I'll never forget the first time I met this brilliant man. I was an undergrad at Oxford and all I had with me was a mathematics textbook and green pen. I asked him to sign it and he happily obliged. I still open it almost weekly to see the short, wonderful message inside. It said "All the best, Joe Rogan".
Rogan is a meathead but he’s fairly close to average intellect, he can recognize when he cannot say anything contributory. It’s not evident with normal guests but Penrose is a genius on Einstein’s level and beyond. Rogan felt it.
No one interrupts when they don't know what a person is talking about, but one who is extremely well versed in a subject will also not interrupt, as they do not need to. We know which side joe is on with regards to maths, but at least he's not like other know it all hosts.
"I've briefly met this man. I had him sign my pre-algebra textbook. Then I completed my undergrad in Math. That textbook sits proudly on my shelf" You have a degree in math...and you are clueless that Penrose made a fool of himself in this video.
@@jengleheimerschmitt7941 To be fair, a mathematician on DMT would be very interesting. Since the geometry that you see seem to be related to the internal structure of the brain as perceived by your consciousness.
Joe Rogan talks to Roger Penrose, what the hellll is going on? Can Joe Rogan understand 1% of what Penrose talking about? It is like Denis Rodman having a conversation with Einstein.
The LOGICAL structure of Godel's proof is simple. He tweaks "This statement is false" to make this: "This statement is unprovable." There are two possible truth-values for this: If the statement is true, then you have a true statement that is unprovable. If the statement is false, then the statement is provable, which means you have proof of a false statement. So any (sufficiently complex) mathematical-logical system is either incomplete (with statements you know to be true but can't prove) or self-contradictory (with false statements you can prove), or both.
So is the philosophic implication that: a "system", may never be able to completly assess it's self, or to word it differently, one can never completely assess a system from within the system? A system being anything from a system of formal logic to a universe.
@@DavidKolbSantosh Yea I think so too. Specifically, the system can't completely assess itself the way it wants to using its own rules. Kind of like the human brain trying to understand itself using the knowledge understood by the human brain.
@@edwardjones2202 Absolutely right. I understand WHAT Godel did to implement his proof, but I don't yet understand HOW he worked it through. In particular, I've wondered whether he proves merely that some self-referential statements (such as the one I stated above; not original with me) entail this uncertainty--falsity vs. incompleteness--or whether his method of proof somehow gets around this limitation so that the proof potentially applies to any meaningful statement that can be asked within the system. Surely though the latter cannot be the case, or Godel's proof would not be the big deal that it is.
I remember going through the proof in my logic class. We spent 2 full long lectures establishing the coding system and learning how it works. Then, in like 15 minutes, the professor writes the code that translates to "This statement cannot be proved." And proves that the fact this statement can be encoded in a system (and very simple systems can encode it) means that the system cannot be complete.
@@ihsahnakerfeldt9280 I think the 200 level course on logic did that. I took the 400 level and we did all the details. Not really sure I got any benefit out of those details though. I'm kind of with you.
@@ihsahnakerfeldt9280 no because you have to use the same symbology as the system you are trying to demonstrate is incomplete. It must be done on its own terms to stand mathematical scrutiny, although a conceptual explanation of the proof is then possible. Its a matter of rigour.
You can mock Joe for not knowing every thing that every guest is really good at. Yet he gives them the space, and gives us the opportunity to see these kind of things.
Your comment made me realize that a lot of people here are slinging hate at Joe because they too have no fucking clue what Penrose is talking about. Makes them feel better, I'm sure. Anyway, great observation!
@@robertwalkley4665 (rather than just conducting them) If Joe actually learned from these interviews as to how they are applicable to daily thought and argument formulation, specifically in the realm of the scientific mathematical reasoning, he would have never start availing himself as some specialized microbiologist epidemiologist virologist….
Not shown here, but I loved how Rogan immediately jumped in with several razor-sharp rebuttals to Penrose’s thesis; especially with regard to the worrying epistemological implications of applying temporal elliptic curves to the integral space-time manifold projected in Hilbert space by the application of algebraic homotopy determinants.
Unless the referee is convinced by the set of steps you take to be true you didn't tap. No one can prove the truthfulness of the set of axiomatic/self-referential intermediate steps using the formal declaration of tapping itself. I think what you're referring to is called the observer effect not the self-referential fiasco that Godel proved.
You might have confused Schrodinger's cat-in-a-box thought experiment with Godel's incompleteness theorem. "Is the cat required to be an observer, or does its existence in a single well-defined classical state require another external observer?"
Especially for a lay person who's really interested in all that but will probably never have the time to really get into the nitty gritty of understanding the ins and outs of Principia and all the logic stuff.
I really get the impression that Penrose is doing his absolute best to explain the theorem to the lay-person (not just Rogan, but also the wider audience) and unfortunately, he completely lost me. I'm no pure mathematician by anyone's reckoning, but I have reasonable training in mathematics at a university level from studying physics and then from taking an interest in it beyond that. I still don't understand what he said. Penrose is a bit of a legend in the physics world, though. Honestly I didn't realise he was still alive to be able to give interviews at all.
I agree. You can't understand Goedel's theorem from Penrose's description. The problem with Goedel's theorem is that there's no simple easily stated example, unlike Fermat's theorem or Goldbach's conjecture, to simply show what the theorem is "about", let alone to show it's truth.
No, actually, it's quite simple. Mathematics is not and cannot ever be complete. Just let that soak in, and it doesn't matter what the proof itself consists of.
I wouldn’t say that Penrose “just explained Gödel’s incompleteness theorem.” He described it, but did not explain it at all. Douglas Hoffstaedter explained it, but maybe not in a way that’s accessible to everyone, in his book “Gödel, Escher, Bach; an Eternal Golden Braid.” It hasn’t aged well, but if you forget that computer languages used to be much more basic it still makes sense. That one book will transform your knowledge and understanding of the world, sciences, arts, and everything else. G-Plot - one of Hoffstaedter’s illustrations - is like witnessing the code of the universe. It’s a book that’s impossible to sum up, but which genuinely tries to make the most complex things simple enough to understand and appreciate.
I agree about that book. Superb and raised my understanding of life and intelligence to a whole new level. Douglas wrote a book about just Goedel that was not so good. After that I tried to find a good book about the theorem and failed. So then I tried to find a good book about the logic behind the theorem and failed. It turned out that at that time no-one had yet formally proven Goedel's theorem, and the logic books were filled with BS handwaving. This whole area of math caused me to take my career in the direction of computers instead of functional analysis, much as Penrose is talking about here. Sure did me a great favor in the long run.
When I remember, it is interesting to me how many of my intellectual interests find their first spark in Godel, Escher, Bach, which I read in 8th grade.
Roger Penrose is actually the first intellectual who got me started on this whole train of logic, incompleteness and halting problems. I was always interested in the study of logic, but by first listening to him talk about this matters is what deepened my curiosity. His books go into more detail and is a pure joy to read for logic/math/science nerds alike. I have immense respect for this man, just for the way he bridges the cutting edge of human thought with simplistic language, a feat not easy for mere mortals like us.
@@pavel672 His book Emperor's New Mind gets into a lot of things. Chapter 2 is almost a complete description of Turing's machine including one exercise for the reader lol.
I think this man is one of the 5 most brilliant people in the world, his papers are deep and profound and his books are little masterpieces of prose and logic.
The simplest ("word") explanation is: whatever system of (mathematical) rules you adopt, there will always be statements that can be constructed in accordance with those rules...and yet be unprovable BY those rules.
2:43 and further: this goes deeper than the average explanation of Godel's theorem: " You see this statement Godel comes up with, is something you can see on the basis of the same understanding that allows you to trust the rules, that it (the statement) is true. But it is not actually derivable by the rules. You see it's true by virtue of your believe in the rules"
Praise to gödel who saved us from the centuries spanning tyranny of mathematicians and logicians who espoused their indifferent, static and abstract proofs as a means to come to deeper knowledge which resulted in the complete disruption of human connectedness with our being en sui. It took a righteous logician, a truth-seeker in the deepest sense, to end their reign.
You can't use logic to defend logic. Wow. This is my new favourite quote. I've been frustratingly trying to explain this to people, and you just summed it up perfectly in 7 words. 👌
And I just realised. Defending logic with logic, is itself a self-reference paradox, that requires something outside of it (belief) this goes deeper than I realised.
I think the Incompleteness Theorem supports the philosophical view of mathematics as something that is constructed. It is more natural and less paradoxical in that conception of mathematics. Since we build things as we go along basically, that is we add axioms to make sense of a new mathematical object or what-not. (And I think this view has been internalized in a way. For example, Fermat's last theorem seemingly required a new kind of machinery in order to prove it. And when confronted with something that is difficult to prove, the mathematician now is convinced that it requires a "new" kind of mathematics. It's also been said in related fields like theoretical physics. How often the physicist would remark that gravity and quantum mechanics require a "new" physics. ) It is only kind of paradoxical in the realist view of mathematics, where the mathematician is conceiving a grand "theory-of-everything" as they used to, from which derives all theorems, in the way of Euclid. That realist view of an absolute conception of mathematics is in some tension with incompleteness. I think Godel's genius thrived in that tension. I couldn't imagine him as anything but a realist.
Godel himself was a platonist, and believed his theorem supported that metaphysics. If Mathematics cannot be exhausted by merely constructing formal systems, then there is some sense in which Mathematics is "beyond" any human construction. Personally, I don't think the theorem really has any effect on metaphysics. You can make up explanations either way.
@@APaleDot The first claim in your first sentence I will not dispute. The second claim, of course, I find interesting whether Godel really thought that way I don't know. But it's plausible. The second sentence is either vague or a non-sequitur. And the third sentence is a hasty generalization.
@@helioliskfire5954 The implication is there: "either mathematics is too big for the human mind, or the human mind is more than a machine." and here: "Intuition is not proof; it is the opposite of proof. We do not analyze intuition to see a proof but by intuition we see something without a proof."
Penrose: "Everything proceeds mathematically" Joe: "Wanna see a video of a one thousand pounds polar bear being killed by four Eskimo midgets using nothing but Inuit axes?"
That's awesome! I believe Kant's revolution in epistemology goes around that insight as well when it shows that theories (mainly philosophical ones) are incompatible as they are because reason is self-sufficient. Independently on which epistemic approach you come up with, no approach to the 'truth' of an object can ever exhaust the intelligibility of such object. Mind blowing eh?
@@Tauan That’s fantastic, thanks for sharing that. I heard a debate on epistemology recently and we’ve always assumed as brains get more complex (early mammals -> primates -> humans, etc) that our conceptual world model would become more accurate, and map ever closer to the actual, objective world. But research has shown that the only driver of brain complexity is reproductive success. So our shared intellectual model of reality has evolved for reproductive advantage only, and there’s little chance the actual, objective world bears any real resemblance to the one we’ve evolved to see (or experience). I can’t cite the source at the moment, so take my account of it with a grain of salt.
@@baTonkaTruck Hoffman found (in a computer simulation) that species that select for commensurability with "reality" die out and species with the most useful models survive.
I always thought of it as a fundamental “issue” with logic itself. Essentially “You cannot prove logic with logic.” It’s circular reasoning. Proving a theorem based solely upon the rules of logic that you’ve constructed. You create a box made of various rules, and you can “prove” things inside of the box, but you cannot prove the box.. without creating another box, outside of that box. .. That’s my understanding of it as best I can explain it, I apply the concept well outside of mathematics and see it as applicable to rational thought as a whole. Which is in part why I’ve embraced absurdism.
yeah, it's about relative perspective similar to the 1-way speed of light problem, or the notion of higher dimensions... you need a higher level perspective to demonstrate a proof
I think my favorite video explaining Goedel's Incompleteness Theorem is Veritasium's, he does a great job walking a layman through it. If you're actually looking to kind of play around with how this is more formally proven, I really like Raymond Smullyan's book To Mock a Mockingbird which is presented as a long series of logic puzzles using birds that culminates in a proof of the Incompleteness theorem. Goedel, Escher, Bach is another classic book that covers how self-reference is used by all three of them in their various masterworks.
or you could just googlescholar "biological annhilation" - the common claim by mathematicians is that "it works" - well if it works so well then why are we accelerating to "biological annihilation"?
