the same time,

the personal and professional

attacks on him...

have become more

and more extreme.

In fact he writes to a friend

saying he's not sure

he can take them anymore.

And indeed, he can't.

By May of that year, he has

a massive nervous breakdown.

His daughter describes how his

whole personality is transformed.

He will rant and rave,

and then fall completely and

uncommunicatively silent.

Eventually, he's brought here...

to the 'Nervenklinik' in Halle,

which is...an asylum.

Today, we would say Cantor suffered

from manic depressive illness.

From Cantor's time,

we have left,

the case notes of

most of his psychiatrists.

In the notes for example,

we see that he, at times,

was quite disturbed,

was screaming...

and see that he was

really suffering from...

severe bouts of mania.

Sometimes he would be angry...

and he would have

ideas of grandeur

and sometimes he had

also ideas of persecution.

After his breakdown, everything

about Cantor is transformed.

He tells a friend he's not sure he'll

ever be able to do mathematics again.

He asks the university if he

can stop teaching maths

and teach philosophy instead.

But interestingly,

during this whole time...

despite having claimed, him not

being able to do mathematics again,

he never stops working on

the Continuum Hypothesis.

It's as if...

he just can't put it down,

can't look away.

You can only think:

i must find the proof!

This i can understand because

when you are a mathematician,

then you are for all the

time a mathematician.

It's a form of...living for you.

You must think

about mathematics, and...

you can't think anything else...

the whole day.

You are thinking and

thinking and thinking.

And you say:

i must find it!

I must, i must, i must!

You can't think anything else.

In August of 1884,

he writes a letter,

to his friend and colleague,

the last man who still

publishes his work.

A man called, Mittag-Leffler.

And the letter is ecstatic.

He says: i've done it!

I've proved the Continuum

Hypothesis. It's true.

And he promises that he'll send

the proof in the following weeks.

But the proof never comes.

Instead, three months later,

a second letter arrives.

And in this one, you can

feel Cantor's embarrassment.

He says: i'm sorry i should never

have claimed that i proved it.

And he says: my beautiful

proof lies all in ruins.

And you can see the wreckage

of his work, in the letter.

But then, three weeks after that,

this letter arrives:

and in it he says:

i've proved that the Continuum

Hypothesis is not true.

And this pattern continues.

He proves that it is true...

and then he's convinced

that it's not true.

Back and forth.

And in fact,

what Cantor is doing...

is driving himself slowly insane.

One of the things that will happen

especially in the early stages and,

the stages just before

a schizophrenic break,

but also in the early stages,

will be that the patient is...

in a way,

looking too hard at the world

and too concentrated away.

As a kind of rigidity of

the perceptual stance.

When he could not solve

the Continuum Hypothesis,

Cantor came to describe

the infinite, as an abyss.

A chasm perhaps,

between what he had seen...

and what he knew must be there,

but could never reach.

What can happen, is that

some object in the world that...

that the rest of us would just...

consider just a sort of

random thing there,

seem somehow symbolic in some way.

There's a way in which in

order to understand something

you have to look very hard at it.

But you also have to be able to

sort of move away from it

and kind of see it

in a kind of context.

And the person who stares too hard

can often lose that sense of context.

Cantor never fully recovered.

For the rest of his life...

he would be drawn back to work

on the problem he could not solve.

And each time,

it would hurt him, profoundly.

In 1899, Cantor had

returned once again

to work on the

Continuum Hypothesys.

And again it made him ill

and he returned to the asylum.

He was just recovering

from this breakdown,

when his son Rudolf died, suddenly.

Four days short of

his thirteenth birthday.

Cantor wrote to a friend,

saying how his son had

had a great musical talent,

just as he had had

when he was a boy.

But he had set music aside,

in order to go into mathematics.

And now with the death

of his son,

he felt that, his own dream

of musical fulfillment

had died with him.

Cantor went on to say,

that he could no longer even

remember why he himself...

had left music,

in order to go into maths.

That secret voice, which had once

called him on to mathematics,

and given meaning

to his life and work,

the voice he identified with God...

That voice too, had left him.

Here we have to leave Georg Cantor,

because if we treat

Cantor's story in isolation,

it makes it into a tragic

but obscure footnote,

to the broader sweep of history.

Where as in fact, the fear that

Cantor had dislodged something,

was part of a much broader feeling,

that things once felt to

be solid, were slipping.

A feeling seen more clearly in the

story of his great contemporary:

a man called, Ludwig Boltzmann.

Just as Cantor had revolutionary

ideas in mathematics and was opposed,

so Boltzmann, his contemporary,

had revolutionary ideas in physics,

and was equally opposed.

This is Ludwig Boltzmann's grave.

And that...carved on it,

is the equation which killed him.

And it did so, because like Cantor...

Boltzmann's ideas were out

of step with his times.

Cantor had undermined the ideal of a

timeless and perfect logic in maths.

Boltzmann's formula and

his destiny...

was to undermine the ideal

of a timeless order in physics.

Together, their ideas were part of

a general undermining of certainty,

in the wider world

outside of maths and physics.

Boltzmann's and Cantor's times

craved certainty,

in politics, in art, as well as

in science and philosophy.

They were times that looked on

the surface, solid and certain...

but felt themselves to be

teetering and sliding.

The old order was dying.

And it was as if they could already

feel disaster's gravitational pull.

