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[Music] you I'm super super excited about this topic this is one of my favorite topics to think about and to talk about which is the topic of strange loops and paradoxes because this gets to the very essence of what reality is and I love to think about the paradoxes that result when the mind is trying to think about reality I've actually been wanting to shoot this episode for over three years now about girdles and completeness theorem but it's difficult to organize all this material takes a lot of work to organize it in a way that's logical and simple that can be understood by ordinary people in a non-technical way and so that's what you're gonna get here today I will explain girl's incompleteness theorem at a high level without getting you bogged down into the technicalities and the math and then the logic of it because it's a pretty technical proof it's dozens of pages long it's very complicated and there's no need to go into that because what we care about here is not the mathematics of the proof which is sound and if you want you can go watch videos specifically about the mathematics of the proof or you can go actually read the proof itself there are many versions of it what we really care about though is we care about the metaphysical and epistemic implications of the proof and a lot of people who are rational and scientifically minded they don't really comprehend the ramifications and in fact many logicians today and mathematicians today and scientists today who are working on the cutting edge you know areas of science and quantum mechanics and cosmology and all this sort of stuff they still haven't really grasped the consequences the philosophical consequences of girdles and completeness theorem because what it's easy to do is easy to say just oh well this is just mathematics this is just logic there are no consequences which applied to broader domains there are no philosophical consequences but actually there are and so that's what we're gonna focus on here today I really want to give a call out to Douglas Hofstadter who is influenced by thinking about this a lot and I already called him out in reality too strange loop by which I mean I talked about his books I talked about the importance of his work and his concept of the strange loop so I'll be referring to that concept here quite heavily and I'm not gonna go into really explaining that concept much because I already have an episode called realities a strange loop which explains that in great detail so go check that out you can consider this as sort of a continuation of that episode that's like a part two we're gonna go into a lot more detail that I couldn't go into there and specifically I want to talk about Douglas Hofstadter's a brilliant book called girdle Ashur and BA or GE B for sure this is a book that's really popular among nerds it's a very technical book but it's also written in a sort of poetic manner which is quite fun and interesting it's it's very thick it's over 700 pages long it'll take you a month to go through it it contains logical proofs and all this sort of heavy stuff but it also talks about Zen it talks about a sure the the painter and the the the draw the drawings and sketches that he did and the etchings that he did talks about BA and it draws all these interesting interconnections between what is the mind what is logic what are these strange loops how does it all interconnect together so definitely go read that book if you're really interested in what I'm talking about here here I'm just gonna give you sort of an overview of the issues that he taught that he touches upon we're going to talk about the limitations of rationality and logic we're going to talk about the profound interconnections between logic and non duality and we're gonna get a better understanding of what paradox is and why paradox exists and why paradox is necessary so let me just refresh your memory about what a strange loop is a strange loop is this odd sort of hierarchy that goes around in a circle such that when you move up the hierarchy eventually you end up precisely where you began and so the prototypical example of a strange loop would be the Penrose triangle which interconnects to itself in the sort of impossible seeming way and it might seem like such a thing can only happen as an illusion inside of a drawing or a video game or something but actually what you'll see here which I hope to convince you of is that these strange loops actually are very fundamental to the very structure of reality itself everything right here that you're seeing this is a strange loop happening to you now you might wonder of course Leo why are you qualified to talk about this topic are you a logician are you a mathematician where's your PhD are you an academic are you a professor and if not then what gives you the right to think that you can understand the the philosophical consequences of of these proofs well here's the thing it's very interesting that people who are skeptics academics and scientists and rationalists that they they love to criticise religious people when religious people make appeals to Authority so if you ask a religious person for example well why do you believe in your religion and in God and in all this other sort of sort of stuff the person will say well it's because my Bible says so and the Bible is the Word of God and you will say well how do you know it's the Word of God well because the Bible says it's the Word of God or because my priest or my religion tells me and my priest you know he's high up he's the Pope up you know in the hierarchy and so he's the authority figure he knows better than I do and in fact maybe he has a connection to God so he can interpret the Bible for me and and speak the truth to me and that's how I know that it's true that's why I believe it now of course the rationalist will quite rightly deride the theist in this case for making this appeal to authority because logically we know that there's no connection between Authority and between truth that these two things are completely independent just because someone has Authority and credentials doesn't mean that they really understand what they're talking about but what's really ironic is that the rationalists and the scientists and the atheists they make this exact same mistake when it comes to their own positions about science and about mathematics and logic so see here you're asking for my qualifications in my PhD and all this in order to listen to what I have to say but that's the exact same mistake because what you're assuming there is that if you had someone up here speaking to you who produced this video for example or this lecture who had a PhD from MIT or from Caltech that because of that you would assume that he understands what reality is and because of that you would assume that he understands all the philosophical metaphysical epistemic implications of girdles and completeness theorem and that's where you would be wrong because there you are also making appeal to Authority just because someone has a PhD just because someone want a Nobel Prize or is a good scientist or a good mathematician does not mean that they really understand the philosophical implications of a thing you see because actually to succeed within academia that is that's a social hierarchy that you're climbing just like a church and scientifically minded people don't appreciate the significance of this that you are climbing this hierarchy and to climb this hierarchy has very little to do with truth to climb the hierarchy that's how well you play the game that's also how good of a technical scientist you are how good do you do lab work how good do you write your research papers so if you're good at doing that you'll climb the hierarchy and you'll get to the top and you'll have authority you'll have a PhD maybe multiple PhDs you'll have all sorts of accreditation zand and prizes and medals and and awards and your work will get published and you will be this big-name celebrity scientist maybe but does that really mean that you truly understand the consequences of your work because see there's a big difference between just doing technical work and having some sort of narrow technical scientific result that's that's fine that's good but then do you understand all the ramifications of that for life at large and for reality as a whole that's a totally another matter and actually what you'll find is that the people who are really good technicians a scientist that actually they're not that good at understanding the full ramifications of their work because it's precisely by being technicians and being hyper specialized and hyper focused that they narrow their focus that they get so good at that but then they lose sight of the bigger picture you see so what I'm asking you here is not to believe me but to just think through these ideas for yourself that's where the truth comes from it doesn't come from an authority figure it comes from you sitting down and thinking through this stuff for yourself and it doesn't come from believing me and in fact I don't want you to believe me and if you're skeptical that's good being skeptical is great I'm the biggest skeptic there ever is and of course what I do is I apply to a skepticism first and foremost to my own theories and to myself and that's something you don't really see in the videos that's something that happens deep inside of me that's something you should also be doing deep inside yourself thinking through all this for yourself and then you see does it really make sense and if you follow that sort of methodology then what she'll discover is that a lot of people whom you might think otherwise are these authority figures that actually they're full of and they don't really understand the bigger picture so that's the qualifications situation now in a sense this episode is going to mirror the episode about quantum mechanics that I did in that episode I talked about how there's this sort of myth of science that goes around and all in our culture we're how people in mainstream culture think of science and what science is and how science works is a very simplified idealized version of how science really works and that in fact signs much more messy more complicated than people think well likewise with logic and reason there is the actual doing of logic and reason the hard-core proofs the stuff that you can study in graduate school which is very very different than what most people in mainstream culture including many atheists and scientific minded people think logic and reason are because in our mainstream culture we have this myth of logic and reason as being the pinnacle of discovering truth it's it's the one true way that you know that something is true is because it's logical and it's reasonable and that logic and reason are somehow the antithesis of faith and belief and then if you if you have faith or belief in something that means it's false whereas if you have a logical proof and you have reasons and you're rational that means that you're on the side of truth and so what I've discovered is I've discovered that rationalist materialists atheist skeptics scientists doctors and professors they fall into this particular trap which is the dogmas of Reason it turns out that rationality itself comes with dogmas just like religion and I've identified half a dozen or so of these dogmas these I call the core dogmas of rationality number 1 the first Dogma is that reason is sufficient to understand the world that is something that rationalist just believe and take on faith they don't prove to themselves that the whole world is subject to their reason and that reason will be sufficient to understand the whole world that's not proved in any empirical way that's just assumed that's why it's a Dogma number two it's assumed that contradiction or paradox means that you've made an error again that's also not something that's actually proven empirically that's something that's just assumed implicitly and never really questioned which is why the dogma number three it's assumed that the laws of nature are reasonable again you didn't go out and actually investigate every single law of nature empirically and then concluded yes they're all reasonable that would actually be a scientific method instead what you did is you just assume that well all the laws of nature must be reasonable