@@Bodyknock Oh so you've never read Stuart Hameroff's quantum biology work in collaboration with Roger Penrose? That's what Penrose calls "protoconsciousness" and as Penrose emphasizes "calculations are not consciousness." Of course if you need to practice self-censorship by creating your own fake boundaries of thinking, that's perfectly common to do. haha.
@@voidisyinyangvoidisyinyang885 Again, Godel's Incompleteness Theorem has nothing, at all, to do with biology. It's a theory about infinite sets of first order logical statements. Biology is an inherently finite system of cells. There's no intersection between the two. (It doesn't even have anything to do with quantum mechanics or the definitions of consciousness either frankly.)
@@Bodyknock So your youtube troll comment just debunked the Royal Society (formally The Royal Society of London for Improving Natural Knowledge formed in 1660, is a learned society and the United Kingdom's national academy of sciences) Science book of the year award in 2016 for the quantum biology book "Life On the Edge" by Professor JohnJoe McFadden and Jim al-khalili? Wow! Impressive.
Great explaination but I think a key point that is not mentioned and that shows why this is mind blowing, is that this can be applied to all knowledge and not only maths, and that nothing can be proved to be true unless you have a set of base “rules” that you believe in , so theres a impossibility for 100% “real” or “checked”knowledge by definition. Similar to Munchausens trilem.
I’ve been slowly reading, comprehending, and trying to fully understand the entirety of this man’s Book “The Road to Reality: A Complete Guide to the Laws of the Universe” since I graduated High School. Fascinating stuff.
There's a pair of books called Godel Escher Bach that discusses this at length. Every time I read it, I thoroughly understand Godel's proof for about 2 hours. There are also other completely unrelated statements that you can prove are true but unprovable, involving (for example) relationships between infinite sets each member of which is an infinite set. The Halting Problem is probably an easier thing to understand but is the same concept.
Consider the sentence: "Roger Penrose does not believe this sentence is true." There are two cases: either Penrole believes it or he does not. Suppose he does, then it is false and Penrose believes a falsehood. Moreover, it is obvious even to Penrose that this is false, and in fact it is false BECAUSE he believes it, which is crazy. So suppose he doesn't believe it. Then it is true, and indeed OBVIOUSLY true, even to Penrose himself; yet he doesn't believe it. Of course, you can do this for anyone. We all have our own Penrose sentence. What Goedel showed was that you can even do it for arithmetic.
@@tomasmcelhoney4054 Sure, but now you are playing a different game (which has nothing to do with Gödel's theorem). My point was that you can take the same logic used by Gödel and apply it to Penrose himself (or anyone else). Penrose has written several long books arguing that it only applies to machines or formal systems, not to humans. Wrong. BTW, you don't escape the Gödel argument by introducing a third 'unknown' value, it just gets a bit more complicated. The question is "Penrose believes this sentence is neither true nor unknown".
Hmm... there is an underlying assumption here that a statement has to be, statically, either true or false. What if the statement doesnt have that dimension at all. It is like asking what color gravity is. It isnt relevant.
@@acobster Title is missleading. Penrose is not really trying to explain Goedel's theorem here, he is kinda summarizing it to make it easier to understand why it changed his way of thinking.
I've been studying Penrose' CCC for more years now, trying to disprove it and I literally can't. I've never come across an idea so bulletproof that I can't poke even a single hole in it. I genuinely think, despite my fear, that Penrose is correct
Weird. I've seen physicists poke holes in CCC all over the place. Not that they claimed to disprove it, but some of the implications of CCC do not seem to easily agree with our current models. That being said, I'm a fan of CCC and if it turns out to be correct it will be the most exciting new discovery in a very long time IMO
@@PetrCobra I'd very much like to know which physicists you're referring to, what they specifically said, and why you beleive that they're correct. Because I'm not entirely sure you've noticed this but every physicist will tell you that the laws of physics break down at a certain point and don't apply in the earliest stages of the universe. Meaning that they have literally zero business using physics to debunk anything outside of the purview of physics.
@@jaxwhyland I'm aware of that (terrible) idea. It would of course be dreadful if all the tragedies of the world played out again. Not to mention the awful thought of how many times I already typed this sentence. But there's no reason to think, and every reason to doubt, that anything would be the same twice? If the universe reemerged then the tiniest difference would change everything.
Respectfully, he is correct but makes it more confusing for non-mathematicians. Here is the gist: There exist true propositions in mathematical systems, at least as complex as arithmetic, but which cannot be proven to be so. Now, one can create a new system by adding axioms and rules to create a path of proof of that previously inaccessible proof, but now, that newly created system will have certain true propositions which cannot be proven to so; and so, ad infinitum. How godel went about proving the above result will require advanced mathematical background, but this deep result is broadly comprehensible by non mathematicians also.
To me, it's analogous to Cantor's diagonal argument for transfinite ordinals; just with respect to mathematic systems as a whole, not just for an individual set of numbers. Essentially, no matter what system of rules you create and impose upon a system, you can always create an ad hoc exception that satisfies the conditions of that system, but doesn't satisfy one or more of the rules of that system.
@@xxFortunadoxx "analogous" might be a bit of a stretch, but you're right in that in follows a similar line of construction. btw: the Turing halting problem is analogous to the incompleteness theorem and both are answers to Hilbert's entscheidungsproblem
He makes it more confusing for everyone (and this wasn't the only instance of it). Penrose, for all his great work in multiple fields (geometry, astrophysics, etc.), is a terrible teacher. He's clearly very good at _understanding_ things, but terrible at explaining them.
@@RFC-3514 hey...think you need to cut the guy some slack,he's not a young man and when he was, there wasn't this need for complicated theorem to be distilled into two minute sound bites...not everyone needs their teacher to be media savvy.
@@user-fb9os7hy2y - He's always been kind of like this, though. Even when talking about his own work (which I'm sure he understands). I think he's just not very good with language.
All reasoning ultimately traces back to faith in something that you cannot prove. Faith and Reason are not enemies. One is absolutely necessary for the other to exist.
Somewhere Godel is smiling because after all these decades people are still debating this and a lot of them are just as uncomfortable with it as his colleagues were when he first stated it.
So basically Penrose is saying : Godel’s first incompleteness theorem doesn’t say that Mathematics is incomplete, it says that either Mathematics is inconsistent, or it is incomplete. In other words, if you believe the system of axioms is consistent, and it is just a believe since according to Godel’s 2nd incompleteness theorem it can’t derive from the axioms, then it is incomplete. Well, this is my understanding of what Penrose just said.
I suspect that the above is less your understanding of what Penrose said than of what he should have said. He didn't mention the word "inconsistent" even once, so I strongly suspect you got that from somewhere else.
@@alqpr Indeed, it's probably very difficult to understand Penrose's explanation if you yourself have not studied first order logic. I take "believing in the system" to mean "assuming the particular axioms you're working with are consistent". One proves that the Gödel sentence is independent of the axioms by assuming the axioms are consistent. Moreover, one proves that the Gödel sentence is true (in the standard model) by assuming the axioms are consistent.
''So it's exactly like my brain interprets words and concepts but my interpretation is wrong'' - How would you ever know? Noone does? Reason and imagnination isnt enough. The most baseless stuff ever - Yet its right?
@@IDMYM8 But that's just a reflection. How sure are you that your reflection accurately gives you a representation (not an image) of your very own eyes?
and for the average joe to understand (and donald rumsfeld), there are knowable knowns, unknowable knowns, known unknowables, and unknown unknowables. basic axioms are not valid or invalid; validity is reserved for statements where inferences are made. axioms are assumed true because if you assume the axiom is false then you can prove anything by default.
@@Sam_on_TH-cam no. axioms are a given and assumed true for otherwise you can always produce a valid argument (using false premises). a sound argument is both valid and assumes the premises are true. cogent, may also mean sound.
@@TrombonePirate in a set that would contain say basic algebra, there are statements outside the set that are known to be true but cannot be proved from within the set (something like that). take the number "pi" for instance. it's outside the set of natural numbers but you "know" it exists but can't be proven true from within the set of naturals. if i'm wrong then please correct me.
@@ophello i hope not and i am pretty much layman too. I was suggested a good book by a good prof btw. Its called the princeton companion to mathematics. You may find it there. It has brief articles with suggested readings.
@@ophello i copied this from another comment. The LOGICAL structure of Godel's proof is simple. He tweaks "This statement is false" to make this: "This statement is unprovable." There are two possible truth-values for this: If the statement is true, then you have a true statement that is unprovable. If the statement is false, then the statement is provable, which means you have proof of a false statement. So any (sufficiently complex) mathematical-logical system is either incomplete (with statements you know to be true but can't prove) or self-contradictory (with false statements you can prove), or both.
The look on Joe's face is priceless, good on him! I recognize it because it was the same look on my face listening to WVO Quine trying to explain the same thing in a talk I attended in 1993.
Yes. I think so. There would be things in the system that the rules of the system cannot determine the truthfulness of or falseness of. This guy here does a better explanation. th-cam.com/video/HeQX2HjkcNo/w-d-xo.html
Another way of understanding Godel's incompleteness theorem is if you look at language as a mathematical concept. Languages can describe many things, but there is always something incomplete in the description. Take the sentence, 'the man was running'; does it describe the vigor of the run, does it describe the speed, or does describe the milieu of the man running. There are many things that are left out with each statement, but the general concept is still understood. Something is true but incomplete in its description.
The best part of Gödel incompleteness theorem is not really the consequences, but the proof itself. He created the very system using natural numbers and arithmetic operations. He made the system talk about itself and constructed the sentence. It's a "proof" of how we should explore self references much more deeply. And mathematicians tried to avoid it as much as possible due to obvious reasons if you're in the field.
"Sir Roger is number one on my list of greatest intelects of all times!" What he said about Godel's Theorem was horribly wrong. That does not matter to hero=worshipping sycophants.
@@kensandale243 At least in this video, what he said about Gödel's first incompleteness theorem is not actually horribly wrong. But to someone not already familiar with the theorem and its proof, it may not be all that clear. I've heard some people claim Penrose has said false things about Gödel's incompleteness theorems in other media, but for this video, at least, what he says can reasonably be interpreted in an accurate manner by someone familiar with the topic.
My favourite example of incompleteness is that it can be used to prove the halting problem (basically, can I show a piece of code to another piece of code to know for sure that the first piece will end rather than go on forever). The one problem I have with incompleteness in general is that it's a very... Abstract issue, it's unclear that the holes in the tapestry of math are anywhere near where we want to go, can go, but the halting problem/would/ have been incredibly powerful if we could have used it!
It is also interesting that the fact that the halting problem is undecidable isn't surprising, counterintuitive, or anything of the sort. On the contrary, I find it quite intuitive. So, why is it that we can easily accept this, yet we are taken aback by Gödel's theorem, even though the two are deeply connected (and perhaps almost equivalent)? I first read about Gödel in Penrose's book 30 years ago. And Penrose is no fool. However, I feel he might be mistaken on this matter. I don't believe his conclusion that human intelligence is something more than what can be formalized in a computer is accurate. It seems more like a mirage stemming from the challenge of interpreting Gödel correctly.
No, it says in any formal system (if it has a certain strength) there will be true statements which you can not show to be true by starting with your axioms and following your rules of inference
you do not assume that they are unfalsifiable, you assume they are valid so you can build upon them when there is no way to inherently prove their validity. By assuming certain initial axioms, you can build a consistent system without ever really proving those initial axioms
But why must there be a true statement that doesn’t follow from the axioms if it’s a closed system? Perhaps I used the word unfalsifiable incorrectly, but does this all have to do with the fact that the axioms themselves cannot be proven to be true - they’re just the building blocks, not something to be proven or disproven
@@jon553 >But why must there be a true Statement that doesn't follow from the axioms if its a closed system Basically, because Gödel showed us a way to construct at least one statement in any formal system which has to be true but has no prove in the system. If you want to know how he did it, or what the way is to construct that statement, I would recommend listening to a 1 hour lecture by Douglas Hofstadter here on TH-cam. > Does it alle have to do with the fact that the axioms can not be proven No. Even if you constructed a formal system based on self-evident truths or truths proven by a meta formal system, if it has a certain strength, Gödels Theorem applies.
