I have been fascinated by Gödel’s incompleteness theorems for years, ever since I was a graduate student in mathematics many years ago. This is one of my favorite lectures on the subject. Well done!
really enjoy this chap's lecture style. accessible, uninflated, folksy and concise- his explanation of the seats to people in the lecture hall as a way of explaining that you don't have to count everything to arrive at accurate numbers is a great example of how different teachers offer different examples, some easier to understand than others. he, thankfully is very helpful and not obscure.
Excellent. It is not just that Godl found a quirky solution to GR, but because there are so few analytic GR solutions it also shows the enormous skill of Godl.
Excellent three video set on Godel. It is worth noting that Godel made significant contributions to relativistic cosmology, circa 1949. Einstein and Morgenstern took Godel to be sworn in as a U.S. citizen. He was brought before a Federal judge to be sworn in. Godel corrected the judge on several minor matters. The judge then stated that the troubles in Europe with Hitler and Nazism could not happen here in the U.S. Godel had studied the Constitution and Bill of Rights, etc., etc., and started to correct the judge stating that it could easily happen here. Einstein, however, tapped Godel's leg and cut Godel off before he got denied citizenship. Many would love to have heard Godel's comments on how a dictator could take over the U.S.
An excellent post. Goedel's ideas on relativity (popular book) are in "A World without time". One interpretation is that time as we usually view it, doesn't exist -- there is no time travel, since time doesn't flow. Of course, we are still *barely* 100 years into the theory of relativity, so much remains to be seen. Also, the whole "I am lie-ing" problem was (to some extent) dealt with by one of Russell's students Spenser Brown in his book "Laws of Form", but by the time he wrote it, Russell's mind had pretty much been broken (by his own admission) from the years of intense work (and prison for protesting WW I ). But, he said that the work was better than the theory of "types" and "classes" that they had come up with to attempt to fix the original paradox. Brown's work postulates that self-referential statements and such systems involve something like "imaginary" numbers except as a sort of logical equivalent in terms of proofs, sets, etc. Onward into the 21st century. Poincare's theorem is solved, on to the next --- ;) -r.
Wonderful presentation. I loved all of it. Being an engineering major with a huge interest in math, I've always wanted an overview of the incompleteness theorems, but found them quite inaccessible in formal mathematics. Now with an intuitive grasp, I think I can read the mathematics behind them, though I have quite a few questions about them. Huge thanks to the uploader!
Science is simply a ‘method’ which functions in a common sense/rationalistic view of the world. I take that only through a commitment to rationalism, can we "demonstrate" science. Quoting Ed Feser (paraphrasing E.A.Burtt): “the thinker who claims to eschew philosophy in favor of science is tempted “to make a metaphysics out of his method,” trying to define reality as what his preferred techniques can measure, rather than letting reality dictate what techniques are appropriate for studying it.”
The pattern to all these logical paradoxes is that they are syntactically correct, but semantically they imply nonsense decision procedures. In each case the decision procedure boils down to an infinite recursion with no base case. Infinite recursions are never computable without a base case. It's like trying to navigate space with a black hole in it. You never get to the other side. You always get sucked into a bottomless well.
you'll never get thru with arguments of computability to mathematicians and logicians that are idealist instead of materialist. it literally took until alan turing, a materialist mathematicial, to actually start the computer revolution.
And he clearly disagreed with Gödel's idealism: he suggested in his reply that Gödel's solutions of the cosmological equations might have "to be excluded on physical grounds". First time I heard of Godel's arguments for time travel very cool!
To be precise, it applies to any system, that contains natural numbers. ZFC (Zermelo-Fraenkel-Set Theory) for example is nowadays usually used to define sets, and within it I can model arithmetic. So the incompleteness theorem holds for ZFC. It is then possible to prove arithmetic consistent within ZFC - but you can't proof, that ZFC itself is consistent, so... yeah. ^^ I love mathematics as well, especially logic, and in fact, Gödel is the reason why I am a mathematics student ;-)
Logic proving logic? No. Faith proving logic? Yes. I was simply pointing out that reason is priori empirical science. You can’t base reality off empiricism alone. Godel demonstrates that nothing is ultimately certain, and that everything hinges on an elusive rationalism, that of which cannot be proven empirically, but which is seemingly and undoubtedly demonstrated through the sciences themselves.
Because it has been proven to be complete ;-) Keep in mind, that being complete for a logical calculus is something different than for a theory. A theory is complete, if for every sentence phi, either phi itself or its negation follow from the theory. A calculus is complete, if, for every theory, every sentence that actually DOES follow from the theory (so, that holds in every model of the theory) is PROVABLE from that theory in this calculus...
