Incompleteness: Rebecca Goldstein on the Life and Work of Kurt Gödel

[Music] incompleteness is part of a series so this is the norton series on great scientific discoveries and the two editors james atlas and jesse cohn had approached me they just wanted me to do a book for them and the idea behind the series was to get people who really know how to write so they didn't go to a mathematician or to a logician and they they tended to go to people who have a great love for interest and have studied um these subjects but it wasn't it wasn't the usual suspects that they went to um and they simply came and asked me um you know whom i would want to write about and you know i mean immediately kurt girdle came to mind [Music] i'm a philosopher i'm a novelist but i'm trained in philosophy my doctors in philosophy i've taught philosophy for most of my life and girdles um theorems they inhabit both mathematics and philosophy it's very very unusual that they're this kind of cross over that they are philosophical theorems that have great philosophical implications one argues about what the implications are but that they have implications seems pretty clear and he certainly thought they had uh philosophical implications he thought of himself as a philosopher of all mathematical results though this is the one that i felt that i was perhaps more competent to deal with because i have a you know i'm a professional philosopher secondly i i actually had um met girdle very briefly when i was a graduate student i speak about it ever so briefly towards the end of uh incompleteness he was notoriously reclusive almost a hermit it was when i was a i was a graduate student at princeton philosophy department uh he was the reigning god einstein was long dead by that time but godel was you know our genius and we had all sorts of we would when we would get together for parties and not just graduate students but professors as well mathematicians physicists we often traded stories about girdle i mean it was having this legend uh living right there and a reclusive legend that we never saw and um i went to a party at the institute for advanced study and was a party for welcoming newcomers and girdle was there there was this brief window in which he was sociable he was gregarious and you know when i doing research for the book and i spoke to people who knew him and said yes there were you know three four months when he came out into the world and i was just lucky that it happened to be at this party and there he was holding court and uh we were all you know young people we were all standing around him and after he left he was very gracious and courtly and old world and wished us all well and and after he left we all said god i wish i had asked him this i wish i'd asked him that and i my whole life there was one question i had wanted to ask him and didn't so but you know so there it was you know so our paths very briefly crossed and third you know i'm also a novelist i'm primarily nowadays a novelist and girdle is this kind of character i you know i could have created i mean that sounds almost arrogant but it's just the kind of improbable and inconsistent personality um that i love to explore in my fiction somebody who is kind of a hero in the world of thought i mean he just and brave and risk-taking he went where no man or woman had ever gone before he went completely against the tide he brought down formalism according to himself and some others some others not i mean here's where the philosophical implications get murky but he was a a voyager and a hero and in his personal life he was pathetic he was clueless he was uh he was prey to delusions he was paranoid he was not heroic at all in his in his personal life and um and that kind of contradiction especially in a logician right everybody called him the successor to aristotle um so here he is this most the greatest location since aristotle but so you know in some sense irrational was killed because of his paranoid uh delusions thought people were poisonous and and so ended up starving to death i mean that kind of inconsistency which to me is so poignant to me it speaks to the whole human dilemma you know that we can be in some ways so developed and so brilliant in another way so pathetic that's exactly what i always try to get at in my fiction and so he is he's just the kind of character i would want to inhabit and understand and in some sense i often have thought maybe that brief crossing of our lives you know and my own obsession with him when i was a graduate student to a certain extent motivated my fiction that i've always been interested in the helpless genius the first incompleteness theorem states that there are true propositions that within a formal system which has to be defined very carefully within a formal system which is also consistent and which is rich enough to express arithmetic most basic uh mathematics with those three conditions a formal system consistency rich enough to express arithmetic there will be uh propositions that we can express and which are true but they can't be proved within that formal system we can actually see that they're true and the proof shows gives us a way of seeing they're true but we're not seeing that they're true by proving them within the formal system so that's the first incompleteness theorem within every consistent formal system rich enough to express arithmetic there are true but unprovable propositions and the second incompleteness theorem is that one of the things that you can't prove within a consistent formal system is the consistency of that system so you can't within that system prove that the system is consistent and that's that's the second incompleteness theorem by providing a model for the system saying this system describes this set of things the system is about for example numbers you can