Dangerous Knowledge Page #8

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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...

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