"Beneath the surface of the world... are the rules of science." "But beneath them, there is a far deeper set of rules." "A matrix of pure mathematics, which explains the nature of the rules of science, and how it is we can understand them in the first place." "To see a world in a grain of sand," "And a heaven in a wild flower," "Hold infinity in the palm of your hand," "And eternity in an hour." "What is the system that... that everything has to adhere to, if there is no God?" "You had these ideas, and..." "and you had to be very careful because at any moment, they would bite you." "They sounded great but they were very dangerous." "But then of course, people get scared." "So they pull back from the edge of the precipice." "Well, this is not a matter of liking it or not..." "You have here this proof and..." "one has to live with it." "This film, is about how a small group of the most brilliant minds, unraveled our old cosey certainties about maths and the universe." "It is also about how once they had looked at these problems, they could not look away... and pursued the questions to the brink of insanity, and then over it, to madness and suicide." "But for all their tragedies, what they saw, is still true." "Their contempories largely rejected the significance of their work, and we have yet to fully inhered it." "Today, we still stand only on the threshold, of the world they saw." "My name is David Malone." "And this is my hommage, to former great thinkers, who without most of us, even having heard of them, have profoundly influenced the nature of our age, and who's stories have, i think," "an important message for us today." "This is Halle." "A provincial town in Eastern Germany, where Martin Luther once preached the reformation." "Our story starts here, at the towns university with a mathematics professor." "A man called:" "Georg Cantor, who started a revolution he never really meant to start." "But which eventually threatened to shake the whole of mathematics and science on it's foundations." "And he started this revolution by asking himself a simple question:" "how big is infinity?" "Cantor is wonderful because it's so crazy." "It's the equivalent of being on drugs." "It's just an incredible feat of imagination." "Georg Cantor is one of the greatest mathematicians of the world." "Others before him, going back to the Ancient Greeks at least, had asked the question." "But it was Cantor, who made the journey no one else ever had, and found the answer." "But he paid a price for his discovery." "This is the only bust there is of Georg Cantor." "It was made just one year before he died, and he died utterly alone, in an insane asylum." "The question is:" "what could the greatest mathematician of his century have seen, that could drive him insane?" "If all that Cantor had seen was mathematics, then his story would be of limited interest." "But from the beginning," "Cantor realised his work had far wider significance." "He believed, it could take the human mind, towards greater, trancendent truth and certainty." "What he never suspected, was that eventually his maths would make that certainty ever more elusive." "Perhaps even destroy the possibility of ever reaching it." "If you want to understand Georg Cantor you have to understand he was a religious man." "Though not in a conventional sense." "He almost certainly came to this church, but that's not his God." "He wasn't interested in a God who's mysteries were redemption and resurrection." "Ever since he was just a boy, he had heard what he called:" "a secret voice, calling him to mathematics." "That voice which he heard all his life, in his mind, was God." "So for Cantor, his mathematics of infinity had to be correct, because God, the 'True Infinite', had revealed it to him." "These things which are now hidden from you,  will be brought into the light." "If you look at Cantor's last major publication, about Set Theory, in 1895." "It starts with three aphorisms, and it's third motto is from the bible, and it's from Corinthians." "And it's, you know:" "that which has been hidden to you, will eventually be brought into the light." "And Cantor i think, really believed he was the messenger." "That this theory had been hidden." "He was God's means, of bringing this Theory of the Infinite to the world." "There is no contradiction for Cantor between his religious thinking and his mathematics." "He understood or he was thinking that, his ideas were a gift of God." "My view was that Cantor was trying to understand God and that this was really... like a mathematical theology that he was doing." "Cantor's God was the 'Creator God'." "The God who set the planets spinning in their orbits." "Who's mysteries were the eternal and perfect laws of motion." "Laws who's discovery had launched the modern world, and allowed us to see the world as curves, trajectories and forces, and which would one day even put men on the moon." "The eternal certainties layed down by God which Newton and Leibniz had discovered." "And it was infinity which lay at the heart of it all." "But, there was a problem with it." "If you look at that beautiful smooth curve of motion, you notice it's not actually smooth." "It's made of an infinite number of infinitesimally small straight lines." "And each line is an instant in which nothing moves." "But like frames of film, if you run one after another, you get motion." "And it works." "The whole thing relies on infinity but it works." "And because it worked, everyone said:" "alright, we don't understand infinity." "Just leave it alone." "Cantor comes along and says no, if this whole thing rests on infinity, we have to understand it." "And now remember Cantor is a religious man." "So for him, that symbol... isn't just a scientific mystery." "It's a religious mystery as well." "Here's the maths that God uses to keep creation in motion." "And at it's heart, lies the deeper mystery of infinity." "But it was a mystery, that had so far defied every mind that had looked at it." "The first Modern Thinker to confront the infinite was Galileo." "And he tried to do it using the circle." "This is how he did it." "He said: first of all, draw a circle." "Now put a triangle inside it, then a square and keep on going." "Keep adding sides." "Eventually he realised that what the circle is... is a shape with infinately many, infinitesimally small sides." "Which seems great." "Now you can hold the infinite and the infinitesimal in your hands." "But as soon as he done that, he realised it actually opened up a horrible illogical paradox." "Because he said: alright, let's draw a bigger circle outside." "And now with an infinitely sharp pencil, draw from the center... infinitely sharp lines." "One for each of the lines on the inner circle." "There's an infinate number of them, that should be enough for the inner circle." "But now extend those lines out till they meet the outer circle." "Now, those lines are diverging... 