Dangerous Knowledge Page #2

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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!

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