you just assume that without actually testing your hypothesis so that's a Dogma because you know you might go out there you might encounter a law which turns out to be unreasonable what do you do then are you going to admit that all the laws of nature aren't reasonable at that point or are you gonna stick to your guns and keep insisting that no everything is reasonable see these things can't be taken for granted you have to actually test all of your dogmas all of your assumptions which even rational people forget to do a fourth Dogma is that logical proof is the highest standard it's just sort of implicitly assumed by people that if you really want a lot you know an airtight argument or you want the truth then there's going to be a logical proof which proves it to you and if you have this logical proof then anything else is below that and that's the enter that we should aspire to and that if you can't meet that standard then you really don't have anything solid you're just speculating you're just doing philosophy or it's just some religious mumbo-jumbo or it's just some new-age wishful thinking if you can't logically prove it number five is that logic ality is equal to deep understanding of a topic dogma number six is that rationality is self-consistent which of course it actually turns out not to be which is where girls incompleteness theorem comes in and we'll get into all those dirty details in a few minutes dogma number seven if a thing is irrational then it must be false dogma number eight is that science and math is rational whereas religion is irrational and therefore what is true and what is false this is just assumed it's never tested it's never put to the test no rational person actually goes and bothers to study and seriously practice every single religion in the world to test whether they're actually true no one does that you see it's very interesting on this works you prejudge the situation you say oh well they can't possibly be true because they sound too crazy it's all stupid nonsense can't possibly be true but that's just assumed you have to go test your assumptions that's the essence of science Dogma number nine is that metaphysics is speculative nonsense and a corollary that document number 10 is that science and math doesn't need metaphysics you can just do science and math metaphysics free because metaphysics takes us into the field of philosophy and you know philosophy we know how philosophy works you can just sit around on your armchair and do philosophy all day long but how do you really know if philosophy is true and we know that people throughout history have had all sorts of crazy philosophies but none of that stuff really affects science and math math and science are hard-nosed activities whereas metaphysics and philosophy well that's that's very speculative and it's psychological and it's prone to errors and you know you can't really make a logical proof for some metaphysical idea whereas in math and science you can you can prove things definitively so that also turns out to actually be a dogma because actually you cannot do science or math without metaphysics and you can discover that if you just question your assumptions the problem with all these assumptions is that they're made unconsciously you make them but you don't know that you've made them and it actually takes a lot of work and thinking and contemplating and experiencing the world from different perspectives to start to see and suss out your assumptions and of course if you're part of a university or academic setting if you're like doing a graduate program or even your undergrad whatever you know your professors they don't haul out these assumptions because these assumptions constitute the paradigm or the culture that you're working in this is all just part of the package when you go and you sign up for university they don't question this stuff even though you would think that as a university you know what should be the highest ideal and aspiration of a university it should be open-minded free thought and the questioning of everything well it turns out that actually there are specific things which the university system doesn't question which is precisely those assumptions on which the University that was founded and why doesn't it question these things well of course precisely because if it did it would undermine itself it would expose its own internal contradiction and of course we wouldn't want that would we so we're gonna dig into this in a lot of detail here but let's start with some history to really lay the the the historical backdrop for girdle's incredible discoveries because it's hard to really appreciate that unless you put it into the historical context so let's rewind about a hundred years ago into the era where logical positivism was the the fad and the rage in all of Western intellectual tradition logical positivism is this sort of movement that happened within philosophy within science and within mathematics which was an attempt to get rid of metaphysics and philosophy from science and for mathematics so there were key leading figures in this movement like Carnap slick Neurath vic and stein russell whitehead and David Hilbert and many many others scientists and philosophers and and academics signed on board to this logical positivism and their idea was that we need to make science more hard-nosed make it really objective make it completely indisputable and if we do that then we're gonna complete the entire enterprise of science and of mathematics and of logic we're gonna formalize everything we're gonna eliminate every single shred of subjectivity intuition and psychology and philosophy from science and for math that was their great idea that's what they tried to do so these guys were serious scientists and philosophers and some of them were serious logicians and mathematicians and they worked really hard to to succeed in this logical positivist agenda so their aim was basically to boil all of truth down to expressions of language what they believed was that all genuine knowledge if we ask ourselves what is knowledge all genuine knowledge is really just about taking nature and expressing it in a single common language to all the sciences so that science basically boils down to statements of fact about the world and that's how most people think about science if you take a a guy off the street and ask him what's science he'll say well it's just just a collection of true statements about the world you can ask for example is snow white and the answer is either yes or no snow is white or no snow is not white is snow black you would say false and so that would be a scientific statement and so what the logical positivist wanted to do is they just want it to boil science down to this kind of very kind of like simple cut-and-dry atomic process or you can just boil and reduce everything down all the complexity down to these little atomic parts and for them the idea was that all of science basically it's done with language so everything can be broken down into little propositions within language and language itself can be sort of simplified and boiled down from complicated language and in fact you could take all of natural languages English and all the statements that we make within English or any natural language and you can boil that down you can formalize it down into logical statements so logic is a sort of language and so you can you can take some wishy-washy sentence in English and you can boil it down to a logical statement and so in theory should be possible to do that with every piece of knowledge we have until we know exactly what language is saying and so what they thought is they thought that a lot of the confusion that happened in philosophy for the last 2,000 years was was people abusing language misusing language talking about terms which they didn't clearly define they didn't really understand what they were talking about and so people and different philosophers and theorists when they talked about an atom or or a object or God or they talked about the self where they talked about mind or body you know all these concepts they didn't really know what they were talking about because these are very kind of fuzzy abstract terms what we got to do is we got to define our terms very carefully the way that a logician would and then we can we can make a real simple sense out of all of reality and they believed that this was possible to do with mathematics as well they wanted to boil mathematics down to logic this particular Enterprise was called logis ism it was the idea that logic it's that that mathematics itself wise mathematics true for example how do you know that one plus one equals two what is the truth of that relying upon how do you justify mathematics what they thought is that you can boil it down to logic mathematical truths are really just logical truths and so they they went on this on this enterprise to try to boil mathematical truths down to logic and they basically thought that all of this can be formalized into a system of equations and if they succeeded in doing that then their idea was that well we don't need metaphysics anymore we don't need philosophy you don't need speculation all of that is just meaningless fuzzy talk when we've got our equations our equations explain everything and then math and science really become the only valid kind of knowledge and everything else is just fuzzy delusional or some sort of wishful thinking New Age type of stuff and so the logical positivists were really adamant about verifiability which means that if you're gonna make a claim about the world and you think that it's true that means your claim must be tested and it must be testable so if you make a claim about the world which cannot be tested under any circumstances that means that really you're not making any legitimate claim about the world at all what you're doing is you're just engaging in metaphysics and just speculation and there was a lot of that that happened in philosophy for for hundreds of years so in a sense you can you can understand where they were coming from they were tired they were sick and tired of people making these sort of arguments like you could say well what makes the moon orbit around the the earth and people say well it might be gravity and then some religious person might say well it's not gravity it's angels there are invisible angels which help keep the moon in place but then a scientific person would say well but angels what is this concept of an angel holding the moon in place how do we make sense of this can we test this and of course you try to test it and you realize well you can't test it and you can ask the the religious person well can we see these angels and he'll say no angels are invisible and then he'll say well can we feel them they do they register on our instruments and the religious person will keep saying no no no you can't you can't measure them at all so to the logical positivists this started to smell like because now they want to say well if you can't actually verify a statement about the world that means you're just speculating you see you're just playing mind games so they were big on verifiability they were very big on pragmatism they don't want just speculative theories and ideas what they want is they want a pragmatic statement about the world which can be tested and you can do something with it and all the other speculation the theory that's not important that doesn't belong within the domain of science and that really what science is is science is not metaphysics science doesn't really tell you what the world is science simply tells you the equations governing nature and those equations are all that there really is that's sort of what pragmatism means it's all about how do we manipulate nature and beyond that there's nothing more to say they were also into reductionism they believed that all things can be reduced down to their atomic fragments so you might ask them what is color what is consciousness what is love what are emotions what is an animal and they will say well there are no such things all of it just boils down to atoms there's no such thing as a dog it's just atoms there's no such thing as love and emotions and conscious they're just atoms and of course logical positivism worked really well with atheism and materialism so materialism atheism and and this logical positive all sort of