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 Poincaré, 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!" "The...for me, the fun..." "Cantor was...he was..." "He was playing on the edge!" "You know, the idea was, you had these ideas, and... and you had to be very careful because at any moment they would bite you." "They sounded great but they were very dangerous." "You see, they were almost self-contradictory." "The notion of the 'Set of Everything' for example, is self-contradictory." "And...it's..." "and people got frightened." "His critics feared Cantor was going to dislodge the certainty and clarity vital, to mathematics and logic, which might not be able to be put back." "It seemed Cantor had opened maths to the very thing it was supposed to save us from:" "irresolvable uncertainty." "Cantor knew the only way to convince his critics, was to make his theory complete." "Could he show there was a logic to his infinities?" "Some system, that bound them all together?" "What he absolutely must decide now, is, what's the relationship between them." "If he can do that, then his theory is perfect." "If he can't, then all he has is bits." "So he has to decide what's the relationship between them." "And that question, is the 'Continuum Hypothesis'." "No matter how isolated he became, the more he was opposed, the more he struggled." "Where another person might have given up, Cantor didn't." "Clinical psychologist, Dr. Louis Sass, suggests it is precisely this ability to be isolated, which is key, to Cantor's genius." "I think, that willingness to step into a realm... you know, beyond the... the taken for granted, is abolutely essential." "But i think if you're a person who takes that step, in a way you're already doomed, to living outside in some way." "So, you know..." "It's not as if it's only the intellectual project itself that takes you out there." "There is something about you as a person, that is just..." "That unnaturalness, so to speak, comes so naturally to you." "Cantor was trapped." "There were too many things that went to the core of who he was, for him to be able to give up." "When Cantor was just a boy, his father sent him a letter... which became his most precious possession, and which he carried with him all his life." "In it, his father told him, how the whole family looked to him, to achieve greatness." "How he would come to nothing, if he did not have the courage to overcome criticism and adversity." "How he must trust in God to guide him, and never give up." "And he never did." "Well i think, here you come to the root of the problem for Cantor of a theory, that he was certain, was correct, in part because he believed that it had come to him as a message from God." "There's a very important religious aspect to Cantor's... struggle to deal with the infinite, and face the problems of... not being able to resolve many of the open questions that he himself raised for the first time." "By 1894, Cantor has been working solidly on the Continuum Hypothesis for over two years." "At the same time, the personal and professional attacks on him... have become more and more extreme." "In fact he writes to a friend saying he's not sure he can take them anymore." "And indeed, he can't." "By May of that year, he has a massive nervous breakdown." "His daughter describes how his whole personality is transformed." "He will rant and rave, and then fall completely and uncommunicatively silent." "Eventually, he's brought here... to the 'Nervenklinik' in Halle, which is...an asylum." "Today, we would say Cantor suffered from manic depressive illness." "From Cantor's time, we have left, the case notes of most of his psychiatrists." "In the notes for example, we see that he, at times, was quite disturbed, was screaming... and see that he was really suffering from... severe bouts of mania." "Sometimes he would be angry... and he would have ideas of grandeur and sometimes he had also ideas of persecution." "After his breakdown, everything about Cantor is transformed." "He tells a friend he's not sure he'll ever be able to do mathematics again." "He asks the university if he can stop teaching maths and teach philosophy instead." "But interestingly, during this whole time... despite having claimed, him not being able to do mathematics again, he never stops working on the Continuum Hypothesis." "It's as if... he just can't put it down, can't look away." "You can only think:" "i must find the proof!" "This i can understand because when you are a mathematician, then you are for all the time a mathematician." "It's a form of..." "living for you." "You must think about mathematics, and... you can't think anything else..." "the whole day." "You are thinking and thinking and thinking." "And you say:" "i must find it!" "I must, i must, i must!" "You can't think anything else." "In August of 1884, he writes a letter, to his friend and colleague, the last man who still publishes his work." "A man called, Mittag-Leffler." "And the letter is ecstatic." "He says: i've done it!" "I've proved the Continuum Hypothesis." "It's true." "And he promises that he'll send the proof in the following weeks." "But the proof never