Infinity

Infinity

What is Infinity

Infinity is really cool

It is about as big as it gets

U cant really put a name to it because that limits it

Infinity as I see it refers more to the true nature of reality. No one & no thing has any kind of separate inherent or intrinsic nature.

I am a librarian so I feel that Infinity is my business. I connect words that were written a thousand years ago. Also words that were written last week.

There is an infinity number of possibilities. Means I have no limits as a composer or performer.

Every single note that I choose to play in a song I could have played any of the other notes. Or the notes between the notes.

U know like in the blues

Infinity is something that

Goes on & on & is continuous

Does not have an end point

I can show you

My experience of Infinity

Infinity means numbers go on & on

Infinity generally speaking is a very precise mathematical concept

Infinity is pure fiction

An optical illusion

Before Einstein people believed that the speed of light is Infinity.

It is Instantaneous & Einstein explained speed of light is finite - It is very very fast the light but it is finite. So the integer is very very big.

Much bigger than the speed of light

But still finite

What we will be talking about we want to call near Infinity

It is something which actually does take place in our finite world

But it is so big

And complicated that complexity of it is approaching Infinity in practical terms

Now that we have computer it is time to get rid of

this illusion & start doing completely purely finite & discreet mathematics

Infinity is here to stay

I have spent years thinking about it

And I am no wiser

I am not an accountant

This is what we do

We deal with stuff like that we deal with questions like that

This is why people like me go into Mathematics

We got these different perspectives

We are going to be investigating tonight

Infinity looked up from the point of view of Philosophy Theology Mathematics & Physics

And we are going to try to wrap ourselves round all of that in the next 90 mins

So sit back & enjoy the ride

Ok we are going to begin with Philip Clayton

Philip is a Philosopher Theologian - He is a Provost of Claremont Lincoln University

He was the Principal Investigator of the Science & Spiritual Quest Program

He has written more than a dozen books in his field including Adventures in the Spirit

In Quest of Freedom & Religion & Science the basic

He has spoken about this all over the world

Philip take us from what we just seen

Infinity what an incredible concept

U can picture the earliest men & women staring up at the innumerable stars

Watching grains of sand slowly slide through their fingers

And think what is this thing without limit

I wanna take u on a brief little stroll through the history of the Infinite

Try to do that in a finite period of time

5 mins but lets see he is gonna watch me

And see if I can find a couple of Philosophers a couple of Thelogians

Maybe a representative of East & West

To give u some sense of the parameters of this concept

If that is not a contradiction in terms

In some ways a vast variety of ways of conceiving the Infinite

In another sense a few core concepts that keep arising

That u will see over & over again

As we move through this evening

Lets start with Zeno of Elea

The Philosopher who gets credit for first introducing the concept of the Infinite in the West

Actually this is not quite right

because Anex & Manda had introduced the concept

a few years earlier of the Boundless

A pyron

The thing that is without limits

A scary kind of notion that the Greeks really disliked

U cant build a Parthenon with the Boundless right

Zeno found that whenever he introduced real infinites into the world

He ran into paradoxes

So lets take the fastest Greek of all Achilles

And lets set him in a foot race

Against a tortoise right

Can we give the tortoise a head start

Is that all right

So the tortoise is out there in front

An hour later he has made a hundred feet right

And here goes Achilles

Zeno says well he has gotta go half way to the tortoise first

And then he has gotta go half way there & the tortoise

And then he has gotta go half way

U get the point

Achilles Zeno argued could never pass the tortoise

If Infinites are real

Because he always has to transverse half the distance & half the distance

To the end

So the Infinite & paradox are linked from the very beginning

Lets move on to Pythagoras

We give Pythagoras credit for inventing the notion of Mathematics

Which by the way in Greek means That which can be learnt

He did it with a Quays eye religious orientation

Pythagoras said that each integer had some sort of philosophical or spiritual meaning

And the ratios between the integers are crucial

Seven eighths one fourth & so forth

So the harmonic progression was for them a religious insight

He launches the idea of the theorem & what is the most famous theorem from Pythagoras

The Pythagorean theorem right

And all of a sudden the whole project goes to hell in a hand basket

Right we have got an equal right angled triangle

One one so we can see one square plus one square equal two

equal c square so c equals the square root of two

But that is an irrational number

To write down the square root of two would take us infinite digits with no repeating pattern right