came together and this sort of created a sort of paradigm in which science was done back in the early 1900s this really started to flower in the early 1900s and lasted for about 20 or 30 years in the 1930s or so until ultimately logical positivism got debunked by by girdle and and a few other theorists so there was a particular man who started off this this this enterprise of logis ism called Gottlob Frege ii he was a brilliant logician and he devoted his entire life to this enterprise of logis ism and he worked with set theory and he was trying to use set theory to basically create a foundation a logical foundation for all of mathematics he wanted to find a small handful of axioms which he could use to create the foundation for all arithmetic so for free gay he was asking a very basic sort of philosophical question about math and specifically about arithmetic which is like the most basic form of math the stuff they teach you in in first grade he wanted to know and he wanted to to completely rigorously prove that all arithmetic 'el truths are true and that this should be proven and not just intuited but rigorously proven and his whole agenda was to get rid of intuition from within mathematics he thought that intuition was like a fudge factor within mathematics which introduced the sort of fuzzy human element into what should otherwise be a very hard-nosed rigid system it should be cold and heartless not intuitive so he went about dedicating his whole life to this process he came up with all his axioms he did all these proofs and finally he figured it out how to do this and he wrote his book and he's just about to publish the book the book is actually in the printing presses it's being printed and he thinks he's found the logical foundation for all arithmetic until his hopes are shattered by a discovery from a fellow logician Bertrand Russell famous mathematician logician who discovered and showed to frag a that there's a paradox at the very heart of his entire scheme and this was called Russell's paradox and so freddie was working with set theory and I'm not going to get into all the weeds there that we can get into about set theory but set theory is just in a very simplistic explanation all it is is just you're you're grouping objects into sets so if you have one object that's one set you can have two objects that's the second set and you just have sets and so basically mathematics boils down to to sort of manipulating these sets and seeing the interrelationships between all these different sets but it's not quite so simple because there's something deeply paradoxical at the very heart of this entire enterprise of trying to boil down reality to simple equations and that is Russell's paradox Russell's paradox is stated as such Russell basically said what if we find a set within set theory that is the set of all possible sets not containing itself and what this leads to is this leads to a sort of strange loop or sort of contradiction where set theory contradicts itself because if you find the set of all possible sets that doesn't contain itself on the one hand it would it would seem like you could find such a set but if you did then that set what negate itself and if you couldn't find a set then that set must exist so it's a sort of odd paradox that happens where at the same time it must exist but it can't exist and if it can't exist then it must exist and this brought frege's entire program to a halt and he got very depressed about this for a gated but to his credit he was honest about he admitted that yes it actually is not possible to boil down arithmetic to logic and to these simple set of axioms because of these inherent contradictions which exist in addition to fray GAE there was a man by name of David Hilbert who was also a logician and mathematician who embarked on what's called Hilbert's program and Hilbert's program was very similar to frege's he was trying to ground all existing theories into a finite complete set of axioms and so he was working on this problem at the same time and Hilbert's program was derailed by Georg Cantor and Garret Cantor is the guy who made some groundbreaking discoveries about transfinite numbers he discovered that there are different orders of infinity within logic and within mathematics and in fact that there are an infinite number of sizes of infinity and he also discovered the absolute infinite set which is sort of like the set of all possible infinite degrees of infinity that's the absolute and he called that God he was a religious man I talked about Georg Cantor in more detail in my episode absolute infinity part to understanding absolute infinity part two so I talked about his Baxter it's very fascinating so if you want to learn more about him go check that out I won't be covering it any more depth here now I want to talk now about a Kurt durdle because Kurt girdle and also a guy by the name of alfred tarski both of them logicians they made a couple of critical contributions which completely derailed all of these efforts at logis ISM and ultimately at logical positivism and so this is what we are what what is the heart of this conversation that we're having here so what is it that girdle and alfred tarski proved they came up with a couple of theories girdle came up with two incompleteness theories tarski came up with one undecidability theorem they're quite technical they're dozens of pages long I'm not gonna actually go into the full details of that here but let me give you a high-level explanation of of girls and completeness theorem so basically what girdle was able to accomplish is he was able to take a logical system and he was able to say okay so here are our axioms in this system in theory we should be able to take these axioms and we should be able to to use them to make any kind of proof of anything that's true within the system of for example arithmetic so if we try to do that there's a special case scenario like what Bertrand Russell discovered is that we can actually use these axioms in a sort of tricky manner to actually create a statement within the entire logical scheme which will contradict itself due to self reference and so what girdle was able to prove is that if a system is sufficiently complex enough that it's able to reflect on itself or otherwise speak about itself for example in the English language the English language is rich enough of a language that I can speak about myself using the English language so in a sense what I'm doing is I'm going meta I'm speaking about the fact that I'm speaking you see sort of self-consciousness that's happening here this is otherwise known as self reflection or as self reference I can speak and reference myself speaking well if a language like logic or English or some other system is able to do that speak about itself what it can do is it can create a paradox or a contradiction because what I can say is I can say the following thing and this is ultimately what girdles and complete center boils down to let's make an analogy between logic and English so you can do the same thing in logic that you can do in English in English what I can do is I can say the following statement everything I say is a lie now notice what that means that's a paradox because if everything I'm saying is a lie that means what I'm saying that statement everything I say is a lie it's a lie so if it's a lie that means that statement is true but everything I say is a lie that means how can it be true that's a contradiction on the other hand if I am speaking the truth all the time everything I say is true then how can I say the statement everything I say is a lie that is also a contradiction so it's a contradiction from both sides it's a contradiction no matter which way you slice it so what girdle was able to show is that you can do that within logic itself as well not just English at least if your logic is complicated enough that it admits of self reference so what is self reference mean in logic it doesn't mean that logic is referencing the self as an ego or as an eye or as a consciousness no it just means that logic is able to reference itself so that means your logical scheme or system has to be able to speak about its own sentences if the logic can reference itself as a logic then what girdle approved is that you can create statements within that logic which are true but which cannot be proved using that logic and so in essence what is proved is that truth is larger than proved ability and this is always true this is a pretty remarkable result and girdle had to go through some ingenious mental maneuvers to figure out how to actually do this with arithmetic so what he was able to do is he was able to come up with this idea called girdle numbering where he was able to to encode self reference with the in arithmetic such that arithmetic could reference itself and start to speak about itself and therefore he could show that because arithmetic can speak about itself that it must ultimately contain more truth in it then can be boiled down to simple logical axioms and that was a pretty amazing result which means that you cannot boil down arithmetic to a simple algorithm that is something that Frey Gaye and David Hilbert were trying to do their idea was that arithmetic and all mathematical truths it should be possible to come up with a certain algorithm that can just run through every possible possibility and come up with all the true statements of arithmetic and if we program a computer for example with this algorithm the computer can go through and can calculate every single possible mathematical truth that there exists and therefore we have solved the problem of mathematics forever and what girdle's proved is that actually you can't do that mathematics is infinite beyond the possibility of any algorithm to to figure it out and that therefore intuition is a necessary component of all mathematical discovery and in fact what girdle proved is that mathematics is technically it's called uncomputable which means that if you had a computer that had infinite resources infinite processing power and could run for an infinite number of time for trillions of years it would still never be able to discover all the truths that are contained within mathematics think about that that's deep that's deep I'm not just saying that your ordinary home computer couldn't do this or some supercomputer I'm saying imagine a computer with infinite resources and it still could not figure out mathematics that's how big mathematical truth is you can't capture it with language you can't encapsulate it in any kind of formal system with simple axioms and rules so another way to think about what gertle discovered or proved is that if the entire domain of truth is a circle like this then what you can prove is a smaller subset of that circle and they are never the same you can never prove all that there is that's true the truth always exceed exceeds what's provable if self-reference is possible so another way to say it is that he proved that if a system is complete it cannot be consistent and he also proved that the consistency of axioms cannot be proved inside of the system you always need to have a meta language to go outside to be able to reflect upon the system that you're using if you want to talk about truth and this goes to the very core of the dogma that many rationalist have which is they assume that if something is true it must be provable but that's not the case and in fact what girdle was able to demonstrate and to prove was that in any logical system which is capable of self reference you can create a statement as follows this statement is unprovable and that will break the entire scheme this is called the girl statement this is sort of the the wrench in the works this is also Russell's paradox is a journal statement so that's sort of like the strict explanation of what the proof is then there are the philosophical and a histological and metaphysical consequences of this which we will talk about but what I want to do here is I want to read to you a little bit of from Wikipedia about both the discoveries of girdle and about David not David Hilbert but um alfred tarski that really precisely explains what they proved because it can be a little bit difficult to sort of explain this in words it's good to have a nice rigorous explanation of it so we're gonna go to Wikipedia here so here's what it says girdles and completeness