Infinity: The Science of Endless

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

Channel: World Science Festival

Views: 981,193

Rating: 4.612639 out of 5

Keywords: Infinity The Science of Endless, what is Infinity, Mathematics, science of infinity, Raphael Bousso, Philip Clayton, Steven Strogatz, Hugh Woodin, Keith Devlin, New York City, NYC, world science festival, full program, World, Science, Festival, 2013, Big Ideas Series

Id: KDCJZ81PwVM

Length: 83min 57sec (5037 seconds)

Published: Thu Jan 16 2014

Reddit Comments

[u/Eme93]

Whenever I watch or read something like this I get this horrible feeling of anxiety, because I'm not studying astrophysics, math or philosophy and know that while studying orthopedics engineering will provide for me a comfortable and good life it wont satisfy my immense hunger for working on these types of questions.

[u/Speedbird_1]

The greatest part of studying physics is getting side tracked during a lecture and having a conversation just like this with professors.

[u/Joey_Jewsacky]

Awesome

[u/Guan-yu]

Interesting! I've not watched it all, as I am pressed for time, but it seemed very interesting indeed, thank you for the link!