In Vienna,

which was called by Karl Kraus:

"Laboratory for Apocalypse",

there was this feeling that...

this political construct

of the Habsburg Empire

couldn't last for much longer.

They were very strange times.

On the one hand,

those in power spent 20 years

building the monuments

of imperial Vienna,

to declare that this order, firm on

it's foundations, would last forever!

The rich man would

always be in his castle;

the poor man always at his gate.

But on the other hand, the empire

was actually on it's last legs.

And the intellectual

tenor of the times,

was summed up by

the poet, Hofmannsthal...

who said that, what previous

generations believed to be firm,

was in fact, what he called:

"das Gleitende".

The slipping, or sliding away

of the world.

I thing that describes

the feeling in Vienna.

In other places too,

but particularly in Vienna in

this capital of an empire that...

hadn't crumbled yet but,

it looked like...set to break down.

This characterizes it very well.

And it was against this background,

that the scientific questions of

Boltzmann's times were understood.

This is the Great Courtyard

of the Univerity of Vienna...

and these are the busts

of all of the greats who

have ever thaugt here.

Boltzmann's is here too.

But many of his contemporaries,

men, more influential in their

day even than he was,

lined up to oppose

him and his ideas.

But their opposition was as much

ideological, as it was scientific.

The physics of Boltzmann's time,

were still the physics of certainty.

Of an ordered universe,

determined from above,

by predictable and timeless

God-given laws.

Boltzmann suggested,

that the order of the world was

not imposed from above by God,

but emerged from below.

From the random bumping of atoms.

A radical idea,

at odds with his times,

but the foundation of ours.

Professor Mussardo, lives and works

in Trieste, on the Adriatic Coast.

Not far from where

Boltzmann's live ended.

He is an expert on Boltzmann, and

works on the same kind of physics.

I think that there were two

reasons why he could not...

get fully accepted and recognized

by the German physicists.

One of them was,

that he based all his theories on

atoms, that people can't see.

And this was the reason of the very,

very strong criticism by Ernst Mach.

One of the most influential...

Philosophers of Science

at that time.

So the criticism of Mach

was simply:

i can't see an atom...

I don't need them,

they don't exist.

So why should we bring

them in the game?

Worse than insisting on the reality

of something people could not see,

to base physics on atoms,

meant to base it on things who's

behaviour was to complex to predict.

Which meant an entirely

new kind of physics.

One based on probabilities,

not certainties.

But then there was a second aspect;

it was revolutionary as well.

And this consists in...

putting forth and emphasizing

the role of probability,

in the physics world.

And people were used to the laws of

physics and science as being exact.

Once established,

they stay there forever.

There is no room for uncertainty.

So, introducing into the game,

two ingredients like invisible

atoms and probabilities,

means there is no certainty.

You can predict what is probably

going on, but not certainly.

Well, this really contrasted

very, very much with the...

the scientific spirit of the time,

and therefore this produced trouble.

Boltzmann's genius, was that

he could accept probability.

This meant he could begin to

understand complex phenomenon,

like fire and water and life.

Things which traditional physics;

the physics of mechanics,

never could.

But because his solution

relied on probability,

and probability

undermines certainty,

no one wanted to hear him.

And so just like Cantor,

he faced implacable opposition

which he too, found extremely

difficult to deal with.

It seems like Boltzmann was just

the wrong man in the wrong place.

Absolutely...absolutely.

Absolutely, it's true.

It's true.

He could just had his idea

twenty years later...

And in England he would have been

the most succesful physicist

of that time, it's true.

Somehow he met all his enemies.

So he met Ernst Mach, often.

Their careers even crossed,

in a very...in a very...

He meets all his enemies

but none of his friends.

Not his friends, it's true.

It is not hard to see

how Boltzmann's ideas

where so radically at

odds with his times.

Especially when applied not just

to physics but to the social world.

Classical science, classical physics...

gives you this image of a

God-ordered creation.

Where everything is set in stone,

according to perfect

and eternal rules.

Everything is predictable.

Everything has it's place and

everything is in it's place.

But when you come here, to the

Central Cemetery of Vienna,

you see the idealised

vision of that idea.

Because here,

everything is predictable.

Everybody does have a place

and everybody is in their place.

But the problem is...

while such certainty might

seem desirable politically,

the real world...the living world,

the world described by

thermodynamics, just isn't like that.

A timeless and perfect world

never changes, but it is dead.

The real world,

the thermodynamic world is alive

precisely because it

is full of change.

But of course...

that life giving change also

brings with it, disorder and decay.

But then the problem arises,

if you then say:

well, Newtonian Mechanics

on which you are depending

is reversible in time.

So how can you derive a law,

which is asymmetrical in time

from basic principals which

are symmetrical in time.

You run the clock backwards,

it's just as good

Newtonian physics as

you run it forwards.

Yet the entropy

increases in the future.

But it is precisely this

accumulation of disorder and decay,

that science calls 'entropy',

which Bolzmann had understood.

In short this is really

the 'arrow of time'.

I mean, you can

measure the arrow of time

just seeing how things become

more and more disordered.

So it's a natural tendency

in the world

that Bolzmann quantified precisely.

Indeed i call him

the genius of disorder.

Boltzmann's work on entropy,

showed why no system can be perfect.

Why there must always

be some disorder.

It also revolutionized

the idea of time in physics.

In classical physics,

everything, including time

can run equally well

forwards as backwards.

