INCOMPLETENESS: The Proof and Paradox of Kurt Godel, Dr. Rebecca Goldstein, Harvard

[Applause] thank you so much for that embarrassing introduction Gary it was really awesome and and thank you all for coming here tonight this is an incredible place to be talking I was so thrilling to walk up here it wasn't so thrilling to see my name out there and the marquee but to see Kurt girdle's name out there was was really thrilling for me so I want you to make sure picture this scene here's a leafy road it's in suburban New Jersey can't really see them here but there are stately old homes behind the trees there and just beyond those elm trees there there's a lush green carpet of a Country Club a golf course and muted voices of men knocking at balls are coming as if from a great distance and picture two men strolling down this quiet street quietly conversing in German and when an older man and one younger and deep in conversation it's not unusual right now at the timeframe that we're talking about which is in the 1940s to hear German and Hungarian and polish and Russian being spoken on this street since this street is actually in Princeton New Jersey and it's the home not only of a great university but of the newly established in these days in the 1940s Institute for Advanced Study so there's a sign for an Institute for Advanced Study and here in the 1940s that were imagining ourselves into in now it's only just recently moved from it's a home in Princeton University in this old gothic building and here's a newspaper clipping and talks about the intellectual paradise at the Institute for Advanced Study is supposed to be it's just moved from the University campus from this building which is actually the old physics building where Einstein had worked to its own independent campus this is fulled Hall part of the Institute for Advanced Study now and the reason now that we're imagining ourselves in the 1940s the reason that Princeton New Jersey is so filled with people speaking German and Hungarian and other languages is that many people many of the scholars are fleeing scenes like this this is actually a scene from the University of Vienna one of the people that I'm going to be talking about in a moment has just left this the don't know if you can see where is my my lasers not working all right but you can sort of make out the the lecturer he sees a front there he's giving the the Nazi salute all of the students are giving the Nazi salute and so many scholars scientists mathematicians are fleeing scenes like this that one of educator said Hitler shakes the tree and I gather the apples and two of the choicest apples are walking down that road right now right here they are up close you of course recognize the older gentleman there's none other than Princeton's most famous inhabitant at this time the man who even while he's still alive has been immortalized as the genius as the apotheosis of the man of genius and that the townspeople have taken to calling this new Institute the Institute for Advanced Study the Einstein Institute but who's the younger guy they're these two men walked back and forth from the Institute to the town every day they walk home together every day and and other people watch them and wonder what it is that they talk about they're always deep in conversation in all sorts of weather here they are in the winter in fact Einstein said I only go to my office so this is in German here it is an English whoops sorry to have the privilege of being able to walk home with girdle with Kurt girdle they said my own work at this late stage in my life no longer means very much to me and I only go everyday to my office at the Institute in order to have the privilege of walking home with girdle so Kurt girdle obviously it's the name of that younger man and it's in the remote nature both of the men and of his work but he wasn't then and he still isn't as famous as his walking partner Albert Einstein but in his own way as I want to tell you a little bit about it today his work was just was was very revolutionary I don't know if I want to say just as revolutionary but it was very revolutionary shaking up the very foundations of his field which wasn't physics but mathematics in fact Time magazine when it was the new century the new millennium had had a roundup of the 100 most important thinkers artists men and women of the last century and a curt girdle made it to the COTS of a hundred most important scientists and thinkers when I wanted to read what they wrote about him curt girdle he turned the lens of mathematics on itself and we're going to talk about what that means the extraordinary thing that he did in turning the lens of mathematics on itself and hit upon his famous incompleteness theorems driving a stake through the heart of formalism very dramatic and we're going to talk about what that meant what formalism was and what did what did girdle do to this of car so so you know he's quite famous but of course his friend Einstein time that he had him as the person of a century the most important person of the whole century Albert Einstein person of the century he was the iconic 20th century scientists the bumbling professor with the German accent a common cliche in a thousand films instantly recognizable like Charlie Chaplin's Little [ __ ] who writes this stuff Albert Einstein's shaggy-haired visage with us familiar to ordinary people as to the matrons who fluttered about him in salons from Berlin to Hollywood yet he was unfathomable a profound the genius among geniuses who discovered merely by thinking about it that the universe was not as it seemed better actually one of the stories is that one of those matrons one of the women of Princeton when she recognized the beautiful old face that she saw walking down that road lost control of her car and rammed it in to a tree so so Einstein you know clearly is you know the most important than most man of the thinker of that last century and Einstein as I said would only go to his office to talk to to walk home with girdle I wanted to read that honorary doctorate that girdle had gotten in 1952 from Harvard Discoverer of the most significant mathematical truth of this century incomprehensible to laymen revolutionary for philosophers and logicians I'm going to try to convince you that Harvard lies but it's not entirely incomprehensible to laymen even though as you can see the insignia thereof of Harvard Veritas truth I think this is not true I think that in fact it is can be made comprehensible to laymen and even maybe even in the short time that we have here so what is it that he did Kurt gödel in 1930 when he was 23 years old here is a young man it proved an extraordinary thing he had produced an extraordinary proof in a branch of mathematics that wasn't even considered altogether respectable in his day he actually made it respectable mathematical logic for something called the incompleteness theorem in fact it's actually two there are two incompleteness theorems to logically related incompleteness theorems they one of the wonderful things is that unlike most mathematical results girdles and completeness theorems can be expressed in normal words in English without any mathematical symbols at all the nitty-gritty of the details of the proof are very technical are formidable technical but happily delightfully the overall strategy of the proof is not it's very elegant and it's it's uh ingenious and it's simple I'm going to give it to you today more or less and also it can be spoken in real in English and that so I hear is it's not gonna sound like real English to you but by the end of the talk you're gonna understand this is from the encyclopedia of philosophies article on Kurt girdle and it opens up with a very terse a very crisp statement of his two theorems that I'm going to read to you by girdle's theorem the following statement is generally met now just notice there's not gonna be any symbolism here it's English more or less in any formal system adequate for number theory there exists an undecidable formula that is a formula that is not provable and whose negation is not provable this statement is occasionally referred to as girdle's first theorem a corollary to the theorem is that the consistency of a formal system adequate for number theory cannot be proved within the system and sometimes this corollary is referred to as either girdle's theorem or as the second theorem these statements are somewhat vaguely this is still friendly article these statements are somewhat vaguely formulated generalizations of results published in 1931 by kurt gödel then in Vienna and this is the title of the the article that he wrote on formally undecidable sentences of the of principia mathematica and other systems of type one which was received for publication November 17th 1930 you know you might not guess from this short terse statement I'm sure incomprehensible at this point even though there are no symbols even though you might not guess from this terse statement of these incompleteness theorems the incompleteness theorems are extraordinary for among other reasons for how much they have to say they are probably the most talkative mathematical theorems in the history of mathematics they go on and on and on there are so much that one can get out of these theorems as I mentioned they belong to a branch of mathematics known as formal logic we're going to talk about that in a moment or mathematical logic and yet his theorems range far beyond formal logic this formal narrow domain addressing such large and vast and messy questions as the nature of truth and knowledge and certainty in general and because our human nature is intimately involved in these discussions of these issues of truth and knowledge and certainty after all in speaking of knowledge we're speaking about knowers about thinkers us right girdles theorems have also seemed to have important things to say about what our minds could or maybe could not be so girdles theorems are unusual not only because they're rendered in more or less plain English we use terms like formal systems consistency incompleteness these are technical terms we'll talk about what they mean but they're not symbolic right because they're these are very rare mathematical creatures not only because we can avoid symbolism and talking about them but because even more interestingly they seem to address themselves however ambiguously and controversial E and very controversial E - the central question of the humanities what is it that's involved in our human what is it for us to have knowledge how can we have knowledge I mentioned they're the most talkative theorems in the history of mathematics there's disagreement about what they say how much they say precisely what they're saying but there's no doubt that they're saying an awful lot beyond mathematics certainly extending into what we call meta mathematics and I'll explain that term in a moment and perhaps even beyond that even