"By the begining of the 1900s the struggle was getting too hard for him" "Boltzman had discovered one of the fundamental equations which make the universe work and he had dedicated his life to it" "The filosopher Bertrand Russel said that for any great thinker this discovery that everything flows in its fundamental laws comes as he described with the overwhelming force of a revelation" "like a palace emerging from the autumn mist as the traveler ascends an Italian hill side" "and so it was for Boltzman but for him that palace was here at Duino in Italy where he hung himself" "In 1906 Boltzman came here to Duino with his wife and daughter on holiday exausted and demoralized his ideas still not accepted while they were out walking he killed himself and left no note of explanation" "Off course we can never know what Boltzman was thinking but I think we have clues" "Boltzman knew what it was to be in the grip of a beautiful and powerful idea he once wrote that what is the poet's lemma holds for the mathematician that he writes his works with 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 Boltzman's major scientific papers he quotes three lines from Goethe's Faust" "print for what is true write it so is clear defend it to your last breath which of course he does but I think there is 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 Boltzman came here to this beautiful place after thirthy 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 controversal thing that Boltzman 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 its final moments wanted time to stop" "So ironically Bolztman was vindicated just after his death if he had waited a little longer Boltzman would have been one of the fathers of the revolution of the twenty century in physics" "yet Boltzman had died as he had lived out of step with its times he had sowed the seeds of uncertainty in physics but no school or followers took up his work" "against all the odds it's Cantor who had uncovered uncertainty in mathematics around whom followers were gathering" "a new generation of mathematicians and philosophers were convinced if only they could solve the problems and paradoxes that had defeated Cantor maths could be made perfect again" "The most proeminent amongst them Hilbert declared the definitive clarification of the nature of the infinite has become necessary for the owner of human understanding itself" "They were so concerned to find some kind of certainty they've come to believe that the only kind of understanding was really worthing a thing was the logical and the provable" "and a measure of how desperate this attempt to find the perfect system of reasoning and logical 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 thrown up" "but despite the Principia there was now the feeling that the logical 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 regions 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 Nervenklink in Halle was in 1917 and he trully did not want to be here he wrote to his wife begging her to let him come home he was one of the only two civilians left here" "the rest of the place was filled with the casualties of world war I but on the sixth of january of 1918, the greatest mathematician of his century died alone in his room, his great project still unfinished" "Cantor had dislodge the paddle which would one day start a land slide for him it had all been held together the paradoxes resolved in god." "What holds our ideas together when god is dead?" "without god the paddle is dislodged and the avalanche is unleashed and world war I had killed god" "here at last it was dislodged" "Hasn't there always been a desire in the history of the west to find certainty?" "or maybe there wasn't such a desire earlier or in earlier eras because the assumption was that we had that" "There was god!" "Even Descartes, despite all of his skepticism assumes for him unproblematicaly that there is a god" "So what happens when that really comes into question after the death of god so to speak and along with the death of god is a lost of faith in some supernatural order which we are a small part" "no one won the great war nothing was resolved at first sight it was nearly an armistice and none of the intelectual crises that preceeded it had been resolved either things like the Principia had nearly paped over the cracks in a way the Principia was like the Versailles treaty" "only a lot more substantial this is basically ten thousand tons of intelectual 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 libray his name was Kurt Gödel and the work that he did here powdered that dream of finding the perfect system of reasoning and logic crashing down" "Gödel was born the year Boltzman died, 1906 he was an unsationably questioning boy growing up in unstable times his family called him Mr. Why" "but by the time he went to the university, world war I was over" "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 the brilliant philosophers political thinkers, poets and scientists known as the Vienna circle" "Chaos was good because it meant there was no central authority that was imposing ideas the 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 coming into a heap you want to define 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 hole again" "but it was not to be he certainly did not start out this trying to explode Hilbert's program in fact I think it came to Gödel" "ultimately as a surprise when he showed that the next step to show the completeness of arithmetic was unachievable there was actualy something very mysterious happening in pure mathematics and it is in one way as misterious as black holes as the big bang" "as quantum uncertainty in the atom and this was Gödel's completeness theorem and at the time there was a mistery there the one place where you don't expect to be a mystery is in pure reason pure reason should be