comes." "Instead, three months later, a second letter arrives." "And in this one, you can feel Cantor's embarrassment." "He says: i'm sorry i should never have claimed that i proved it." "And he says: my beautiful proof lies all in ruins." "And you can see the wreckage of his work, in the letter." "But then, three weeks after that, this letter arrives:" "and in it he says:" "i've proved that the Continuum Hypothesis is not true." "And this pattern continues." "He proves that it is true... and then he's convinced that it's not true." "Back and forth." "And in fact, what Cantor is doing... is driving himself slowly insane." "One of the things that will happen especially in the early stages and, the stages just before a schizophrenic break, but also in the early stages, will be that the patient is... in a way, looking too hard at the world" "and too concentrated away." "As a kind of rigidity of the perceptual stance." "When he could not solve the Continuum Hypothesis," "Cantor came to describe the infinite, as an abyss." "A chasm perhaps, between what he had seen... and what he knew must be there, but could never reach." "What can happen, is that some object in the world that... that the rest of us would just... consider just a sort of random thing there, seem somehow symbolic in some way." "There's a way in which in order to understand something you have to look very hard at it." "But you also have to be able to sort of move away from it and kind of see it in a kind of context." "And the person who stares too hard can often lose that sense of context." "Cantor never fully recovered." "For the rest of his life... he would be drawn back to work on the problem he could not solve." "And each time, it would hurt him, profoundly." "In 1899, Cantor had returned once again to work on the Continuum Hypothesys." "And again it made him ill and he returned to the asylum." "He was just recovering from this breakdown, when his son Rudolf died, suddenly." "Four days short of his thirteenth birthday." "Cantor wrote to a friend, saying how his son had had a great musical talent, just as he had had when he was a boy." "But he had set music aside, in order to go into mathematics." "And now with the death of his son, he felt that, his own dream of musical fulfillment had died with him." "Cantor went on to say, that he could no longer even remember why he himself... had left music, in order to go into maths." "That secret voice, which had once called him on to mathematics, and given meaning to his life and work, the voice he identified with God..." "That voice too, had left him." "Here we have to leave Georg Cantor, because if we treat Cantor's story in isolation, it makes it into a tragic but obscure footnote, to the broader sweep of history." "Where as in fact, the fear that Cantor had dislodged something, was part of a much broader feeling, that things once felt to be solid, were slipping." "A feeling seen more clearly in the story of his great contemporary:" "a man called, Ludwig Boltzmann." "Just as Cantor had revolutionary ideas in mathematics and was opposed, so Boltzmann, his contemporary, had revolutionary ideas in physics, and was equally opposed." "This is Ludwig Boltzmann's grave." "And that...carved on it, is the equation which killed him." "And it did so, because like Cantor..." "Boltzmann's ideas were out of step with his times." "Cantor had undermined the ideal of a timeless and perfect logic in maths." "Boltzmann's formula and his destiny... was to undermine the ideal of a timeless order in physics." "Together, their ideas were part of a general undermining of certainty, in the wider world outside of maths and physics." "Boltzmann's and Cantor's times craved certainty, in politics, in art, as well as in science and philosophy." "They were times that looked on the surface, solid and certain... but felt themselves to be teetering and sliding." "The old order was dying." "And it was as if they could already feel disaster's gravitational pull." "In Vienna, which was called by Karl Kraus:" ""Laboratory for Apocalypse", there was this feeling that... this political construct of the Habsburg Empire couldn't last for much longer." "They were very strange times." "On the one hand, those in power spent 20 years building the monuments of imperial Vienna, to declare that this order, firm on it's foundations, would last forever!" "The rich man would always be in his castle;" "the poor man always at his gate." "But on the other hand, the empire was actually on it's last legs." "And the intellectual tenor of the times, was summed up by the poet, Hofmannsthal... who said that, what previous generations believed to be firm, was in fact, what he called:" ""das Gleitende"." "The slipping, or sliding away of the world." "I thing that describes the feeling in Vienna." "In other places too, but particularly in Vienna in this capital of an empire that... hadn't crumbled yet but, it looked like...set to break down." "This characterizes it very well." "And it was against this background, that the scientific questions of Boltzmann's times were understood." "This is the Great Courtyard of the Univerity of Vienna... and these are the busts of all of the greats who have ever thaugt here." "Boltzmann's is here too." "But many of his contemporaries, men, more influential in their day even than he was, lined up to oppose him and his ideas." "But their opposition was as much ideological, as it was scientific." "The physics of Boltzmann's time, were still the physics of certainty." "Of an ordered universe, determined from above, by predictable and timeless God-given laws." "Boltzmann suggested, that the order of the world was not imposed from above by God, but emerged from below." "From the random bumping of atoms." "A radical idea, at odds with his times, but the foundation of ours." "Professor Mussardo, lives and works in Trieste, on the Adriatic Coast." "Not far from where Boltzmann's live ended." "He is an expert on Boltzmann, and works on the same kind of physics." "I think that there were two reasons why he