Just the fact that Joe has world-class-mathematician guests who discuss things like _Incompleteness_ is uplifting. As other guests have said, including Jordan Peterson and Elon Musk, we don't know for sure what will come of all this access to knowledge... but it will almost certainly be something amazing.
Uhm, why? Not every talk show was Late night with David Letterman before podcasts. Penrose has appeared on many talk-shows on TV as well as radio-shows to discuss his work and philosophy. The knowledge was always there, available to you. The interest was obviously lacking.
@@Bollibompa i don't think it was merely just uninterest. a smart phone nowadays can be had for like $50. use free McDonalds Wifi, and a lot of people who might not have had the money necessary to get cable can watch things on TH-cam. while there's obviously the "read a book" club, many people find videos to be a lot more ingestible. the simplification of complex topics is not just the "everyone seems dumberzz" phenomenon, but also leads a lot of people who did not have the resources prior to developing an interest in topics they wouldn't have had otherwise. granted, it leads more people to misinformation, but i really do think people are generally more intelligent when misinformed than uninformed. granted i wish it was neither, but i do think its a step up.
@@ethanstump I don't really think that the stoner tidbits dished out on this caveman-show do anything but mildly amuse most. An actual interest in Penrose's teachings requires applying yourself. Being spoon-fed one-liners and easily digestible nuggets of pure genius is not profound.
@@Bollibompa it's not profound, no, but it's not nothing. while there is active learning, there is passive learning even from encountering something new. sure, most will get nothing out of it, but even if only one Pakistani Stem highschooler gets something out of it, that's still something. i personally stopped watching rogan maybe a good four years back, but even watching this was a step up from reading the bible with my Christian fundelmentalist family when i was younger. i would much rather Rogan have Penrose again than Jordan Peterson.
“Is God willing to prevent evil, but not able? Then he is not omnipotent. Is he able, but not willing? Then he is malevolent. Is he both able and willing? Then whence cometh evil? Is he neither able nor willing? Then why call him God?” ― Epicurus
@@LOCATIONREDACTED Even the supposed text doesnt have that meaning. Epicurus was of the position that God/s dont interfere in earthly matters. That was the point of the riddle.
(GODEL) "On Formally Undecidable Propositions of Principia Mathematica and Related Systems," in 1931. Here is a summary of his results: Any consistent axiomatic system of mathematics will contain theorems which cannot be proven. If all the theorems of an axiomatic system can be proven then the system is inconsistent, and thus has theorems which can be proven both true and false.
I mean, I have a maths degree, I like high level maths, and this barely made sense to me. The concept of Godel's theorem may be tricky (and widely misunderstood) but too many people are impressed by what sounds complicated, rather than what is actually insightful, which requires you to understand. The most brilliant people can put things in simple terms a layman can understand. Rogan probably has no idea what Goldbach's conjecture is or Fermat's Last Theorem, so how would he follow the rest? Politely listening is not the same as understanding.
Heh well to be fair I don't know if many people are claiming that Rogan understands what is being said here... but that said, I'd also say that this video does not encompass an explanation of Godel's incompleteness theorems... so the only people who really understand what's being said here already know the subject. But Penrose as always is still doing a good job of mentioning what is necessary to give a layman a rough intuition of how the ideas connect.
@@jhansenhlebica6080 I disagree. I think his explanation is atrocious. Being smart does not guarantee being a good teacher. Honestly he seems terrible, based on this clip. It is a confusing topic though and seldom gets explained well anyway.
@@stevekru6518 Feynman wasnt on Godels' or Einsteins level. He was a genius spefically on his field, a one trick pony. Philosophy was beyond him, a very smart brute he was.
A negating self-referencing proposition is incomplete or contradictory since the logical outcome implied in the proposition either confirms the fact that the proposition can't be proven or contradicts the intent of the proposition.
I am no fan of Joe Rogan, but I might have to check this interview out. Penrose seems like precisely the lovely and intelligent person I’d imagined him to be from his mathematical discoveries!! He’s too often overlooked in my opinion
Read "Godel, Escher, Bach: an Eternal Golden Braid" by Douglas Hofstadter. First edition perhaps 1979. While it may further twist your mind and make you more hungry greater, he has several books to follow up!
The famous Gödel number (G) that Gödel came up with in his proof seems to correspond to: "This statement can't be proven." Assume that G is false. Then, G must be provable and hence, is true. The only assumption we made resulted in a contradiction. Therefore, we accept the inverse of the false assumption: G is true. That makes sense. But, how come the statement refers to itself (self-reference)?
The complete statement is "There is no proof for the statement of Godel number g", and it turns out that the Godel number of this statement is exactly g. If g is provable it means that g is true, and then it means that "There is no proof for the statement of Godel number g" which is a contradiction. If g is not provable, it means that g is here the incompleteness theorem is proven; there is a true statement that is not provable. "This statement can't be proven." is as you pointed out leads to a trivial contradiction (because of trivial self-referencing) that doesn't need all the hustle, however it is not an organic statement that could be (simply) written in the Godel system. So the genuineness is finding the g that hold this self-referencing idea that is encompassed in your naturally expressed as "This statement can't be proven.".
@@truebomba thanks so much! This clears up one major obstacle: A statement can refer to any other statement (or itself) using its unique number. If you wouldn't mind, here is a follow up question: how can a statement express the fact that a statement S "cannot be demonstrated?" I can only guess that somehow it is shown that the rules of deduction wouldn't reach S from any true statements. Any further insights on this?
@@BulentBasaran What is proved is that No consistent system of axioms whose theorems can be listed "by a sequence of symbols" is capable of proving all truths about the arithmetic of natural numbers. As a disclaimer, I kinda explain things loosely because I never read all the details of Godel's proof. After his proof, there are many other proofs and ways to see this exact same result. Morally you can imagine the following; You have m mathematical symbols that are the bases of all possibles (listed) statements within our axiomatic system, theses all possible statements (true or false) and could be represented as polynomial P_n(x_1,....,x_m), where n is the n em statement on our listed propositions. Now we have to define provability, which is kinda difficult to present here, but you can imagine that as stepping a layer above (it is like passing from the category of sets to the category of Powersets, but keeping in mind that we enforce "countability" on this upper layer as well because that's what ensure that our proofs are sequential/"algorithmic" series of statements that leads to the statement we want to prove). With all this rigorous organization of our axiomatic system, a listing of our all possible propositions (within the system), which btw also contains the statements that could be considered as proofs (according to our rules of proofs) for eventually other statements within the same list (this is where the part that the statements are made on natural numbers is important), and our rules of proofs, Godel proved that regardless of the starting axioms or how many there is always a number g that holds the self-referencing property I explained. In another way, we have a statement P_g(x_1,....,x_m) listed somewhere in our list of statements. P_g(x_1,....,x_m) has a formal expression that could be found in this paper for example www.ams.org/notices/200604/fea-davis.pdf
In simpler terms: The law of non-contradiction can not be proven because a proof would assume the law itself. Aristotle did it for logic millennia ago, Kurt Godel did for mathematics in the 20th century. In other words, Mathematics has assumed principles based outside of mathematics itself, and outside of science itself as well.Since, science uses math to measure statements.
No, that's a variant of the Liar's Paradox encoded in a formal language, creating a system that's either contradictory, or consistent but incomplete. It has nothing to do with the fact that you can't prove your own logical axioms, that's why they are axioms.
@@nektariosorfanoudakis2270 The Liar's paradox is no different than speaking of a square-circle, sir. The law of non-contradiction is not a logical absurdity, but a demonstration of the limits of reality, and how reality is based out of the law of non-contradiction.
In grammar school in the late 50s I was very good at mathematics. The headmaster wanted me to apply to Manchester University, his alma mater I think, to study for a mathematics degree there. I had set my sights on a career in Engineering and followed that degree route instead. Of course, my mathematical bent was a great help to me in my Engineering career, which I enjoyed, spanning 50 years but I often think about my decision back then. The idea of sitting in my university study as a Professor surrounded by untidy piles of books seems like an attractive proposition, but I'm sure it's not like that at all. I'm not even sure if I would have been able to grasp some of the concepts that geniuses like Roger Penrose talk about if I'd made that different decision. Most Engineering mathematics involves the use of Newtonian mechanics plus other science and mathematics that had their foundations taught at school. I say "most" because the really clever stuff probably requires more brainpower than I possess.
This is an extremely nuanced and precise, concise explanation of the theorem, but I don’t think it’s the best explanation for someone who has no prior experience with higher math.
it's like... mathematics is a system for checking results. in that system, you can create subsystems for checking subsets of results. however, if you try to create a subsystem that checks all results, then that subsystem cannot be checked. so, our primary knowledge of mathematics must be primitive, not the product of any system.
It's so crazy to me that Joe Rogan, of all people, had Roger Penrose on his podcast. Or maybe it's crazy to me that Roger Penrose would do a podcast like this. What a world we live in.
Gödel's Incompleteness Theorem: A Mathematical Corollary of the Epistemological Münchhausen Trilemma Abstract: This treatise delves into the profound implications of Gödel's Incompleteness Theorem, interpreting it as a mathematical corollary of the philosophical Münchhausen Trilemma. It elucidates the inherent constraints of formal axiomatic systems and mirrors the deeper epistemological quandaries underscored by the Trilemma. --- In the annals of mathematical logic, Kurt Gödel's Incompleteness Theorem stands as a seminal testament to the inherent constraints of formal axiomatic systems. This theorem, which posits that within any sufficiently expressive formal system, there exist propositions that are true but unprovable, has profound implications that reverberate beyond the confines of mathematical logic, resonating in the realm of philosophy. Specifically, Gödel's theorem can be construed as a mathematical corollary of the Münchhausen Trilemma, a philosophical paradigm that underscores the dilemmas in substantiating any proposition. The Münchhausen Trilemma, named after the Baron Münchhausen who allegedly extricated himself from a swamp by his own hair, presents us with three ostensibly unsatisfactory options for substantiating a proposition. First, we may base the substantiation on accepted axioms or assumptions, which we take as true without further substantiation, a strategy known as foundationalism or axiomatic dogmatism. Second, we may base the substantiation on a circular argument in which the proposition substantiates itself, a method known as coherentism or circular reasoning. Finally, we may base the substantiation on an infinite regress of reasons, never arriving at a final point of substantiation, a path known as infinitism or infinite regress. Gödel's Incompleteness Theorem, in a sense, encapsulates this trilemma within the mathematical world. The theorem elucidates that there are true propositions within any sufficiently expressive formal system that we cannot prove within the system itself. This implies that we cannot find a final substantiation for these propositions within the system. We could accept them as axioms (foundationalism), but then they would remain unproven. We could attempt to substantiate them based on other propositions within the system (coherentism or infinitism), but Gödel's theorem demonstrates that this is unattainable. This confluence of mathematical logic and philosophy underscores the inherent limitations of our logical systems and our attempts to substantiate knowledge. Just as the Münchhausen Trilemma highlights the challenges in finding a satisfactory basis for any proposition, Gödel's Incompleteness Theorem illuminates the inherent incompleteness in our mathematical systems. Both reveal that there are boundaries to what we can prove or substantiate, no matter how powerful our logical or mathematical system may be. In conclusion, Gödel's Incompleteness Theorem serves as a stark reminder of the limitations of formal axiomatic systems, echoing the philosophical dilemmas presented by the Münchhausen Trilemma. It is a testament to the intricate interplay between mathematical logic and philosophy, and a humbling reminder of the limits of our quest for knowledge. As we continue to traverse the vast landscapes of mathematics and philosophy, we must remain cognizant of these inherent limitations, and perhaps find solace in the journey of exploration itself, rather than the elusive, final destination of absolute truth. GPT-4
I think that, this incompleteness theory is fundamental to understanding a little bit more about the way in which Humans make discoveries on things that we cannot see. Basically the magic of turning mathematical abstraction to real life applications. Also explains well how there is so much that we don’t know yet, in our journey to have more control over earth and the universe.