theorems published in 1931 showed that Hilbert's program was unattainable four key areas of mathematics in his first theorem girdle showed that any consistent system with a computable set of axioms is capable of expressing it that is capable of expressing arithmetic can never be complete it is possible to construct a statement that can be shown to be true but that cannot be derived from the formal rules of the system so what this means is that you cannot have a complete formalization of of mathematics and really you cannot have a complete formalization of anything because if we're talking out the truth the truth is always going to be bigger than the system of formal rules that you use to to talk about the truth and then it goes on to say quote in his second theorem he showed that such a system could not prove its own consistency so it certainly cannot be used to prove the consistency of anything stronger with certainty this refuted Hilbert's assumption that a finite istic system can be used to prove the consistency of itself and therefore anything else end quote so it was assumed that logic can sort of prove itself but then girdle was actually able to prove using logic that logic cannot prove itself which if you think about is very interesting so what girdle was able to do is he was able to sabotage logic from the inside out and what's significant about this result to me and I think to our work here is that it's one thing for example to criticize logic from the outside from outside the system for example you might have a very logical person and then you might have a religious person the religious person might criticize the logical person oh well you're too logical you're too stuck in your head why are you so logical maybe you need to have more faith but of course the logical person is not gonna buy that because he's gonna have the position that all faith faith is just some some nonsense New Age stuff what we what we need is we need more logic the problem with the world is that we're not being logical enough if everybody in the world was just more logical everything would be beautiful so you're not gonna be able to convince a logical person who's stuck in his logic that his logic is limited using appeals to faith or religion or intuition or other outside stuff but was what girdle was able to do is he was able to to to prove to the law to the logical person from inside of his own system that his own system is inconsistent that his own logic undermines itself logic taken to its ultimate conclusion shows its own limits which is a remarkable result and if you think about it it should have been obvious to these logicians that this would be the case because after all if you just think about very simplistically think about like this if there's a person who murdered somebody and he's on trial now the fact that he murdered someone is let's say true he did in that case but does that mean you can prove it no there's no reason why you would be able to prove it he could have murdered somebody and whether you can prove it or not well that depends maybe you have some some video footage on some security camera maybe you have some text messages maybe you have some audio maybe you have some circumstantial evidence or whatever pointing like you have some footprints or you have some blood splats or whatever maybe you've got that but maybe he committed this murder so perfectly that you can't prove it does that mean he didn't murder the person no it just means you can't prove it but he still murdered him you see so truth is larger than what you can prove that's sort of a an analogy that we can make with with a sort of a more human example to bring this down to earth now this seems sort of reasonable you would seem like well it would seem like well of course that that that seems like common sense yes but logicians didn't get it for a long time they didn't get it many of them still don't really get it because they still insist on using rationality and logic to to figure out reality and what girdle was showing is the limitations of that now let's move on to alfred tarski this is also from Wikipedia and it says alfred tarski zunda findability theorem shattered all hopes of reducing reducing mathematics to logic informally the theorem states that arithmetic 'el truth cannot be defined in arithmetic so you need something beyond arithmetic to define the truths of arithmetic see the truths of arithmetic are not merely grounded in the symbols of arithmetic there's sort of a deeper layer or a metal layer that you need to appeal to this is my commentary here now so what's important about tarski is tarski came after girdle he came out right on the heels of girdle and while girdles result was more limited to two specific to a specific logical system tar skis result was more general and it became more broadly applied to to language and to logic so that's why tarski is significant although really it was later shown that girdle also derived the same result that our ski did but he just didn't publish it as as early so wikipedia goes on to say the theorem tarski theorem applies more generally to any sufficiently strong formal system showing that truth in the standard model of the system cannot be defined within the system the undefined ability theorem shows that encoding girdles and coding cannot be done for semantic concepts such as truth it shows that no sufficiently rich interpreted language can represent its own semantics a corollary is that any meta language capable of expressing the semantics of the language that is talking about the object language must have expressive power exceeding that of the object language the meta language includes primitive notions axioms and rules which are absent from the object language so that there are theorems provable in the meta language but not provable in the object language end quote so maybe that confused you it's a little technical but let me explain what this means it's quite simple if you have a language like English I can use that language to talk about stuff normally how we use languages we use it just to talk about stuff like people and cars and cats and dogs and business and and so forth but the language is also capable of talking about itself and that's the meta language see we can use English to talk about English and we can use English to talk about letters and I can even say that right now I'm speaking words and the word W I mean the word the word word starts with a w the word word has four letters in it see I'm actually now starting to talk in a self-conscious way about the actual language and so to do that I need to appeal something to something deeper higher I need to sort of meta language to talk about the language and the same thing with logic if you are using logic to just do logical deductions that's fine no problem there but if you're trying to use logic to talk about logic itself or to talk about the truths that logic is producing you can't do that at the level of logic what you need is you need meta logic logic about logic a meta language and so what tarski demonstrated is that in order to accomplish what frey gay and and hilbert and logical positive we're trying to accomplish what they didn't understand is that they need a meta language to accomplish that and you might say well okay so Leo so what's the problem so let's have let's create a meta language a meta logic which we can use to talk about logic and so therefore we've we've sort of used logic to justify and explain logic but you can't do that either because see the system you're using to talk about some other system always has a set of primitive notions axioms and rules and assumptions metaphysical assumptions which must be made for the system to get up and running in the first place so you can use a metal language to talk about the first-order language but you can't use the meta language to talk about the meta language so you need a meta meta language to talk about the truths of the meta language but then how do you prove the meta meta language you need a meta meta meta language and so on to infinity you see you can never escape the need for making ungrounded axioms and ungrounded assumptions never ever if you're gonna have any kind of system of explaining the world you're always going to need primitives primitive things which are just taken on faith which is why reason itself is in us inextricably connected to faith reason is not the arch enemy of faith as is conventionally assumed but rather you need faith in reason you need faith in your reasonable axioms you need faith that reason is accurately describing reality if you didn't have that you wouldn't be able to use your reason and it's precisely because of that that your reason is inconsistent because your reason assumes that everything is just reasonable except of course your very reason is unreasonable if reality was perfectly reasonable you actually couldn't have reason that's the paradox if the human mind was just a machine with no intuitive capacities it could actually not have mathematical insights that these mathematicians are having so the deep irony here is that these logicians and mathematicians who are making these groundbreaking discoveries within logic and mathematics nothing wrong with that but what they're not understanding is that they're actually using their intuition they're using and connecting to infinite intelligence to make their discoveries they're having creative insights it's not a mechanical process it's deep intelligence and this deep intelligence is not formal izybelle into a set of algorithms or simple rules the way that people assumed the way the logical positivists wanted and the deeper metaphysical and epistemic lesson here is that you cannot capture reality and encapsulate it into a formal system because reality is infinite and so it will always escape any attempt to encapsulate it as it must because reality is one thing it is the self the self is capable of self reference and when you're capable of self reference this necessarily leads to Kipp paradox into contradiction and this is not a problem this is not something that we need to be afraid of or that we need to consider an error or a mistake no that's correct that's precisely correct because all of your reasoning is actually grounded upon something that's beyond reason existence itself is beyond reason because reality and existence itself is groundless and it is a strange loop because it is itself reality is itself it is the ultimate chicken-and-egg problem it is the ultimate self-reference problem you are a self within reality you are the universe looking at itself thinking about itself reasoning about itself wondering how did I get here what am i doing here what is existence well what is that that's existence asking itself what is existence and the deep irony is that existence doesn't actually exist so ironically you're standing here asking what is existence I am existence asking myself what is existence and is the same as non-existence and so this is the Ouroboros this is the snake eating its own tail and what girdle showed is that logic is a little microcosm of the larger macrocosm of this Ouroboros logic eats itself by its own tail just like the snake as it must because we have absolute infinity which is all of reality as a whole but then absolute infinity includes all the minor infinities as Georg Cantor discovered every object including logic because logic is a part of reality he's also infinite and mathematics is also infinite and the chair is infinite and the dog the cat the tree the bird in the sky it's all infinite all of it is a different form of infinity and then all of it of course is contained within absolute infinity which is the whole thing given all at once but see you cannot cut encapsulate reality into any formal system because the formal system is not separate from reality as many scientists and rationalist like to assume it's a part of it the scientist is reality science is itself reality so science has to as its doing science ultimately has to explain itself and for most of Sciences history science was done simply externally it was like you were just you were just looking at the world as though you're not connected to the world you're disconnected you're kind of standing outside the universe and you're trying to judge it objectively but then with quantum mechanics for example and with with these with these theorems