could not... get fully accepted and recognized by the German physicists." "One of them was, that he based all his theories on atoms, that people can't see." "And this was the reason of the very, very strong criticism by Ernst Mach." "One of the most influential..." "Philosophers of Science at that time." "So the criticism of Mach was simply:" "i can't see an atom..." "I don't need them, they don't exist." "So why should we bring them in the game?" "Worse than insisting on the reality of something people could not see, to base physics on atoms, meant to base it on things who's behaviour was to complex to predict." "Which meant an entirely new kind of physics." "One based on probabilities, not certainties." "But then there was a second aspect;" "it was revolutionary as well." "And this consists in... putting forth and emphasizing the role of probability, in the physics world." "And people were used to the laws of physics and science as being exact." "Once established, they stay there forever." "There is no room for uncertainty." "So, introducing into the game, two ingredients like invisible atoms and probabilities, means there is no certainty." "You can predict what is probably going on, but not certainly." "Well, this really contrasted very, very much with the... the scientific spirit of the time, and therefore this produced trouble." "Boltzmann's genius, was that he could accept probability." "This meant he could begin to understand complex phenomenon, like fire and water and life." "Things which traditional physics;" "the physics of mechanics, never could." "But because his solution relied on probability, and probability undermines certainty, no one wanted to hear him." "And so just like Cantor, he faced implacable opposition which he too, found extremely difficult to deal with." "It seems like Boltzmann was just the wrong man in the wrong place." "Absolutely...absolutely." "Absolutely, it's true." "It's true." "He could just had his idea twenty years later..." "And in England he would have been the most succesful physicist of that time, it's true." "Somehow he met all his enemies." "So he met Ernst Mach, often." "Their careers even crossed, in a very...in a very..." "He meets all his enemies but none of his friends." "Not his friends, it's true." "It is not hard to see how Boltzmann's ideas where so radically at odds with his times." "Especially when applied not just to physics but to the social world." "Classical science, classical physics... gives you this image of a God-ordered creation." "Where everything is set in stone, according to perfect and eternal rules." "Everything is predictable." "Everything has it's place and everything is in it's place." "But when you come here, to the Central Cemetery of Vienna, you see the idealised vision of that idea." "Because here, everything is predictable." "Everybody does have a place and everybody is in their place." "But the problem is... while such certainty might seem desirable politically, the real world...the living world, the world described by thermodynamics, just isn't like that." "A timeless and perfect world never changes, but it is dead." "The real world, the thermodynamic world is alive precisely because it is full of change." "But of course... that life giving change also brings with it, disorder and decay." "But then the problem arises, if you then say:" "well, Newtonian Mechanics on which you are depending is reversible in time." "So how can you derive a law, which is asymmetrical in time from basic principals which are symmetrical in time." "You run the clock backwards, it's just as good" "Newtonian physics as you run it forwards." "Yet the entropy increases in the future." "But it is precisely this accumulation of disorder and decay, that science calls 'entropy', which Bolzmann had understood." "In short this is really the 'arrow of time'." "I mean, you can measure the arrow of time just seeing how things become more and more disordered." "So it's a natural tendency in the world that Bolzmann quantified precisely." "Indeed i call him the genius of disorder." "Boltzmann's work on entropy, showed why no system can be perfect." "Why there must always be some disorder." "It also revolutionized the idea of time in physics." "In classical physics, everything, including time can run equally well forwards as backwards." "Yet in thermodynamics, while everything else is reversible, time moves inexorably forward, like an arrow." "The idea of entropy, had a profound philosophical and political significance." "Entropy is what changes the ticking of a clock into the destroyer of all things." "It is wat underlies the inexorable passage from youth to old age." "Entropy is decay, and with the decay nothing lasts forever." "Boltzmann had in essence, captured mortality in an equation." "Physics now declared... that no order, not even a God-given one, will last forever." "That there was no natural order that God had set in stone, had already been pointed out by the scientist Boltzmann most admired:" "Charles Darwin." "In place of timeless perfection, was a dance of evolution and extinction." "With his equation of entropy," "Boltzmann brought this picture of constant change, into the very heart of physics itself." "Did Boltzmann understand the similarity?" "Almost certainly." "When Boltzmann was asked how his century would be remembered, he did not chose a physicist." "He said it would be the century of Darwin." "So he likes in Darwin, the momentum, the evolution of life, that is not static, the fact that he's progressive." "Progress sometimes has a jump in it." "And the fact that he can adress a... a "life aspect", with ideas of science, that before was kind of an ideological ground." "Bolzmann's ideas, like Cantor's and Darwin's were revolutionary, even though he did not think of them that way." "But his times were frightened times." "Times when people felt new ideas, could upset societies fragile structure... and bring it down." "At the end of the 19th century," "Viennese society was searching for some certainty, some principal... wether