black and white it should be really clear" "but in pure reason the clearest thing there is was revealing that there was things that were unclear" "this is one of the cafes where the vienna circle used to meet regularly" "late summer of 1930" "Gödel came to the cafe with two eminent coleagues 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 will always be somethings which 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 set of axioms, in arithmetics there would always be statements that are true but cannot be proved" "no matter how much data you have to build on you will never proof all true statements" "what this meant was that the great renaiscence dream that one day maths and logic would be able to prove all things and give us a god like 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 his coleagues understood what he'd just told them it was as if there was an explosion but the blast wave hadn't hit them yet" "unaware of what happened in the cafe the very next day, Hilbert, now the grand old man of mathematics stood up and gave a lecture in Konigsberg in which he said we must know, we will know" "the irony was that the very day before" "Gödel had proved that there were somethings we will never know some didn't like it, in particular for instance Hilbert," "in the beggining was quite annoyed and 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 Gödel's argument?" "no, there are not it was a perfect argument, cristal 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" "here we are again in the great courtyard of Vienna university with the bust of all the great thinkers, except for Kurt Gödel there is no bust to Gödel here" "and I can't help but feel that at least part of the reason that he is not here, is simply due to the nature of his ideas" "you see, nobody wants to face it in my opinion, nobody wants to face the consequences of Gödel basically people want to go ahead with formal systems anyway ...as if Hilbert had it alright" "In my opinion, Gödel explodes that formalist view of mathematics that you can just mechanicaly grind your way on a fixed set of concepts so even though I believe" "Gödel pulled out the rug from underneath intelectually nobody wants to face that fact so there is a very ambivalent atitude to Gödel even now, a century after his birth, very ambivalent atitude" "On the one hand he is the greatest logician of all times so logicians will claim him but on the other hand people who are not logicians will talk about the consequences of Gödel's work because the obvious conclusion 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 its 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 it 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 theory 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" "clean and logical except for this one this is the workbook where he is 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... and perhaps there is also a danger at the more existencial or personal, psycological level... if you are a person who is already prone to a kind of exagerated" "intelectual self-reflection, self-concioussness." "You may find that your intelectual work is exagerating, exacerbating that tendency 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 sanatorium just like Cantor." "We are talking about people here who of course are capable of and maybe afflicted with the capacity to care very much about things that are very abstract, to lose themselves in these intelectual problems." "One of the sanatorium that Gödel spent some time in is here the Purkersdorf sanitorium just outside of Vienna." "The Purkersdorf itself was built to embody the philosophy that the calm, smooth lines of rationalism are the cure for madness." "Ironic thing that Gödel driven mad by pushing the limits of rationalism should come here to recover" "While the man who'd 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 demagog who promissed 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 this pre-war years that another brilliant young man, Alan Turing, enters our story." "Turing is most famous for his war time 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 pratical man that Gödel" " I think it relates here but I can't find it 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 is an absolutely devastating result from a philoshophical point of view we still haven't absorbed but the proof was too superficial it didn't get to 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 am unprovable, 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 be some problems they would never solve a machine fed with 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 a 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 would be the unprovable problems" "How do you know when to stop?" "you will never know when the problem you are working on is simply extraordinarily difficult or if it is fundamentaly 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 is there in his paper, he didn't speak of it in those terms but the ideas are really there, in his original paper" "and this is so concrete and down to earth computers are physical devices you start running and there are two possibilities ...a self-contained program running with no input/output it is just there, running on a computer and one possibility is it is going to stop eventualy," "finishes the work, come up with an answer and stop done... finished the other possibility is he is going to be searching forever, never find what it is looking for, never finish the calculation just go on forever" "it is one or the other the problem is how can we tell that a program is never going to stop?" "and the answer is there is no systematic general way to do it and this is Turing's version of incompleteness" "Turing gets incompleteness Gödel profound discovery he gets it as a corolary of something more basic which is uncomputability, things