Yet in thermodynamics, while

everything else is reversible,

time moves inexorably forward,

like an arrow.

The idea of entropy,

had a profound philosophical

and political significance.

Entropy is what changes

the ticking of a clock

into the destroyer of all things.

It is wat underlies the inexorable

passage from youth to old age.

Entropy is decay, and with the

decay nothing lasts forever.

Boltzmann had in essence,

captured mortality in an equation.

Physics now declared...

that no order, not even a

God-given one, will last forever.

That there was no natural order

that God had set in stone,

had already been pointed out by the

scientist Boltzmann most admired:

Charles Darwin.

In place of timeless perfection,

was a dance of

evolution and extinction.

With his equation of entropy,

Boltzmann brought this

picture of constant change,

into the very heart

of physics itself.

Did Boltzmann understand

the similarity?

Almost certainly.

When Boltzmann was asked how

his century would be remembered,

he did not chose a physicist.

He said it would be

the century of Darwin.

So he likes in Darwin, the momentum,

the evolution of life,

that is not static,

the fact that he's progressive.

Progress sometimes

has a jump in it.

And the fact that he can adress a...

a "life aspect",

with ideas of science,

that before was kind of

an ideological ground.

Bolzmann's ideas, like Cantor's and

Darwin's were revolutionary,

even though he did not

think of them that way.

But his times were frightened times.

Times when people felt new ideas,

could upset societies

fragile structure...

and bring it down.

At the end of the 19th century,

Viennese society was searching for

some certainty, some principal...

wether it was in politics,

philosophy, the arts or science.

But there didn't appear

to be any philosophy,

capable of holding

everyone together.

Upon which everything

else could be based.

So when the university

commissioned Gustav Klint,

to paint a ceiling to

celebrate philosophy,

this is what they got:

Such was the outrage,

that twenty professors petitioned,

to have the painting removed.

Now whatever else it is,

it's not a celebration of certainty.

The radicals of Bolzmann's times,

knew, the old order, with it's

worn-out certainties was doomed.

But Viennese culture, was not

ready to embrace the new.

And Boltzmann,

was caught in the middle.

As a scientist, his personality

entered deeply into the game,

because he was very stubborn.

Not self-ironic.

He could not take criticism.

He always took it personally,

and Boltzmann was definately

a passionate man.

He used to swing rapidly from

incredible joy to deep depression.

As Boltzmann got older, and more

exhausted from the struggle,

these mood swings became

more and more severe.

More and more of

Boltzmann's energy,

was aborbed in trying

to convince his opponents,

that his theory was correct.

He wrote:

no sacrifice is too high for

this goal, which represents

the whole meaning of my life.

In the last year of Boltzmann's life,

he didn't do any research at all.

I'm talking about

the last ten years.

He was fully immersed in dispute,

philosophical dispute...

Tried to make his point,

writing books,

which were most of

the time the same,

repeating the same

concept and so on.

So you can see he was in a loop...

that didn't go ahead.

But by the beginning of the 1900's,

the struggle was

getting too hard for him.

Boltzmann had discovered one

of the fundamental equations

which makes the universe work,

and he had dedicated his life to it.

The philosopher Bertrand Russel

said that for any great thinker,

this discovery that everything

flows from these fundamental laws,

comes, as he described it,

whith the overwhelming

force of a revelation.

Like a palace, emerging

from the autumn mist,

as the traveller ascends

an Italian hillside.

And so it was for Boltzmann.

But for him,

that palace was here,

at Duino in Italy,

where he hung himself.

In 1906,

Boltzmann came here to Duino,

with his wife and

daughter on holiday.

Exhausted and demoralised,

his ideas still not accepted.

While they were out walking,

he killed himself,

and left no note of explaination.

Of course we can never know

what Boltzmann was thinking,

but i think we have clues.

Boltzmann knew what it was,

to be in the grip of a

beatiful and powerful idea.

He once wrote that,

what the poet laments,

holds for the mathematician:

that he writes his works,

whith the blood of his heart.

So we know that he

was a passionate man.

But i think there is another clue.

At the start of one of Boltzmann's

major scientific papers,

he quotes three lines

from Goethe's Faust:

"Bring forth what is true."

"Write it so it's clear."

"Defend it to your last breath."

Which of course he does.

But i think there's

something deeper here.

Why quote Faust, at the

start of a scientific paper?

The pact, that Faust

makes with the devil,

is that the devil will give

him all of the knowledge

and all of the experience

that he wants,

so long as he never asks to stay,

in any one moment.

And i think when

Boltzmann came here,

to this beautiful place,

after thirty years of fighting

for what he believed in,

he simply said:

i want to stay here, in this

perfect, beautiful moment.

I don't want to have to leave.

I want time, for me, to stop.

The great and controversial thing

that Boltzmann had done,

was to introduce,

into the unchanging perfection

of classical physics,

the notion of real time.

Of irreversible change.

And yet it was this man,

who in his final moments,

wanted time to stop.

So ironically, Boltzmann was

vindicated just after his death.

If he would have

waited a little longer,

Boltzmann would have been

one of the fathers of the

revolution of the

twentieth century fysics.

Yet Boltzmann died as he had lived:

out of step with his times.

He had sawn the seeds of

uncertainty and fysics,

but no school of followers

took up his work.