it was in politics, philosophy, the arts or science." "But there didn't appear to be any philosophy, capable of holding everyone together." "Upon which everything else could be based." "So when the university commissioned Gustav Klint, to paint a ceiling to celebrate philosophy, this is what they got:" "Such was the outrage, that twenty professors petitioned, to have the painting removed." "Now whatever else it is, it's not a celebration of certainty." "The radicals of Bolzmann's times, knew, the old order, with it's worn-out certainties was doomed." "But Viennese culture, was not ready to embrace the new." "And Boltzmann, was caught in the middle." "As a scientist, his personality entered deeply into the game, because he was very stubborn." "Not self-ironic." "He could not take criticism." "He always took it personally, and Boltzmann was definately a passionate man." "He used to swing rapidly from incredible joy to deep depression." "As Boltzmann got older, and more exhausted from the struggle, these mood swings became more and more severe." "More and more of Boltzmann's energy, was aborbed in trying to convince his opponents, that his theory was correct." "He wrote:" "no sacrifice is too high for this goal, which represents the whole meaning of my life." "In the last year of Boltzmann's life, he didn't do any research at all." "I'm talking about the last ten years." "He was fully immersed in dispute, philosophical dispute..." "Tried to make his point, writing books, which were most of the time the same, repeating the same concept and so on." "So you can see he was in a loop..." "that didn't go ahead." "But by the beginning of the 1900's, the struggle was getting too hard for him." "Boltzmann had discovered one of the fundamental equations which makes the universe work, and he had dedicated his life to it." "The philosopher Bertrand Russel said that for any great thinker, this discovery that everything flows from these fundamental laws, comes, as he described it, whith the overwhelming force of a revelation." "Like a palace, emerging from the autumn mist, as the traveller ascends an Italian hillside." "And so it was for Boltzmann." "But for him, that palace was here, at Duino in Italy, where he hung himself." "In 1906," "Boltzmann came here to Duino, with his wife and daughter on holiday." "Exhausted and demoralised, his ideas still not accepted." "While they were out walking, he killed himself, and left no note of explaination." "Of course we can never know what Boltzmann was thinking, but i think we have clues." "Boltzmann knew what it was, to be in the grip of a beatiful and powerful idea." "He once wrote that, what the poet laments, holds for the mathematician:" "that he writes his works, whith the blood of his heart." "So we know that he was a passionate man." "But i think there is another clue." "At the start of one of Boltzmann's major scientific papers, he quotes three lines from Goethe's Faust:" ""Bring forth what is true."" ""Write it so it's clear."" ""Defend it to your last breath."" "Which of course he does." "But i think there's something deeper here." "Why quote Faust, at the start of a scientific paper?" "The pact, that Faust makes with the devil, is that the devil will give him all of the knowledge and all of the experience that he wants, so long as he never asks to stay, in any one moment." "And i think when Boltzmann came here, to this beautiful place, after thirty years of fighting for what he believed in, he simply said:" "i want to stay here, in this perfect, beautiful moment." "I don't want to have to leave." "I want time, for me, to stop." "The great and controversial thing that Boltzmann had done, was to introduce, into the unchanging perfection of classical physics, the notion of real time." "Of irreversible change." "And yet it was this man, who in his final moments, wanted time to stop." "So ironically, Boltzmann was vindicated just after his death." "If he would have waited a little longer," "Boltzmann would have been one of the fathers of the revolution of the twentieth century fysics." "Yet Boltzmann died as he had lived:" "out of step with his times." "He had sawn the seeds of uncertainty and fysics, but no school of followers took up his work." "Against all the odds, it was Cantor, who had uncovered the uncertainty in mathematics, around whom followers where gathering." "A new generation of mathematicians and philosphers were convinced:" "if only they could solve the problems and paradoxes that had defeated Cantor, maths could be made perfect again." "The most prominent amongst them, Hilbert, declared:" "the definitive clarification of the nature of the infinite, has become necessary for the honour of human understanding itself." "They were so concerned to find some kind of certainty, they had come to believe that the only kind of understanding that was really worth anything, was the logical and the provable." "And a measure of how desperate this attempt to find the perfect system of reasoning and logic had become, is this:" "three volumes of the Principia Mathematica, published in 1910." "It takes a huge chunk of this volume, just to prove, that one plus one equals two." "And a large part of that proof, revolves around the problems of the finite and the infinite, and the paradoxes that Cantor's work had trown up." "But despite the Principia, there was now the feeling that the logic of maths, had undone itself, and it was Cantor's fault." "As the Austrian writer, Musil wrote at the time:" "suddenly mathematicians, those working in the innermost region, discovered that something in the foundations, could absolutely not be put in order." "Indeed, they took a look at the bottom, and found that the whole edifice, was standing on air." "Cantor had stretched the limits of maths and logic to breaking point, and paid for it." "Much of the last twenty years of his life, was spent in and out of the asylum." "The last time that Cantor came here to the Nervenklinik in Halle, was in 1917, and he truly did not want to be here." "He wrote to his