As with all formal proofs, this is not about how humans do things, or even about how this universe does things. Formal proofs, if correct, are correct in any universe and no universe.
No, you obviously don't understand it at all. Firstly a statement in a logical system can be true and provable and still unkown. A statement having no proof in the system doesn't mean you can't know it. And science is not just mathematics. You don't know something about the real world, because you are simply lacking information about the real world and you need to go out and collect it. That has absolutely nothing to do with Incompleteness.
I'm not sure this really qualifies as an explanation of "Godel" "incompleteness" or "theorems" but I bet it's an interesting listen for people interested in this man and familiar with it
Omg. God Bless your Heart Roger Penrose. I love you like the uncle I never wanted. It took you three minutes to say what's obvious to all after 90 years of consideration on the topic. We cannot prove irrational or infinite queries. There are many queries that while finite still cannot be effectively calculated do to the sheer computational requirements to do so. Chaos, or the math of iteration, does not allow for vast computation for a number of reasons. One, is precision. The three body problem. The other is the sheer rate in which decision trees branch. Iteration in which each step produces exponential growth of the system. TSP NP-Hard problems.
I think you need a bit more time to explain this in a way that the layman (eg me) can understand. I remember seeing this explained with cards by I think veritasium, and it clicked with me in a way that this explanation didn't. Not Mr Penrose's fault, it's not just something you can explain in 3 minutes IMHO, at least not to someone who isn't a mathematician.
With something like this theorem, you want a good visual to make it understandable. Once you understand the visual, you understand Penrose's explanation
The absence of even an attempt to state Godel's theorem makes this video less valuable. There are actually two incompleteness theorems that relate to each other. It is an elaboration of the famous "Liar's Paradox". I also missed a concise summary of the implication of Godel's theorem. Let me make an attempt: "In any formal system powerful enough to be interesting, there are either true theorems that cannot be proven or theorems that may be proven both true and false." A formal system that has even one theorem that may proven both true and false is "inconsistent" -- an absolutely fatal flaw, because that one inconsistent theorem allows any theorem at all to to be constructed and proven "true". Thus, the more interesting outcome is that there is truth outside the realm of proof of every formal system, including of course all mathematics. This concise little gem was Mr. Godel's response to Bertrand Russell, who was attempting to prove that mathematics itself was both formal and complete. The philosophical dispute has to with "reality", lived experience, and observation. In Bertrand Russell's formulation (shared by great many others), we live in a "clockwork" universe where no observation is never needed or trusted -- once the initial conditions are set, the universe unfolds from that moment forward by simply following its rules. Like clockwork. Any observation that contradicts the formalization of the system is a false observation -- and so observation is irrelevant to truth. This was anathema to Godel, and his marvelously terse incompleteness theorem devastatingly destroys the edifice Bertrand Russell and his colleagues were attempting to construct.
@@talastra : Oh yes indeed I am. To me, it is miraculous and inspiring to see this result emerge completely from within a formal system (any formal system!) with no need for magical beings.
@@thomasstambaugh5181 Formal systems are the opposite of "objective" reality, and the more formal they are, the more this is true. If incompleteness is the case, and it is, then objective reality is unprovable. Even the statement, "O yes, I am" becomes untenable. :)
@@talastra : Understood. I'm talking about the various demos (many derived from GEB) that show the limits of any formal system. I think, though I'm not sure, that this is related to the various demos that show "random" vs "chaotic" processes. In biology, it can be seen in the way that lipid bilayers form themselves in a liquid. Then the way that autocatalytic sets operate within the microenvironment created by that membrane. The very building blocks of life itself, emerging from deterministic processes -- no magic required.
@@kensandale243 Just because his explanation left much to be desired, it doesn't make him a fool? The guy's resume speaks for itself. So fucking cringy when people, like yourself, feel the need to speak in such an inflammatory way
@@andrewmcpherson8441 "ust because his explanation left much to be desired, it doesn't make him a fool? " No, the fact that he says foolish things makes him a fool. "The guy's resume speaks for itself." You are impressed by things you should not be. If someone is an idiot but has an "impressive" resume, it does not mean he is not an idiot. It means resumes are not always accurate. Einstein's resume right after grad school was that he graduated in the lower half of his class, and was the only student who could not find a job. So, logically, you should have thought he was worthless. Are you willing to be logical? What matters is the quality of a person's thought, not his popularity among his peers.
@@kensandale243 Your logic is flawed in suggesting that I would dismiss Einstein as a fool based on his school attainment. By Penrose's resume I'm simply meaning his body of work; y'know, the things he's legitimately achieved?! My opinion of Penrose has sweet F.A. to do with his popularity among his peers. All you did was come bumbling in, dismiss him as a fool, without offering any elaboration as to why you think he is such. Might I suggest you actually explain why you think he's wrong? Bonus points for doing it with a bit of respect, rather than just insulting someone with no explanation supplied. I'm always a willing ear for someone actually giving an explanation about WHY they think someone's work or thinking is problematic, but when someone just calls someone a fool and leaves it at that, wtf do I have to work with?
@@andrewmcpherson8441 "By Penrose's resume I'm simply meaning his body of work; y'know, the things he's legitimately achieved?! " Oh really? Are you on a position to evaluate Penrose's work? How well do you know Relativity?
I enjoyed getting enough understanding about Godel's incompleteness theorem from a gifted lecturer so I could pass an undergraduate mathematical logic exam in the 1970s. But I've never understood anyone else's explanation of what Godel was doing. And now in my late 60s it's far too late to worry about it. But I still wonder how I could understand enough as an undergraduate to have some kind of a handle on what was going on.
This reminds me of that time when I read Shakespeare to a pigeon.
🤣
Roger Penrose Spoke about Gödel's Proof for Three Minutes would be an accurate title. He "explained" nothing.
lmao
This might be the greatest comment I’ve ever seen
@@meofamily4 he did tho
I'll never forget the first time I met this brilliant man. I was an undergrad at Oxford and all I had with me was a mathematics textbook and green pen. I asked him to sign it and he happily obliged. I still open it almost weekly to see the short, wonderful message inside. It said "All the best, Joe Rogan".
I got belaired so bad I'll never escape the shame cube.
Well done.
xd
@@dexterwestin3747 Indeed, what was yours again?
So that is true but we can't prove that it's true?
Throughout that entire speech, All Joe is thinking is - "I could take this dude, easily."
😄
🤣😂
Throughout that entire speech, all Joe is thinking is about a bj he got from a DMT entity
Haha cmon Roger Penrose
was 89 during this interview.
@@Fascistbeast makes it easier for Joe
If there is one nuanced intellectual on the planet with which you want to discuss high level pure mathematics, it's Joe Rogan.
LOL
I feel Joe would approve of that joke at his expense.
He has had several interviews with and about Penrose's work, you prancing dunce 🙄
Exactly! Because Joe Rogan puts it on TH-cam for all of us to enjoy!
I've never seen Joe so lost for words in this interview !
Actually: If there's one nuanced intellectual ... , *then* it's Joe Rogan.
I appreciate that Joe just lets Penrose talk uninterrupted to complete his thought. The interview is about the interviewee.
You can’t interrupt someone if you don’t know what they’re talking about.
@@roccodimeo3271 comment of the century.
@@roccodimeo3271 hahahaha!
Rogan is a meathead but he’s fairly close to average intellect, he can recognize when he cannot say anything contributory. It’s not evident with normal guests but Penrose is a genius on Einstein’s level and beyond. Rogan felt it.
No one interrupts when they don't know what a person is talking about, but one who is extremely well versed in a subject will also not interrupt, as they do not need to. We know which side joe is on with regards to maths, but at least he's not like other know it all hosts.
I've briefly met this man. I had him sign my *pre-algebra* textbook. Then I completed my undergrad in Math. That textbook sits proudly on my shelf.
Everyone thinks you mean Roger Penrose, but I know you mean Joe Rogan. 😁
Lucky.
"I've briefly met this man. I had him sign my pre-algebra textbook. Then I completed my undergrad in Math. That textbook sits proudly on my shelf"
You have a degree in math...and you are clueless that Penrose made a fool of himself in this video.
@@kensandale243
How did he make a fool of himself?
I saw him once giving an excellent keynote talk and still insisted on using those old projectors with hand drawn slides.
Penrose: *finishes explaining mathematical theorems*
Joe: I too think that Conor will win the trilogy
... do you want to come over and do some DMT?
@@jengleheimerschmitt7941 To be fair, a mathematician on DMT would be very interesting. Since the geometry that you see seem to be related to the internal structure of the brain as perceived by your consciousness.
😂😂😅
Joe Rogan talks to Roger Penrose, what the hellll is going on? Can Joe Rogan understand 1% of what Penrose talking about? It is like Denis Rodman having a conversation with Einstein.
These interviews are for Jamie who got an A in physics.
This is like when you accidentally wander into a zone 90 levels too high in an mmo
True unless you are smurfing in that case it can't be proven
lmfao
Hahahhahaa
Joe’s face in the first few seconds says it all.
hahahah
that's the face of breathing through your mouth
He looks like he's about to fall asleep!
I lost it
hahahah!
The LOGICAL structure of Godel's proof is simple. He tweaks "This statement is false" to make this: "This statement is unprovable." There are two possible truth-values for this: If the statement is true, then you have a true statement that is unprovable. If the statement is false, then the statement is provable, which means you have proof of a false statement. So any (sufficiently complex) mathematical-logical system is either incomplete (with statements you know to be true but can't prove) or self-contradictory (with false statements you can prove), or both.
Well that's the easy bit. The ingenuity for which he is celebrated lies in making such statements equivalent to statements of number theory.
So is the philosophic implication that: a "system", may never be able to completly assess it's self, or to word it differently, one can never completely assess a system from within the system? A system being anything from a system of formal logic to a universe.
@@DavidKolbSantosh Yea I think so too. Specifically, the system can't completely assess itself the way it wants to using its own rules. Kind of like the human brain trying to understand itself using the knowledge understood by the human brain.
This is way better than Roger Penrose explanation
@@edwardjones2202 Absolutely right. I understand WHAT Godel did to implement his proof, but I don't yet understand HOW he worked it through. In particular, I've wondered whether he proves merely that some self-referential statements (such as the one I stated above; not original with me) entail this uncertainty--falsity vs. incompleteness--or whether his method of proof somehow gets around this limitation so that the proof potentially applies to any meaningful statement that can be asked within the system. Surely though the latter cannot be the case, or Godel's proof would not be the big deal that it is.
I remember going through the proof in my logic class. We spent 2 full long lectures establishing the coding system and learning how it works. Then, in like 15 minutes, the professor writes the code that translates to "This statement cannot be proved." And proves that the fact this statement can be encoded in a system (and very simple systems can encode it) means that the system cannot be complete.
th-cam.com/video/SOWt2fBI1VI/w-d-xo.html
But the question is why go through the effort of concocting this coding system? Couldn't this proof have been just conceptual?
@@ihsahnakerfeldt9280 I think the 200 level course on logic did that. I took the 400 level and we did all the details. Not really sure I got any benefit out of those details though. I'm kind of with you.
Thats a better explanation.
@@ihsahnakerfeldt9280 no because you have to use the same symbology as the system you are trying to demonstrate is incomplete. It must be done on its own terms to stand mathematical scrutiny, although a conceptual explanation of the proof is then possible. Its a matter of rigour.
You can mock Joe for not knowing every thing that every guest is really good at. Yet he gives them the space, and gives us the opportunity to see these kind of things.
You get it.
Your comment made me realize that a lot of people here are slinging hate at Joe because they too have no fucking clue what Penrose is talking about. Makes them feel better, I'm sure. Anyway, great observation!
Quite, Joe is great in the chair
This is when Joe Rogan is and was at his best. We're getting less and less of it over time though on the JRE.