beyond metamathematics girdles theorems are so unusual because they're mathematical theorems that seem to escape the limits of mathematics they seem to speak both from inside and outside mathematics they have the precision of mathematics and the reach of philosophy so you could understand why philosophers find them so fascinating I used that term mehta mathematics and I should explain it this prefix meta comes from the Greek and it means after Beyond and it suggests the view from outside as it were all sorts of different cognitive fields present meta questions questions such as how is it possible for this ear area of knowledge to be doing what it's doing so we can ask meta questions of science we can ask meta questions of art we can ask meta questions of mathematics mathematics is a field of knowledge which is in many very important respects unique and so it presents very pointed very special meta questions questions about what it is that we are doing when we're doing and how is it possible that we're doing it the rigor and the certainty of mathematics is arrived at a priori whoops I skipped ahead a priori I should have had a slide up there I thought I did saying what a priori it is and what it means basically a priori is that we know these truths somehow independent of experience it doesn't mean that we're born knowing them it doesn't mean that they're an 8 but it needs that no experience is going to count against them so in some sense their status is independent of experience so for example 5 plus 7 equals 12 what would happen if I were to count five things and seven things and not get 12 things get 13 if that were going to have if that should happen and actually it has happened to me well what would I do I would recount if I still got 13 if I continue to get 13 would I say that 5 plus 7 equals 12 had been invalidated had been falsified so I would say either it doubled something doubled or I'm seeing double maybe something is wrong with me maybe I'm dreaming right I would go perhaps that is that is possible or even if I continued to convince myself that all was well and I was awake even that I was going mad right this is something of what we mean when we say that 5 plus 7 equals 12 and in fact all the truths of mathematics are a priori this truth is used to evaluate our experiences of counting and not the other way around right we don't evaluate the truths of Mattox by our experiences somehow these truths stand outside of experience they are a priority so mathematics just because it's unique using a priori methods to establish it's often astounding even though certain and incorrigible results has always forcefully presented theorists of knowledge those who study knowledge core knoweth epistemologists that's an epistemology stand epistemologists think about thinking all right the mathematics has always presented a pistol ologists with very special questions how can the likes of us you know who are thrown up by the random thrashings about of evolution attained any sort of infallibility which we seem to do in mathematics any sort of certainty how can we know things a priori and to press this real it might help to think about marks only not that marks that was right who said that he wouldn't belong to any country club that would accept the likes of him well similar a similarly some fretted that if mathematics is really so certain how can it be known by the likes of us how could we have gained entry into so restricted a cognitive country club these are many questions meta mathematical questions many questions about a field say about science or mathematics or the law all of which raise meta questions are not normally questions that are contained in the field itself so a meta question of science is not itself a scientific question meta questions in law are not something that can be answered through the methodology of the law rather their philosophical questions that's the domain of philosophy philosophy of science philosophy of the law philosophy of math Bertels theorems are spectacular exceptions to this general rule these are mathematical mathematically proved theorems a priori proved theorems after all this is a map a piece of mathematics and yet they establish a meta conclusion it's as if so he's addressing within mathematics itself he's addressing many mathematical questions and this is amazing this is this is unique in the history up until then of mathematics it's as if someone had painted a picture that manages to address the basic questions of art what is it to be beautiful and art and maybe even answer how it is why it is that art is able to move us the way does this is the closest I could come an Escher picture to a question that seems to get out of the framework and to address the question of art itself girdles theorems say something about the limits of proof the limits of formal systems use this term we'll explain it in a moment what the first theorem states is that all formal systems are either inconsistent or incomplete all formal systems that are rich enough to Express arithmetic so you have a choice with formal systems either inconsistency or incompleteness take your pick you can't have both you can't have a formal system that complete and that's consistent so I've used three technical terms here formal systems consistency completeness and now it's time for me to explain them let's talk about formal systems first these are big big semesters worth of ideas right so I hope to give you only enough understanding so that you grasp the extraordinary thing that Kurt gödel did with these theorems formal system is basically a system in which everything is done by rules it's a it's systematized rules there is nothing that is not done by rules in a formal system it's all a kind of mechanical procedure the sort of thing that could be programmed into a computer formal system basically has three sorts of rules the first sort specifies just what's the alphabet of the symbols what are the symbols that are used in the formal system what are the symbols that can be used that can appear in the formal system the second sort specifies how these symbols can be combined with one another what is a meaning according to the rules of combining these symbols into what we call well-formed formulas and the third sort of rule tells you which combinations of symbols follow from which combinations of symbols right so it's all very rule-oriented three types of rules what are the symbols how can they be combined what's the most follow from what symbols what combination of symbols follow from what combinations of symbols these rules of a formal system go as far as defining what the symbols and what the well formula well-formed formulas mean in the formal system the way that we put this in mathematical logic or philosophy is that in a formal system meaning is all syntactical it all has to do with the rules the rules of the grammar of the system three types of grammatical rules that I've just given you there it's a matter of rules and nothing but rules so very abstract talk here's an example here's a symbol right you think you know what it means right you have a and right you think this ampersand means and here in a formal system are the rules that define the meaning of this term right and if this term means nothing over and above these rules here's one rule if you've got P saw an ampersand Q you can you can deduce P right so this is if you've got P ampersand Q you can deduce Q if you've got P then you've got Q what can you deduce thank you right and these are the syntactical rules that completely exhaust what that symbol and means in a formal system and that's what a formal system is everything is done by rules you don't ask outside the rules what does it all mean so a formal system has no meaning outside of the rules that constitute it formal systems run computers run by formal systems and here's a joke sort of a comic that's supposed to get the idea of a formal system that absorbers are restricted to formal systems I think therefore I am he comes to some sort of meaning Here I am I am I think therefore I am a live of life with life and thought and my sweet consciousness and immortal soul portable systems this can't happen in formal systems right formal systems are used to prove theorems what girdle's incompleteness first incompleteness theorem says is that all formal systems that are rich enough to express arithmetic very basic mathematics are either inconsistent or incomplete we're going to talk about what that means in a moment the whole reason for formal systems is to produce proofs in a very mechanical fashion why they're so important improves whether they're so important for proofs whether proof proof comes down to nothing but proving in a formal system this is the meta question that lies at the heart of girdles and completeness theorems it's actually the meta question motivating the whole girl incompleteness issue what the whole issue was about was whether or not all of mathematics can be reduced to the sort of mechanical procedures that we do in formal systems so remember the slide that we saw from Time Magazine's 100 do I have it yes here we are Kirt girdle he turned the lens of mathematics on itself and hit upon the famous and completeness theorems driving a stake through the heart of formalism formalism is the view that all of mathematics can be reduced to formal systems that great mystery of how can the likes of us get into this restricted country club how can the likes of us attain such incorrigible and fallible knowledge what formalism tried to say is that we create these formal systems it's kind of a game that we play and we carry out the implications the entailments of this formal system how do we get these intuitions where do they come from or finite creatures how can the likes of us know about infinity such that no empirical evidence would count against it you would call us on that I would rather say I was insane than that I had it some sort of experience that falsified 5 plus 7 equals 12 how does this happen it's all a matter of formal systems there's really nothing very mysterious about it for example we know that every natural number has a successor right we're never going to run out of natural numbers we know that there is an infinite number of prime numbers we have a proof of that they're never going to we're never going to get to the highest prime number how can we who are so small and finite and how do I know up there beyond any any number that we've counted to there is it some highest prime number how can I know this how how can we attain this kind of infinite knowledge this is this great minna question made of mathematical question and the fact that we could have some sort of intuitions of it if somehow that we finite creatures can have these intuitions of infinity can sound like a kind of mysticism right that we can somehow attain to this knowledge of of infinity that we have intuitions of it that somehow we see