wife, begging her to let him come home." "He was one of only two civilians left here." "The rest of the place, was filled with the casualties of World War I." "But of the 6th of January 1918, the greatest mathematician of his century, died alone in his room, his great project still unfinished." "Cantor had dislodged the pebble, which would one day start a landslide." "For him, it had all been held together." "The paradoxes resolved, in God." "But what holds our ideas together, when God is dead?" "Without God, the pebble is dislodged, and the avalanche is unleashed, and World War I, had killed God." "Here at last, was the slippage." "Well, hasn't there always been a desire in the history of the West to find certainty or...maybe, there wasn't so much a desire in earlier era's because, the assumption was that we had that." "You know, there was God!" "And, you know even Descartes, despite all of his scepticism, assumes... for him unproblematically, that there is a God." "So what happens when that really, really comes in to question?" "After the death of God, so to speak." "And along with the death of God is a...is a loss of faith in some... supernatural order, of which we are a small part." "No one won the Great War." "Nothing was resolved at Versaille." "It was merely an armistice." "And none of the intellectual crises that proceded it, had been resolved either." "Things like the Principia, had merely papered over the cracks." "In a way, the Principia was like the Versaille Treaty, only a lot more substantial." "This is basicly ten thousend tonnes of intellectual concrete poured over the cracks in mathematics." "And for a while, it looked like it really might hold." "But then a young man came here to the university of Vienna, to this library." "His name was Kurt Gödel." "And the work that he did here, brought that dream of finding the perfect system of reasoning and logic, crashing down." "Gödel was born the year Boltzmann died: 1906." "He was an insatiably questioning boy, growing up in unstable times." "His family, called him: "Mister Why"." "But by the time he went to university," "World War I was over." "But Austria like the rest of Europe, was in the grip of the depression, and Hitler was forming the National Socialist Party." "Gödel for his part, became one of a brilliant group of young philosophers, political thinkers, poets and scientists, known as 'The Vienna Circle'." "Chaos was good because it ment that there was no central authority that was imposing ideas so individuals could come up with their own ideas." "The chaos around them, on the one hand had a liberating effect." "And on the other hand they were desperately searching for ideas, that they could believe in because everything else around them was crumbling in a heap." "So you'd want to find some beautiful ideas that you could believe in." "Though Gödel was surrounded by radicals and revolutionary thinkers, he was not one himself." "He was an unworldly and exact man, who believed, like Hilbert, that maths at least, could be made whole again." "But it was not to be." "He certainly did not start out, with trying to explode Hilbert's program also." "In fact, i think it came to Gödel... ultimately as a surprise when he showed that the next step, to show the completeness of arythmetic, was unachievable." "There was actually something very mysterious happening in pure mathematics." "In it's own way as mysterious as black holes, the big bang, as quantum uncertainty in the atom." "And this was Gödel's Incompleteness Theorem." "And at that time, there was a mystery there." "The one place where you don't expect there to be mystery is in pure reason!" "Because pure reason should be black and white." "It should be really clear." "But, pure reason, the clearest thing there is, was revealing that there were thing that were unclear." "This is one of the cafés where the Vienna Circle used to meet regularly." "Late summer of 1930," "Gödel came to the café with two eminent colleagues." "Towards the end of their conversation, he just mentioned an idea he'd been working on, which he called the 'Incompleteness Theory'." "And what he told them, was that he had just proved, that all systems of mathematical logic, were limited." "That there would always be some things wich while true, would never be able to be proved to be true." "What Gödel showed in his Incompleteness Theorem, 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 café, the very next day," "Hilbert, now the grand old man of mathematics, stood up and gave a lecture in Königsberg, in which he said:" ""We must know!" "We will know!"" "The irony was, that the very day before," "Gödel 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 Gödels argument?" "No, there are not." "This was a perfect argument." "This argument was so crystal clear and obvious." "Gödel 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 Gödel." "There's no bust to Gödel 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 Gödel." "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, Gödel explodes that formalist view of mathematics." "that you can just mechanically grind away on a fixed set of concepts." "So even though i believe Gödel pulled out the rug out from under it intellectually, nobody wants to face that fact." "So there's a very ambivalent attitude to Gödel." "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 Gödels work, because the obvious conclusion from Gödels work" "is that logic is a failure." "Let's move on to something else." "And this would destroy the field." "Gödel 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 Gödel 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 Gödel'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." "Gödel, 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 Gödel 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 Gödel, 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." "Gödel's madness passed." "Austria's didn't." "In 1939, Gödel 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 Gödel's already devastating Incompleteness