1. the books "Gödel's Proof" and "Gödel, Escher, Bach" contain proofs. 2. Roughly: Take the Sentence G="This sentence is not provable". If the system is consistent, G must be true. Therefore, if I could prove the consistency, it follows, that G is true, therefore I would have proven G. However that would mean, that G is wrong and my system is therefore inconsistent - contradiction. 3. It could theoretically exist - most mathematicians assume, ZFC (and Peano Arithemtic specifically) is consistent
I think it has a groundbreaking implication outside mathematics, and that is it may just show that we can never ever find a so called physical theory of everything. A framework that can derive all physical truths abiut the universe we live in.
Computers don't have faith. Our faith on the axiom of choice made us accomplish very interesting results like Hahn-Banach theorem, Banach-Alaoglu theorem, Baire category theorem, open mapping theorem, the closed graph theorem...
Godel's solutions of Einstein's equation could serve _limitatively._ as an example of fictional supernatural spacetimes to which the AdS in AdS/CFT might have to be compared. In Headline form: "AdS/CFT holography a Godel-encoding of Godel's own closed-wordlines solutions to Einstein's Equations?" Or, less ambitiously: "AdS/CFT holography as an analogue to the closed worldlines in Godel's disk-shaped models of Einstein's equations"
Given Godel's proof that 'it is impossible to demonstrate, within a mathematical theory, that this very theory is non-contradictory,' Alain Badiou argues that: a reasonable ethic of mathematics is to not wish to force this point; to accept that a mathematical truth is never complete. But this reasonable thing is difficult to maintain. As can be seen with scientism, or with totalitarianism, there is always a desire for the omnipotence of the True. philpapers . org / archive / BABCPO.1.pdf
Mathematicians are underrated only in the minds of mathematicians. Everyone else recognizes that the "problems" of mathematics are almost totally of the mathematicians' own making. They are the penultimate in ivory tower academicians, philosophers being the ultimate.
>>> As a follow up to my recent posts on (Goedel’s incompleteness theorem) the architecture of materiality and that of the realm of abstraction, the two structurally linked, which prohibit the formulation of conceptual contradictions, I present the following for critique. After watching several video presentations of Geodel’s incompleteness theorems 1 and 2, as presented in each I have been able to find, it was made clear that he admired Quine’s liar’s paradox to a measure which inspired him to formulate a means of translating mathematical statements into a system reflective of the structure of formal semantics, essentially a language by which he could intentionally introduce self-referencing (for some unfathomable reason). Given that it is claimed that this introduces paradoxical conditions into the foundations of mathematics, his theorems can only be considered as suspect, a corruption of mathematic’s logical structure. The self-reference is born of a conceptual contradiction, that which I have previously shown to be impossible within the bounds of material reality and the system of logic reflective of it. To demonstrate again, below is a previous critique of Quine’s liars paradox. Quine’s liar’s paradox is in the form of the statement, “this statement is false”. Apparently, he was so impacted by this that he claimed it to be a crisis of thought. It is a crisis of nothing, but perhaps only of the diminishment of his reputation. “This statement is false” is a fraud for several reasons. The first is that the term “statement” as employed, which is the subject, a noun, is merely a placeholder, an empty vessel, a term without meaning, perhaps a definition of a set of which there are no members. It refers to no previous utterance for were that the case, there would be no paradox. No information was conveyed which could be judged as true or false. It can be neither. The statement commands that its consideration be as such, if true, it is false, but if false, it is true, but again, if true, it is false, etc. The object of the statement, its falsity, cannot at once be both true and false which the consideration of the paradox demands, nor can it at once be the cause and the effect of the paradoxical function. This then breaks the law of logic, that of non-contradiction. Neither the structure of materiality, the means of the “process of existence”, nor that of the realm of abstraction which is its direct reflection, permits such corruption of language or thought. An entity cannot be at once here and there. Likewise, one cannot claim that he can formulate a position by the appeal to truths, that denies truth, i.e., the employment of terms and concepts in a statement which in its very expression, they are denied. It is like saying “I think I am not thinking” and expecting that it could ever be true. How is it that such piffle could be offered as a proof of that possible by such a man as Quine, purportedly of such genius? How could it then be embraced by another such as Goedel to be employed in the foundational structure of his discipline, corrupting the assumptions and discoveries of the previous centuries? Something is very wrong. If I am, I would appreciate being shown how and where. All such paradoxes are easily shown to be sophistry, their resolutions obvious in most cases. What then are we left to conclude? To deliberately introduce the self-reference into mathematics to demonstrate by its inclusion that somehow reality will permit such conceptual contradictions is a grave indictment of Goedel. Consider; As mentioned above, that he might introduce the self-reference into mathematics, he generated a kind of formal semantics, as shown in most lectures and videos, which ultimately translated numbers and mathematical symbols into