it in a way akin to sense perception the idea that that's what mathematics is which is opposed to formalism or formalism is opposed to this idea of some kind of mystical insight into an extra sensible extra physical world somewhat seemingly mystical idea there thinker with whom this idea that there really are independent mathematical truth and the mathematician is discovering them not inventing them in formal systems but discovering them is most associated with this thinker he's the the the note the the point of view at posed to formalism it's known as Platonism and here's the first clayton Asst none other than good old plato who lived in the 5th century bc formalism is the attempt to nip this kind of mystical mathematical view of play of Platonism in the bud formalism wanted to banish the whole idea of intuitions from mathematics according to formalism all of mathematics is just a matter of rules it's formal systems right it's those three kinds of rules that kind of thing we can program into computers we make up our rules and then we follow them and we see what they lead to in formalisms telling mathematics becomes chess raised to a sort of higher order of intricacy there is all of us would agree no objective chess no objective world of chess some reality that the system of chess captures in chess the stipulated rules constitute the whole truth of chess similarly similarly according to formalism which is most associated with this man David Hilbert who was the most important mathematician of the generation prior to girdles according to formalism Hilbert the stipulated rules of a formal system constitute the whole truth of mathematics away with all of this mysticism this intuition this super sensible reality of Plato mathematics is a game and we win in mathematics by proving theorems that is by showing some uninterpreted string of symbols follows from some other uninterpreted string of symbols using the agreed-upon rules of inference and we don't bring in any extra meanings the rules provide all the meaning there's no external truth against which mathematics has to measure itself mathematics is not contra Plato describing some trans empirical reality all of mathematics can be reduced to formal systems which are consistent and complete ok so you have a feel for what formal systems are more or less and what formalism was now for these other terms complete what does that mean so formalism means need needs that all of mathematics can be reduced to formal systems that are consistent and that are complete what does complete means a formal system is complete basically you can prove everything in it that you want to prove in it given any well formed formula anything you can express in that system you can show that either it or its negation are provable that there are no undecidable propositions with within a propositions that can't be proved true and can't be proved false and then inconsistent you probably know what that means but we'll go up what does it mean a formal system is if inconsistent if you can prove a contradiction in it if you can derive within that system using the rules of that system both P and not P for some proposition P P and not B this is very very bad very bad you do not want this we can live with incompleteness in our formal systems we're living with it we have to live with it ever since girdle we can't live with inconsistency in a formal system what's so wrong with inconsistency in a formal system I'll tell you what's so wrong it's just that it makes the entire system completely worthless right throw it away it's worthless an inconsistent system is one in which you can prove a contradiction and from a contradiction you can prove formerly from a contradiction anything at all follows any stupid proposition you could throw at me anything right can be proved from a contradiction if you have a contradiction anything follows from it so as the whole point of formal systems is is to prove if it is it's an inconsistent system that makes it way too easy you can prove anything any worthless string of not not nonsense it has to be meaningful within the system but you can prove anything from a contradiction so an inconsistent system is to complete right it can prove anything that it can Express if nothing ruins a system like inconsistency you don't want inconsistency so now already despite what Harvard said in awarding Goodall his honorary doctorate we're beginning to see what girdle did and what why it's so men and mathematically important why it was that he drove a stake in the heart of formalism formalism claimed that mathematics is nothing but formal systems formal systems which are of course consistent we're only interested in consistent systems inconsistent systems don't want those formalism claim for example that a field like arithmetic can be reduced to a formal system that's what it means for it to be true provable in a formal system there is acclaimed a consistent formal system of arithmetic formalism claims this that contained all the truths of arithmetic and here's what girdles first incompleteness theorem states no formal system of arithmetic is both consistent and complete for any formal system rich enough to express arithmetic this is not his language but my there can be shown to be errant medical truths which are not provable provable within that system notice he says truth we can actually see that these propositions are true but they're not drivable within that system you can add them as an axiom to and make your system bigger but then girdle shows you a way of constructing another proposition which we can see is true but is improvable within that system so you can already see why he's making trouble for formalism I know his second incompleteness theorem claim that the consistency of a formal system cannot be proved rich enough to express arithmetic I have to say cannot be proved within that system so when we're using a system we can't unless we go out of that system and provide an interpretation of talk about what talk about what that system is talking about in a way that formalism doesn't want right there way that kind of pulls the plug on formalism if we stay within the formal system if we're talking about nothing but the formal system we can't even prove that that system is consistent and maybe one of those horrific inconsistent systems that work operating within this is very very bad for formalism right all right and who did this it's actually it's one of my favorite pictures of him here he is sometimes in Vienna in his hiking clothes any of you have spent time in I know all the Austrians go hiking all the time when there's girdle one motivated girdle and he did this sometime before he was 23 girdle no zipper girdle was a plate inist in fact he was a passionate plainest he apparently became one when he was very young as a young undergraduate at the University of Vienna he actually entered the University of Vienna intending to study physics he was always most interested in learning about reality the nature of reality that was always his burning motivation just like the man who would become his best friend many years later on the other side of the world both of them refugees from the madness of Nazism in New Jersey but some time as an undergraduate think about this as an undergraduate sometime in the 1920s he took an introductory course in philosophy with a professor Gunn parents Heinrich Gunn Paris gun parents years later in New Jersey many many years later after Einstein had died and girdle who really had only wanted to talk to I'm Stein and once I'm Stein died he treated more and more into his own private world and became unfortunately quite he had always been somewhat paranoid but he became even more paranoid something of a recluse a sociologist named burke cron jean who had been trying to interview kurt gödel for a long time in girdle did in to answer and finally out of frustration the sociologist sent kurt gödel a questionnaire and he asked him all sorts of questions wanted to know among other things who were the most important info on girdle who were the philosophical influences on girdle and the sociologist listed a bunch of philosophers to make it easy for girdle ladies all I had to do was put a check or a or an X and asked him just to say you know which of these and girdle made the found if that he made it out twice he didn't send it off the first time a grand gene sent it on the second time how do I know all this because like so much else in the girdle remains in the nasty archives he never sent it off right it's all down there in the archives of Princeton University right and so along with all sorts of other things is flips from books he took out as an undergraduate deliveries of coal from Vienna he saved everything in there I found this questionnaire from branching and Gerda listed just three philosophers as having influenced him he seemed by the way to take particular offense and one of the philosophers that grandjean had offered as perhaps being a great influence on Unger at all Ludwig Wittgenstein and in this questionnaire that girdle did not send off but which I was lucky enough fortunate enough to find down there girdled actually acts a little offended and says Vicki Stein had nothing to do with my ideas in in metamathematics in fact he never understood my proofs and if he did he would have said the things he did so it really took offense at this the three philosophers he listed were Plato live Nets he had a very very close relationship with a lightness as a matter of fact after good line Stein died and Colonel seemed to be not really talking to any of his contemporaries seem like his closest colleague was was liveness and professor golf parents he lists professor gone paris i couldn't find a picture of gone parrots but I found this little sketch of him and that these were the three philosophers who had most influenced him and he said that he took an introductory course in philosophy with professor gone parrots um professor compares his father had been uh Theodore Godric's had been a famous or historian of ancient philosophy her three books he wrote on ancient philosophy and why did Goethe list professor Gompertz as being such an important influence on him apparently while he was a student in this introductory course girdles life was was was more or less changed he he more or less became galvanized by these ideas of Plato with this fundamental idea in Plato that this spatio-temporal world that we experience is really a projection out of an abstract manifold a level of reality that's graspable only through reason in the way that we grasp mathematics and that's as real but actually even more real than the spatio-temporal world that it explains in fact that the spatio-temporal world enjoys whatever reality it does because of the participation in this more abstract realities the essence of plato and girdle seemed wall an undergraduate to have become passionately interested and devoted