Right in the middle of the religion of the Infinite

Breaks out the impossible

The irrationality the religion of rationality

Shot to hell right at the

outset of the story

Lets step over to India for a moment & lets see how things look different over there

We go to the great founder of the Jain religion U know Ahimsa Martin Luther King Ghandi Mahavira

The Jains weren't famous just for that though

They also did incredible speculation on the nature of Infinite

As far as we know

By 400 BC the Jains had introduced a notion that hadn't been used before

The difference between that which go on without ending

And the truly limitless

The truly endless

They also began to distinguish orders of Infinite

So there is Infinite in length

Infinite in area

Infinite in volume

And Infinite perpetually

Get the distinction

They are trying to give us orders of Infinite as they go

And the last thing for which we credit the Jains

When they distinguished between Enumerable Innumerable & Infinite

They said the highest Enumerable number

And they defined it the way that as we will hear in a moment

Cantor defined Aleph naught the first number of the transfinites

So amazing comprehension in India some 2400 years ago

All right Lets step back to the Greeks

And come to Aristotle the Father of some two dozen scientists

That crucial thinker

Aristotle realizes that with actual Infinities we are in deep doo doo

Basically Physics is not gona work well

So what he decides is that there exists no actual Infinite

We have to banish that concept from the physical world

If we are gona do good Science

So he says it is fine that we can have a potential Infinite

It just cant ever be actual

So for example the potential Infinite is start counting upward

U can go as long as u want to

But in a finite period of time

U only make it a finite distance

The world is only a finite number of years old he said

Therefore there are no actual Infinites

Take a quantity & start dividing U can divide it as much as u want to

But at some point u will run out of time As I am about to

And u have to stop

Aristotle launched this notion of potential of Infinity All right lets finish up with three theologians & three radically different understandings of the Infinite

Lets go to the High Middle Ages the Scholastic Period & the greatest of thinkers Thomas Aquinas

Thomas Aquinas says as a Theologian I am interested in a different question

U all are interested in quantity I am interested in a quality of existence

What mode of existence would God have that divine

God would be qualitatively Infinite

Which means God would be Insperfect thee si mum The most perfect being

It is a mode of existing a way of existing that sets God apart from anything else

Not quantity So he bifurcates between the mathematical Infinite & the religious or philosophical Infinite

But still Aristotle rears his ugly head

Because even this Infinite God couldn't create an Infinite object

Because anything in the world must be by nature finite

Moving on from Aristotle to the 15th century we find the Bishop of Cologne Nicholas of Cusa

And here is a Theologian who decides he is gona go with the Infinite all the way

No restraint

So the Infinite what is that

It is that thing which has no limits

Therefore every thing must be included within the Infinite

Nothing can be excluded

The world itself must be within God

It might cost him his job

Mathematics love Nicholas de Cusa because he used Mathematical examples to describe the God world relation

So God is the circle whose center is everywhere

And whose circumference is therefore nowhere

Nothing is outside of God

And he said if u get that right it means a coincidence of opposites

God world God human kind linked together

And he said we are the animal that stands at the boundary between the eternal & the finite

And finally let me close with that great heretic

The one that the modern thought love to hate Baruch Spinoza

A Jewish fascinating Jewish philosopher kicked out of the synagogue at aged 14 for allegedly saying that God has a body

Spinoza decided to write his metaphysic in the guise of geometry

Ethica More Geometrico - Ethics in the guise of Geometry

And u guys it is overwhelming compelling argument If God is Infinite God is the One substance

There can be no other substances besides this One

The substance must have Infinite attributes

And we finite little beings cant be separate beings so we must be modes of the One

God however is not the transcendent God the personal God who does stuff

He used the phrase Deu Siva Latora God that is Nature

God & Nature become absolutely one for Spinoza

And the highest ethic is for u to live in accordance with the laws of Nature

To be a part of that One unfolding Infinite hold is our fate

Half the philosophers revere him as Science friendly

Half the philosophers revere him as Theological

For him it is the intellectual love of God to know nature to know yourself as part & parcel of Nature

And to behave together with Nature

The Infinite then focuses Spinoza from Theology back into the natural world

So Keith a short history of Infinity

Applause

Makes me feel nervous we are live streaming this I am not quite sure who is listening in to this broadcast