give a model theoretical proof for the consistency of the system but to specify a model is is to go outside strict you know the strict proof theoretic system and it is in fact to say that the system is about something and that is precisely what the formless wanted to say wasn't true so the only way to say that the system is consistent is to say well it's about something it's about numbers for example specify a domain that it's describing the system itself is inadequate to prove its own consistency and it's inadequate to prove all the true propositions that you can express within the system so the system that purely formal system seems to be inadequate for doing some things for people who wanted to reduce all of mathematics to formal systems formalism that was a problem uh and and so that's why it has all of these implications because the big issue in mathematics is what is it we're doing when we're doing mathematics right what is it what is this about and it's always baffled us it's our our most certain domain uh it's the only place where we can actually prove things and prove really interesting things prove things that surprise us you know prove things about infinity you know and we can actually get our little finite hands on infinity we can it's so powerful and it's certain and it's conclusive but what are we doing when we're doing it are we describing something are we making up a game and following rules and that's is just a higher form of chess the whole question of what is it that we're doing we're doing mathematics that belongs to the philosophy of mathematics and curdles and completeness theorems may or may not have something to do with what it is that we're doing when we're doing mathematics the most important mathematician of the previous generation before godel uh was david hilbert and he was a formalist and he was interested in this question what is it we're doing when we're doing mathematics and basically saying look we create these formal systems they are transparent we know exactly what we're doing we lay down these rules we lay down these axioms kind of stipulate them and it's kind of game and then we see what follows and it's not some independent reality that we're discovering just as we're not discovering a chess reality when we play chess but we're just seeing the implications of these stipulated rules and that is a that is the great value of dispelling the mystery i mean it's a it's a it's a very pleasing um answer to what is it that we're doing when we're doing mathematics it makes it really kind of understandable and there were certain things that had to be proved and that david hilbert said we must prove in order for this hilbert program this formalist program to go forward and one of the things was the consistency to be able to prove within a formal system the consistency of arithmetic and that's why um girdle's second incompleteness theorem you know really sort of sent a shock through the mathematical world he was saying that can't be done and so but the formulas basically were saying you know what is it that we're doing we're doing mathematics we are creating formal systems and seeing the implications um and that it's not that there are no formless i don't want to make it sound as if girdle you know just sort of drove dow formalism into its grave although some people do talk that way i don't i don't i don't feel competent to say that i think things are murky um but he made it harder to be a formalist you have to dance around a lot post-girdle in the way that you didn't have to before a girdle the first thing he proved now let me just go back a little bit he he was very interested in philosophy as an undergraduate he first you know thought he might major in philosophy and then he thought he would go into physics but it was when he he took a philosophy course and he became a plaintiff somebody who believes abstract ideas really exists and so mathematical reality really exists and what are we doing when we're doing mathematics we're discovering this extra spatiotemporal non-physical abstract beautiful perfect eternal world great appeal to him as an undergraduate first thought he was going to go into number theory he formed this audacious ambition to prove something mathematically that would vindicate this philosophical view of platonism i mean that's like audacious nobody's ever done this and he decided he was going to do this he first decided thought he would go into number theory that that would be the place to do it sometime he decided and we don't know when he was very very private reticent man um it was going to go into logic um that that was going to do it logic was a kind of iffy sketchy field of mathematics at that point a lot of mathematicians didn't take it terribly seriously was too close to philosophy he developed mathematic i mean mathematical logic so it's you know now taught in uh branches in in departments of math before it was portioned to the philosophy departments so sometime when he was an undergraduate he had an inkling that logic could do this as a graduate student his dissertation what he proved was the completeness of predicate calculus which is not as strong enough to express arithmetic that's a formal system you can't express arithmetic in at some point we don't know when he came up with this proof for the incompleteness of a formal system rich enough to express arithmetic the first inkling king was um he was at a conference on gutenberg the same city kant lived in all his life and it was a conference on the foundations of mathematics and all of the favorite competing groups were represented there not platonism because nobody took platonism seriously at that point there was um somebody who represented formalism hilbert wasn't there but johann fandoyman the great great young pio hansa neumann um had come in from princeton