Theorem, even worse." "Turing was a much more practical man than Gödel." "And simply wanted to make Gödel'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 Gödel's Incompleteness Theorem, more concrete." "Because for many, Gödel'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," "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." "Gödel 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 Gödel, 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;" "Gödel'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 Gödel'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 Gödels 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 Gödel'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... despite all these theorems we know about non-computability, it still might be, that we are computational entities, and then point out:" "well, because of this and this loophole and so on." "And maybe he... came to believe those loopholes were sufficient to get him out." "But yet, he did do these things like looking at oracle machines which were sort of super Turing machines; went beyond them." "They're not machines that you could see any way of constructing out of ordinary stuff." "But nevertheless, as a theoretical entity, these devices were... theoretical things which would go beyond, standard computers." "This tension, between the human and the computational, was central to Turing's life." "And he lived with it, until the events which led to his death." "After the war, Turing increasingly found himself drawing the attention of the security services." "In the Cold War, homosexuality was seen, as not only illegal and immoral, but also a security risk." "So when in March 1952, he was arrested, charged and found guilty of engaging in a homosexual act, the authorities decided, he was a problem that needed to be fixed." "They would chemically castrate him by injecting him with the female hormone estrogen." "Turing was being treated as no more than a machine, chemically reprogrammed, to eliminate the uncertainty of his sexuality, and the risk they felt it posed, to security and order." "To his horror, he found the treatment affected his mind and his body." "He grew breasts, his moods altered, and he worried about his mind." "For a man who had always been authentic, and at one with himself, it was as if he had been injected, with hypocrisy." "On the 7th of June 1954," "Turing was found dead." "At his bedside, an apple... from which he had taken several bites." "Turing had poisoned the apple, with cyanide." "Turing was dead, but his question was not." "Whether the mind was a computer, and so limited by logic, or somehow able to transcend logic, was now the question that came to trouble the mind of Kurt Gödel." "Gödel was now working in America, at the institute for advanced study, where he continued to work, as obsessively as he ever had." "Of course, Gödel recovered from his time in the sanatorium, but by the time he got here to the Institute for Advanced Study in America, he was a very peculiar man." "One of the stories they tell about him, is if he was caught in the Commons, with a crowd of other people, he so hated physical contact, that he would stand very still so as to plot the perfect course out," "so as not to have to actually touch anyone." "He also felt he was being poisoned by what he called "bad air", from heating systems and air conditioners." "And most of all, he thought his food was being poisoned." "He insisted his wife, taste all his food for him." "He would sometimes, order oranges, and then send them straight back claiming they were poisoned." "Peculiar as Gödel was, his genius was undimmed." "Unlike Turing," "Gödel could not believe we were like computers." "He wanted to show how the mind had a way of reaching truth outside logic, and what it would mean, if it couldn't." "In principal you can have a machine grinding away, deducing all the consequences of a fixed set of principles and mathematics would be static and dead." "I mean, it would just be a question of mecanically... deducing all the consequences." "And so... and so mathematicians in a sense would just be...machines." "I mean, Turing did think that he was a machine." "I think he did." "And i think... that paper on the imitation game... shows that." "And Gödel, clearly did not think that he was a machine." "He thought that he was divine." "You know, that human beings have a...devine spark in them that enables them to create new mathematics i think." "Why was Gödel, so convinced humans had this spark of creativity?" "The key to his believe, comes from a deep conviction he shared with one of the few close friends, he ever had." "That other, Austrian genius, who had settled at the institute:" "Albert Einstein." "Einstein used to say that he came here, to the Institute for Advanced Study, simply for the privilege of walking home with Kurt Gödel." "But what was it that held this most unlikely of couples together?" "Because on the one hand, you've got the warm and avuncular Einstein and on the other, the rather cold, wizened, and withdrawn Kurt Gödel." "And the answer i think, comes from something else that Einstein said." "He said that," "God may be subtle but he's not malicious." "What does that mean?" "What it means for Einstein, is that however complicated the universe might be, there will always be beautiful rules, by which it works." "Gödel believed the same idea from his point of view to mean, that, God would never have put us into a creation, that we could then not understand." "The question is, how is it that Kurt Gödel can believe that God isn't malicous?" "That it's all understandable?" "Because Gödel is the man who has proved, that some things can not be proven logically and rationally." "So surely, God must be malicious." "The way he gets out of it, is that Gödel, like Einstein, believes deeply in intuition." "That we can know things, outside of logic, because we just...intuit them." "And they believe it because they have both felt it." "They've both had their moments of intuition." "Just