Against all the odds,

it was Cantor,

who had uncovered the

uncertainty in mathematics,

around whom followers

where gathering.

A new generation of mathematicians

and philosphers were convinced:

if only they could solve the

problems and paradoxes

that had defeated Cantor,

maths could be made perfect again.

The most prominent amongst them,

Hilbert, declared:

the definitive clarification of the

nature of the infinite,

has become necessary for the honour

of human understanding itself.

They were so concerned to

find some kind of certainty,

they had come to believe

that the only kind of understanding

that was really worth anything,

was the logical and the provable.

And a measure of how desperate this

attempt to find the perfect system

of reasoning and logic

had become, is this:

three volumes of the Principia

Mathematica, published in 1910.

It takes a huge chunk

of this volume,

just to prove, that one

plus one equals two.

And a large part of that proof,

revolves around the problems

of the finite and the infinite,

and the paradoxes that

Cantor's work had trown up.

But despite the Principia,

there was now the feeling

that the logic of maths,

had undone itself,

and it was Cantor's fault.

As the Austrian writer, Musil

wrote at the time:

suddenly mathematicians, those

working in the innermost region,

discovered that something

in the foundations,

could absolutely not

be put in order.

Indeed, they took

a look at the bottom,

and found that the whole edifice,

was standing on air.

Cantor had stretched the limits of

maths and logic to breaking point,

and paid for it.

Much of the last

twenty years of his life,

was spent in and out

of the asylum.

The last time that Cantor came

here to the Nervenklinik in Halle,

was in 1917, and he truly

did not want to be here.

He wrote to his wife,

begging her to let him come home.

He was one of only

two civilians left here.

The rest of the place, was filled

with the casualties of World War I.

But of the 6th of January 1918,

the greatest mathematician

of his century,

died alone in his room,

his great project still unfinished.

Cantor had dislodged the pebble,

which would one day

start a landslide.

For him, it had all

been held together.

The paradoxes resolved, in God.

But what holds our ideas together,

when God is dead?

Without God,

the pebble is dislodged,

and the avalanche is unleashed,

and World War I, had killed God.

Here at last,

was the slippage.

Well, hasn't there always been a

desire in the history of the West

to find certainty or...maybe,

there wasn't so much a desire

in earlier era's because, the

assumption was that we had that.

You know, there was God!

And, you know even Descartes,

despite all of his scepticism,

assumes...

for him unproblematically,

that there is a God.

So what happens when that really,

really comes in to question?

After the death of God,

so to speak.

And along with the death of God

is a...is a loss of faith in some...

supernatural order,

of which we are a small part.

No one won the Great War.

Nothing was resolved at Versaille.

It was merely an armistice.

And none of the intellectual

crises that proceded it,

had been resolved either.

Things like the Principia, had

merely papered over the cracks.

In a way, the Principia was

like the Versaille Treaty,

only a lot more substantial.

This is basicly ten thousend

tonnes of intellectual concrete

poured over the

cracks in mathematics.

And for a while, it looked

like it really might hold.

But then a young man came here

to the university of Vienna,

to this library.

His name was Kurt Gdel.

And the work that he did here,

brought that dream of finding

the perfect system of reasoning

and logic, crashing down.

Gdel was born the year

Boltzmann died: 1906.

He was an insatiably

questioning boy,

growing up in unstable times.

His family,

called him: "Mister Why".

But by the time he

went to university,

World War I was over.

But Austria like the rest of Europe,

was in the grip of the depression,

and Hitler was forming

the National Socialist Party.

Gdel for his part,

became one of a brilliant group

of young philosophers,

political thinkers,

poets and scientists,

known as 'The Vienna Circle'.

Chaos was good because it ment

that there was no central authority

that was imposing ideas

so individuals could come

up with their own ideas.

The chaos around them,

on the one hand

had a liberating effect.

And on the other hand they were

desperately searching for ideas,

that they could believe in because

everything else around them

was crumbling in a heap.

So you'd want to

find some beautiful ideas

that you could believe in.

Though Gdel was surrounded by

radicals and revolutionary thinkers,

he was not one himself.

He was an unworldly and exact man,

who believed, like Hilbert,

that maths at least,

could be made whole again.

But it was not to be.

He certainly did not start out,

with trying to explode

Hilbert's program also.

In fact,

i think it came to Gdel...

ultimately as a surprise when

he showed that the next step,

to show the completeness of

arythmetic, was unachievable.

There was actually something

very mysterious happening

in pure mathematics.

In it's own way as mysterious as

black holes, the big bang,

as quantum uncertainty in the atom.

And this was Gdel's

Incompleteness Theorem.

And at that time,

there was a mystery there.

The one place where you don't

expect there to be mystery

is in pure reason!

Because pure reason should be black

and white. It should be really clear.

But, pure reason,

the clearest thing there is,

was revealing that there were

thing that were unclear.

This is one of the cafs

where the Vienna Circle

used to meet regularly.

Late summer of 1930,

Gdel came to the caf

with two eminent colleagues.

Towards the end

of their conversation,

he just mentioned an idea

he'd been working on,

which he called

the 'Incompleteness Theory'.

And what he told them,

was that he had just proved,

that all systems of

mathematical logic, were limited.

That there would always be

some things wich while true,

would never be able to

be proved to be true.

What Gdel showed in

his Incompleteness Theorem,

is that, no matter how large

you make your basis of reasoning,

your axioms, your set of axioms,

in arythmetic, there would

always be statements

that are true

but can not be proven.