originally hungarian he was there and and others but you know and sort of arguing this out girl i think gave some little talk nothing he didn't talk about it then the third day i believe was the third day when they were sort of rehashing what had been done the general discussion what had been done during the conference girdle said and this was the announcement right he said it took maybe 30 seconds and and i would imagine that he mumbled it it might be possible to be able to form a arithmetical proposition on the order of goldbach's conjecture a very famous unsolved uh problem still unsolved that we can show is true but not provable and we might be able to have a formal proof for that and nobody paid any attention to it you know it was just you know how when you're all talking about one thing and somebody you know it's just totally on another wavelength you just don't even hear it you just you know it's just and he's just a graduate student well what happened was it and who knows what he would have done if this had not happened would he have ever published i don't know who knows but johnny von neumann was there and he button told him afterwards he said were you saying what i thought you were saying you know and he questioned him about this and girdle gave him the outlines of the proof he gave him something vanoymund went back to princeton thought a little bit more wrote back to girdle and said you know an implication it seems to me if what you're saying is right an implication would be that we can't prove the consistency of arithmetic within the formal system of arithmetic and girl wrote back he said indeed yes and he had of course the whole worked out proof and it was van royman who had paid attention to this who had caught it even though it went completely against what he believed he was representing hilbert there he was a complete formalist but he and fel neumann was fantastic amazing and um he got it and he's the one who publicized it and talked about it brought eventually brought uh girdle over to princeton where he you know spent the rest of his life and but that's that's how it was made and it's just so typical characteristic of this strange man a man named john lucas a philosopher had published in the early 60s maybe i think 1962 a paper claiming basically the kernel of what roger penrose would go on and develop in two books the emperor's new mind and i can't remember the second shadows of consciousness or i can't remember the the the second um book basically he had argued this um he said the implications of our of girdles and completeness um is that our minds can't be computers we can't be digital computers in any case because the the one place where you would think perhaps we're being computers isn't in knowing mathematics i mean forget you know writing novels and writing poetry and music but i mean the one place where you would think where really a a computer could duplicate what we're doing would be in mathematics you say but girl has shown that that isn't the case computers run on formal systems that's what a you know that a computer is a formal system made physical and and in fact you know some people claim that because of girdle the work that girdle did and then alan turing an english logician who took off from girdle's work and did another theorem on undecidability the computer especially because of during the computer was actually developed out of these logicians work because they made clear precisely what a formal system is by showing the limitations of a formal system that it can prove everything that where there will always be things that we can know that escape a formal system they made the notion of a formal system clearer and we needed that to create computers but in any case what the art how the argument goes is by showing clearly what a formal system is and showing that there are certain truths that we know that we can always see and we can see or eluding the formal system we are we can't be a computer a computer couldn't duplicate what we're doing a computer running on a formal system can't duplicate what we're doing that's basically how the argument goes i think it was quite persuasive i wish i had asked colonel what he thought of john lucas's work you know and that's it's it's bothered me you know the last 30 40 i mean however how many years it's been like it's not that many it's been many decades and it's bothered me that i i never have i he spoke a great deal to a logician named hao wong and hao wong um who was at rockefeller university he was used to come in and speak uh to girdle and how wong published quite a few books at least three that i can think of right now on his conversations with girdle and so so much of what we know you know those of us who didn't have the opportunity or weren't brave enough to ask him when we when we were with him you can you could go to how long and so there is a place in which you know how long it does ask him about this and it's fascinating what girdle says he said if we really have the mathematical knowledge that we have that we think we have rather but we may be deluded about it but if we really do then we can't be computers so it's a sort of disjunctive proposition he's not saying we're not computers but he is saying that we might be deluded computers to computers with delusions of grandeur who think that we know things that we really don't that we can't prove um and in that case you know we we are we are computers we're just diluted computers and that also i think is just so interesting given that he in fact suffered from paranoia from very severe paranoia what he's sort of worrying is that we may be paranoid um computers and if you are you can never we could never discover that and that of course is one of the great tragedies about paranoia you never if you're going through a proof that you're not