like Cantor had had his." "He talks about new principals..." "that the mathematician... closing your eyes, tuning out the real world, you can try to perceive, directly by your mathematical intuition, the platonic world of ideas, and come up with new principles, which you can then use to extend the... the current set of principles in mathematics." "And he viewed this as a way of getting around, i think, the limitations of his own theorem." "I don't think he thought there was any limit to the mathematics that human beings were capable of." "But, how do you prove this?" "The interpretation that Gödel himself drew, was that... computers are limited." "He certainly tried again and again, to work out that... the human mind transcends the computer." "In the sense that he can understand things to be true, that can not be proven, by a computer program." "Gödel also was wrestling with some... finding means of knowledge, which are not based on experience and on mathematical reasoning, but on some sort of intuition." "The frustration for Gödel, was getting anyone to understand him." "I think people very often, for some reason, misunderstand Gödel." "Certainly his intention." "Gödel was deliberatly trying to show, that, what one might call "mathematical intuition"." "He referred to, what he called, "mathematical intuiton", and he was... demonstrating, clearly in my mind demonstrated, that this is outside just following formal rules." "And, i don't know..." "Some people... picked up on what he did and said, well, he's showing there are unprovable results and therefore beyond the mind." "What he really showed, was that for any system that you adopt, which, in the sense the mind has been removed from it, because you..." "The mind is used to lay down the system." "But from thereon, it takes over." "And you ask what's it's scope?" "And what Gödel showed, is that it's scope is always limited." "And that the mind can go beyond it." "Here's the man who has said certain things can not be proved, within any rational and logical system." "But he says, that doesn't matter, because the human mind isn't limited that way." "We have intuition!" "But then of course the one thing he really must prove to other people, is the existence of intuition." "The one thing you'll never be able to prove." "He has these drafts of papers where he expresses himself very strongly." "But he didn't..." "He wasn't satisfied with them." "Because he couldn't prove a theorem about creativity or intuition." "It was just..." "a gut feeling that he had." "And he wasn't satisfied with that." "And so Gödel, like Cantor before him, had finally found a problem, he desperatly wanted to solve, but could not." "He was now caught in a loop." "A logical paradox, from which his mind could not escape." "And at the same time, he slowly starved himself to death." "Using mathematics, to show the limits of mathematics, is... is psychologically very contradictory." "It's clear in Gödel's case, that he appreciated this." "His own life has this paradox." "What Gödel is, is the mind thinking about itself, and what it can achieve at the deepest level." "Someone used the phrase:" ""the Vertigo of the Modern"." "You can be led into that particular reflexive whirlpool where you're beginning to think about thinking about thinking... about thinking about thinking... and you find yourself entangled in your own...in your own thoughts." "Well that seems to me, almost the quintessence of the Modern moment because there you have a..." "what you could call a paradox of self-reflection." "The kind of madness that you find associated with Modernism, is the kind of madness that's bound up with, not only rationality, but with all the paradoxes that arise from self-consciousness." "From the consciousness contemplating it's own being as consciousness or from logic contemplating it's own being as logic." "Even though he's shown that logic has certain limitations, he's still, so drawn to that, to the significance of the rational and the logical, that he desperately want's to prove whatever is most important," "logically." "Even if it's an alternative to logic." "How strange." "And what a testimony to his.." "his inability to separate himself, to detach himself from the need for logical proof." "Gödel of all people..." "At the beginning of our story, Cantor had hoped, that at it's deepest level, mathematics would rest on certainties." "Which for him, were the mind of God." "But instead, he had uncovered uncertainties." "Which Turing and Gödel then proved, would never go away." "They were an inescapable part, of the very foundations of maths and logic." "The almost religious belief, that there was a perfect logic, which governed a world of certainties, had unraveled itself." "Logic, had revealed the limitations of logic." "The search for certainty, had revealed uncertainty." "I mean, there's a fashionable solution to the problem, which is basically, in my opinion," " people are going to hate me for this - is sweeping it under the carpet." "But you see, the problem is:" "i don't think you want to solve the problem." "I think it's much more fun to live with the problem." "It's much more creative!" "This notion of absolute certainty..." "There is no absolute certainty in human life." "But our knowledge, our possible knowledge of this world of ideas, can only be incomplete and finite, because we are incomplete and finite." "The problem is that today, some knowledge, still feels too dangerous... because our times are not so different, to Cantor, or Boltzmann, or Gödel's time." "We too, feel things we thought were solid, being challenged... feel our certainties slipping away." "And so, as then... we still desperately want to kling to a believe in certainty, that makes us feel safe." "At the end of this journey, the question i think we are left with... is actually the same as it was in Cantor and Bolzmann's time:" "are we grown up enough, to live with uncertainties?" "Or will we repeat the mistakes of the 20th century, and pledge blind allegiance, to yet another certainty?"