As a simple Mathematician I am feeling quite nervous there

And I am gona very rapidly bring it back to my own comfort level Coz my next guest is a regular practicing Mathematician like myself

It is Steven Strogatz one of the world's most highly cited Mathematician

His honours include the Presidential Young Investigator Award

A Lifetime Achievement Award for the communication of Mathematics for the general public

And membership in the American Academy of Arts & Sciences

He is currently a Professor of Applied Mathematics at Cornell

And his publications include Non Linear Dynamics The Calculus of Friendship & wait for it The Joy of X

Steven

Thank you

Applause

Thank you very much Keith & thank you all for coming out

I like to as Keith said bring it back to Mathematics & for us in the Math world the really great transcendent hero of Infinity is Georg Cantor

Cantor lived in the Mid 1800s & came up with ideas that are so mind blowing

And so stunning that we are still kinda reverberating from his insights--=

We will be hearing about the disturbing things that he thought of

They are still so counter-intuitive that the Mathematicians & Physicists are really grappling with them even today

So the big insight from Cantor the biggest of all is that there are different kinds of Infinity

Some Infinities can actually be bigger than others

And so I wana walk u through a few of his Mathematical arguments because they are so beautiful & so elegant

that if u haven't seen them I think they will change your life

And if you have seen them this is like listening to the greatest song your favourite song a second time I don't think u will mind

So but to set this up I want to

I am gona show u some visual version of what Cantor came up with

But first I think it is worth mentioning a little bit about his life

Because it was as with any great maverick there were people who opposed him

He was not regarded necessarily as a hero to all Mathematicians

In fact one of the greatest Mathematicians of his era

described Cantor's work as a disease

Ouch

Another one of his colleagues a Mathematician named Kronecker who was in a position of power in Berlin

Where Cantor very much wanted to be a Professor

He never got to be a Professor in Berlin

Kronecker wouldn't let him

Kronecker described Cantor as not only a charlatan but a corrupter of youths

Now I don't know what that makes me since I see there are some young people here & I will be talking about his work

Erm so yes corrupter of youth

Well actually this was no joke for Cantor because he as it turns out suffered from mental illness

It seems that he had depression I mean we know that he did have very significant bout of depression

May have had bipolar disorder

And he spent quite a few stints in sanatoria

It used to be thought that it was because of these attacks from Mathematicians that he was driven to this

But in more modern thinking probably he just had he suffered from depression

But didn't help that he was taking his withering criticism from so many people

On the other hand some of the other great Mathematicians of the time like a man named David Hilbert probably equalled to greatness as the two greatest of that era

Hilbert described Cantor's work as a paradise

And said nothing will ever expel us from this paradise Cantor has created

So let me introduce to u Cantor's paradise first by showing u some of the paradoxes

And really startling things that come about when u start thinking about Infinity mathematically

And the way we will do this is with a charming video that was produced by the Open University a couple of years ago

60 Seconds Adventures in Thought

Number 4 - Hilbert's Infinite Hotel

A grand hotel with an Infinite number of rooms

And an Infinite number of guests in those rooms

That was the idea of German Mathematician David Hilbert friend of Albert Enstein & enemy of chambermaids the world over

To challenge our ideas of Infinity he asked what happens if someone new comes along looking for a place to stay

Hilbert's answer is to make each guest shift along one room

The guest in room 1 moves to room 2 & so on

So the new guest will have a space in room 1 & the guest book will have an Infinite number of complaints

But what about a coach containing an Infinite number of new guests pulls up

Surely he cant accommodate all of them

Hilbert frees up an Infinite number of rooms by asking the guests to move to the room number which is double their current one

Leaving the Infinitely many odd numbers free

Easy for the guest in room 1

Not so easy for the man in room eight million six hundred thousand five hundred & ninety seven

Hilbert's paradox has fascinated Mathematicians Physicists & Philosophers even Theologians

And they all agree u should get down early for breakfast

It is a brilliant video I hope u caught the main Mathematical points at the risk of over explaining I am just gona remind u what u saw

That in Hilbert hotel which is this parable of Infinity

It has Infinitely many rooms & so as it was explained if another guest shows up u can still make room for that guest by shifting everyone over one room

And there is always room because there is Infinitely many rooms for them and then the new guest can go in room one

If a bus or the video has a coach even a bus with Infinitely many new guests is still not enough to cause trouble at the Hilbert hotel