@@robertwalkley4665 (rather than just conducting them) If Joe actually learned from these interviews as to how they are applicable to daily thought and argument formulation, specifically in the realm of the scientific mathematical reasoning, he would have never start availing himself as some specialized microbiologist epidemiologist virologist….
glad he cleared that up
Gosh I love this comment hahA
lmao
kek
Lmao
Not shown here, but I loved how Rogan immediately jumped in with several razor-sharp rebuttals to Penrose’s thesis; especially with regard to the worrying epistemological implications of applying temporal elliptic curves to the integral space-time manifold projected in Hilbert space by the application of algebraic homotopy determinants.
Full video where?
Next semester youll being talking about Gordon Wood...
@@DG-kr8pt But sure, embarrass my friend and go park the car in Harvard Yard.
He spent the entire 3:38 holding in the hit of DMT he took right before.
I had Bertrand Russell in da back ov my fucking cab de ovver day ,,,,off to Kensington of course!!
Joe's thoughts: 'for the life of me I'm not even sure if he's speaking english!'
🤣🤣
He sounded like he was speaking in some kind of British dialect
thousand cock stare
Joe is by no means a brilliant man, but he was smart enough to listen and not interrupt.
He was asleep... didn't you hear the snoring in the background ? 🙄😂
Joe Rogan : So you are saying that you can tap someone out but you can not really prove it unless the referee is watching it?
this fucking ended me
Unless the referee is convinced by the set of steps you take to be true you didn't tap. No one can prove the truthfulness of the set of axiomatic/self-referential intermediate steps using the formal declaration of tapping itself. I think what you're referring to is called the observer effect not the self-referential fiasco that Godel proved.
I'm dying. Lol
Penrose could have tried this for Joe: There are legal moves in a match that can't be foreseen from the rules.
You might have confused Schrodinger's cat-in-a-box thought experiment with Godel's incompleteness theorem. "Is the cat required to be an observer, or does its existence in a single well-defined classical state require another external observer?"
Veritasium owned this in his video. Best explanation I've seen.
Facts. Great video.
Came here after seeing that video
Especially for a lay person who's really interested in all that but will probably never have the time to really get into the nitty gritty of understanding the ins and outs of Principia and all the logic stuff.
@@spacevspitch4028 Have you read Principia? 😯
@@atomknife9106 Hell no 😄. I've feathered through the pages before. It's a gorgeous tome but damn 🤯
I really get the impression that Penrose is doing his absolute best to explain the theorem to the lay-person (not just Rogan, but also the wider audience) and unfortunately, he completely lost me. I'm no pure mathematician by anyone's reckoning, but I have reasonable training in mathematics at a university level from studying physics and then from taking an interest in it beyond that. I still don't understand what he said.
Penrose is a bit of a legend in the physics world, though. Honestly I didn't realise he was still alive to be able to give interviews at all.
I agree. You can't understand Goedel's theorem from Penrose's description. The problem with Goedel's theorem is that there's no simple easily stated example, unlike Fermat's theorem or Goldbach's conjecture, to simply show what the theorem is "about", let alone to show it's truth.
Physicist here too, still a bit of a hard time following Penrose's explanation on this.
"You see it is true by virtue of your belief in the rules."
@@BrucknerMotet "It depends what you mean by true"
JBP
No, actually, it's quite simple. Mathematics is not and cannot ever be complete. Just let that soak in, and it doesn't matter what the proof itself consists of.
I wouldn’t say that Penrose “just explained Gödel’s incompleteness theorem.” He described it, but did not explain it at all. Douglas Hoffstaedter explained it, but maybe not in a way that’s accessible to everyone, in his book “Gödel, Escher, Bach; an Eternal Golden Braid.” It hasn’t aged well, but if you forget that computer languages used to be much more basic it still makes sense. That one book will transform your knowledge and understanding of the world, sciences, arts, and everything else. G-Plot - one of Hoffstaedter’s illustrations - is like witnessing the code of the universe. It’s a book that’s impossible to sum up, but which genuinely tries to make the most complex things simple enough to understand and appreciate.
Sounds like a bunch of hokum to me. A bit like those post modernist clowns. You know, Derrida and whatnot.
I agree about that book. Superb and raised my understanding of life and intelligence to a whole new level. Douglas wrote a book about just Goedel that was not so good. After that I tried to find a good book about the theorem and failed. So then I tried to find a good book about the logic behind the theorem and failed. It turned out that at that time no-one had yet formally proven Goedel's theorem, and the logic books were filled with BS handwaving. This whole area of math caused me to take my career in the direction of computers instead of functional analysis, much as Penrose is talking about here. Sure did me a great favor in the long run.
When I remember, it is interesting to me how many of my intellectual interests find their first spark in Godel, Escher, Bach, which I read in 8th grade.
@@talastra you must be super smart
@@Hubertusthesaint Maybe you! :)
Roger Penrose is actually the first intellectual who got me started on this whole train of logic, incompleteness and halting problems. I was always interested in the study of logic, but by first listening to him talk about this matters is what deepened my curiosity. His books go into more detail and is a pure joy to read for logic/math/science nerds alike. I have immense respect for this man, just for the way he bridges the cutting edge of human thought with simplistic language, a feat not easy for mere mortals like us.
Hey, wouldn't have expected Penrose to have written on logic and halting problems. Any reading recommendations where he gets into the topics?
@@pavel672 His book Emperor's New Mind gets into a lot of things. Chapter 2 is almost a complete description of Turing's machine including one exercise for the reader lol.
@@abhishekshah11 Great! Thx :)
It's gives me goosebumps when I think about Gödel Church Turing and how computer science came to be
@@Artaxerxes. Hilbert: Mathematics is complete, consistent, and decidable!
Gödel Church and Turing: We're about to fuck this mans day up big time...
I think this man is one of the 5 most brilliant people in the world, his papers are deep and profound and his books are little masterpieces of prose and logic.
What work of his would you recommend as an introduction?
Rodger Penrose was quite interesting too😁
Who would you say are the other 4?
He is the smartest along with Higgs
@@Hates-handleStuart Hameroff
Peter Shor
David Deutsch
Gregory Chaitin
...imho 🤗
I greatly enjoy Penrose's style of discourse and presentation. A venerable thinker, calm and polite.
The simplest ("word") explanation is: whatever system of (mathematical) rules you adopt, there will always be statements that can be constructed in accordance with those rules...and yet be unprovable BY those rules.
2:43 and further: this goes deeper than the average explanation of Godel's theorem:
" You see this statement Godel comes up with, is something you can see on the basis of the same understanding that allows you to trust the rules, that it (the statement) is true. But it is not actually derivable by the rules. You see it's true by virtue of your believe in the rules"
Yes exactly, you can't use logic to defend logic.
Praise to gödel who saved us from the centuries spanning tyranny of mathematicians and logicians who espoused their indifferent, static and abstract proofs as a means to come to deeper knowledge which resulted in the complete disruption of human connectedness with our being en sui. It took a righteous logician, a truth-seeker in the deepest sense, to end their reign.
Most people miss this. Belief trumps Logic. Lol
You can't use logic to defend logic. Wow. This is my new favourite quote. I've been frustratingly trying to explain this to people, and you just summed it up perfectly in 7 words. 👌
And I just realised. Defending logic with logic, is itself a self-reference paradox, that requires something outside of it (belief) this goes deeper than I realised.
I think the Incompleteness Theorem supports the philosophical view of mathematics as something that is constructed. It is more natural and less paradoxical in that conception of mathematics. Since we build things as we go along basically, that is we add axioms to make sense of a new mathematical object or what-not.
(And I think this view has been internalized in a way. For example, Fermat's last theorem seemingly required a new kind of machinery in order to prove it. And when confronted with something that is difficult to prove, the mathematician now is convinced that it requires a "new" kind of mathematics. It's also been said in related fields like theoretical physics. How often the physicist would remark that gravity and quantum mechanics require a "new" physics. )
It is only kind of paradoxical in the realist view of mathematics, where the mathematician is conceiving a grand "theory-of-everything" as they used to, from which derives all theorems, in the way of Euclid. That realist view of an absolute conception of mathematics is in some tension with incompleteness.
I think Godel's genius thrived in that tension. I couldn't imagine him as anything but a realist.
Godel himself was a platonist, and believed his theorem supported that metaphysics.
If Mathematics cannot be exhausted by merely constructing formal systems, then there is some sense in which Mathematics is "beyond" any human construction.
Personally, I don't think the theorem really has any effect on metaphysics. You can make up explanations either way.
@@APaleDot The first claim in your first sentence I will not dispute. The second claim, of course, I find interesting whether Godel really thought that way I don't know. But it's plausible.
The second sentence is either vague or a non-sequitur.
And the third sentence is a hasty generalization.
@@helioliskfire5954
The implication is there:
"either mathematics is too
big for the human mind, or the human mind is more than a machine."
and here:
"Intuition is not proof; it is the opposite of proof. We do not analyze intuition to see a proof but by intuition we see something without a proof."
Maybe the universe is just a paradox and that's all there is to it
There's a chapter in Penrose's book The Emperor's New Mind dedicated to Göedel's theorem, worth reading. Not only the chapter, the whole book.
The follow-up book, Shadows of the Mind, goes more deeply into the argument mostly to fend of the critics of the first.
What you see...a rather gentle old man.
What you don't see...one of the greatest minds of his generation.
Books and covers...and all that.
I know man, I love Joe Rogan too. Too bad we couldnt see him all that much because of the old man.
Penrose: "Everything proceeds mathematically"
Joe: "Wanna see a video of a one thousand pounds polar bear being killed by four Eskimo midgets using nothing but Inuit axes?"
Where's that video?
"It's entirely possible."
Penrose : "fuck yes, and hoi me a blunt over you Bogart."
Alan Watts talked about Goedel, and described his theorem very elegantly: “No system can define all of its own axioms.”
That's awesome! I believe Kant's revolution in epistemology goes around that insight as well when it shows that theories (mainly philosophical ones) are incompatible as they are because reason is self-sufficient. Independently on which epistemic approach you come up with, no approach to the 'truth' of an object can ever exhaust the intelligibility of such object. Mind blowing eh?
@@Tauan That’s fantastic, thanks for sharing that. I heard a debate on epistemology recently and we’ve always assumed as brains get more complex (early mammals -> primates -> humans, etc) that our conceptual world model would become more accurate, and map ever closer to the actual, objective world. But research has shown that the only driver of brain complexity is reproductive success. So our shared intellectual model of reality has evolved for reproductive advantage only, and there’s little chance the actual, objective world bears any real resemblance to the one we’ve evolved to see (or experience). I can’t cite the source at the moment, so take my account of it with a grain of salt.
@@baTonkaTruck Hoffman found (in a computer simulation) that species that select for commensurability with "reality" die out and species with the most useful models survive.
that.. has nothing to do with the theorem
@@98danielray For a thread about a theorem that involves self-referential statements, your "that" appears to have no clear referent at all!
note the camera didnt pan over to joe having a stroke trying to remember the third word penrose said
I read most of the previous comments until your cough me unprepared. I've laughed so hard and I don't understand why 🤣🤣🤣
I always thought of it as a fundamental “issue” with logic itself. Essentially “You cannot prove logic with logic.” It’s circular reasoning. Proving a theorem based solely upon the rules of logic that you’ve constructed. You create a box made of various rules, and you can “prove” things inside of the box, but you cannot prove the box.. without creating another box, outside of that box. .. That’s my understanding of it as best I can explain it, I apply the concept well outside of mathematics and see it as applicable to rational thought as a whole. Which is in part why I’ve embraced absurdism.
yeah, it's about relative perspective
similar to the 1-way speed of light problem, or the notion of higher dimensions... you need a higher level perspective to demonstrate a proof
@Jay Steg - "You cannot prove logic with logic" - thank you sir. That one statement finally explained it to me.
Love this!!
You cannot prove logic with logic, and you can't also prove that you can't prove logic with logic, with logic.
@@theunicornbay4286 that would be the logical conclusion, yes. 😂
I think my favorite video explaining Goedel's Incompleteness Theorem is Veritasium's, he does a great job walking a layman through it.