from girdle and from tarski what's what's being realized that when you go really deeply into your inquiry because science is an inquiry into the nature of reality when you go really deeply into that inquiry what you discover is that wait a minute if I'm inquiring what is the chair what is a cat what is a tree what is logic what is science ultimately have to enquire what am i what am i what is consciousness what is my self and then when when you inquire deeply into that you realize that I am the stuff that I was looking at I am a part of reality I'm not separate from it and so the subject and object distinction breaks down it collapses and when that happens then you become the stuff that you were investigating see you can't just assume that logic is something separate from you or that logic is separate from intuition or logic is separate from emotion not at all all of these things must be ultimately interconnected into a unity and that's what non-duality tells us and so really to fully appreciate girdle's discovery you have to understand non duality because ultimately girdle's theorems and tarski theorems they were pointing to non duality again they cannot actually make non duality perfectly clear to you because for that you actually need to become non dual you actually need to have an Enlightenment experience but they were pointing to the fact that there is a truth which is beyond language beyond logic beyond your mind so that's the significance now of course they don't teach you any of this in school when you're taking your physics classes your math classes your logic classes and even in university they don't teach you this stuff nobody talks about this stuff this is our sort of the hidden stuff the hidden under Melly underbelly the hidden structures of these scientific and mathematical methods because if you notice when you were learning mathematics or science or history or whatever very little attention was actually paid on the process of science on the process of history on the process of logic or math you weren't really doing metamathematics you weren't doing meta logic you weren't doing meta philosophy you weren't doing meta physics you weren't doing meta history you just sort of assumed that you could use history you could use philosophy you could use science or whatever to just to get your results but the deeper you go and the more you use those tools eventually have to ask yourself wait a minute what is this tool that I'm using where is it coming from how is the tool connected to myself and to my own being to my own consciousness and to reality as a whole and when you do that that's when the self reference problems come up and academia does a really good job of hiding all these self reference problems because when you really explore the self reference problem to its ultimate root you discover that there is no separate self everything is the self and you might wonder well what's the problem with that well the problem with that is that it destroys all systems it destroys your ability to encapsulate and to capture reality into a little box which is the whole premise of science and the whole premise of philosophy and the whole premise of theory and the whole private premise of modeling and the whole premise of of language of of logical understanding the whole premise of that is that you can capture it into some kind of system but you can't because the system is smaller than the whole and the system is a part of the whole and the map is not the territory and now I want to read you some excerpts of quotes that I've compiled from a collection of various authors about girdle and his discoveries and also tar skis discoveries they do a great job of eloquently sort of distilling down all the complexity into into little concise snippets and metaphors and images that that do a great job of explaining the significance of this so first we have Douglas Hofstadter and he says quote the paraphrase of girdle's theorem says that for any record player there are records which it cannot play because they will cause it's indirect self-destruction and quote I love that image that's the perfect metaphor for what we're talking about here and you can see how now it starts to connect to non-duality into self-actualization work so what he's talking about here is a record player so imagine old school records those sort of photographs which have that needle in that groove record so now imagine if you could design a record that you can put on that player such that when the needle went around the circle's there was something encoded in the record that would actually cause that needle to to vibrate and finally the whole thing was just selfish self implode that would be the equivalent of what girdle accomplished he found a way to do that using arithmetic and using logic that's pretty cool and he was able to prove he wasn't just able to come in he was able to prove that any system which is capable of self reference will have this Achilles heel that will cause its own self destruction because that's what self reference must do and of course the thing that he himself never realized and that that even Douglas Hofstadter really never fully realized even though they understood this stuff in theory they never actually got to the point where they realized that actually my own physical existence as a self and as a mind as a body there is there exists a record so to speak that can be played inside my mind which will cause my own death think about that there is an idea let's say the the analogy here's a record for record player and idea for the mind there exists an idea such that if we feed it into your mind it will destroy you that's pretty amazing that's pretty amazing you might wonder what would that idea be well it's not necessarily just as simple as one specific idea rather the way it really works in reality is that there can be some level or some realization that your mind can reach where it realizes that it itself never really existed and when it realizes that and it realizes that reality never really existed and then like it it has a full sort of overarching meta understanding of itself the entire thing collapses and is seen to be a house of cards and that's enlightenment pretty cool but also notice that if such a thing really were possible then ideas are far more dangerous than most people assume see most people think that all ideas ideas are just ideas yeah you could read ideas in a book you can hear ideas on TV but it's not like an idea can kill me or can it could this be why people and minds are so ideological and they cling to ideas so much have you noticed this how much Minds cling to ideas you cling to ideas your your family claims to ideas your friends cling to ideas that's why you argue with them so much people on TV politicians and religious people scientific people logical people they all clink to ideas they all have pet theories and ideologies why is that could it be because deep down inside the mind knows that if it if it's not careful and it explores the wrong ideas eventually it will run into an idea that will kill it now there's an idea there's an idea that might be one of the first ideas in this chain of ideas that will ultimately lead to your self-destruction to even be open to the possibility that there might exist an idea which will lead to your own self-destruction and to not reject that idea and to take it seriously that already is a part of this whole self-destructive sort of ideology see which is precisely why most people would reject what I'm saying here what I'm saying here is not commonly accepted it's not talked about in mainstream media it's not talked about in universities nobody talks about this stuff and if they do they're very careful to make it just an idea so that it's just a theory enlightenment is just a possibility some philosophy or some some hypothetical thing but it's never actualized because it's dangerous for the brain it's dangerous for the mind to take these ideas really seriously that's why when you're sitting and you're doing self inquiry one of most important things you have to do is you have to you have to take the possibility of eradicating yourself very seriously because that's what you're doing in self inquiry that's what you're trying to do that's what you're trying to do with meditation that's what you're trying to do with yoga people sit down they they're doing this process but they don't really grasp the significance of what they're doing and of course really you can't because you can't fathom the significance of it until after the self destruction has really started to occur Douglas Hofstadter also says quote prove ability is a weaker notion than truth and quote so we've talked about that and my ideas about probability and truth were greatly shaped by Douglas Hofstadter I really recommend you check out his book girdle assure and Boff where he goes into a lot of detail in a very beautiful way at the same time it's both technical and challenging it's one of the most challenging books you'll ever read but also it's is very accessible and it's written in this very playful and sort of fun and interesting style so if you're nerdy and you want to go really deep on this topic then I recommend you check it out one of my favorite books of all time it's just a delightful to read it's a total mindfuck especially when you combine it with the things you have learned if you've been watching actualised org that takes Earl girdle Escher and Bach and just takes it to the enth degree cuz Douglas Hofstadter didn't really realize the significance of what he was writing about he never did even though he wrote about zen and he wrote about enlightenment he still didn't really understand what he was talking about he was still it was still just a concept it wasn't really actualized so if you're a fan of gertle Escher and Bach and maybe that's why you stumble upon this episode my piece of advice to you is that there's something deeper if you love gertle Escher and Bach if you love the work of douglas hofstadter go deeper go deeper that's just the tip of the iceberg what you're invited to do here is to actually find that record which will destroy you as the record player and actually actualize that let's go on to the next quote by George Gilder he says quote girdle demonstrated that every logical scheme including mathematics is dependent upon axioms that it cannot prove and that cannot be reduced to the scheme itself girdle was reportedly concerned that he might have inadvertently proved the existence of God a faux pas in his Viennese and Princeton circle end quote the Viennese in Princeton circle that they're talking about in this quote is the logical positivists they were otherwise known as the vienna circle so why did girdle think that he proved the existence of God perhaps well because he saw the the larger consequences here he saw that reality is such a thing that you can't ever encapsulate it and what does that mean reality is that means it's infinity the next code is from Perry Marshall and he says quote in 1931 this young Austrian mathematician Kurt girdle published a paper that once and for all proved that a single theory of everything is actually impossible end quote people have been trying to come up with theories of everything for a long time and I'm sure that there are people sitting in universities all around the world today working really hard dedicating their entire careers to finding a theory of everything and they will never succeed because what they don't understand is is that everything is infinite what theory are you gonna come up with that's infinite your theory is never gonna be infinite your theory is always finite so you're always gonna fail your theory of everything will actually not be everything your theory of everything is is an oxymoron it's a contradiction in of itself a theory is not everything you can't have a theory of everything you can have a theory which is something but it's not everything or you can have everything but you can't have both because everything is not a theory everything is if everything was a theory that means this would be a theory right here but it's not this is not a theory this is what we call reality the gist between theory and realities precisely that theory is not all of reality that's why the idea of a theory is useful because it's more limited than reality because reality is too much it's too much