Because u can shift all the current guests into the even rooms leaving the odd number rooms for the guests

So the question starts to become well if Infinity is really that big at the Hilbert hotel

Is there any number of guests that u couldn't accommodate at the hotel

I mean are there Infinities so big that even the Hilbert hotel would not fit them

And so Cantor worried about this & came up with an attempt to find an Infinity too big even for I mean he didn't think in terms of the Hilbert hotel

But let me illustrate that way by imagining a situation that if u are kinda a nasty Contrarian

U might think what if an Infinite number of buses each carrying an Infinite number of people show up

What will the night manager do with this

So let me illustrate in fact that problem can be solved at the Hilbert hotel that is no problem at all

And so here is the solution to it

Lets just visualise these people by showing a picture which will have a row containing all the Infinitely many guests

That would correspond to bus number one And then there is bus number two

Each with its Infinite number of guests which I am gona depict as faces

And then there is an Infinite number of of these buses

Its not supposed to be just 4 by 4 u are supposed to picture it with Infinitely many guests in each row & Infinitely many buses

(Keith ) The audience probably cant see but from my position I can see dots at the end of those rows

Yes thank u right

So those three dots are supposed to indicate that both the rows & buses go on in all directions

The first inexperienced manager might attempt to solve the problem this way

I mean what the goal is is to assign each passenger a room

And the problem will be solved if for any given passenger u can say

That person is in a room with a finite number a specific number

Like imagine a guest list & I can say oh yes Mr Smith u are gona be in room 133

As long as everybody has a room everybody is happy

Here is the first attempt to put people into rooms

Let me put Person 1 from Bus 1 in Room 1

And then I will put the second person from that same bus in Room 2

Third person in Room 3 u can see the this is not a good solution

Do u see what the problem is

No one in Bus 2 is gona be happy

Because the whole hotel will just be filled with passengers from Bus 1 & so u have infinitely many bus loads of infinitely many unhappy people

This is not what u want at your hotel

So that is not a good solution to the problem

But there is a much more clever way of doing it which then does accommodate everyone

What u do notice this zig zagging pattern of arrows

U start with the first person

That person gets Room 1 the second person is at the end of the first arrow

So it would be passenger 2 from Bus 1

Then u go down to Bus 2 Passenger 1 that person gets Room 3

Just follow the arrows & that is the order which people are placed into rooms

Can u sort of see what is happening how it is fanning out from the corner

And that will have the property that u can pick any person in that Infinite array & that person will be served in a finite time

They will have a room

So this is a way of actually accommodating Infinitely many people in Infinitely many buses in the Hilbert hotel

This raises the by the way I should say that there is the abstract version of this which is that this shows that the fractions

The positive fractions

Starting u know like well not starting but think about any fraction one third one fifth two eighth

U know don't worry about lowest terms

Four seventh Four seventeenth

Any fraction u can think of would correspond to one of these passengers

Like the person in Bus 4

A passenger 17

That might be what we would call four seventeenth

I mean that is a correspondence between the people in these diagrams & the positive fractions

And so this argument that I have given u here is essentially Cantor's argument for showing that the positive fractions can be counted

In the sense that they can be put into what Cantor called the one-to-one correspondence

With the natural counting number the numbers 1 2 3 4

There is a way of listing all positive fractions so that we can say this is the first one this is the second one this is the third one

And every positive fraction will be at some finite place in that list

They are not ordered by size by the way they are ordered in this way that I just described in this zig zagging pattern

So the rational numbers the fractions are countable meaning they can be put into one-to-one correspondence with the natural numbers 1 2 3 4 & so on

And this raised the question in Cantor's mind it seems like everything he could think of is countable

Every Infinity he could think of was countable or to put in another way

The Hilbert hotel really lives up to its model - There is always room at the Hilbert hotel

Well actually though in 1873 Cantor discovered an amazing example showing that that was not true

That there was an Infinity that is so big that it would defeat the Hilbert hotel no matter how clever the manager is

And this would has to do with the amount of Infinity contained in a continuous line

If u imagined the number line the u know traditional number line that was tagged on the wall with your elementary school

And shows the integers 1 2 3 4 & there is all the space in between where all the fractions & irrational numbers are