language, producing the statement, “this statement cannot be proved”, it being paradoxical in that in mathematics, all statements which are true have a proof and a false statement has none. Thus if true, that it cannot be proved, then it has a proof, but if false, there can be no proof, but if true it cannot be proved, etc., thus the paradox. If then this language could be created by the method of Goedel numbers (no need to go into this here), it logically and by definition could be “reverse engineered” back to the mathematical formulae from which it was derived. Thus, if logic can be shown to have been defied in this means of the introduction of the self-reference into mathematics via this “language” then should not these original mathematical formulae retain the effect of the contradiction of this self-reference? It is claimed that this is not the case, for the structure of mathematics does not permit such, which was the impetus for the development and employment of this language in the first place. I would venture then that the entire exercise has absolutely no purpose, no meaning and no effect. It is stated in all the lectures I have seen that these (original) mathematical formulae had to be translated into a semantic structure that the self-reference could be introduced at all. If then it could not be expressed in mathematical terms alone and if it is found when translated into semantic structures to be false, does that not make clear the deception? If Quine’s liar’s paradox can so easily be shown to be sophistry, how is Goedel’s scheme any less so? If the conceptual contradiction created by Goedel’s statement “this statement has no proof” is so exposed, no less a defiance of logic than Quine’s liar’s paradox then how can all that rests upon it not be considered suspect, i.e., completeness, consistency, decidability, etc.? I realize that I am no equal to Goedel, who himself was admired by Einstein, an intellect greater than that of anyone in the last couple of centuries. However, unless someone can refute my critique and show how Quine’s liar’s paradox and by extension, Goedel’s are actually valid, it’s only logical that the work which rests upon their acceptance be considered as invalid.
@jimbo: Yes, that talk would further differentiate the power of the human "mind" from that of the machine. Perhaps the speaker did not want to further highlight those differences? @5:30 he virtually dismisses these differences and the significance of Goedel's work in illustrating them.
I think one can give similar arguments that a machine cannot create another machine. The only way to do this is the human mind programs the first machine to create the second machine. That is, machines cannot create algorithms. Or in other words, it is impossible to give an algorithm which creates another algorithm. This leads to the fact that "thinking" is not an algorithmic process.
That is so cool! Thanks. So this argument applies not only to arithmetic but to any system which can encode every mathematical statement? If so that is way more awesome than I thought. I just love mathematics! ^_^
+Ian Jackson Godel's inconsistency/incompleteness stems from the liar paradox, which includes the opposite incorporated into the proposition like the true statement 'I am not a liar'. Turing using 'self-reference' to overcome incompleteness and prove ingeniously, the halting problem in computers.
there are provably consistent systems of course, however, they are not powerful enough to even model basic arithmetic. For example take the set {r,p,s} with a relation R, such that R(r,s), R(s,p) and R(p,r) are true. r,p and s stand for rock, paper and scissors and R is the relation that tells you, what beats what. It's obviously consistent, but you can't do a lot of math with it ;-)
But for that to work, G must be expressable WITHIN the system. Gödel achieved this, by encoding formulas as numbers, requiring arithmetic. He then was able to talk about formulas and proofs just by talking about numbers with certain properties. This makes it possible to say "there is no proof for this sentence" by saying "there is no number c, such that c has the Property P" - and that sentence itself has a certain number assigned to it that can be put into the sentence itself... boom ^^
Present theories are known to have contrary conclusions if taken far enough, so there is no way to get remotely in the range of the idealizations that concern axiomatic logic.
@packe777: Yes, I must admit that I have been troubled that Goedel is a Platonist. Can you provide more info on "original Goedel interpretation" that you speak of. Thanks.
Really good talk, but I don't think (& it wasn't expected) the speaker took into account the autopoiesis of the individual as a sub-set of it's environment (rhizome) when trying to compare the human mind (second order cybernetics) to that of a computer (first order). That would be an interesting discussion to hear.
@OptimusPanzer: YES ! ! ! ! A unified theory of physics is impossible. I would hope that people can start to see how dangerous Gödel's finding's were to men with power, especially those attempting to promote science as the new god. Perhaps Gödel's fear of being poisoned by food were not so illogical.
It's not odd that a humanly constructed model that mimics some aspects of human minds seem to be oddly reminiscent of human reasoning. It's about as unsurprising that a lifelike statue is oddly reminiscent of what a human looks like. Anyone who thinks computers think is practicing a modern form of Pygmalionism, doing the same thing children do before they learn that their teddy bears and dolls aren't alive.
Interesting question... I had that same thought. I'm nowhere near versed enough in math or logic to have my head around these ideas in one viewing (or likely 100)- but it seems to me be a fair question from a logical standpoint. I'm sure that this fellow would have an interesting answer.