to this idea he told another logician how long many years later that under this influence he changed his major from physics to mathematics that for him a thematics now became a way of discovering reality he was very committed to the view that mathematics is not an invention but a discovery he's a plainness he's not a formless he first went into number theory he said that's where he first thought that he would find some kind of truths that would reflect on this meta math of mathematical reality so he was he's a funny thing about kernel he went into the math for the philosophy that's what he was interested in he wanted mathematical results that would somehow be able to reflect on the interpretation of mathematics in general and he knew which interpretation he was looking for which mathematical view he was looking for platonism so this was his motivation um he wanted to show that arithmetic was about numbers right that set theory was about sets that it was about things outside of the formal system that it was about things he wanted to show that there was a platonic sphere of abstract entities that constituted what we call the models for our formal systems and that our formal systems are true to the extent that they truly describe the entities of their intended models this gives one a glimpse into the really outsized ambitions of this rather strange undergraduate at the University of Vienna there is who was outwardly almost interestingly almost absurdly timid and cautious it was it is powered like prey to all sorts of absurd unfortunately fears sometimes darkened into gen genuine paranoia of such intensity that he would sometimes have to be hospitalized here's a haha an asylum even in Austria where he suffered these these bouts of extreme paranoia but who let's get off of this bad picture I don't like this picture will stay here with Hilbert but who although outwardly cautious fearful paranoid insofar as his intellectual ambitions went and his intuitions went he was fearless he was a kind of hero he had such a heroic confidence and intuitions it's a truly audacious ambition for an undergraduate who's bothered by meta mathematical questions and who learns a Plato in an introductory course from Professor Khan parents - ensign himself the task of discovering a mathematical conclusion that would simultaneously be a meta mathematical result that would support mathematical realism Platonism that mathematics is about something and it was even warned as this is amazing enough but if you actually look at the historical context in which he was then living it's even more astounding that he had this ambition this point of view because this was completely out of fashion at the time formalism was the reigning school of meta mathematics with the most influential mathematician of the day David Hilbert behind formalism and why was formalism so popular just then well first of all there's there's the spooky aspect right this whole commitment to an extra sensible world this platonic world a world of abstract objects there's always a reason to try to eliminate that Occam's razor right you try to get rid of all extraneous ontological committing those words that philosophers use ontological commitments commitments about what exists but also remember that formalism must was trying to vanish intuitions from a thematic intuitions when you you just know that something is true right it's that gut feeling you know it's true even though you can't prove it you know it and you know it with certainty and there are always reasons for people to be suspicious of intuitions right people claim to have intuitions about all sorts of things some often one person has an intuition or P and another person has an intuition for not P sometimes people are so certain of their intuitions that they wait you know they wage holy wars in support of their intuition so even outside of mathematics there's always a reason to be suspicious of these intuitions right these intuitions of certainty but in mathematics as well and the late 19th century the early 20th century had had some very dramatic evidence of the shakiness of our mathematical intuitions with the discovery of first of all non Euclidean geometry and also with the discovery of what we call the paradoxes of set theory there were several discoveries of paradox right within set theory right there within mathematics and I'm going to tell you about one of them there's there's Cantor's paradox of the set of all sets that can't exist I'm going to tell you about one really serious paradox of set theory known as Russell's paradox it was discovered by formalized mathematics starting with axioms definitions deriving everything from there there's our kernel there's someone I haven't mentioned right next to kernel that's that's hans hahn who was kurt girdles those of you who are mathematicians know of hahn the famous panache Kahn theorem that Hahn was girdle's dissertation adviser and he had no idea interestingly enough what this young man was cooking up right he had no idea until girdle announced very quietly that he had proved the first incompleteness theorem that he had derived a proposition that we could see is true but is unprovable within any formal system within that's normal system of arithmetic Hahn had no idea what Goethe was up to so there's Hahn there's guilt Hilbert there's there's Bertrand Russell so here's the set of all mathematicians mentioned in my talk today so you can also have properties of assets so and you can ask so sometimes sets can be members of other sets right so I can ask for example of this set of all mathematicians whether it's a member of itself is the set of all mathematicians a member of itself clearly it's not a member of itself because the set of all mathematicians is not a mathematician it's a set right it's a but the set of all say mathematical constructions is that set of a member of itself yes because the set of all mathematical constructions is itself a mathematical construction so is a member of itself so let's think about this particular property of sets not being a member of its self there's a property not a holds of sets all sets which are not members of themselves and we're going to form this set the set which has as its members all the sets which aren't members of itself all right it's that set it's a perfectly good description I've explained it to you let's form the set is it a member of itself is the set of all sets that aren't members of itself itself a member of itself either it is or it isn't that's just logic folks well if it is a member of itself then it's not a member of itself because this set contains only sets that are members of itself so if it's a member of itself it's not a member of itself okay so it's not a member of itself but if it's not a member of itself then it's a member of itself because this set contains all the sets that aren't members of itself right so this set is a member of itself it's a member of itself and it's not a member of itself the set of all sets that are members of themselves both isn't isn't a member of itself contradiction right we don't want contradictions right there's this is the promise right we don't want contradictions so this was the fact that we can form this set in set theory that set theory as it then existed was able to produce a contradiction was was pretty damn bad the set of all cells that aren't members of themselves it doesn't exist it can't exist we know it can't exist because it produces contradictions but we have the intuition wherever you have a description you can form this set here is proof intuitions can go awry intuitions are not all together reliable even in mathematics let alone in FX or in politics or all the other places religion all the other places that people claim intuitions even in mathematics are most exact science intuitions can go awry Hilbert was very upset by this that the formation of as you can imagine right of of the paradoxes within set theory and here's a quote from him admittedly the present state of affairs where we run up against the paradoxes is intolerable just think the definitions and deductive methods which everyone learns teaches and uses in mathematics lead to absurdities if mathematical thinking is defective where are we to find truth and certitude this comes from David Hilbert an article on the infinite the most important thing is to show that formal or thermal systems are consistent that they can prove any contradictions because as I've already explained to you a formal system that's inconsistent is worthless you can prove anything it's to complete so Hilbert told his cadre of mathematicians in a pep talk that he gave them in 1900 that they must go out and prove fourth with me kiss me the consistency of arithmetic Hilbert himself had already shown that geometry was conditionally consistent meaning what he had shown is that geometry is consistent if arithmetic is it was conditional on proving the consistency of the formal system of arithmetic and this is why of course he urged his mathematicians to go out and prove the consistency of enter girdle there he is that's that's me but the Hilbert's program was to formalize everything get rid of those sketchy intuitions show that we don't need them all depended on being able to formalize arithmetic to show that arithmetic their formal system that could express a risk meticulous both complete and consistent and girdle showed that this was impossible and he showed even further that with his second incompleteness theorem that we can't even within the system of arithmetic ever proved the consistency of that system we have to go out and provide what we call a model for it show that it's about something so he delivers a double stake in the heart of formalism just as the Time magazine said it's a double stake actually where did I get this all right what I wanted to do was show you just a little bit it's such a beautiful beautiful proof like I obviously can't you know get into it here it's a very it's a it's quite short it's very very dense a tremendous amount packed into it in fact a lot of a lot of mathematics the whole notion of a recursive function and recursion theory model theory came out of what girdle did in in that proof and in fact in trying to show the limitations of the formal sis of a formal system he sharpened the notion of a formal system beyond any measures that had had been gone to before so he really made the notion of a formal system much clearer in trying to to show its limitations in mathematics but I wanted to give you justin Lee's brief last minutes a little feel for what he does here this ingenious proof which is just so beautiful it's all worked out extraordinary what it's fantastic what he manages to do through this amazing detailed work something that we now call girdle numbering though of course modest girdle didn't would never have called it by his own name is that he gets arithmetic 'el statements to kind of talk about