If you're actually looking to kind of play around with how this is more formally proven, I really like Raymond Smullyan's book To Mock a Mockingbird which is presented as a long series of logic puzzles using birds that culminates in a proof of the Incompleteness theorem.
Goedel, Escher, Bach is another classic book that covers how self-reference is used by all three of them in their various masterworks.
or you could just googlescholar "biological annhilation" - the common claim by mathematicians is that "it works" - well if it works so well then why are we accelerating to "biological annihilation"?
@@voidisyinyangvoidisyinyang885 Godel's Incompleteness Theorem has literally zero to do with biology. Are you posting in the wrong thread?
@@Bodyknock Oh so you've never read Stuart Hameroff's quantum biology work in collaboration with Roger Penrose? That's what Penrose calls "protoconsciousness" and as Penrose emphasizes "calculations are not consciousness." Of course if you need to practice self-censorship by creating your own fake boundaries of thinking, that's perfectly common to do. haha.
@@voidisyinyangvoidisyinyang885 Again, Godel's Incompleteness Theorem has nothing, at all, to do with biology. It's a theory about infinite sets of first order logical statements. Biology is an inherently finite system of cells. There's no intersection between the two. (It doesn't even have anything to do with quantum mechanics or the definitions of consciousness either frankly.)
@@Bodyknock So your youtube troll comment just debunked the Royal Society (formally The Royal Society of London for Improving Natural Knowledge formed in 1660, is a learned society and the United Kingdom's national academy of sciences) Science book of the year award in 2016 for the quantum biology book "Life On the Edge" by Professor JohnJoe McFadden and Jim al-khalili? Wow! Impressive.
Great explaination but I think a key point that is not mentioned and that shows why this is mind blowing, is that this can be applied to all knowledge and not only maths, and that nothing can be proved to be true unless you have a set of base “rules” that you believe in , so theres a impossibility for 100% “real” or “checked”knowledge by definition. Similar to Munchausens trilem.
Ergo, the Earth is flat.
it is kind of redeeming to learn that wittgenstein did not wanted to accept godel and penrose don't know how to explain it concisely
I’ve been slowly reading, comprehending, and trying to fully understand the entirety of this man’s Book “The Road to Reality: A Complete Guide to the Laws of the Universe” since I graduated High School. Fascinating stuff.
There's a pair of books called Godel Escher Bach that discusses this at length. Every time I read it, I thoroughly understand Godel's proof for about 2 hours. There are also other completely unrelated statements that you can prove are true but unprovable, involving (for example) relationships between infinite sets each member of which is an infinite set. The Halting Problem is probably an easier thing to understand but is the same concept.
Consider the sentence: "Roger Penrose does not believe this sentence is true." There are two cases: either Penrole believes it or he does not. Suppose he does, then it is false and Penrose believes a falsehood. Moreover, it is obvious even to Penrose that this is false, and in fact it is false BECAUSE he believes it, which is crazy. So suppose he doesn't believe it. Then it is true, and indeed OBVIOUSLY true, even to Penrose himself; yet he doesn't believe it.
Of course, you can do this for anyone. We all have our own Penrose sentence. What Goedel showed was that you can even do it for arithmetic.
Russell's paradox
@@vincentrusso4332 Not quite, but closely related. Both Russell's and the Goedel result can be seen as adaptations of the old liar paradox.
@@tomasmcelhoney4054 Sure, but now you are playing a different game (which has nothing to do with Gödel's theorem). My point was that you can take the same logic used by Gödel and apply it to Penrose himself (or anyone else). Penrose has written several long books arguing that it only applies to machines or formal systems, not to humans. Wrong.
BTW, you don't escape the Gödel argument by introducing a third 'unknown' value, it just gets a bit more complicated. The question is "Penrose believes this sentence is neither true nor unknown".
@@patrickhayes2516 Very good explanation "this statement is false" - is this true or false? Either way its bs.
Hmm... there is an underlying assumption here that a statement has to be, statically, either true or false. What if the statement doesnt have that dimension at all. It is like asking what color gravity is. It isnt relevant.
Veritasium explains it better. To follow Penrose you have to understand the basics.
I agree. I don't think this is a very good explanation.
@@acobster Title is missleading. Penrose is not really trying to explain Goedel's theorem here, he is kinda summarizing it to make it easier to understand why it changed his way of thinking.
I agree, very nice explanation for people who don't speak mathemathese
So the ultimate validity of a formal system cannot be achieved by the system itself.
I've been studying Penrose' CCC for more years now, trying to disprove it and I literally can't. I've never come across an idea so bulletproof that I can't poke even a single hole in it.
I genuinely think, despite my fear, that Penrose is correct
Why do you fear this idea?
Weird. I've seen physicists poke holes in CCC all over the place. Not that they claimed to disprove it, but some of the implications of CCC do not seem to easily agree with our current models. That being said, I'm a fan of CCC and if it turns out to be correct it will be the most exciting new discovery in a very long time IMO
@@PetrCobra I'd very much like to know which physicists you're referring to, what they specifically said, and why you beleive that they're correct. Because I'm not entirely sure you've noticed this but every physicist will tell you that the laws of physics break down at a certain point and don't apply in the earliest stages of the universe. Meaning that they have literally zero business using physics to debunk anything outside of the purview of physics.
@@mrb7094 Nietzsches "eternal recurrence" is the best way of putting it
@@jaxwhyland I'm aware of that (terrible) idea. It would of course be dreadful if all the tragedies of the world played out again. Not to mention the awful thought of how many times I already typed this sentence. But there's no reason to think, and every reason to doubt, that anything would be the same twice? If the universe reemerged then the tiniest difference would change everything.
Roger Penrose's younger brother, Well-known British Chess Grandmaster Jonathan Penrose passed away earlier this year. May his soul rest in peace.
man i read 'cheese-gradnmaster' at first
Jonathan Penrose beat Tal…
Respectfully, he is correct but makes it more confusing for non-mathematicians.
Here is the gist:
There exist true propositions in mathematical systems, at least as complex as arithmetic, but which cannot be proven to be so.
Now, one can create a new system by adding axioms and rules to create a path of proof of that previously inaccessible proof, but now, that newly created system will have certain true propositions which cannot be proven to so; and so, ad infinitum.
How godel went about proving the above result will require advanced mathematical background, but this deep result is broadly comprehensible by non mathematicians also.
To me, it's analogous to Cantor's diagonal argument for transfinite ordinals; just with respect to mathematic systems as a whole, not just for an individual set of numbers. Essentially, no matter what system of rules you create and impose upon a system, you can always create an ad hoc exception that satisfies the conditions of that system, but doesn't satisfy one or more of the rules of that system.
@@xxFortunadoxx "analogous" might be a bit of a stretch, but you're right in that in follows a similar line of construction. btw: the Turing halting problem is analogous to the incompleteness theorem and both are answers to Hilbert's entscheidungsproblem
He makes it more confusing for everyone (and this wasn't the only instance of it). Penrose, for all his great work in multiple fields (geometry, astrophysics, etc.), is a terrible teacher. He's clearly very good at _understanding_ things, but terrible at explaining them.
@@RFC-3514 hey...think you need to cut the guy some slack,he's not a young man and when he was, there wasn't this need for complicated theorem to be distilled into two minute sound bites...not everyone needs their teacher to be media savvy.
@@user-fb9os7hy2y - He's always been kind of like this, though. Even when talking about his own work (which I'm sure he understands). I think he's just not very good with language.
Joe: That’s crazy, man. Have you ever done DMT?
I went to school with his nephew, Mathew . An absolute child genius, now a professor of mathematics
I never would have guessed Joe's nephew would be a professor of mathematics.
Something in the water over there when you look at the Penrose family.
All reasoning ultimately traces back to faith in something that you cannot prove. Faith and Reason are not enemies. One is absolutely necessary for the other to exist.
I don't blame Joe for being silent through this: Roger is a very gifted teacher able to express the fundamentals of a difficult concept like this one.
Not this time.
How? This was one of the worst expositions of the theorem that I have ever seen..
Somewhere Godel is smiling because after all these decades people are still debating this and a lot of them are just as uncomfortable with it as his colleagues were when he first stated it.
Gödel went mad.
Joe understood that so completely that, unusually for him, he didn't have any questions to ask or supporting comments to make.
So basically Penrose is saying :
Godel’s first incompleteness theorem doesn’t say that Mathematics is incomplete, it says that either Mathematics is inconsistent, or it is incomplete.
In other words, if you believe the system of axioms is consistent, and it is just a believe since according to Godel’s 2nd incompleteness theorem it can’t derive from the axioms, then it is incomplete.
Well, this is my understanding of what Penrose just said.
Penrose uses it to conclude that brains are more than computers. Dan Dennett debunks this view.
uh?
In the context of the first order theory of Robinson Arithmetic, yes 👍
I suspect that the above is less your understanding of what Penrose said than of what he should have said. He didn't mention the word "inconsistent" even once, so I strongly suspect you got that from somewhere else.
@@alqpr Indeed, it's probably very difficult to understand Penrose's explanation if you yourself have not studied first order logic. I take "believing in the system" to mean "assuming the particular axioms you're working with are consistent". One proves that the Gödel sentence is independent of the axioms by assuming the axioms are consistent. Moreover, one proves that the Gödel sentence is true (in the standard model) by assuming the axioms are consistent.
So it's kinda like using my eyes to look at stuff but not being able to use my eyes to look at my eyes.
nice !
''So it's exactly like my brain interprets words and concepts but my interpretation is wrong'' - How would you ever know? Noone does? Reason and imagnination isnt enough. The most baseless stuff ever - Yet its right?
Mirror
@@IDMYM8 think about it
@@IDMYM8 But that's just a reflection. How sure are you that your reflection accurately gives you a representation (not an image) of your very own eyes?
"But, and hear me out, could a chimp smoking weed maybe help disprove this?"
and for the average joe to understand (and donald rumsfeld), there are knowable knowns, unknowable knowns, known unknowables, and unknown unknowables.
basic axioms are not valid or invalid; validity is reserved for statements where inferences are made. axioms are assumed true because if you assume the axiom is false then you can prove anything by default.
In logic, we use "sound" or "unsound" for axioms and "valid" or "invalid" for proofs thay lead to conclusions. Different words for a reason.
@@Sam_on_TH-cam no. axioms are a given and assumed true for otherwise you can always produce a valid argument (using false premises). a sound argument is both valid and assumes the premises are true. cogent, may also mean sound.
How are there unknowable knowns
@@TrombonePirate in a set that would contain say basic algebra, there are statements outside the set that are known to be true but cannot be proved from within the set (something like that). take the number "pi" for instance. it's outside the set of natural numbers but you "know" it exists but can't be proven true from within the set of naturals. if i'm wrong then please correct me.
@András Belina that's true. We know spies are out there but will never know who they are and doesn't much matter.
Godel's Proof is amazing, as Sir Roger stated.
Can you explain it in terms that make it obvious to a layman?
@@ophello i hope not and i am pretty much layman too. I was suggested a good book by a good prof btw. Its called the princeton companion to mathematics. You may find it there. It has brief articles with suggested readings.
th-cam.com/video/SOWt2fBI1VI/w-d-xo.html
@@ophello i copied this from another comment. The LOGICAL structure of Godel's proof is simple. He tweaks "This statement is false" to make this: "This statement is unprovable." There are two possible truth-values for this: If the statement is true, then you have a true statement that is unprovable. If the statement is false, then the statement is provable, which means you have proof of a false statement. So any (sufficiently complex) mathematical-logical system is either incomplete (with statements you know to be true but can't prove) or self-contradictory (with false statements you can prove), or both.
The look on Joe's face is priceless, good on him! I recognize it because it was the same look on my face listening to WVO Quine trying to explain the same thing in a talk I attended in 1993.
Does this mean that a system can have properties that can’t be derived from the rules of the system?
Yes. I think so. There would be things in the system that the rules of the system cannot determine the truthfulness of or falseness of. This guy here does a better explanation. th-cam.com/video/HeQX2HjkcNo/w-d-xo.html
Yes; importantly, though, the properties must be consistent with the rules even if they are not derivable from them.
Wait until he finds out about Tarski’s undefinability theorem.