so we subdivided into components do you want a theory of everything I can give it to you right now this is about as simple as a theory of everything we'll ever get that's it it's one symbol just draw the infinity symbol and that's everything that's the closest theory of everything yet it's still not really everything it's a concept it's just a theory but you know what come up with a better theory everything that infinity that's what it is you don't need complex complex equation mathematics and you don't need to do anything just infinity that's it but of course that doesn't satisfy the scientist because the scientist can't build a career off of off of infinity well he can but to build a scientific career off of infinity what you have to do is you have to dedicate your whole life to subdividing and fragmenting and atomizing infinity forever and that's what scientists have done see scientists are very clever people they have invented a career for themselves which will never ever end scientists will be in business for a trillion years doing science subdividing infinity into infinite number of formulas over and over and over again and that's that's what they're gonna do and their job will never end and see in a sense to realize that everything is just infinity well it's too simple it's so simple you can't really build a career around it you can't really write a book about it I mean people do people try people still sell many books that car totally wrote wrote and sold millions of books based on based on infinity but uh but you know if you really want to be truthful about it there's nothing you can say it's just infinity that's it this next quote is by John Barrow he says quote as with geometries so with logics there are an infinite number of consistent schemes of logical reasoning that can be constructed there is no such thing as an absolute truth in logic and mathematics the best one can do is talk about the truth of statements given a set of rules of reasoning it is quite possible to have statements that are true in one logical system but false in another endquote this is very consequential because that myth of reasoning that we've talked about in the in the very beginning that mainstream myth of logic well it assumes that there's only one logic that there's only one way to reason but actually that's not true at all if you go to graduate school in philosophy or within logic or within mathematics you will quickly discover that there are dozens and hundreds of different logical systems and that all of these different systems have their own sets of axioms their own rules and some of these logics are very counter intuitive and twisted and not like normal logic and so you might ask well which logical system is the right one but of course there isn't one because your logical system is taken on faith which logic you use which axiom constitutes your logic every logical system needs to have a set of axioms well those are arbitrary you just choose them and then it becomes an empirical question whether your logical system will correspond to reality or to the physical universe maybe it will maybe it won't maybe in in quantum mechanics it'll work but in everyday life it won't work or vice-versa for example ordinary logic that we traditionally call logic that we use just around everyday life you know we could say well I saw an animal on the road and all and then I can ask you well was it a cat or was it a dog and you can say it was both but of course that doesn't work in everyday logic you can't have a cat and a dog in one it was either a cat or a dog or it was neither but it can't be both if it was just one animal but then see if we try to take this logic and we start to apply it for example to quantum mechanics to light we can ask a light well is light a wave or is it a particle and you might think if you're trying to apply ordinary logic of cats and dogs to light and two particles and waves you might say well he's either gonna be a particle or a wave can't be both but the answer turns out that it's both or maybe that it's neither see but that had to be empirically tested and many people when they first saw those experiments they didn't want to believe it took a lot of work of genius people to really to square that circle you see so you can't really rely on rationale in your logic because there's actually an infinite number of logical systems that you could create so which one are you going to use how are you going to determine which logical system is the right one to apply in which situation are you going to use logic to determine that no you can't you see because that's just a recursion of the problem you might say well there's a meadow logic okay but then there's an infant number of meta logics which meta logy are gonna use how are you gonna determine that well by using a meta meta logic how many meta logics are you gonna have how many men a meta meta meta logics are you gonna have where does it end it doesn't end it can't end because it's infinite even logic itself is infinite even logic itself can't confine itself to one set of axioms you see logic itself is very diverse it's so diverse that you can't even control it it spills out that's the key feature of realities that always spills out that's the key feature of infinity it has no end it has no boundary it just kind of keeps going and going and going and going here's another quote by John Barrow I love this one quote if a religion is defined as a system of ideas that contains unprovable statements then gertle taught us that mathematics is not only a religion it is the only religion that can prove itself to be one and quote I love that that is the delicious irony the poetic justice if you will of rational people criticizing religious people for believing in unprovable things well logic itself is unprovable so who are you criticizing it's the pot kettle calling the kettle black that criticism just doesn't work if you are aware of what your own system is see the only way that a rationalist can really criticize a religious person is by not being aware of his own internal self contradictions which is precisely why rational people like to debate religious people because when they're debating religious people they can just attack outwardly but they're not really looking and self reflecting inwardly to self-reflect you have to just be silent you can't be in a debate you can't be arguing you can't be discussing you have to actually be silent and turn inwards and see your own inherent self contradictions now I want to talk about more I want to talk more about girdles a man because he he's a very interesting figure and the reason I want to do this is important strategic reason because see rational people here will listen to what I'm saying and then they will they will come up with a sneaky little objection they'll say AHA leo but but girdles ideas you know girdle was just a logician all he cared about was just formal mathematics it wasn't philosophical there's nothing philosophical or metaphysical or epistemological or theological or spiritual or non dual about any of girdles work you're projecting that on to girdle that's what they'll tell me and they'll say Oh leo this is just like with quantum mechanics you know people all sorts of New Age people love to use quantum mechanics to justify their religious beliefs and all their spiritual nonsense but all of that is just it's not really correct and this is where I need to again I need to use the evidence against you I need to use girdle against you in this case because your mind is being tricky it's being sneaky you're not letting yourself actually experience the the full ramifications here you're wrong so for this we go to girdles biographer Hao Wang he was a Chinese professor who interacted with girdle throughout his life and he wrote a book sort of by a biography of girdle and girdles ideas and girls philosophy so I'm gonna quote to you from this book and here's a couple of key things that are made very clear in this book I was quite surprised reading this firstly it's revealed that girdle was first and foremost a philosopher not a mathematician not a logician but a philosopher he cared deeply about metaphysics girdle was not a realist or a materialist he was a plate inist or an idealist and he was really resonating with Leibniz his philosophy and with who Cyril and liveness of course was a famous tidy list and also he was a of a very adamant believer in God and his philosophy and his metaphysics and his pista mala JEE all sort of revolve around God and who Cyril was sort of the father of phenomenology which is the study of first-person experiences so what that tells you is that girdle was resonating with these rather than with the logical positivists girdle believed in a personal God girdle in fact invented logical proofs for the existence of God I'm not saying those proofs are valid but I'm saying he did attempt it so that just kind of shows you where he's coming from and the thing that girdle was most interested in in his life is creating a complete philosophy of reality he wants to understand what the whole universe was what is the connection between metaphysics and epistemology and between logic and mathematics he want to understand everything he wanted the big picture that was his goal but he never succeeded in completing his philosophy even though he had some of these great breakthroughs within within the field of logic and mathematics so here's some quotes from from the biography quote when girdle was about four he was nicknamed mr. Y by family and friends because he always wanted to get to the bottom of things with his intensive questioning end quote that kind of reminds me of myself also quote I single out the quest for a worldview as girdle's central goal his aim in life was to make the greatest possible contribution that he could to the ideal of finding and justifying the correct and true worldview and quote you see so girdle was not just a technical mathematical or logical hack he wanted to understand the full ramifications of of what's going on with mathematics and with logic also quote girdles interest was less in physics itself than in his philosophical foundations and significance and quote and also quote girdle told me that his general philosophical theory is a life Nietzschean mana dalla G with the central monad being God although he also stressed that liveness had not worked out the theory end quote so if you're not familiar with the philosophy and metaphysics of live Nets and this idea of monads or mana dalla G it's a little bit arcane but basically liveness had this metaphysical idea that reality was not composed of particles or atoms but these things called monads and this was perhaps the earliest idea for a holographic universe these monads were little Holograms so what a monad was was just a particle but this particle contained inside of it all the other particles sort of like a reflection of them and so there were many of these monads but every monad contained inside of it all the other monads also sort of a strange loop and that that's ultimately what God was is that God was this monad sort of like a singularity so it's a little bit wonky the details of his of his of his theory were a little bit wonky I wouldn't go and buy into all of it but I'm just saying that Leibniz was onto something there because he intuitively sensed that that there's something more to the universe than just plain stupid materialism and so did girdle they just weren't really able to to fully articulate it because they were still operating on a level of duality they were trying to understand non duality using duality and so they never quite succeeded here are a few quotes from the biography from girdle himself quote the plate mystic view is the only one tenable thereby I mean the view that mathematics is describing a non sensual reality which exists independently both of the axe and the dispositions of the human mind and is only perceived and probably perceived very incompletely by the human mind end quote so when we say Platonism what that means is Plato's metaphysics and Plato's metaphysics was not material but more idealistic so what Plato believed is that these sense perceptions that we see here the objects cars trees dogs and cats and people that this isn't what's really true about reality what's true is the ideas the concepts the forms these abstract entities for example