themselves there's a kind of double speak going on so that these propositions have both a straightforward arithmetic 'el meaning they say something about the relationship of numbers right it's just natural numbers but they're also talking about themselves at the same time they're talking about their own proved ability whether they can be proved in the system or not they're saying something arithmetic 'el and they're say something met a mathematical something extra formula something about themselves and their own proof ability it's all worked out extraordinarily carefully beautifully it's kind of heart-stoppingly beautiful and here's what he comes up with it is an arithmetic 'el statement that's both saying something about numbers and it's also saying that it itself is not provable the self referential stuff right that we saw with the set of all sets is that a member of itself girdle tells us and trying to give us a little help heuristic help in understanding this proof which was very unlike any proof that had come before he tells us to compare it to this statement P this very sentence is false that's this sentence the P is this sentence it is saying that it itself is false this is known as an ancient paradox it's known as the liars paradox so what's wrong with this sentence I bet you can already say is this sentence true is P true well if P is true then P is false right okay so it's false well if it's false then it's true right because that's what it says so P is both true and false I mean it is this self referential paradox girdle's strange arithmetic all arithmetic all propositions says something analogous and it's doublespeak it cooks up a proposition in a system that says something arithmetic all but it also says G says the G is this very proposition and it says that G is improvable in the system so here's an arithmetic 'el sentence through the magic and it feels like magic believe me in the proof but it's not it's a proof it all works through that cunning of the proof this proposition which is a straight very strange arithmetic proposition is also sane about itself that it is unprovable in the system the negation of G is that G is provable in the system because G says it's not provable in the system so what not G says the negation of G is that G is provable in the system is G provable in the system if G is provable then its negation not G which after all says that she is provable in the system would be true so if G is provable in the system its negation is true get it but if the negation of a proposition is true then the proposition itself is false right P or not P so if G is provable in the system then it's false but if G is provable in the system then it's also true if the system is consistent right so that is a condition of this whole proof if the system is consistent then it's incomplete so we're assuming the consistency of the system so if G is provable then G is both true and false right if the system is consistent after all what is a proof show assuming of course the system is consistent then that approved proposition is true so assuming the consistency of the system if G is provable then it's both true and false what do we conclude from this that's a contradiction therefore G is not provable we've just proved that G is not provable right we've proved and because that's exactly what she says that G is not provable is exactly what she says therefore G is true but not provable yet he's proof here is it therefore G has both unimprovable and true which is precisely what the famous conclusion of girdles proof says that there is a true of an unprovable proposition expressible in the system if the system is complete is it consistent and because G also has a straight forwardly arithmetic Ulm eating which of course is true of Jesus true because it is G girdle's proof shows that there are errant medical truths for example G that can't be proven in the formal system assuming the system to be consistent right the formal system is either inconsistent or incomplete that's the first incompleteness theorem well anyway we've we've done something very important we've proven Harvard wrong right you all understand you're laymen you all understand right basically what happened here I just want to end with this Baron Munchausen and Kurt hurdle the poet this isn't my idea a post poet instance verga compared girdle's argument with the tale about Baron Munchausen famous liar children's story and one of the stories of him is he manages to heave himself and his horse out of a swamp by pulling on his own ponytail right and this wonderful poet who also tried to see a girdle I found that letter in the North us also compares what girdle does here within the formal system proving that meta mathematical result of the of the limits of the formal system within the formal system kind of compares it to this tall tale of Baron Munchausen but the thing is that Baron Munchausen was a liar and Kurt gödel had a proof though thank you very much you is the fact that 0 times 1 is equal to 0 but yet 0 divided or 1 divided by 0 which is the same thing as 0 times 1 is undefined is that also an example of gödel I don't think so that actually the truths that you're mentioning here really follow from the rules of arithmetic right I mean it really is completely within arithmetic following from the basic rules that we use the axioms you know of arithmetic arithmetic has been acts minute eyes right and so this that kind of so that's a kind of really arithmetic codes within the system of arithmetic it doesn't have that strange kind of inside outside feature of girdles incompleteness theorems it's really that it's really arithmetic oh it's not meta mathematical maybe you're seeing something it seems you know there are there are many things that we deduce within mathematics that seem strange and paradoxical so that many many of the results that we come to I mean the whole notion of an irrational number you know that the Greeks had is in some sense a strange in fact that the the classifications the way that these these new advances in mathematics get named often register the deep surprised when we deduce certain results within mathematics all right so we call irrational numbers and you know imaginary numbers and rational numbers as opposed to real numbers that these that the very classification I think some registers the kind of historical surprise that when we deduce some of the patience of our arithmetic but that's a different kind of surprise than in this surprise of that girdle delivered G itself to be an arithmetic 'l axiom if we decide to assume that instead of trying to prove it excellent question for and that's of course the next rational move to do you just add G right as an axiom and then this goes on ad infinitum girdle shows you how you cook up another G for the new system so no matter how many times you add in your axioms he's given you a recipe within that system for cooking up something that escapes the your formal system I mean one way of putting this result there is I just think you know it's here's the proof what it actually means what it's actually showing us Mehta mathematically about about mathematics is is certainly up for debate mathematics is clear but the interpretation of mathematics is not right so that whether or not this really shows as girdle thought as girdle actually says that he what motivated him and what he actually thought followed from this was mathematical realism was the fact that mathematics was descriptive was the unlimite bility of intuitions that intuitions that when we do mathematics as baddest and as risky as intuitions are with intuitions can lead us awry but that they can't be eliminated from mathematics that we can't just make it a mechanical formal purely syntactic procedure you know that is certainly the way girdle interpreted these these these these conclusions I mean some sense what he's telling us that you know mathematics we can't eliminate the risk right that it's it's risky we may go terribly awry we may be using intuitions that are going to lead ultimately to into two to two problems two paradoxes but that they are they can't be eliminated so in some sense it's interesting you know that anything or at least mathematics I like to universalize it to say anything worth doing is risky you know but mathematics as well but there's a certain amount of risk and if formalism was trying to minimize the risk in fact eliminate it anyway but that's what happens if you try to add it there's a way of cooking up the system so that you can create another G I am sort of a two-fold question I don't really don't want to open up a can of worms here but can you give an example of G like is there really a equation or something that really says that it itself is not provable somehow and then the other thing that I wanted to know was really what is the implications all this of all this philosophically you know okay so if math can't prove everything does that just mean that okay we don't know everything or that we can't prove everything or what is it that it's really saying about us thank you okay so the G that's actually cooked up here which we know to be true but unprovable is a very strange arithmetic all proposition right I'm not gonna you know however girdle says that these types of of true but unprovable propositions are of are of the following sort so there is and he actually mentions a famous as yet unsolved problem Goldbach's conjecture he actually mentions this in the forward to his proof so what Goldust conjecture is is that every even number higher than two is the sum of two prime numbers right and so this was so this mathematician Goldbach conjecture this he doesn't have a proof right so that every even number higher than two is the sum of two prime numbers we don't have a proof for this right every even number that we've checked so far but you know what there there are an infinite number of them we're not gonna check them off every one we've checked so far this is true for every even number is the sum of two primes that proposition is either true or false if it's false that every even number is a sum of two primes then in principle we could discover that because somewhere out there in infinity there exists a counter example there exists an even number which is not the sum of two primes with prime numbers are numbers that can only be divided by themselves and one and by nothing else right that you can't it doesn't have any divisor except itself and one and here's like you know here's an amazing thing go home and check it right for every even number you'll see that you can come up if you there it's the sum of two prime numbers so if it's false in principle we could discover that because you know we'll keep at it long enough we'll discover a counter example but let's say it's true that every even number is the sum of two prime numbers and yet there is no way to prove it there's no way to get from our axioms of arithmetic