Dangerous Knowledge

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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...

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