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?