Another way of understanding Godel's incompleteness theorem is if you look at language as a mathematical concept. Languages can describe many things, but there is always something incomplete in the description. Take the sentence, 'the man was running'; does it describe the vigor of the run, does it describe the speed, or does describe the milieu of the man running. There are many things that are left out with each statement, but the general concept is still understood. Something is true but incomplete in its description.
The best part of Gödel incompleteness theorem is not really the consequences, but the proof itself. He created the very system using natural numbers and arithmetic operations. He made the system talk about itself and constructed the sentence. It's a "proof" of how we should explore self references much more deeply. And mathematicians tried to avoid it as much as possible due to obvious reasons if you're in the field.
Gödel went mad himself.
Roger Penrose is a really interesting guy. It's cool of Rogan to just sit back and let him talk, because that's when he really shines IMO.
Formal logic professor here. Yup this still holds. It's typically used to show the limitation of science as well.
You don’t need to be a logic professor to understand the implications of the incompleteness theorem you arrogant twat
What does it have to do with limitations of science? Scientific beliefs don't follow from some fixed axioms.
Does A + B = B + A? Is it true, not true or unprovable?
Is it a human-invented rule?
@@user_375a82 that depends on how you define addition, equality, and so forth. The rule you stated is typically taken as an axiom
@@absolutezero6190 Which also shows the limitations of science.
the tension is killing me. how will Joe respond once he stops talking lmaooooooooooo
Mortals listening to an immortal speak...
Sir Roger is number one on my list of greatest intelects of all times!
"Sir Roger is number one on my list of greatest intelects of all times!"
What he said about Godel's Theorem was horribly wrong.
That does not matter to hero=worshipping sycophants.
@@kensandale243 At least in this video, what he said about Gödel's first incompleteness theorem is not actually horribly wrong. But to someone not already familiar with the theorem and its proof, it may not be all that clear.
I've heard some people claim Penrose has said false things about Gödel's incompleteness theorems in other media, but for this video, at least, what he says can reasonably be interpreted in an accurate manner by someone familiar with the topic.
My favourite example of incompleteness is that it can be used to prove the halting problem (basically, can I show a piece of code to another piece of code to know for sure that the first piece will end rather than go on forever). The one problem I have with incompleteness in general is that it's a very... Abstract issue, it's unclear that the holes in the tapestry of math are anywhere near where we want to go, can go, but the halting problem/would/ have been incredibly powerful if we could have used it!
It is also interesting that the fact that the halting problem is undecidable isn't surprising, counterintuitive, or anything of the sort. On the contrary, I find it quite intuitive. So, why is it that we can easily accept this, yet we are taken aback by Gödel's theorem, even though the two are deeply connected (and perhaps almost equivalent)? I first read about Gödel in Penrose's book 30 years ago. And Penrose is no fool. However, I feel he might be mistaken on this matter. I don't believe his conclusion that human intelligence is something more than what can be formalized in a computer is accurate. It seems more like a mirage stemming from the challenge of interpreting Gödel correctly.
In other words, in a formal system... there are, at the bottom, things (axioms) you must assume which are unfalsifiable?
No, it says in any formal system (if it has a certain strength) there will be true statements which you can not show to be true by starting with your axioms and following your rules of inference
you do not assume that they are unfalsifiable, you assume they are valid so you can build upon them when there is no way to inherently prove their validity. By assuming certain initial axioms, you can build a consistent system without ever really proving those initial axioms
But why must there be a true statement that doesn’t follow from the axioms if it’s a closed system? Perhaps I used the word unfalsifiable incorrectly, but does this all have to do with the fact that the axioms themselves cannot be proven to be true - they’re just the building blocks, not something to be proven or disproven
@@jon553 >But why must there be a true Statement that doesn't follow from the axioms if its a closed system
Basically, because Gödel showed us a way to construct at least one statement in any formal system which has to be true but has no prove in the system.
If you want to know how he did it, or what the way is to construct that statement, I would recommend listening to a 1 hour lecture by Douglas Hofstadter here on TH-cam.
> Does it alle have to do with the fact that the axioms can not be proven
No. Even if you constructed a formal system based on self-evident truths or truths proven by a meta formal system, if it has a certain strength, Gödels Theorem applies.
@@obakillaking5643 Fascinating. Thank you.
Just the fact that Joe has world-class-mathematician guests who discuss things like _Incompleteness_ is uplifting. As other guests have said, including Jordan Peterson and Elon Musk, we don't know for sure what will come of all this access to knowledge... but it will almost certainly be something amazing.
Uhm, why? Not every talk show was Late night with David Letterman before podcasts. Penrose has appeared on many talk-shows on TV as well as radio-shows to discuss his work and philosophy.
The knowledge was always there, available to you. The interest was obviously lacking.
Incompleteness is a sensation felt by almost all human beings, its what drives life's struggle.
@@Bollibompa i don't think it was merely just uninterest. a smart phone nowadays can be had for like $50. use free McDonalds Wifi, and a lot of people who might not have had the money necessary to get cable can watch things on TH-cam. while there's obviously the "read a book" club, many people find videos to be a lot more ingestible. the simplification of complex topics is not just the "everyone seems dumberzz" phenomenon, but also leads a lot of people who did not have the resources prior to developing an interest in topics they wouldn't have had otherwise. granted, it leads more people to misinformation, but i really do think people are generally more intelligent when misinformed than uninformed. granted i wish it was neither, but i do think its a step up.
@@ethanstump
I don't really think that the stoner tidbits dished out on this caveman-show do anything but mildly amuse most. An actual interest in Penrose's teachings requires applying yourself. Being spoon-fed one-liners and easily digestible nuggets of pure genius is not profound.
@@Bollibompa it's not profound, no, but it's not nothing. while there is active learning, there is passive learning even from encountering something new. sure, most will get nothing out of it, but even if only one Pakistani Stem highschooler gets something out of it, that's still something. i personally stopped watching rogan maybe a good four years back, but even watching this was a step up from reading the bible with my Christian fundelmentalist family when i was younger. i would much rather Rogan have Penrose again than Jordan Peterson.
George boolos once explained the second incompleteness theorem using only one syllable words
This is the most interesting concept I have ever heard.
that is really awesome, that was my first time hearing Gödel's theorem, and just as you roger im now amazed.
“Is God willing to prevent evil, but not able? Then he is not omnipotent.
Is he able, but not willing? Then he is malevolent.
Is he both able and willing? Then whence cometh evil?
Is he neither able nor willing? Then why call him God?”
― Epicurus
Is this a direct translation? Was he a monotheist? Not being a turd, genuinely interested.
@@LOCATIONREDACTED This is a supposed Epicurus riddle, there is no original text.
@@mskidi Thanks. Steven Pressfield writes in similar terms in his book Gates of Fire, wondered if I've been missing something re Greek theology.
@@LOCATIONREDACTED Even the supposed text doesnt have that meaning. Epicurus was of the position that God/s dont interfere in earthly matters. That was the point of the riddle.
Roe Jogan just sitting there like he’s the missing link
(GODEL) "On Formally Undecidable Propositions of Principia Mathematica and Related Systems," in 1931. Here is a summary of his results:
Any consistent axiomatic system of mathematics will contain theorems which cannot be proven.
If all the theorems of an axiomatic system can be proven then the system is inconsistent, and thus has theorems which can be proven both true and false.
Joe knows how to get great guests and he is great at letting them talk. He may not be a Math PhD but he’s a great interviewer.
Lol
I didn't find this to be a concise explanation of Goedel's theorem, though he did touch upon the halting problem.
Was the halting problem around in Godel's day, or is it a computer language problem?
I mean, I have a maths degree, I like high level maths, and this barely made sense to me.
The concept of Godel's theorem may be tricky (and widely misunderstood) but too many people are impressed by what sounds complicated, rather than what is actually insightful, which requires you to understand. The most brilliant people can put things in simple terms a layman can understand.
Rogan probably has no idea what Goldbach's conjecture is or Fermat's Last Theorem, so how would he follow the rest? Politely listening is not the same as understanding.
Heh well to be fair I don't know if many people are claiming that Rogan understands what is being said here... but that said, I'd also say that this video does not encompass an explanation of Godel's incompleteness theorems... so the only people who really understand what's being said here already know the subject. But Penrose as always is still doing a good job of mentioning what is necessary to give a layman a rough intuition of how the ideas connect.
@@jhansenhlebica6080 I disagree. I think his explanation is atrocious. Being smart does not guarantee being a good teacher. Honestly he seems terrible, based on this clip. It is a confusing topic though and seldom gets explained well anyway.
Goedel's theorem is pure bs. (like, snow is white = unprovable its a human-rule)
The explanation was crap...
imagine having a conversation like this with kimmel or fallon
You know Roger is a genius, but you can’t prove it
Read his works. They prove it.
I can prove Einstein, Godel and Feynman were geniuses. I can’t prove Penrose is a genius, but perhaps others can
@@stevekru6518 Feynman wasnt on Godels' or Einsteins level. He was a genius spefically on his field, a one trick pony. Philosophy was beyond him, a very smart brute he was.
A negating self-referencing proposition is incomplete or contradictory since the logical outcome implied in the proposition either confirms the fact that the proposition can't be proven or contradicts the intent of the proposition.
I am no fan of Joe Rogan, but I might have to check this interview out. Penrose seems like precisely the lovely and intelligent person I’d imagined him to be from his mathematical discoveries!! He’s too often overlooked in my opinion
His interview with Lex Fridman is pretty great too, if you want to check it, he's a computer scientist so the dialogue is a bit more fluent.
this video has twisted my mind into a pretzel and now I'm hungry dammit!
Read "Godel, Escher, Bach: an Eternal Golden Braid" by Douglas Hofstadter. First edition perhaps 1979. While it may further twist your mind and make you more hungry greater, he has several books to follow up!
The famous Gödel number (G) that Gödel came up with in his proof seems to correspond to: "This statement can't be proven."
Assume that G is false. Then, G must be provable and hence, is true. The only assumption we made resulted in a contradiction. Therefore, we accept the inverse of the false assumption: G is true.
That makes sense. But, how come the statement refers to itself (self-reference)?
The complete statement is "There is no proof for the statement of Godel number g", and it turns out that the Godel number of this statement is exactly g.
If g is provable it means that g is true, and then it means that "There is no proof for the statement of Godel number g" which is a contradiction. If g is not provable, it means that g is here the incompleteness theorem is proven; there is a true statement that is not provable.
"This statement can't be proven." is as you pointed out leads to a trivial contradiction (because of trivial self-referencing) that doesn't need all the hustle, however it is not an organic statement that could be (simply) written in the Godel system.
So the genuineness is finding the g that hold this self-referencing idea that is encompassed in your naturally expressed as "This statement can't be proven.".
But G is gangsta
@@truebomba thanks so much! This clears up one major obstacle: A statement can refer to any other statement (or itself) using its unique number.
If you wouldn't mind, here is a follow up question: how can a statement express the fact that a statement S "cannot be demonstrated?" I can only guess that somehow it is shown that the rules of deduction wouldn't reach S from any true statements. Any further insights on this?
@@BulentBasaran What is proved is that No consistent system of axioms whose theorems can be listed "by a sequence of symbols" is capable of proving all truths about the arithmetic of natural numbers.
As a disclaimer, I kinda explain things loosely because I never read all the details of Godel's proof. After his proof, there are many other proofs and ways to see this exact same result.
Morally you can imagine the following; You have m mathematical symbols that are the bases of all possibles (listed) statements within our axiomatic system, theses all possible statements (true or false) and could be represented as polynomial P_n(x_1,....,x_m), where n is the n em statement on our listed propositions. Now we have to define provability, which is kinda difficult to present here, but you can imagine that as stepping a layer above (it is like passing from the category of sets to the category of Powersets, but keeping in mind that we enforce "countability" on this upper layer as well because that's what ensure that our proofs are sequential/"algorithmic" series of statements that leads to the statement we want to prove).