That continuous line what we think of it as the real numbers

If the real numbers showed up at the hotel

There wouldn't be room for all of them

And here is the proof

This is Cantor's famous diagonal proof

He said imagine there was a roster a sort of a guest list

And this was a guest list it is gona be approved by contradiction

I will show that there is a real number that doesn't have a room

Here is a putative list

Where room 1 assigned to this real number that is being shown on here .42 etc

And room 2 is assigned to some other real number don't worry about the details of number

The idea is just to imagine a whole list of all the real numbers just like we could list all the rational numbers

Maybe we could list all the real numbers

Well Cantor's argument is no because if there were such a list and here is an attempt to show what it might look like

U then circle the first digit of the first number I don't know if u can see there is a little circle around that 4

And then u circle the second digit of the second number 2

And the third digit of the third number 1

So from that u construct the real number which in this case would be point 4 2 1 and so on

That is not the problem that is not the bad number

What we then do is change each of those digits in that number

Like the 4 that begins this number

Maybe make it a 3

Or just whatever u want just change it

Or the 2 that is the second digit make it something else make it a 7

Just have some system

But the rule is u construct this number down the diagonal & u systematically change its nth digit

To something else

That is if it is the nth number the nth digit gets changed

And so by doing that Cantor constructs a number which is not in the list

Because it cant be the first number in the list coz it differs from the first number in the first place

It is not the second number in the list coz it differs from the second number in the second place

And so on

So by systematically going down the diagonal & then changing every digit

He has now produced a number that is not on the list

And this argument

Shows that in fact that real numbers cannot be counted

There is no list that can embrace them all

And so with that comes the question then

This very nagging question that we are gona be talking about next

Which is

Well if there are now 2 kinds of infinity

The countable kind & this continuous kind which is bigger

Are there other Infinities

And really more to the point

Is there some Infinity that is sandwiched in between them

Is there some Infinity smaller than the continuous Infinity but bigger than the Infinity of the 1 2 3 variety

That is called the Continuum Hypothesis

The idea that there is no such Infinity in between

It was Cantor's great conjecture

But he never proved it

Maybe Keith will tell us a little about his own experience with the Continuum Hypothesis

Yeah because earlier on in my career I actually went into set theory coz I got seduced by what happened next

And here is what happened next

In 1940 an Austrian Mathematician called Kurt Godel showed this practised conjecture Continuum Hypothesis that there is no Infinity between the natural numbers & the real numbers

Godel showed that that conjecture could not be proved false

OK u might say

It means it is gona be true right

Well no because in 1963 an American Mathematician called Paul Cohen showed u couldn't prove it could be true

Cant prove it is false

Cant prove it is true

it is independence of the axioms that we take of Mathematics

And as a graduate student beginning at the very end of the 1960s

I think this was significant that this was in the 1960s - People were hiring all sorts of all things

High enough to try to go into Infinity

There was a question of

We soon realized we had a technique called Forcing that allowed us to construct alternative set theories

Alternative mathematics

In which u could prove things true not true or neither of them

U could prove things independent

Un like many mathematicians of my generation I got seduced by that kind of question

And we used cohens technique in particular & sort of Godels technique

In order to investigate all sorts of universities

And it was a wild period

But it was a period that was eventually gona get tamed

And I think it was gona get tamed by I would say it has been tamed

To a large extent by our next guest

Who is William Hugh Woodin

Set theorist at the University of California Berkeley

Large cardinal in my business

Does not mean a large portly gentleman in a uniform

That is his business

A large cardinal is an infinite an infinity

So the infinity of a very large size

Hugh Woodin has a large cardinal named after him called The Woodin Cardinal

He has made many notable contributions to the theory of Inner models & Determinacy

I think of him as a mathematical equivalent of Physics

Who has invented 21st century mathematics

Dropped into the end of 20th century

Hugh take us from there

Applause

Thank u for that introduction

Thank u all for being here

So I spent my life trying to solve the Continuum Hypothesis

Now we just heard that u cant prove it true

U cannot prove it false

It doesn't mean it doesn't have a solution

It just means we don't have all the principles yet

So that is the question

Are we missing the key principles or the principles defined

That will enable us to determine whether the Continuum Hypothesis is true or false

Or maybe there is a much stronger version of Collins Theorem that shows there really is no answer

So really the fundamental question is this

We have this conception of Mathematically Infinity

And it is embodied in our conception of the universe of sets