Dangerous Knowledge Page #7

---

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,

Rate this script:0.0 / 0 votes