No matter how much data

you have, to build on,

you will never...

prove all true statements!

What this meant, was that the

great Renaissance dream,

that one day, maths and logic

would be able to prove all things

and give us a godlike knowledge.

That dream was over!

But this idea was so far away,

from what anyone else

was working on,

what anyone else even suspected,

that neither of these colleagues

understood what he had just told them.

It was as if...

there was an explosion, but the

blast wave hadn't hit them yet.

Unaware of what had happened

in the caf, the very next day,

Hilbert, now the grand old

man of mathematics,

stood up and gave a

lecture in Knigsberg,

in which he said:

"We must know!"

"We will know!"

The irony was,

that the very day before,

Gdel had proved, that there were

some things, we would never know.

Some, didn't like it.

Some...

In particular for instance, Hilbert.

It seems that at the beginning, he

was quite annoyed and even angry.

This is not a matter

of liking it or not...

You have here this proof and...

one has to live with it.

Are there any holes

in Gdels argument?

No, there are not.

This was a perfect argument.

This argument was so

crystal clear and obvious.

Gdel had joined Hilbert,

in trying to solve the paradoxes,

uncovered by Cantor.

Instead, he had just proved,

that would never happen!

His work, springing directly

from Cantor's work on infinity,

proved, the paradoxes

were unsolvable,

and there would be more of them.

But being right,

didn't make him popular.

So here we are again in the Great

Courtyard of Vienna University

with the busts of all

of the great thinkers...

except for Kurt Gdel.

There's no bust to Gdel here.

And i can't help but

feel that at least

part of the reason

that he's not here,

is simply due to the

nature of his ideas.

Ah! Well you see,

nobody wants to face him.

In my opinion nobody wants to

face the consequences of Gdel.

You see, basically people want to

go ahead with formal systems anyway,

as if Hilbert had it all right.

You see?

And in my opinion, Gdel explodes

that formalist view of mathematics.

that you can just mechanically grind

away on a fixed set of concepts.

So even though i believe

Gdel pulled out the rug

out from under it intellectually,

nobody wants to face that fact.

So there's a very ambivalent

attitude to Gdel.

Even now, a century after his birth.

A very ambivalent attitude.

On the one hand, he's the

greatest logician of all time

so logicians will claim him,

but on the other hand,

they don't want,

people who are not logicians

to talk about the consequences

of Gdels work, because the obvious

conclusion from Gdels work

is that logic is a failure.

Let's move on to something else.

And this would destroy the field.

Gdel too, felt the effects

of his conclusion.

As he worked out the true

extent of what he had done,

Incompleteness began to

eat away at his own beliefs

about the nature of mathematics.

His health began to deteriorate,

and he began to worry about

the state of his mind.

In 1934,

he had his first breakdown.

But is was after he

recovered however,

that his real troubles began,

when he made a fateful decision.

Almost as soon as Gdel has

finished the Incompleteness Theorem,

he decides to work on the great

unsolved problem of modern mathematics:

Cantor's 'Continuum Hypothesis'.

And this is the effect

that it has on him.

These are some pages from one

of Gdel's workbooks

and they all, look like this.

Beautifully neat,

beautifully logical.

Except for this one.

This is the workbook,

where he's working on

the Continuum Hypothesis.

Gdel, like Cantor before him,

could neither solve the

problem, nor put it down.

Even as it made him unwell.

There could be a danger...

a danger in it.

And perhaps there's also a danger

in it at the more existential

or personal, psychological level.

If you're a person,

who is already prone to

the kind of exaggerated...

intellectual, self-reflection,

self-conscienceness...

you may find that your,

intellectual work is

exaggerating, exacerbating

that tendency, which...

which of course can make

life more difficult to live.

He calls this the

worst year of his life.

He has a massive

nervous breakdown,

and ends up in a sanitorium,

just like Cantor.

We're talking about people

here who, of course are...

are capable of, and

maybe afflicted with,

the capacity to care

very, very much

about things that are

very, very abstract.

To really lose themselves in

these intellectual problems.

One of the sanatoria that Gdel

spent some time in, is here:

the Purkersdorf Sanatorium,

just outside of Vienna.

The Purkersdorf itself, was

build to embody the philosophy

that the calm,

smooth lines of rationalism,

are the cure for madness.

Ironic then,

that Gdel, driven mad by

pushing the limits of rationalism,

should come here to recover.

But while the man who had

proved, there was a limit

to rational certainty,

was in the sanatorium,

outside, a greater

madness was unfolding...

as a nation threw itself into

the arms of a demagogue

who promised, there was certainty.

Gdel's madness passed.

Austria's didn't.

In 1939, Gdel himself was

attacked by a group of Nazi thugs.

That same year, he reluctantly

left Austria, for America.

It was during these pre-war years,

that another brilliant young man,

Alan Turing, enters our story.

Turing is most famous, for his

wartime work at Bletchley Park,

breaking the German Enigma code.

But he is also the man,

who made Gdel's already

devastating Incompleteness Theorem,

even worse.

Turing was a much more

practical man than Gdel.

And simply wanted to make Gdel's

theorem clearer, and simpler.

How to do it, came to him,

as he said later...in a vision.

That vision...was the computer.

The invention that has

shaped the modern world,

was first imagined

simply as the means,

to make Gdel's Incompleteness

Theorem, more concrete.