With all this rigorous organization of our axiomatic system, a listing of our all possible propositions (within the system), which btw also contains the statements that could be considered as proofs (according to our rules of proofs) for eventually other statements within the same list (this is where the part that the statements are made on natural numbers is important), and our rules of proofs, Godel proved that regardless of the starting axioms or how many there is always a number g that holds the self-referencing property I explained. In another way, we have a statement P_g(x_1,....,x_m) listed somewhere in our list of statements. P_g(x_1,....,x_m) has a formal expression that could be found in this paper for example
www.ams.org/notices/200604/fea-davis.pdf
@@truebomba Thanks so much for the detailed explanation and the link to the PDF!
In simpler terms: The law of non-contradiction can not be proven because a proof would assume the law itself. Aristotle did it for logic millennia ago, Kurt Godel did for mathematics in the 20th century.
In other words, Mathematics has assumed principles based outside of mathematics itself, and outside of science itself as well.Since, science uses math to measure statements.
No, that's a variant of the Liar's Paradox encoded in a formal language, creating a system that's either contradictory, or consistent but incomplete. It has nothing to do with the fact that you can't prove your own logical axioms, that's why they are axioms.
@@nektariosorfanoudakis2270 The Liar's paradox is no different than speaking of a square-circle, sir.
The law of non-contradiction is not a logical absurdity, but a demonstration of the limits of reality, and how reality is based out of the law of non-contradiction.
Happy birthday Sir Penrose! Thank you for the talk today!
Either you wind up with an infinite regress or you affirm one or more axioms
Think he could explain it for another 3 hours and I’d still look like I failed high school Physics again.
In grammar school in the late 50s I was very good at mathematics. The headmaster wanted me to apply to Manchester University, his alma mater I think, to study for a mathematics degree there. I had set my sights on a career in Engineering and followed that degree route instead. Of course, my mathematical bent was a great help to me in my Engineering career, which I enjoyed, spanning 50 years but I often think about my decision back then. The idea of sitting in my university study as a Professor surrounded by untidy piles of books seems like an attractive proposition, but I'm sure it's not like that at all. I'm not even sure if I would have been able to grasp some of the concepts that geniuses like Roger Penrose talk about if I'd made that different decision.
Most Engineering mathematics involves the use of Newtonian mechanics plus other science and mathematics that had their foundations taught at school. I say "most" because the really clever stuff probably requires more brainpower than I possess.
The most fascinating speculation for all us, what if I'd taken a different road.
This is an extremely nuanced and precise, concise explanation of the theorem, but I don’t think it’s the best explanation for someone who has no prior experience with higher math.
it's like... mathematics is a system for checking results. in that system, you can create subsystems for checking subsets of results. however, if you try to create a subsystem that checks all results, then that subsystem cannot be checked.
so, our primary knowledge of mathematics must be primitive, not the product of any system.
It's so crazy to me that Joe Rogan, of all people, had Roger Penrose on his podcast. Or maybe it's crazy to me that Roger Penrose would do a podcast like this. What a world we live in.
Gödel's Incompleteness Theorem:
A Mathematical Corollary of the Epistemological Münchhausen Trilemma
Abstract: This treatise delves into the profound implications of Gödel's Incompleteness Theorem, interpreting it as a mathematical corollary of the philosophical Münchhausen Trilemma. It elucidates the inherent constraints of formal axiomatic systems and mirrors the deeper epistemological quandaries underscored by the Trilemma.
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In the annals of mathematical logic, Kurt Gödel's Incompleteness Theorem stands as a seminal testament to the inherent constraints of formal axiomatic systems. This theorem, which posits that within any sufficiently expressive formal system, there exist propositions that are true but unprovable, has profound implications that reverberate beyond the confines of mathematical logic, resonating in the realm of philosophy. Specifically, Gödel's theorem can be construed as a mathematical corollary of the Münchhausen Trilemma, a philosophical paradigm that underscores the dilemmas in substantiating any proposition.
The Münchhausen Trilemma, named after the Baron Münchhausen who allegedly extricated himself from a swamp by his own hair, presents us with three ostensibly unsatisfactory options for substantiating a proposition. First, we may base the substantiation on accepted axioms or assumptions, which we take as true without further substantiation, a strategy known as foundationalism or axiomatic dogmatism. Second, we may base the substantiation on a circular argument in which the proposition substantiates itself, a method known as coherentism or circular reasoning. Finally, we may base the substantiation on an infinite regress of reasons, never arriving at a final point of substantiation, a path known as infinitism or infinite regress.
Gödel's Incompleteness Theorem, in a sense, encapsulates this trilemma within the mathematical world. The theorem elucidates that there are true propositions within any sufficiently expressive formal system that we cannot prove within the system itself. This implies that we cannot find a final substantiation for these propositions within the system. We could accept them as axioms (foundationalism), but then they would remain unproven. We could attempt to substantiate them based on other propositions within the system (coherentism or infinitism), but Gödel's theorem demonstrates that this is unattainable.
This confluence of mathematical logic and philosophy underscores the inherent limitations of our logical systems and our attempts to substantiate knowledge. Just as the Münchhausen Trilemma highlights the challenges in finding a satisfactory basis for any proposition, Gödel's Incompleteness Theorem illuminates the inherent incompleteness in our mathematical systems. Both reveal that there are boundaries to what we can prove or substantiate, no matter how powerful our logical or mathematical system may be.
In conclusion, Gödel's Incompleteness Theorem serves as a stark reminder of the limitations of formal axiomatic systems, echoing the philosophical dilemmas presented by the Münchhausen Trilemma. It is a testament to the intricate interplay between mathematical logic and philosophy, and a humbling reminder of the limits of our quest for knowledge. As we continue to traverse the vast landscapes of mathematics and philosophy, we must remain cognizant of these inherent limitations, and perhaps find solace in the journey of exploration itself, rather than the elusive, final destination of absolute truth.
GPT-4
Never saw Joe add so little of his mustard
Geee I wonder why...
I think that, this incompleteness theory is fundamental to understanding a little bit more about the way in which Humans make discoveries on things that we cannot see. Basically the magic of turning mathematical abstraction to real life applications. Also explains well how there is so much that we don’t know yet, in our journey to have more control over earth and the universe.
As with all formal proofs, this is not about how humans do things, or even about how this universe does things. Formal proofs, if correct, are correct in any universe and no universe.
No, you obviously don't understand it at all.
Firstly a statement in a logical system can be true and provable and still unkown. A statement having no proof in the system doesn't mean you can't know it. And science is not just mathematics. You don't know something about the real world, because you are simply lacking information about the real world and you need to go out and collect it. That has absolutely nothing to do with Incompleteness.
I have an hypothesis on movement of the human body. telle UW Lyon
I’m all ears!
I'm not sure this really qualifies as an explanation of "Godel" "incompleteness" or "theorems" but I bet it's an interesting listen for people interested in this man and familiar with it
Omg. God Bless your Heart Roger Penrose. I love you like the uncle I never wanted. It took you three minutes to say what's obvious to all after 90 years of consideration on the topic. We cannot prove irrational or infinite queries. There are many queries that while finite still cannot be effectively calculated do to the sheer computational requirements to do so. Chaos, or the math of iteration, does not allow for vast computation for a number of reasons. One, is precision. The three body problem. The other is the sheer rate in which decision trees branch. Iteration in which each step produces exponential growth of the system. TSP NP-Hard problems.
I think you need a bit more time to explain this in a way that the layman (eg me) can understand. I remember seeing this explained with cards by I think veritasium, and it clicked with me in a way that this explanation didn't. Not Mr Penrose's fault, it's not just something you can explain in 3 minutes IMHO, at least not to someone who isn't a mathematician.
With something like this theorem, you want a good visual to make it understandable. Once you understand the visual, you understand Penrose's explanation
The absence of even an attempt to state Godel's theorem makes this video less valuable. There are actually two incompleteness theorems that relate to each other.
It is an elaboration of the famous "Liar's Paradox". I also missed a concise summary of the implication of Godel's theorem. Let me make an attempt:
"In any formal system powerful enough to be interesting, there are either true theorems that cannot be proven or theorems that may be proven both true and false."
A formal system that has even one theorem that may proven both true and false is "inconsistent" -- an absolutely fatal flaw, because that one inconsistent theorem allows any theorem at all to to be constructed and proven "true".
Thus, the more interesting outcome is that there is truth outside the realm of proof of every formal system, including of course all mathematics. This concise little gem was Mr. Godel's response to Bertrand Russell, who was attempting to prove that mathematics itself was both formal and complete.
The philosophical dispute has to with "reality", lived experience, and observation. In Bertrand Russell's formulation (shared by great many others), we live in a "clockwork" universe where no observation is never needed or trusted -- once the initial conditions are set, the universe unfolds from that moment forward by simply following its rules. Like clockwork. Any observation that contradicts the formalization of the system is a false observation -- and so observation is irrelevant to truth.
This was anathema to Godel, and his marvelously terse incompleteness theorem devastatingly destroys the edifice Bertrand Russell and his colleagues were attempting to construct.
I suspect that, despite that, you are a proponent of "objective" reality in some correspondence sense.
@@talastra : Oh yes indeed I am. To me, it is miraculous and inspiring to see this result emerge completely from within a formal system (any formal system!) with no need for magical beings.
@@thomasstambaugh5181 Formal systems are the opposite of "objective" reality, and the more formal they are, the more this is true. If incompleteness is the case, and it is, then objective reality is unprovable. Even the statement, "O yes, I am" becomes untenable. :)
Maybe you are sufficiently epistemological, but few people manage it.
@@talastra : Understood. I'm talking about the various demos (many derived from GEB) that show the limits of any formal system. I think, though I'm not sure, that this is related to the various demos that show "random" vs "chaotic" processes. In biology, it can be seen in the way that lipid bilayers form themselves in a liquid. Then the way that autocatalytic sets operate within the microenvironment created by that membrane. The very building blocks of life itself, emerging from deterministic processes -- no magic required.
Penrose: *Explains Godels theorem*
Rogan: “I like roundhouse kicks! Wooo!”
"Penrose: Explains Godels theore"
He got the explanation horrifically wrong. The guy is a fool.
@@kensandale243 Just because his explanation left much to be desired, it doesn't make him a fool? The guy's resume speaks for itself. So fucking cringy when people, like yourself, feel the need to speak in such an inflammatory way
@@andrewmcpherson8441 "ust because his explanation left much to be desired, it doesn't make him a fool? "
No, the fact that he says foolish things makes him a fool.
"The guy's resume speaks for itself."
You are impressed by things you should not be. If someone is an idiot but has an "impressive" resume, it does not mean he is not an idiot. It means resumes are not always accurate.
Einstein's resume right after grad school was that he graduated in the lower half of his class, and was the only student who could not find a job. So, logically, you should have thought he was worthless. Are you willing to be logical?
What matters is the quality of a person's thought, not his popularity among his peers.
@@kensandale243 Your logic is flawed in suggesting that I would dismiss Einstein as a fool based on his school attainment. By Penrose's resume I'm simply meaning his body of work; y'know, the things he's legitimately achieved?! My opinion of Penrose has sweet F.A. to do with his popularity among his peers.
All you did was come bumbling in, dismiss him as a fool, without offering any elaboration as to why you think he is such. Might I suggest you actually explain why you think he's wrong? Bonus points for doing it with a bit of respect, rather than just insulting someone with no explanation supplied. I'm always a willing ear for someone actually giving an explanation about WHY they think someone's work or thinking is problematic, but when someone just calls someone a fool and leaves it at that, wtf do I have to work with?
@@andrewmcpherson8441
"By Penrose's resume I'm simply meaning his body of work; y'know, the things he's legitimately achieved?! "
Oh really? Are you on a position to evaluate Penrose's work? How well do you know Relativity?
I enjoyed getting enough understanding about Godel's incompleteness theorem from a gifted lecturer so I could pass an undergraduate mathematical logic exam in the 1970s. But I've never understood anyone else's explanation of what Godel was doing. And now in my late 60s it's far too late to worry about it. But I still wonder how I could understand enough as an undergraduate to have some kind of a handle on what was going on.
"You see it's true by virtue of your belief in the rules". Nietzsche said we killed god, Godel said he hid behind reason.