to to a proof of this it just happens to be a fact about about numbers that every even number is the sum of two primes a formalist would say well then it just has no truth value either it's false so in principle it say you know we can discover it or if it if it just doesn't follow from our formal system if no proof follows it's not true it's just it has no it has no truth fell to gur it all to a plainness look it's either true or false either every even number is the sum of two prime numbers or there exists a counter example but that girdle gives us as a possible he says that there are true but unprovable propositions of the order of gold offs conjecture and that you know a famous conjecture that we can show our true you know but but unprovable so and what is the what are the implications for this well here are some of the implications that some people I mean it seems it does seem to follow that mathematics cannot be reduced to formal systems that does seem to fall there are still formulas around there are still people who are formulas you know who dance around girdles and completeness theorems one way or the other but it's become much harder to be a formless since girls and completeness there so I wouldn't even say that formalism is you know he drove a stake through the heart of formalism I mean I actually I would say but so it's not even you know so in terms of metamathematics what follows certainly difficulties for formalism what more follows so you ask girdle what girdled didn't conclude from this that we a limitation on mathematical knowledge he who he argued from this a limitation on formal systems so girdles way of understanding this is that we had we have mathematical knowledge that can't be formalized he wasn't saying that there are there's mathematics that escapes us in fact we actually can see that this proposition is true although on unprovable and the elegance of that proof that I just gave you can see that it's it's true but unprovable so girdle is not saying the incompleteness of mathematics he's saying the incompleteness of formal systems right that we can't we can't there are formal systems can't exhaust even all of our mathematical knowledge 10 our mathematical knowledge exhaust all mathematical reality given what girdle says that there is a mathematical reality you know he he actually sometimes allows themselves to talk about this there's a famous article called what is girdles what is Cantor's Continuum Hypothesis in which he really puts forth his his platonism and he says so here's a here's another undecidable it's too much to explain but Cantor's Continuum Hypothesis is as another one of these theorems that doesn't follow from set theory to set theoretic here's what it basically says is I can't do it I can't do it too well alright so we have the seven here's here's an amazing thing that happened in 19th century mathematics with with Cantor Georg Cantor fantastic mathematician he showed that there are orders of infinity you can have an infinite set you can have another infinite set it's kind of bigger I mean let's again this is sound so paradoxical this is one of the you know the surprising results in mathematics I mean infinity is infinity right you would think I remember reading about this when I was a kid I think this is what got me started in mathematics when I read gamma 1 2 3 infinity I don't know if any of you have read it and tossed up but a wonderful book I mean you should all read this book but he talks about these orders of infinity so there is the set of natural numbers counting numbers right and then there's set of real numbers right the rationals and Irrational's right and there are more real numbers than there are natural numbers right there's a fantastic argument to show this that there are actually one infinite set is something's bigger than the other meaning that you can put in one-to-one correspondence you can pair up every natural number with a real number and they're going to be real numbers left over that's what that means so we have these two sets we have the natural numbers we have the set of natural numbers we have the set of real numbers and what we want to know is is there some set in between that's bigger than the natural numbers and smaller than the real numbers and that was Cantor's Continuum Hypothesis and I think he said that there wasn't I can't remember what the hypothesis was either there was one in between or there was it and girdle together with the mathematician Paul Cohn together approved that the Continuum Hypothesis can't be proved within set theory both it and it's negation are compatible with the axioms of set theory but it's either true or false girdle says it's either true or false it's beyond our mathematical knowledge we can't get to it from our axioms and what girdle says is eventually hopefully we'll have more precise bigger more expansive axioms that will allow us to get to it but maybe not maybe we'll never get to it but it's either true or not there is either as a set between those two sets or there isn't so what whether he thought that the the what followed here our is that the limitations of our mathematics how much mathematics can I know all that he was really proving was mathematics can't we know much more than can be contained in formal systems one more thing that has been on sorry that's such a good question that you asked to go on like this one more thing that has been claimed is that that the fact that our mathematical knowledge is bigger than formal systems that it can never be contained in formal systems shows that our minds are not computers even when we're doing mathematics much less we're writing poetry or something like this but even when we're doing mathematics right the most formal thing that we do our minds cannot be digital digital computers this argument with made when girdle was still alive in 1962 by the English philosopher John Lucas and it has since been made extremely popular and it forms the basis of Roger Penrose is two of Roger Penrose a--'s best-selling works the emperor's new clothes and shadows of the mind in which he argues he's also he's he's a real good alien platon s he completely buys girdles argument that girdles and completeness shows the truth of mathematical reality and he argues that girdles and completeness theorems show something about the mind that the mind is not a digital computer the mind is something if the might week when we're doing mathematics we're doing something beyond a formal system beyond something that can be programmed into a computer so all sorts of very large claims have been urged on the basis of girdles incompleteness theorems they that's why I say they are the most talkative mathematical results in the history of mathematics it's not clear what they're saying they're saying a lot for height they're going beyond mathematics yes first I want to thank you for your time in your research but what I was wondering if I was kind of picturing myself on that street and you have god--all and you have Einstein talking and with this theory as you as I understand it from today and then I think of Einstein and his theory he would my understanding as he was working on kind of like a universal theory something that would explain everything and to me that seems contradictory so I'm wondering what your research might have found about what kind of conversations they might have had in that sense oh yeah everybody yes thank you for that question everybody wonders about this and I interviewed when I was writing the book I interviewed old-timers I got to them just in time they were all to hide white people right after I spoke to them and they and they a one in particular on Mon Burrell a a mathematician told me you know that he would watch them every day were walking back and forth and he said to me I can still hear him they only would talk to each other they didn't want to talk to anybody else it was you know fascinating he said we all wondered what is it that they talked about and I speculate in my book very much what it was that they talked about and here's the the interesting thing about both of them about I mean one thing we know that they spoke about with physics girdle learned a lot of physics from Einstein and for his Einstein's 67th birthday a Festschrift source celebrate celebratory volume was put together of all scholars writing writing on Einsteins work and girdle contributed something which is so fascinating it's it's an interpretation it's a model of relativity theory in which time is cyclical actually time runs in a loop you can go back into your past girdle was fascinated by this idea and so and this is published in in the specialist so we know and and and all the physicists were amazed at how well girdle understood physics where did he learn it he only learned it from the master he learned it from Einstein so one of the things they certainly spoke about there was physics was relativity theory was quantum mechanics you all know Einstein didn't like quantum mechanics girdled it like what mechanics either and both men were very committed realists they were both Einstein in physics girdle in mathematics and in a physics girdle was a super realist they were very committed to the view that their particular fields were descriptive of reality and interestingly enough I mean that's a strange view in mathematics one could say I mean this sort of extra sensible transcendental world although girdle made it hard to brush that under the rug after the incompleteness theorems but in physics you would think well of course everybody believes that physics is a description of reality you know that it somehow does it but in fact in girdle and Einstein's day and even in our own day that's not such a popular view because of paradoxes and quantum mechanics the measurement problem all of these these problems in a in quantum mechanics and both Einstein and girdle interestingly enough felt themselves marginalized in their respective communities which is really fascinating to important thinkers of their day right all that would cause such profound revolutions in their chosen field so that their field had to reconfigure itself around them right around relativity theory and Einstein also had a lot to do with the beginnings of you know his ideas in quantum mechanics as well he certainly but an internal as well cause a tremendous revolution in you know David Hilbert's whole formalism program was ditched after curdle so here are these guys who you would think are this central center players in their field and they felt marginalized why because they're the meta interpretations that they gave their discoveries both of them were committed realists I am committed to physics being descriptive of objective reality girl was committed to mathematics being descriptive of objective reality they were both super realists and in their particular day that was that was not that wasn't fashionable so I think that a lot to do with why they