which cannot be calculated things which no computer can calculate and in certain domains most things cannot be calculated" "But that is your work, isnt'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 Turing's paper although he doesn't emphasizes it" "Startling as the halting problem was the really profound part of incompleteness for Turing was not what it said about logical 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 died tragically young" "Turing never recaptured that first pure love but never let go of the memory of what it had been" "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 a fantastic idea that one day computers would be more than machines" "they would be like children, capable of learning, thinking and comunicating 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 the infinity in Gödel's hands became a question about the limits of logic and now with Turing it comes into focus as a question about us" "and the nature of our minds" "This is sort of standard view that Turing was a computationalist and certainly in certain stage of his life he did take this point of view maybe you can make one of these machines imitate human mind but he was off course well aware of these limitations of computers" "and that was one of the proven results of his own" "I think he may have vacillated a bit ...having one view and then another but then when he really developed the computers as actual machines he sort of took off and thought maybe these really are... going to... it is a kind of when you get into a scientific thing you get totally to think" "that maybe this is solving all problems but then you realize the limitations that are there and which you are part of its own theories" "Turing understood that Gödel and its 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 does't have to be consistent with its personality." "There is this work on artificial inteligence where I think he does believe that machines could become inteligent just like people, better or different but inteligent if you look at his first paper when he points out that machines have limits because they are numbers, in fact most numbers" "cannot be calculated by any machine he showed the power of the human mind to imagine things that escaped what an machines could ever do that may go against his own philosophy." "He may think of himself as a machine but his very first paper is smashing machines he is creating machines and then he is pointing out their devastating limitations" "Turing was well aware of these problems but desperately wanted to prove he could get the formulas 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 stucked to all his life was to be free from hypocrisy, compromise and deceit" "Turing was a homosexual when it was both ilegal 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 vacillated, they could have one view and then wonder about this, is this really right?" "and then have another view and play around, you know..." "I think good scientists, they will do that they won't just doggedly follow one point of view" "I suspect Turing vacillated rather but I think in a lot of his analysis and criticisms of other people who criticized his view" "he would show the flaws in their arguments and say look, may still be despite all these theorems about uncomputability it still might be that we are computational entities and then point out because this and this loophole and maybe he came to believe those loopholes were sufficient to get him out" "but yet he did do these things looking at oracle machines which were sort of super Turing machines they are not machines you can see anywhere or construct them out of ordinary stuff but nevertheless it is a theoretical entity his devices were theoretical things that go beyond standard computers" "this tension between the human and the computational was central to Turings life and he lived with it until the events which lead to his death" "after the war Turing increasingly found himself drawing the atention of the security services in the cold war homosexuality was seen as not only ilegal and imoral but also a security risk" "so when in march 1852 he was arrested, charged and found guilty of engaging in an homosexual act." "The authorities decided he was a problem that needed to be fixed they would chemicaly castrate him by injecting him with female hormone, estrogen." "Turing was being treated as no more than a machine chemicaly reprogramed 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 mood altered and he worried about his mind" "for a man who had always been authentic and one with himself it was as if he had been injected with hypocrisy." "On the 7th of june of 1954 Turing was found dead at its bed side an apple from which he had taken several bites" "Turing had poisoned the apple with cianide" "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" "off course Gödel recovered from its 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 hatted physical contact that he would stand very still so as to plot the perfect course out" "so as to not have to actualy touch anyone he also felt that 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 that his wife tastes 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 untamed 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 he couldn't." "In principle you could have a machine grinding away deducing all the consequences of a fixed set of principles and mathematics would be static and dead." "It would just be a question of mecha- nicaly deducing all the consequences and so mathematicians in a sense would just be machines" "Turing did things that he was a machine" "I think he did and I think that paper on the imitation game shows that" "Gödel clearly did not think he was a machine he thought that he was divine for him all human beings had a divine spark that enables them to create new mathematics, I