paranoid but you're the whole proof is infected with your mental illness the proof is no good right it's it's it's infected as well so that you know that even that answer somehow is poignant in the context of his story but that's the answer that he gives ernest nagle was a philosopher of science he was my professor when i was an undergraduate at columbia and he's so it's a wonderful it's a and it goes into i give a very very um impressionistic version of the of of the of the proof you know enough so you can get the flavor of it um and they give you know really far more detail so that is a wonderful place to go to uh to to to understand the proof the proof itself is extraordinarily dense i had studied the paper itself when i was a graduate student and then you know left it and you know did other things and then when i decided i would write this book you know i had to go back and re-immerse myself in all of that and that was itself very you know painful because i had to work so hard to recover um having been away from it for so long but it's a very very dense paper and so many things fall out from it he makes he makes logic mathematically respectable he puts it uh you know he really puts it on the map as a genuine respectable mathematical branch and so there's recursion theory he defines what a recursion cursive function is recursion theory comes out of their model theory proof theory so much has come out of um out of that incredibly dense brilliant paper so the fun neumann was talking up you know this extraordinary work and comes as a guest i ought to say that you know this is this was a time of great upheaval in in uh austria uh where he lived he was uh in vienna university of vienna hitler had come to power girdle wasn't jewish which is worth saying actually because everybody thinks he was um and even birch russell uh writes because you know burst and russell when he came to um to the states he uh came to to princeton and he had discussions with girdle and he writes about these uh um expatriates these these immigrants and you know the the austrian jew girdle and um and then i think colonel said oh you know he says just for the sake of truth i am going to correct bertrand russell i have not to be jewish you know it's not that it doesn't matter one way or the other but that's interesting the reason he he his group in in um in vienna had been been associated with a group of the vienna circle of logical positivists he was not a positivist he was very anti-positivistic but he did associate with them i think they were about 15 and nine of them were jewish and his dissertation uh hans han his dissertation advisor was jewish and so he did associate with with many jews and he kind of looked jewish and he was beaten up in vienna um by the brown shirts who mis took it for granted as bertrand russell did that he was jewish and his uh wife is uh every doubt of a wife um fought the brown shirts off with a umbrella but he did want to leave he did they had recently bought and refurnished refurbished a an apartment in vienna but he did want to leave and it was hard to get him out the director of the institute had van neumann who had to work very hard to get him out the he had never joined the nazi party but he wasn't not approached he was very very apolitical but they they managed to to to get him takami came very very long way first sail to san francisco and then they made their way over land and then he finally did come to to the institute and remain there pretty much for for the rest of his life so the another very important uh thing that he did is there's another there's a hypothesis called the continuum hypothesis as a hypothesis in set theory i don't know how much detail you want me to go into he contributed um to a very important conclusion uh which is that the continuum hypothesis can't be proved given our set theory they're you know the current uh system of set theory so here was again here is a proposition that was unprovable it's either true or it's false and we can't approve it and he he contributed one half of the proof that the continuum hypothesis is unprovable he showed that it is consistent with the axioms of set theory and its negation is consistent with the axioms of the set theory so both the condemned hypothesis and its negation are consistent with the essence of that theory therefore you can't prove it you know you can't prove it you can't disprove it and he did one half of it i think proving that the negation of it was consistent and paul cohn a great logician did the other half of it and that that's another very important result but here's the thing about girdle he only was interested in mathematics for the philosophy that's very very odd i mean this great great mathematician good season but he thought of himself as primarily a philosopher and he was interested only in doing mathematics that was going to have philosophical implications so he set the bar extraordinarily high and he did think of himself as a failure we see in how long's many interviews with him because he judged himself not in regard to other mathematicians but by plato leibniz you know descartes he wanted a something that would completely change our philosophical view of the world and and judged by those lights he considered himself a failure but also he was only looking for mathematics that could do that you see so it's very that's very very difficult to find him it's amazing he found it at all right but this was an unusual sort of mathematician when i was doing the research for the book i went to the institute and spoke to mathematicians who were there which was fascinating they've all since died i got them just at the nick of time you know they told me stories about watching these two walking back and forth every morning every evening they would go together i think they would walk home together