they come to each other their personalities were completely different and that's why also people wondered about it and people I spoke to say you know I mean I'm Stein was very sane none neurotic and sage and kind of philosophical in that sort of popular sense of the word you know and girdle was an erotic you know and very mistrustful of everybody and of common sense and and yet on the mental level there was complete agreement between them and so I think that's what cemented that relationship but but people still wonder and you know of course it would be the conversation I would most want to listen to I should say I did get to meet girdle once he was by the time I was at Princeton he was a recluse and nobody could really get to him but he had a window was sure I shouldn't say that I mean there were people that he that he spoke to that he and he sort of kept up with keeping up with with the feel David there are certain people that I spoke to that he want to know what are the the newest developments in in logic and would keep up with but in general he was inclusive sadly so and and how long he spoke to her quite a lot but but there was this brief period of sociability lasted about three months and I was fortunate enough to be at a party at the Institute for Advanced Study for newcomers and there was Kurt curdle and it was you know it was amazing and he was sort of all of the young magicians were surrounding him and he was very courtly in old-world and gracious but we were all tongue-tied you know and after he left you know he wished us all luck in our future research after he left you know we all belong the fact that we were too you know shy to ask him a question and the question I most wanted to ask him and wish to this day I had asked him although now I think I know how he would answer is what did he think of that John Lucas argument claiming that the implications of his theory of his theorems is that our minds camp Computers did he buy that did he think it had it had such implications but I was too young and too shy to ask sure dr. goldstein is it in my correct in thinking that girdle serums deal with mathematical statements that are self referential I mean it seems like these are you know mathematical statements that are saying something about themselves yes you know you know as you were presenting this I was I was struck by other other systems outside of mathematics that deal with you know self referential statements every ordinary human language you know DNA is a information containing molecule that says something about itself and I think you've just kind of touched on this with a John Lucas argument but could you speak to what kind of implications and applications there might be for girdle's theorems beyond mathematics so yes first of all you know it you're absolutely right that the meta mathematical what this sighs these these these propositions are cooked up so that they're speaking they're speaking both within the system and outside of the system it's it's incredible right and meta that that meta mathematical statement is self referential it's talking about itself it's cooked up in such a way so that what it is actually seen under that interpretation is that it itself is unprovable so it has that self referential aspect to it though it also has a straightforward arithmetic 'el interpretation and as I had mentioned in the answer to another question he said that you know just straightforward mathematical propositions like Goldbach's conjecture could be also of this sort oh and one of the things also there are certain things which are true but unprovable in the system and one of them is this statement of the consistency of the system itself that is also if it's true its unprovable if you can prove the consistency of a system it's inconsistent right that's that's what that means right if you've gone and you've proved this the consistency of a system rich enough to express arithmetic I keep adding that proviso because girdle himself in his PhD dissertation had proved the consistency and the completeness of what we call predicate logic which is not a rich enough system to express arithmetic so there are systems in logic which are consistent in which are complete and ironically enough it was kurt gödel who proved that right so so that's kind of interesting but anyway so consistency itself is something that you can't prove within the system what the implications are beyond mathematics is first of all it seems to sell something about mathematics itself which is something you know it's about mathematics it's a metamer it's in philosophy of mathematics it seems to tell us something about our knowledge of mathematics that we have knowledge of mathematics that can't be formalized it seems to tell us that we have to rely on intuitions that intuitions can't be eliminated from mathematics this is all very interesting about mathematics it might also tell us something you know about the human mind what the capacities of the human mind is maybe I'm much less comfortable with the the Penrose argument the Lukas argument that from girdle's implicate theorems we can actually prove that the mind is not a digital computer girdled himself shied away from this group there is one place in which how long who was a logician who got three books out of his conversations with kurt gödel he persisted he tracked him down and he spoke to him and he interviewed him a great deal and there are three books in which he talks about girdles ideas and what girdle says about this or this argument does it do his theorems show that we are not computers our minds what we do in mathematics cannot be duplicated by a computer answered very very interestingly he answered with a disjunction he said either it shows that or we don't have the mathematical knowledge that we think we have we may not we may not have this mathematical knowledge that escapes formal formalization and that we can't actually prove that we do have that knowledge that escapes formal proof because it escapes formal proof so that we so he answered with this kind of disjunction what he sort of was saying was either we're not computers or were computers with delusions of mathematical grandeur and that so it doesn't quite follow and I'm happy with that answer that was good enough for a girdle that's good enough for me that that that disjunction oh okay I'm sorry I know less than the normal 25 year old because I had been trained to dismiss all such questions as so much froth so much fluff so much meaninglessness and so it put me into the state of thinking what is it a person to whom these ideas Platonism girdle quantum interpretations of quantum mechanics on it it's the it's the the center of their life what's that do to their life to the other things the other concerns and and that's when I and I couldn't write about that as a philosopher because my first class training kicked right in and the methodology itself didn't allow me to ask it in sort of a free way that I wanted to ask it and I could only do it in a novel and and and and so that's how I I wrote that that that first novel and that's always been an interest with me this question okay here it is I I do believe that thinking is a passion you know it is for me it is the people that I'm interested it's not that there's passion on one side is thinking on the other thinking is a passion as an intoxicating passion but what is it so I'm interested in you know thinking as it's done by people and what does it do to their life and what is it to be the kind of person for whom thinking is a passion and I'm interested in looking at that and the novels and I'm interested in you know these last two non-fiction books that I've done one Spinoza the most abstract thinkers but what it wasn't like to be them you know what is it like to be good Oh what was it like to be Spinoza so ideas embedded in lies is for me very interesting you know in that regard you know I think that I've become less of a serious philosopher by doing I've certainly though sir specifically the profession considers me to be less of a serious philosopher for for doing this for looking at this but it is it is of interest to better maybe maybe a better philosopher but but that's what interests me you know I just that that is what interests me I mean it is after all people who are doing this thinking and should we care so much about these things you know or not you know and what it's what does it do to a life to have this as as one central concern is is to me my central concern as you were talking early on and everything isn't right you know and and in some sense the that's the human predicament we are the creatures who realize that we are incomplete that everything is incomplete right you know that that's what we are right and we're constantly trying to solve it to be or not right yeah we are constantly trying to fix it we're constantly coming up with systems that are going is in some sense it's the beginning of philosophy right to try you know to recognize we are leaking into the univer incomplete right and and to fix it all right and it all and we always would come up with the system so we always try to fix the pendant we always bring awake and that is and you know there's a certain point in thing and that bad I guess is to get back to that answer I mean I find that's poignant I you know there's there's something poignant in our passion for completeness and our constant discovery that we are incomplete yeah true true but the ultimate incompleteness is that we're going to die I mean that is the other the other maybe you ever read this book there was this book in the 60s that we will all read leave the denial of death honks Becker and you know he you know was you know very pivotal book right and so you need to find you know that he just talks there that we are the animals who realize that were animals you know the way I like to think about it no we are the incomplete creatures who realize that we're incomplete but it's the same sort of thing but there's something you're something non tragic it is it is it is it is it is the source of all beauty liberating source of art but even the attempt to overcome it structure is of course of course yeah structure is always be structures at the heart of this beauty and we certainly had a lot to do with the beauty right the formal name the structural yeah was what the mental illness necessary for the incompleteness theorems or was it a result so that's everybody I read about it but you know this was an amusing common but they never seem to me amusing it seems I'm very well they're very real and very very tragic and very real but when you do read this proof it's such an extraordinary proof yeah we're so you know it's so because it combines one of the interesting things about good old is that part of it he was so cautious and careful there's something like a bookkeeper there you know keeping track of everything in the girdle numbering is like such oh