what's more true for Plato is the idea of a dog in the abstract rather than a dog in particular now that's a bit weird and I'm not saying that that's the right way to look at reality he was still looking at reality very conceptually and that's a problem and that same problem basically also then got passed out to girdle which is why girdle had his problems because he was still not recognizing the distinction between direct experience and concepts but nevertheless what girdle was sort of thinking was that reality is is mathematical that's not correct quite correct but he was still basically thinking that what's most real about reality is the higher level sort of like the higher spiritual levels that the mind can reach only occasionally and only get little glimpses of not this physical gross reality so he sort of believed in a subtle reality and he thought that that's what he was accessing when he was doing these proofs and that's why you need intuition is to access that and that's why you can't just use a mechanical process because the mechanical process at this gross physical level you need something that actually reaches out into the subtle spiritual level from which insights come and those insights are what the mathematicians are using to to really power their work he also said quote it was the anti platonic prejudice which prevented people from getting my results this fact is the clear proof that the prejudice is a mistake and quote so there he's specifically addressing the question of why wasn't girls incompleteness theorem discovered earlier and the reason is is because people assume that everything is just mechanistic but because girdle didn't assume that reality and math was mechanistic because he intuited that there was an important component and role for intuition to play because he was a plate inist that because that he was able to derive his results he also said quote you cannot understand what mathematics is without understanding knowledge in general and you cannot understand what knowledge is without understanding the world in general end quote so see here he's he's saying that you can't just understand all these subjects in isolation you can't just be a mathematician and think that you're gonna understand mathematics without understanding knowledge epistemology or metaphysics or the world see because it's all interconnected you can't assume that your little field is a separate little bubble that lives off by itself that assumption doesn't hold he also said quote when I entered the field of logic it was 50% philosophy and 50% mathematics it is now 99% mathematics and 1% philosophy even the 1% is bad philosophy I doubt whether there is really any clear philosophy in the models of modal logic and quote so here he's criticizing what's been happening over the last hundred years in academia academia has become less and less philosophical less concerned about metaphysics less concerned about epistemology and just more concerned about getting the right result because science and academia has just become the quantification of everything if you can quantify something if you can model it and you can get the right results and then you could just be very pragmatic about application of your scientific formulas to make predictions about the future then that's it you're done that's what modern science sort of works like it's lost that deep philosophical connection that it used to have and asked in large part because the logical positivists even though they were ultimately debunked and disproven and logical positivism has largely fallen out of favor people who study this stuff know that logical positivism is is a load of it's it's completely inconsistent but nevertheless this has sort of permeated the culture and has just sort of become part of the scientific and rationalist paradigm and the materialist paradigm which operates at a much deeper level than then even most scientists are able to fathom it's just part of the culture it's way beyond you you're just programmed with it and now you know Goerdeler was already seeing that starting to happen in his time and he didn't like that because he cared about the philosophy of it he didn't think philosophy was nonsense she also said quote the answer to the question what am I would then be that I am something which is of itself has no properties at all rather like a clothes hanger on which one may hang any garment one wishes and quote so see even girdled was starting to understand the importance of the question what am i and he was starting to see that really there's no self inside what I am is nothing but nevertheless he never became enlightened this was still conceptual for him and then his biographer goes on to say quote more than once he had said that the present age was not a good one for philosophy end quote and also quote gourdel was in favor of metaphysics as opposed to positivism and quote by which he means logical positivism and then girdle himself said quote the discovery of metaphysical truth will benefit mankind end quote and then his biographer said quote the superiority of mind is undoubtedly important for his ontological idealism which sees mind as prior to matter and quote so ontological idealism is just the the idea that what's more real is mind over matter matter is something that happens within mind mind is not something that happens as a phenomena of matter he also said quote girdle regarded contemporary science as mistaking the part for the whole and quote that's exactly right that's exactly what's happening this is the whole problem of confusing the map for the territory the map is the is the part the territory is the whole the part is a part of the whole but it's not the whole he also said quote girdle proposed a doctrine according to which time and change were not objectively real end quote and lastly girl himself said quote matter will be spiritual eyes when the true theory of physics is found and quote it's pretty illuminating huh so I don't want to hear from you rationalist objecting that girdle was just a mathematician or just a logician that his results have no metaphysical consequences or epistemic consequences or philosophical consequences yes they do or at the very least he certainly believed they did maybe it's use not seeing the deeper consequences here because you've been so blindsided to just look inside you're a little technical sub domain of your field you got to go beyond logic that's what girls and complete this theorem is telling you so you might wonder what happened to girdle in the end he was alive during the heyday of quantum mechanics and of course he was understanding all the stuff that was going on in quantum mechanics and that probably helped them to to make some of his discoveries and and to see that that reality was was not material but ultimately what happened to girdle was quite sad at the end of his life he became mentally unstable in fact he was so paranoid of being poisoned that he would only eat food that his wife prepared for him when his own wife became ill and she was not able to prepare food for him anymore then he actually starved to death and he died sixty-five pounds of body weight that's how much he weighed he died from starvation a rather sad end to a rather ingenious individual I think there's a lesson to be drawn even from the way that his life ended just like with quantum mechanics with David Bohm I told you about how David Bohm became depressed even though he was this brilliant physicist who understood basically non duality he understood in theory but again he didn't actualize it and that's the same problem I think that happened with girdle now this is a little bit of speculation on my part I don't really know why he became mentally unstable but I can speculate with some degree of reasonableness and likelihood that what happened with him is that as he get older and older and he's hard to face the prospect of death he started to think about well what is my legacy did I really accomplish what I wanted to in my life with my life purpose and his life purpose was to figure out what the hell is going on why is there reality what does all of this mean what is what is the metaphysical truth what is the right philosophy and he was never able to piece it all together because he was trying to do it still coming from the conceptual domain what he didn't fully understand is that you have to go beyond logic yes but also beyond concepts beyond ideas beyond mathematics beyond philosophy beyond language beyond symbols beyond the mind entirely and he was never really able to do that and I think that that might have driven him crazy just like with Georg Cantor he sort of had the same problem as well is that Garret Cantor understood that his logic and his mathematical proofs were pointing to this transcendental truth that that he called God and that girdle called God but he was never actually able to actualize it and to see it for himself and he probably wanted to very badly that was his whole life purpose but he never made it and that's because ultimately I think he got stuck in reasoning even though he made these proofs he still got stuck in his own mind and there's a really great quote that I think hits at the heart of this problem and it is this quote a madman is not someone who has lost his reason but someone who has lost everything but his reason end quote by GK Chesterton and I think that's precisely what happened with girdle and many of these other theorists like Cantor and to a certain extent David Bohm they were still stuck within reason even though they went post rational and they knew the limits of reason but it's not enough to know the limits of reason you have to transcend reason which is another level altogether now let's tackle some objections hmm now let's tackle some objections but Leo girdle's theorem only applies to strict formal systems and what you're doing is you're bastardizing it and you're taking it and you're running with it you're being way too loose and metaphysical well that's precisely why I wanted to quote to you from his biographer and some of girdle's quotes from the biography himself so that you can get an insight into the man himself and how he thought hopefully those quotes have convinced you that he wasn't just strictly a logician it's not that I'm being too loose and metaphysical it's that actually you're being too rigid and too limited in your scope which you might think that that's actually a good thing because you're trying to protect science and to protect knowledge and to protect your web of belief because you want you want it to be strict and you want it to be just hard-nosed but actually in doing that you actually limit your knowledge you limit your understanding of the bigger picture you need to allow yourself to start to see the metaphysical and epistemic consequences of this stuff that's not loosey-goosey that's that's exactly what's necessary otherwise you fall into the trap of logical positivism or for example in psychology they had a similar movement which was coming from the same sort of intention to make science overly strict which was behaviorism and that also didn't pan out very well it just doesn't explain enough reality is more rich and more complicated and more nuanced and more interesting and more way more fascinating than just simple reductionism would lead you to believe so I really encourage you to to think through these consequences for yourself very deeply and try to see how your insistence on being rational is actually fragmenting reality and when you're fragmenting reality you're not seeing the larger whole and to really understand what reality is as a whole well by definition you're talking about the whole and not some little fragmentary part sure yes strictly speaking girdle's theorem only applies to some strip little formalism within logic but you know there's more there's more realities not just a little fragment logic is not some isolated hermetically sealed field from the rest of reality logic is occurring inside of your your brain and Union in your mind so that means it's happening to you as a human being you can't really separate logic from emotions you can't separate the left brain from the right brain they actually work together and actually when you're doing logic and when you're analyzing and you're reducing reality and you're fragmenting reality into these conceptual categories that's actually