Because for many, Gdel's proof

had simply been too abstract.

It's an absolutely

devastating result,

from a philosophical

point of view,

we still haven't absorbed.

But the proof was too superficial.

It didn't get at the real heart

of what was going on.

It was more tantalizing

than anything else.

It was not a good

reason for something so...

devastating and fundamental.

It was too clever by half.

It was too superficial.

It said: i'm unprovable.

You know, so what?

This doesn't give you any insight

into how serious the problem is.

But Turing, five years later...

his approach to Incompleteness...

that, I felt...

was getting more

in the right direction.

Turing recast Incompleteness,

in terms of computers.

and showed, that since

they are logic machines,

Incompleteness meant,

there would always some problems

they would never solve.

A machine,

fed one of these problems,

would never stop.

And worse...

Turing proved,

there was no way

of telling beforehand,

which these problems were.

Gdel had proved,

that in all systems of logic,

there would be some

unsolvable problems.

Which is bad enough.

Then Turing comes along,

and makes matters much worse.

At least with Gdel,

there was the hope,

that you could distinguish

between the provable

and the unprovable,

and simply leave the

unprovable to one side.

What Turing does,

is prove that in fact

there is no way of telling

which will be

the unprovable problems.

So how do you know,

when to stop?

You'll never know whether

the problem you're working on

is simply

extraordinarily difficult,

or if it's

fundamentally unprovable.

And that...

is Turing's 'halting problem'.

But Turing makes it

very down to earth,

because he talks about machines,

and he talks about whether

a machine will halt or not.

It's there in his paper.

He didn't call it...

didn't speak of it in those terms

but the ideas are really

there in his original paper.

That's where i learned them.

And this sounds so

concrete and down to earth.

You know, computers are

physical devices and you just...

You started running, and...

there are two possibilities:

if you start a program running,

a self-contained program running,

you know, with no input-output.

It's just there!

It's running on a computer.

And one possibility is

it's going to stop, eventually,

saying, i finished the work.

Come up with

an answer and stop...

Done!...Finished!

The other possibility is,

it's going to be searching forever

and never find what it's looking

for, never finish the calculation.

Just go on forever.

It's one or the other.

The problem is...

How can we tell that a

program is never going to stop?

And the answer is: there's no

systematic, general way to do it.

And this is Turing's

version of Incompleteness.

Turing get's Incompleteness;

Gdel's profound discovery,

he get's it as a corollary of

something more basic

which is uncomputability.

Things which, can not be calculated.

Things which no

computer can calculate.

In certain domains, most things

can not be calculated.

But that's your work isn't it?

You come along and make it worse, again!

I do my best.

As if the news wasn't bad enough!

Yeah, i do my best.

Some of it is already

contained there in...

in Gdel's...in Turing's paper

although he doesn't emphasize it.

Startling as the

halting problem was,

the really profound part of

Incompleteness for Turing,

was not what it said

about logic or computers,

but what it said about us,

and our minds.

Were we,

or weren't we, computers?

It was the question that went

to the heart of who Turing was.

Turing was a man

of two great loves.

The first, was for a young man:

Christopher Morcom.

The second,

was for the computer which

he felt he had

brought into this world.

His love for Christopher,

had a unique place in his life,

because Christopher had died,

tragically young.

Turing never recaptured

that first pure love,

but never let go of the memory,

of what it had been.

But when Turing developed

the idea of the computer,

he began to fall in love

in a very different way,

with the sheer power,

of what he had imagined.

He fell in love,

with the fantastic idea,

that one day, computers

would be more than machines.

They would be like children,

capable of learning,

thinking and communicating.

And the scientist in him,

could also see, that if our

minds were like computers,

then here, in our hands, was the

means to understand ourselves.

What started with Cantor,

as a question from

pure mathematics,

about the nature of infinity,

in Gdels hands,

became a question

about the limits of logic.

And now with Turing,

it comes into focus

as a queston about us,

and the nature of our minds.

There is this sort of standard view

that Turing was a computationalist.

And certainly, in a

certain stage of his life,

he did take that point of view.

He said: well, maybe you can

make one of these machines,

imitate the human mind.

But he was of course well aware

of these limitations of computers

and that was one of his

important results of his own.

I think he may have

shifted his view...

he may have vacillated a bit,

and had one view and then another

but then, when he really developed

the computers as actual machines,

he sort of took of and thought,

maybe these really are,

going to...

It's a kind of...

When you get into a scientific

thing, you get...totally...

You think, you know,

maybe this is solving all problems

but without realising the

limitations that are there,

and which are part of

his own...his own theories.

Turing understood,

that Gdel's and his own work,

said that if our

minds were computers,

then Incompleteness

would apply to us,

and the limitations of logic,

would be our limitations.

We would not be capable of leaps

of imagination, beyond logic.

Turing's personality is one thing.

His mathematics doesn't have to

be consistent with his personality.

There is his work on

artificial intelligence,

where i think he...

he does believe that...

machines could become

intelligent...just like people,

or better or different

but intelligent.

But if you look at his first paper,

when he points out

that machines have limits,

because there are numbers...

In fact most numbers,

can not be calculated

by any machine.

He's showing the power of the human

mind to imagine things that...

escape what any machine

could ever do...you see?

So that may go against

his own philosophy,

he may think of

himself as a machine,

but...his very first paper is...

is smashing machines.