it's a bookkeeper type of thing you know matching that matching that and so much oh so much for trivial little detail that you have to keep track of that he had to keep track of then there's this sort of Alice in Wonderland mind of you know you're in another world right just a playfulness but Allison wonder I was organizing a that Alice in Wonderland you know that there's something that you've got these propositions speaking about themselves and you know stalking orekhova take the time at something else then the two things brought together are such a strange thing you know I'm maybe you didn't have to have a something you know I mean not in it not everybody could yeah it's not just the brilliant it's there's something all these that gets me when I read through that proof that's unsettling actually there's something that's settling it's both beautiful moving and I'm settling and but anyway yeah I you know I so his mental illness fascinates me the other aspect that fascinates me about him is that he had Mike Spinoza mean it's interesting that plane Gretl who were such legendary friends were both when asked to describe that for all soften outlook went back to the 17th century rationalist you know was mine obsessed trades so Spinoza Einstein described himself as a Spinoza and girl described himself as a like knits yet so these were these two seventeenth century yeah so it's fascinating actually that they both their closest allies besides each other were in the 17th century of the rational and girdle I found in the non floss you know this incredible resource at Princeton because he saved everything so there was this little card a little index card which he wrote in German his 14 principles the things that he most believed in and the first one was developed as phonetic the world is intelligible this notion that there's all you know always an explanation on what life that's called the principle of sufficient reason and that's very interesting also because it also shaded into his particular form that is mental illness took which was paranoia and he always people I spoke to and also from people who wrote about him he always you always had an explanation for everything everything had an explanation but sometimes this took very paranoid form you know so that and that psychiatrists doctors said that you know in some sense paranoia is rationality run amok you know that you're just there's always an explanation there's always an oscar mormons during a great economist and game theorist who was was also a confidant of of jeridolp they are at the Institute wrote in his journal you know that I you know I speak to ticker Idol he seats plots everywhere there are always plus there's always always an explanation so that also just sort of interests may that in these these various aspects in which the genius is the pulse of shades off into the mental illness I'm not an advocate of the view that all geniuses or mad or that love must be mad to Beijing yes Einstein certainly was not you know what many or not and the interesting thing also with with Google getting back to Carol is that um he was right everybody did was interpret but almost everybody did misinterpret his his incompleteness there he passionate plainest that he is right he wants to audaciously discover mathematical proof that will you know have this implication and then everybody interprets it as oh there's also uncertainty and there's relativity and there's a lightness and everything's objective I mean subjective there's no objective truth and it's all one big postmodern free-for-all right exactly Ortiz or he's interpreted as a you know still people will say that this was the greatest outcome of the VNO positivist circle a logical problem quite quite quite the opposite again he hated positivism right positives believe that all mathematics is syntactical right that this was this was he hated this he wanted to in fact prove the exact opposite so you know and just as we would say in the sixties just because you're paranoid doesn't which is very funny because the book is being translated right now into Chinese and I gave a funniest funniest the the two translations of my kernel book that have given me the most trouble was a German when the German was because I translated all this German stuff into English and they want him they don't want to trust like my translation back into German so they wanted the original but I was no longer living in Princeton so it was very very hard for me to get the original book so they gave me a lot of trouble on the Chinese is so interesting because I uh so for example this last part when I think just because yes we used to say in the sixties right so he raced pani says I don't understand are you saying that because girdle is an old man in his sixties that he was [Laughter] personally I can't read Chinese and see how this book is gonna turn out so man is no trouble Oh fascinating question fascinating question he so Vic in Stein was the reigning god of the logical positivist I mean it's really very very interesting how they worship and they really they know Schlick and they worship this man in the sky nice men it was just this guy Wiseman would begin every he was writing above on the truck tata soar and no I think I've been confused on mathematics and he would report at every meeting of the vienna circle like circle he would begin with reporting the latest the latest changes in thinking Stein's ideas are this although the Kingston has renounced responsibility for my transmission of them and you know so uh and then they took them so seriously that some philosophers in Vienna theorized that there really was no dr. Vik and Stein let this one thing they took their imagination to give them some gravitas theory but anyway so but they really and then they spent two years of reading the truck Tatars which which argues that all mathematics is in tactical and they actually they read it the way people always think not nine of the fourteen of them work we're Jewish of the theological positive this and well of course thing we're not believing I mean all these shortly meaning must write biological standards what was it it's almost the ways we're reminding me the way they studied and read and discussed the trough taught us the way Jews read the Torah portion every week create a new reading to go through in the whole year and you managed to finish it and you know really reminded me of the way I know reading parsha achieve Ori reading the Torah portion of the week and so so then he shines a real God there and my fear and and very few of them were allowed to meet with him slip was allowed Wiseman was allowed to and then karna asked too many questions and he was banished and then Michael liked Cornett too much so that they could shine got angry at Bible anyone left he's a real prima donna and um I think I think that girdle plaintiffs among the positivists that he was right was pissed as all hell at the worship of of the consign and and you know and I and I think he never met bekenstein but he has various again down there in the knock Lots he didn't send them off angry letters about thickest I and so thicken Stein in foundations of mathematics talks a lot about girdle and says and says the girl couldn't possibly prove what he calls it logical magic tricks and he really kind of denigrates them and you know picking Stein I mean and girdle reacts to this and sins of very angry Larry sends a very angry letter to carl menger's the matter father and and so yeah so I think that in fact actually I say in the book that each of them were thorns deep in each other's metamathematics neither neither of them could accept the others so I think that the annoying plays with an important role but it but this is their evidence and you know actually both of those last non-fiction books I was asked to write they were both appeared in series one the girl book appeared in the Norton series on great scientific discoveries and they simply came to me and they said who do you want to write about thing both of these series wanted people who knew how to write I don't tell us story this is the new thing with abstract ideas you know to put it into a narrative frame and so both of it that's what both of these series we're looking for and so it was very easy when Norton's asked me to write and that's a great scientific discovery that I wanted to do ger it all and I've been obsessed with girl since and and didn't ask him the question I wanna ask him and one of the reasons I called it the trains Spinoza was that I do look at photos in a way that I never looked at in before and that goes against my philosophical training certainly goes against Spinoza's philosophy I look at him is coming out of that Jewish history that particular community which was a extraordinary community and I actually see the interesting thing that happened was that I saw his philosophy at the top it's been on this since I was 26 years old I think I understand that inside out I actually saw the system changing as I looked at it from that historical perspective looked at it from the viewpoint of Jewish history and saw it as very much a kind of rethinking of what personal identity is because of his community's obsessions with identity and Jewish identity and what's essential to be in a person whether his Jewishness was essential he argues it isn't to Cathy but it gives it so the whole slant changed but it can't be very interested in this whole science religion controversy that was so intense in the 17th century was pre enlightenment but I really do believe that Spinoza had a lot to do with creating the Enlightenment that he really this thinking his way out of Jewish history of trial and ultimately trying to solve the problem that he felt very keenly of Jewish suffering this historic suffering that he himself witnessed that these were these were refugees from the worst Jewish calamity of that tongue of the Portuguese Spanish Inquisition that that he thought his way into this you know universalism the secularism and he was bashed you know once he was excommunicated by the Jews he was attacked by all of Europe of Christian Europe and but just this war between science and religion and faith and reason and look how they were slugging it out in the 17th century and then I take my head out of the 17th century and I look at what's coming on right now in the world and in America and the separation of church and state and it's like oh I thought we solved these problems Spinoza showed us away and and there was and and that seemed to me fascinating you know that somehow these issues are Center there's there once again so I'm ready to novel about that that's a very long

Channel: Linus Pauling Memorial Lecture Series

Views: 12,649

Rating: 4.8192091 out of 5

Keywords: Gödel, Incompleteness, paradox, mathematics, classical logic, positivism to postmodernism, Gödel's theorem

Id: QjFyKUDmUhU

Length: 118min 32sec (7112 seconds)

Published: Sat Sep 29 2018