and i would produce some pictures in the book of them walking across the field at the institute they were both refugees from the nazi terror uh i was jewish german uh god out and einstein actually said towards the end of his life you know he things he said the end of einstein's life he was working on the unified field theory and things were not coming out he was kind of a little marginalized from uh the community of physicists and he said that um his own work doesn't mean all that much to him anymore he wrote this in german i go to the office each day to have the privilege to walk home with girdle it was extra it was a very very important relationship for both of them certainly for for girdle i mentioned before that griddle was interested in mathematics that intersected with philosophy and that you know he put the bar very hard high for himself einstein was also a philosopher physicist he was also interested in the philosophical implications of his physics the nature of time the nature of space and nature of reality both men held points of view that were quite oppositional to the reigning philosophical point of view to the velden chiang of their time they were both very anti-positivist they both believed that their fields were descriptive einstein believe that physics abstract physics theoretical physics describes an external world out there it's not just a matter of our experiences it's not just making predictions of our experiences that his theories and relativity theory in particular is is giving us the mathematical structure of reality this went quite against the green because of the various problems in quantum mechanics some of the um the most popular interpretation of why we get the weirdness in quantum mechanics is it's not describing anything objectively real it's just predictive of our experiences and this is this is so einstein we all know it's part of the whole legend rejected quantum mechanics his wasn't just that he didn't he didn't like probability he the stochastic nature it wasn't just that god doesn't play dice with the world it was big it was this whole notion that in order to save the quantum phenomena if if to save this theory we have to do away with external reality the hell with the theory that was basically i mean he saw physics as descriptive of realities with deep realists about his his subject girdle was a deep realist about mathematics as well he was you know that mathematics is descriptive of what's real our formal systems can capture everything because truth is too big for our formal systems the formal that the truth is is out there both of them believe the truth is out there and what fired them up in their fields was this that we have access through these fields to be to go beyond our own experience to to to get access to objective reality so they were both there was a deep philosophical affinity between the two of them and so people said to me um i remember aman barel a mathematician who had been at the institute said you know we all wondered what did they talk about what did these two talk about watching them go and he said to me they only wanted to talk to each other they didn't want to talk to anybody else dartle was a young man einstein was an old and disappointed man when einstein died you can say that girdle lost the only person he really wanted to talk to you know he had his wife but he didn't talk about her to her without philosophy or objective reality or or mathematics that wasn't her role that wasn't the kind of marriage that they had and he really stopped talking to people and it was amazing when i spoke to people who had been his colleagues they're the mathematicians how i got a sense of how isolated he was they didn't they they they were they were mathematicians they weren't logicians some of them were so still had that prejudice i mean somebody said to me i won't say who said to me oh you're not the man i was surprised when i started to talk to him he knew mathematics he didn't know his mathematics he wasn't only a logician i mean just to say that seemed to me you know i could sense what he had to overcome and he was an odd odd man he was a stickler for rules he drove them crazy and then finally you know the big thing at the institute is the appointments the permanent appointments you have the temporary visitors at the institute for advanced study and you have the permanent and you know to be a permanent visitor it's like okay you've arrived on mount olympus you are a god this is so they take it extremely extremely seriously so there's and it can also often be quite uh um i don't want to stay violent but you know the discussions can be very heated about who gets to be a god and or goddess and and whom not and you know apparently you know girdle could be such a stickler and so obstinate that finally they decided to make him a department of one he was no longer in the mathematics department he was in the logic department he was the only logician and he he didn't read the folders of the people who were applying to the institute he had no part of this anymore and that's how they solved their problem but you know it meant he was all the more isolated and more all the more i guess free to indulge in these terrible paranoid delusions you know that eventually killed him so when he lost einstein he lost a lot he really lost his tie to to the world of people it seems and um and einstein was very amused by him you know and also people say they were such different kinds of people because einstein was so down to earth and funny and full of common sense and girl had no common sense you know he was just a very you know he had very strange views all over the place and there's this one story where uh this guy meets uh einstein and einstein said oh this is in the 1950s you know like einstein died about 1954. he said girdle he's completely