think" "Why was Gödel so convinced humans had the spark of creativity?" "the key to his belief 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 Studies simply for the privilege of walking home with Kurt Gödel" "Why was in hell this most unlikely of couples together?" "On the one hand you got the warm and avuncular Einstein while on the other hand the rather cold, weaseled and withdrawn Kurt Gödel" "and the answer I think comes from something else that Einstein said he said that god may be subtle but it is not malicious" "what does that mean?" "what it means for Einstein is that however complicated the universe might be there would always be beautiful wombs by which it works" "Gödel believed the same ideas from his point of view to mean god would never have put us into a creation that we could not understand the question is how is it that" "Kurt Gödel can believe that god isn't malicious?" "that is is all understandable because Gödel is the man who has proved that somethings cannot be proven logicaly and rationaly" "so surely god must be malicious the way he gets out of it is" "Gödel like Einstein believed deeply in intuition that we can know things outside of logic because we just intuited them and they believed it because they have both felt it they have both had their moments of intuition just like Cantor had his" "he talks about new principles that the mathematician closing your eyes tuning out the real world you can try to perceive directly by your mathematical intuition in the platonic world of ideas and come up with new principles which you can then use to extend" "the current set of principles in mathematics and he viewed this as a way of getting around 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 to 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 trancends the computer in a sense he can't understand things to be true that cannot be proved by a computer program" "Gödel also was wrestling for finding means of knowledge which are not based on experience and mathematical reasoning but on some sort of intuition the frustation for Gödel was getting anyone to understand him" "I think people very often for some reason misunderstand Gödel in his intention Gödel was deliberatly trying to show that what I may call mathematical intuition" "I here refer to what he called mathematical intuition and he was demonstrating clearly in my mind demonstrated that this is outside just following formal rules" "some people picked up on what he did and said he is showing their unprovable results and therefore beyond the mind what he really showed was that for any system that you adopt in the sense that the mind has been removed from it" "because the mind is used to lay down the system but from there on it takes over and you ask what is its scope?" "and what Gödel showed is that the scope is always limited and that the mind can go beyond them he is the man who has said certain things cannot 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 off course the one thing he really must prove to other people is the existence of intuition the one thing you will never be able to prove" "he has these draft of papers where he expresses himself very strongly but he wasn't satisfied with it because he couldn't prove a theorem about creativity and intuition." "It's just a gut feeling he had and he wasn't satisfied with that and so Gödel like Cantor before him had finaly found the problem he desperately 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 psicologicaly very contradictory" "it is clear in Gödel's case that he appreciated this his own life has this paradox where" "Gödel is the mind thinking about itself and what it can achieve at the deepest level" "Someone used to phrase that the vertigo of the modern can be lead into that particular reflexive whirlpool where you are thinking about thinking... and you find yourself entangled in your own thoughts" "that seems to me almost the quintessence of the modern moment because there you have what you could call a paradox of self reflection the kind of madness that you find associated with modernism is a kind of madness that is bound up with not only rationality" "but with all the paradoxes that arise from self-conciousness from the counciousness contemplating its own being as counciousness or from logic contemplating its own being as logic" "even though he's shown that logic has certain limitations he is still so drawn to that to the significance of the rational and logical that he desperately wants to prove whatever is most important logicaly even if it is an alternative to logic" "how strange!" "and what a testimony to his inability to separate himself to detach himself from the need for logical proof" "Gödel of all people." "At the begining of our story Cantor had hoped that at its 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 inascapable part of the very foundations of maths and logic" "the almost religious belief that there was a perfect logic which governed the world of certainties had unravelled itself logic had revealed the limitations of logic the search for certainty had revealed uncertainty" "there is a fashionable solution to the problem which is basicaly in my opinion people are gonna hate me for this is sweeping it under the carpet but the problem is I don't think you want to solve the problem" "I think it is much more fun to live with the problem it is 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 diferent to Cantor, Boltzman or Gödel's time" "We too feel things we thought were solid being chalenged feel our certainties sliping away and so as them we still desperately want to cling to a belief in certainty to make 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 Boltzman's time" "Are we grown up enough to live with uncertainties?" "Or will we repeat the mistakes of the twentieth century and pledge blind allegiance to yet another certainty?"