happening thanks to the fact that you have intuitive capacities and you're actually tapping into unbeknownst to you infinite intelligence to do that to actually sit down and make sense of a complicated proof to have some deep insight within some mathematical theorem that requires infinite intelligence you have to have an aha moment that comes from outside of your ego outside of your little selfish mind outside of rationalism that's where it comes from and then you use that to actually build up your rational frameworks start to see that start to self-reflect as you're doing logic start to notice and observe and be mindful of yourself doing the logic or doing your reasoning start to carefully observe the internal processes rather than just using it externally to solve some sort of problem out in the world or some problem that your professor gave you and then ultimately if you start to have mystical experiences and you start to glimpse absolute infinity for yourself directly by going beyond your mind it'll all interconnect in a very beautiful and profound way for you the other objection here of course is the the perennial one Leo how is any of this stuff practical what does logic have to do with my personal life well actually quite a bit a lot more than you would assume because if you grew up in Western modern 21st century culture then you have grown up with this myth of rationalism this myth that things can be proven to you in some sort of airtight way and hopefully many of these myths have been shattered here when you're stuck in rationalism that is a very practical thing you are stuck and it does affect your ability to think creatively it affects your ability to theorize to do proper philosophy to understand science to be able to open your mind to new ways of looking at the world and new perspectives this is extremely important the problem with rationalism is that it it calcifies your mind it makes your mind rigid and dogmatic it's the exact same problem that religious people have religious fundamentalists same exact problem when the mind becomes rigid it's not able to be truly creative it's not really thinking for itself it's not opening itself up to infinite intelligence it's not opening itself to new diverse perspectives you know here's a very practical thing I can give you a life changing book about yoga for example and I can tell you here go take this book go practice it go read it within a year it will completely transform your life the quality of your life nothing theoretical about it all practices exercises go do them and if you're a rationalist or you're a scientist or you're an academic or whatever you're not going to be able to get five pages into this book why not because the paradigm that this book will be coming from will be so radically different from your rationalist logical materialistic Western but the paradigm that it's not gonna it's not gonna compute for you you're gonna get disgusted you're gonna throw this book away you can say oh this is just nothing this is just metaphysics it's just speculation this is just some New Age stuff look Lea they're talking about chakras and they're talking about out-of-body experience all this this is all nonsense we all know that those are just hallucinations in the brain we all know that consciousness just happens in the brain that's just nonsense I'm not gonna do these exercises Leo you want me to waste a year doing these exercises no way see that's how your mind will trick you and that's very practical you're limiting yourself you're really limiting yourself not only are you limiting yourself with some of these New Age techniques that are real that can really benefit you which you poopoo but you also limit yourself as a theorist and as a thinker to be a genius theorist a genius scientist a genius mathematician a genius engineer a genius thinker of any kind to make original genius and valuable contributions to mankind you need to cultivate a certain kind of intellect not a rationalist intellect that's not gonna do it that's just gonna make you a technician and a hack you need to cultivate a very open minded creative intellect where your mind becomes playful and you're able to jump between different perspectives and look at the world this way and then that way from a new-age perspective from a religious perspective from a scientific person from illogical all of it you need to be able to jump around comfortably and not to take sides and not to get offended and not to get emotional not to get attached to any of these perspectives you need to be able to question all your assumptions because that's the essence of thinking outside the box how do you think outside the box well first of all you have recognized that you're always in some kind of box when you're thinking that helps and then you can learn tools and methods and you can do various practices you can do visualization exercises you can do yoga exercising you do meditation self inquiry phillup philosophical actually you can contemplate the nature of almost anything and as you do that and you can also self reflect and you can observe your own process you can observe how you use language how you use logic how I use rationality as you do all this you become more conscious the more conscious you become the more of a vessel for infinite creative intelligence you become the more you purify yourself the more you purify your chakras your creativity will just go through the roof if you think you're a good scientist right now or a good mathematician or a good logician or a good theorist right now man wait till you spend five or ten years doing these practices and really self actualizing and opening up your mind in the ways that I'm talking about your your creativity will go to levels that are superhuman your problem is gonna be that you'd have so much creativity you're not gonna have enough time to actually make all your ideas a reality there's going to be an overabundance of brilliant ideas an overabundance of a brilliant business ideas brilliant ways to transform society to improve the world to help yourself to help your family it's gonna be too much I have so much creativity because my mind is so open and I've been doing these practices my mind is so creative at this point that it's actually exhausting it's exhausting because I'm pummeled every week I'm puddled by by so many new ideas that I could follow up on business ideas and how I could transform the world how I can earn millions of dollars it's just it's too much I don't have enough time to actualize all that I would need ten lifetimes for it I have to be very selective about that see so this extremely practical and we're just we're just getting to the to the tip of the iceberg of the practicality of it and I haven't even mentioned your health for example your psychological health or your your physical health all of that is affected and if you're a theoretician one of the things you face is you face the danger of depression of getting stuck and mired in all your concepts because your understanding so much about reality but even though you understand all this stuff it's just turning into the sort of brain fog you're walking around with all these concepts but your body isn't really growing along with all the stuff that's in your head so you're sort of disconnecting yourself from your emotions you're disconnecting yourself from your physical body you're stressed out you're not able to relax you're not able to be at peace your mind is always in a monkey mind mode you're always thinking in the future always in the past can't be present see so there are tons and tons of of practical ramifications here that we haven't even begun to to elucidate the ball but even though we're wrapping up part one here there's gonna be a part two so stay tuned for part two that will be released soon and in part two there's still more to talk about I really want to focus on how Western intellectual tradition actually fears paradox there's a rich tradition in the West of fearing paradox and actively hiding paradox and I want to talk about why that's the case why it happens the deep reasons and I want to show you examples of how that works and I'm gonna show you examples of how in science and in math actually Western rational thinkers have been sweeping these paradoxes under the rug hiding them from you precisely because they pose some some threats to the entire establishment of science and rationality and philosophy so stick around for that that's it for here please click the like button for me go check out actualize org my blog the resources are there life purpose course book list the forum is there you can ask questions discuss all this stuff and stick with me as your mind is becoming more open by some of these philosophical topics that I cover what that will do is that will then make you sort of fertile soil in which seeds can be planted which can then be nurtured and grown i have already shared many practices with you which you can start to practice to transform your life in radical ways if if you want something very practical go to actualize that our own slash start and you will see a list of practices there and lots of videos to get you started in this process so there's a lot of practical stuff there but also i'll be sharing a lot more practical stuff practical techniques that you can use in future one of the challenges i have when one of my sort of missions and visions for use i want to share techniques with you that are so radical that you are not going to be able to do these techniques if you're operating under the materialist rationalist paradigm so a lot of the time that I spend shooting these long complex philosophical videos is just to open your mind up because I know how stuck you are in your mind we to open your mind up first then once we've done that then we can start to introduce techniques to you and then you'll be more open to those techniques and then as you start to practice those techniques it's gonna create this sort of positive feedback loop you see because when you have a rigid mind with rigid theory that prevents you from actually going out there and practicing radical techniques that can transform your mind but when your mind opens that allows you to now go and explore new techniques so you go explore those techniques and practices and those in turn reflect back and change your mind and change your ideology and open your mind even more so as you practice more of these new techniques that opens your mind more but as your mind opens more you open yourself up to even more powerful and advanced techniques and so it just creates this positive feedback loop you see so the theory of the practice of course they all feed into each other there's no difference between theory and practice that's a duality that people use to keep themselves stuck and all dualities of course by this point you should realize our gonna break down including the one between theory and practice and as you do more of the practices I really want you to start doing the practices because if you like the theory you gotta start doing the practices because if you're just gonna ignore the practices and just listen to the theory eventually what's gonna happen is that at some point you're gonna get disgusted with the theory because it's gonna start to seem like only all you do is just talk about theory what's the point well the point is if you're doing your practices the theory I talked about will deepen your practices and deepen your insights and deepen your understanding in ways that are gonna be very beautiful and very profound and very enriching for your entire practice so the theory is sort of like a treat for those people who are doing the practices but then people who don't do any practices just learn the theory and then they engage in mental masturbation and they don't take any action well those people are gonna suffer so don't be one of those people start doing the practices now I've already told you what all the practices are stop asking me what the practices are you know what the practices are you just don't want to do them cuz you're lazy I know because I I've been there I know how that works theory is much easier than practice because you can just sit back and read books or just watch videos for hours on hours on end that's fun and you need a certain amount of that when you're getting started but then start doing the practices too you