lost he's completely crazy now and the guy says uh what did he do now he said he voted for eisenhower he voted for eisenhower for for for president and so you know there was this uh you know they were just very different kinds of people but it's it's poignant i think philosophically very close he had suffered from paranoia even when he was still in vienna he had been in a in an asylum for a while one of the professors the mathematicians who had been one of his professors i think he said is it in the nature of the work that makes him mentally unstable or must you be on mentally unstable in order to do this kind of work so there had been um instability before of a you know a kind of paranoid nature when he lost einstein and became really a very lonely man a very isolated man who saw his theorems being misused this was an interesting thing he felt that his theorems were not understood the way that he that mathematically everybody understood them but the philosophy that he was trying to argue for behind he saw that you know the exact opposite uh conclusions were often drawn so that instead of seeing uh people saying we have mathematical knowledge that um goes beyond the formal system that people often interpret is oh we don't have this mathematical knowledge everything is uncertain everything is you know it's all even mathematics is uncertain that he wasn't saying anything like that and so and this could feed into his paranoid delusions and he just decided there was a kind of conspiracy to keep people stupid like we used to say in the 60s just because you're paranoid doesn't mean that they're not after you and it was kind of you know true with with with with colonel and he felt you know it it fed into the the delusions that he didn't like he didn't like the way his his theorems were being used and he still disliked positivism very much logical positivism very much and always protested loudly when his theorems were seen in that context so he and the interesting thing the thinker he felt closest to was leibniz who was a 17th century rationalist who also believed that sort of everything could be deduced through a priori reason and he felt very he felt very very close after after einstein died that was his closest contact with somebody who died you know 300 years before he was born and he had actually thought that einstein's work went you know that had been people had um connived to to hide some of his basic work i mean it all fed into this kind of uh delusion and people who knew him can't think of his name right now it's morgan val i can't think of it right now but he he knew him back from vienna and you know he was sort of uh somewhat involved in in his life and he writes with tremendous pain he himself was dying of cancer very painfully and but he's he's saying it's such a it's such a um a burden on me this i'm i'm one of his few ties to to to the world of people and then you know and he died and he was just a very lonely man and he one of the i've read i did a little research on paranoia you know when i was trying to understand him a very classic symptom is the delusion that people are trying to poison you and he had always been very fastidious about his food and there the delusions were often involved with poison poison gases from the refrigerator poison food and and his wife had to go she would taste his food for him she had to go into the hospital for her own surgery and while she was away he just uh shriveled away into nothing how long came his wife how long his wife had roasted a chicken and he came from new york from rockefeller to bring uh girdle the chicken and girdle wouldn't open the door to how wong this one person who we had trusted and given all these interviews to and how long certainly worshipped uh how long a you know an important logician in his own right worshiped girdle he stared out from the door he wouldn't open the door so how long finally just left the chicken you know on the doorstep and went away by the time that girdle's wife came home you know they got him into the hospital he was just he was 75 pounds and he died of starvation so it's it's tragic and it's sad one psychiatrist whom i consulted said that in some sense paranoia is like logic run amok there's a reason for everything and girdle always thought there's always a reason for everything leibniz says this but he wasn't paranoid you know there's a way to interpret this so it's just rationalism it's just 17th century rationalism but it it's somehow it bled into and it reinforced his own um unstable mentality and so when he saw reasons for everything he saw conspiracies he saw plots he saw human machinations and so it uh it it was partly this kind of intellectual viewpoint but then it there was just it intersected with his own mental weaknesses and and and it was tragic but it's a it's an extraordinary story and we have this beautiful beautiful proof one of the most beautiful proofs to actually get a feel for it people always talk about beauty and mathematics and outsiders how can math be beautiful this is an example of something where when you get to the certain point where he actually brings everything together your heart stops really is heart-stoppingly beautiful and it seems a little bit like sleight-of-hand like magic how how did he make all of this come about um so it's an extraordinary proof and he's left us that so this is my book and it's called incompleteness the proof and paradox of kurt girdle uh it was for in a series the norton uh series on great scientific discoveries and it's about a mathematical logician named kurt girdle who proved when he was very very young to very important perhaps the most important mathematical theorems of the last hundred years [Music] you

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Length: 41min 17sec (2477 seconds)

Published: Mon Oct 19 2020