It's creating machines and then

it's pointing out

their devastating limitations.

Turing was well aware

of these problems,

but desperately wanted to prove,

he could get the fullness

of the human mind

from mere computation.

And it wasn't just the scientist

in him, that wanted to do this.

Turing's personal philosophy,

which he stuck to all his life,

was to be free from hypocrisy,

compromise and deceit.

Turing was a homosexual,

when it was both illegal

and even dangerous.

Yet he never hid it,

nor made it an issue.

With computers, there are

no lies or hypocrisy.

If we were computers,

then we were the kind of creature,

Turing wanted us to be.

People could vacillate here.

They can have one view and

then wonder about this.

Is this really right?

And then have another view,

and play around.

If they're good scientists

they will do that.

They won't just doggedly

follow one point of view.

So i suspect Turing,

vacillated rather.

But, i think...

in a lot of his analysis

on criticisms of other people

who criticize his view,

he would show the flaws

in their arguments and say:

well look, you see:

it may still be...

despite all these theorems we

know about non-computability,

it still might be, that we

are computational entities,

and then point out:

well, because of this and

this loophole and so on.

And maybe he...

came to believe those loopholes

were sufficient to get him out.

But yet, he did do these things

like looking at oracle machines

which were sort of super

Turing machines; went beyond them.

They're not machines that you

could see any way of constructing

out of ordinary stuff.

But nevertheless,

as a theoretical entity,

these devices were...

theoretical things which would

go beyond, standard computers.

This tension, between the

human and the computational,

was central to Turing's life.

And he lived with it,

until the events

which led to his death.

After the war, Turing

increasingly found himself

drawing the attention

of the security services.

In the Cold War,

homosexuality was seen,

as not only illegal and immoral,

but also a security risk.

So when in March 1952,

he was arrested,

charged and found guilty

of engaging in a homosexual act,

the authorities decided, he was

a problem that needed to be fixed.

They would chemically castrate him

by injecting him with the

female hormone estrogen.

Turing was being treated

as no more than a machine,

chemically reprogrammed,

to eliminate the uncertainty

of his sexuality,

and the risk they felt it posed,

to security and order.

To his horror,

he found the treatment

affected his mind and his body.

He grew breasts,

his moods altered, and he

worried about his mind.

For a man who had always been

authentic, and at one with himself,

it was as if he had been

injected, with hypocrisy.

On the 7th of June 1954,

Turing was found dead.

At his bedside, an apple...

from which he had

taken several bites.

Turing had poisoned

the apple, with cyanide.

Turing was dead,

but his question was not.

Whether the mind was a computer,

and so limited by logic,

or somehow able

to transcend logic,

was now the question that came

to trouble the mind of Kurt Gdel.

Gdel was now working in America,

at the institute for advanced study,

where he continued to work,

as obsessively as he ever had.

Of course, Gdel recovered

from his time in the sanatorium,

but by the time he got here

to the Institute for Advanced

Study in America,

he was a very peculiar man.

One of the stories

they tell about him,

is if he was caught in the Commons,

with a crowd of other people,

he so hated physical contact,

that he would stand very still

so as to plot the

perfect course out,

so as not to have to

actually touch anyone.

He also felt he was being poisoned

by what he called "bad air",

from heating systems

and air conditioners.

And most of all, he thought

his food was being poisoned.

He insisted his wife,

taste all his food for him.

He would sometimes,

order oranges,

and then send them straight back

claiming they were poisoned.

Peculiar as Gdel was,

his genius was undimmed.

Unlike Turing,

Gdel could not believe

we were like computers.

He wanted to show

how the mind had a way

of reaching truth outside logic,

and what it would

mean, if it couldn't.

In principal you can

have a machine grinding away,

deducing all the consequences

of a fixed set of principles

and mathematics would

be static and dead.

I mean, it would just be

a question of mecanically...

deducing all the consequences.

And so...

and so mathematicians in a

sense would just be...machines.

I mean, Turing did think

that he was a machine.

I think he did.

And i think...

that paper on

the imitation game...

shows that.

And Gdel, clearly did not

think that he was a machine.

He thought that he was divine.

You know, that human beings

have a...devine spark in them

that enables them to create

new mathematics i think.

Why was Gdel, so convinced

humans had this spark of creativity?

The key to his believe,

comes from a deep conviction

he shared with one of the

few close friends, he ever had.

That other, Austrian genius,

who had settled at the institute:

Albert Einstein.

Einstein used to say

that he came here,

to the Institute for Advanced Study,

simply for the privilege of

walking home with Kurt Gdel.

But what was it that held this

most unlikely of couples together?

Because on the one hand,

you've got the warm

and avuncular Einstein

and on the other,

the rather cold, wizened,

and withdrawn Kurt Gdel.

And the answer i think,

comes from something

else that Einstein said.

He said that,

God may be subtle

but he's not malicious.

What does that mean?

What it means for Einstein,

is that however complicated

the universe might be,

there will always be beautiful

rules, by which it works.

Gdel believed the same idea

from his point of view to mean,

that, God would never

have put us into a creation,

that we could then not understand.

The question is,

how is it that Kurt Gdel can

believe that God isn't malicous?

That it's all understandable?

Because Gdel is

the man who has proved,

that some things can not be

proven logically and rationally.

So surely, God must be malicious.

The way he gets out of it,

is that Gdel, like Einstein,

believes deeply in intuition.

That we can know things,

outside of logic,

because we just...intuit them.

And they believe it

because they have both felt it.

They've both had

their moments of intuition.

Just like Cantor had had his.

He talks about new principals...

that the mathematician...

closing your eyes,

tuning out the real world,

you can try to perceive,

directly by your

mathematical intuition,

the platonic world of ideas,

and come up with new principles,

which you can then

use to extend the...

the current set of

principles in mathematics.

And he viewed this as a way

of getting around, i think,

the limitations of his own theorem.

I don't think he thought

there was any limit

to the mathematics that

human beings were capable of.

But, how do you prove this?

The interpretation that

Gdel himself drew,

was that...

computers are limited.

He certainly tried again and

again, to work out that...

the human mind

transcends the computer.

In the sense that he can

understand things to be true,

that can not be proven,

by a computer program.

Gdel also was

wrestling with some...

finding means of knowledge,

which are not based on experience

and on mathematical reasoning,

but on some sort of intuition.

The frustration for Gdel,

was getting anyone to understand him.

I think people very often, for

some reason, misunderstand Gdel.

Certainly his intention.

Gdel was deliberatly

trying to show,

that, what one might call

"mathematical intuition".

He referred to, what he called,

"mathematical intuiton",

and he was...

demonstrating, clearly in

my mind demonstrated,

that this is outside

just following formal rules.

And, i don't know...

Some people...

picked up on what he did and said,

well, he's showing there are

unprovable results and

therefore beyond the mind.

What he really showed, was that

for any system that you adopt,

which, in the sense the mind has

been removed from it, because you...

The mind is used to

lay down the system.

But from thereon, it takes over.

And you ask what's it's scope?

And what Gdel showed,

is that it's scope

is always limited.

And that the mind

can go beyond it.

Here's the man who has said

certain things can not be proved,

within any rational

and logical system.

But he says, that doesn't matter,

because the human mind

isn't limited that way.

We have intuition!

But then of course the one thing

he really must prove to other people,

is the existence of intuition.

The one thing you'll

never be able to prove.

He has these drafts of papers where

he expresses himself very strongly.

But he didn't...

He wasn't satisfied with them.

Because he couldn't prove a theorem

about creativity or intuition.

It was just...

a gut feeling that he had.

And he wasn't satisfied with that.

And so Gdel,

like Cantor before him,

had finally found a problem,

he desperatly wanted to solve,

but could not.

He was now caught in a loop.

A logical paradox, from which

his mind could not escape.

And at the same time,

he slowly starved himself to death.

Using mathematics, to show

the limits of mathematics, is...

is psychologically

very contradictory.

It's clear in Gdel's case,

that he appreciated this.

His own life has this paradox.

What Gdel is,

is the mind thinking about itself,

and what it can achieve

at the deepest level.

Someone used the phrase:

"the Vertigo of the Modern".

You can be led into that particular

reflexive whirlpool where you're

beginning to think about

thinking about thinking...

about thinking about thinking...

and you find yourself entangled

in your own...in your own thoughts.

Well that seems to me, almost the

quintessence of the Modern moment

because there you have a...

what you could call

a paradox of self-reflection.

The kind of madness that you find

associated with Modernism,

is the kind of madness

that's bound up with,

not only rationality,

but with all the paradoxes that

arise from self-consciousness.

From the consciousness contemplating

it's own being as consciousness

or from logic contemplating

it's own being as logic.

Even though he's shown

that logic has certain limitations,

he's still, so drawn to that,

to the significance of the

rational and the logical,

that he desperately want's to

prove whatever is most important,

logically.

Even if it's an

alternative to logic.

How strange.

And what a testimony to his..

his inability to separate himself,

to detach himself from

the need for logical proof.

Gdel of all people...

At the beginning of our story,

Cantor had hoped,

that at it's deepest level,

mathematics would

rest on certainties.

Which for him,

were the mind of God.

But instead, he had

uncovered uncertainties.

Which Turing and Gdel then

proved, would never go away.

They were an inescapable part,

of the very foundations

of maths and logic.

The almost religious belief,

that there was a perfect logic,

which governed a

world of certainties,

had unraveled itself.

Logic, had revealed

the limitations of logic.

The search for certainty,

had revealed uncertainty.

I mean, there's a fashionable

solution to the problem,

which is basically,

in my opinion,

- people are going to

hate me for this -

is sweeping it under the carpet.

But you see, the problem is:

i don't think you want

to solve the problem.

I think it's much more fun

to live with the problem.

It's much more creative!

This notion of absolute certainty...

There is no absolute

certainty in human life.

But our knowledge, our possible

knowledge of this world of ideas,

can only be incomplete and finite,

because we are incomplete and finite.

The problem is that today,

some knowledge,

still feels too dangerous...

because our times

are not so different,

to Cantor, or Boltzmann,

or Gdel's time.

We too, feel things

we thought were solid,

being challenged...

feel our certainties slipping away.

And so, as then...

we still desperately want to

kling to a believe in certainty,

that makes us feel safe.

At the end of this journey,

the question i think

we are left with...

is actually the same as it was

in Cantor and Bolzmann's time:

are we grown up enough,

to live with uncertainties?

Or will we repeat the

mistakes of the 20th century,

and pledge blind allegiance,

to yet another certainty?