Well, wouldn't an experimental physical proof be a measure of our physical world, which is a system that is external to theoretical physics? And given that our physical world is not a formal system I don't think Goedel would apply. So yeah, if I understand your comment correctly, I believe your first point is accurate.
Thanks. But, I don't understand why any system that can model arithmetic must not be provably consistent. Do you know where can I find Godel's proof? But another question is then, can a system capable of modeling arithmetic be (unprovably) consistent. I am simply asking if such a system could theoretically exist, not if we can find it (because the answer is obviously no to that).
I have a question: has it been proven whether there are any systems that are consistent? I understand that no system can prove that it itself is consistent, but do we know if a consistent set is even logically possible? sorry if this is a stupid question.
@ShopSmartShopSMart1: Are you able to expand on this comment (within TH-cam's posting constraints)? How would one "unify all information". Can you refer us to any reference material, articles, etc.? Thanks.
The brain could be computational AND be something which transcends computation, they're not mutually exclusive as this fellow seems to imply. See R. Penrose.
I really dont know. What cannot be proven formally will need an experimental proof. But, on what formal basis will they explain and connect the results of thoses experiments with the unified theory...?
The "human mind is not a machine" is not the conclusion. It only implies that the human mind -might- not be a deterministic Turing machine. My only criticism. Otherwise great lecture.
Godel's theorem being applicable or not doesn't say anything about existence of a theorem. If it's applicable to a given physics theorem say string theory for example then it says that string theory can't prove it's self. But people can still write down string theory and scratch their heads about it.
Thanks for the answer! Can't experimental proofs of physical theories kick physics out of Godel theorem applicability range? Or considering physics laws and experimental results as axioms means that Godel theorem applies to physics?
Proof by experiment in physics is a form of induction. Proof in mathematics has to be a logical deductive proof with axioms only. Proof by induction in physics is never true proof in the mathematical sense. An example would be newton's mechanics to be disproved by Einsteins relativity. But logically Einsteins theory could be wrong and it only represents our understanding of physical world through observation. It is not the same as logical truth.
I was not able to understand the thinking that produced the paradox (that may not be the best word to use) in "This sentence is true." or "...false", because those 4 words are not a sentence, they don't convey any information. Just 4 words strung together. Also had difficulty with Cantors' uncountable numbers, the tortoise and the hare race where some kind of concept of infinity emerges. What is the sum of 1 infinity and 2 infinities? Or 1/2 infinity + 3/4 infinity? 😉
Maybe the problem is that BiPolar Logic does not apply in reality. Perhaps, Spectral Logic (some call it Fuzzy Logic) will yield a more realistic set of Math Systems which better model "reality".
Hmm so I've been thinking about Godel's incompleteness theorem, and i am not a mathematician or physicist, but I think that taking his theorem a little further, we can explain why it is human nature to believe in an omnipotent god. I am not saying this theorem proves or disproves the existence of said god. I believe that at the core of human cognition, is simply pattern recognition, we see patterns in everything, we even see complete randomness as a form of a pattern, a lack of a pattern if you will. Mathematics is the art of quantifying these patterns that we perceive. Mathematics is a man made system, a system that at least attempts to describe everything in the known universe, as well as everything in the imaginable universe. but for the sake of the point i'm trying to make, let's say that mathematics, in one way or another, can quantify, or describe the entire universe as we know it, from the smallest particle to the entirety of the universe, however, in order to prove that the system of mathematics is true, one must go outside the system, but the system already includes the entirety of the universe, including the entirety of human cognition, so how then, do we prove that the mathematical system, and everything included within the system, to be true? We cannot actually prove the system that we are a part of, which is where an omnipotent god comes in. we imagine something that is infinitely powerful, that is essentially infinity raised to an infinite power, raised to an infinite power to infinity, and that something is what we label as god, however i believe that "god" is simply a placeholder for the system which proves the entirety of our currently known, quantifiable system. so in summary, the human mind sees patterns in everything, literally, and feels the need to quantify these patterns, however it does not stop with patterns we can actually observe, but goes further to try to quantify what we can not yet rationally quantify, which results in an abstract idea, which is often named god. so then by that logic, once we discover the system that explains everything, including our entire known system, then the need to ascribe the label of god to anything, will vanish? So then is string theory the theory that will replace god? all this is simply ideas rolling around in my head and I realize that my logic may be somewhat fallacious, however I'd love legitimate feedback, either positive or negative on everything i just said.
I find your thoughts on god interesting. If I am not mistaking you are basically saying that god is 'the unknown'. I prefer to keep the unknown for what it is and dont replace it by some sort of entity but I can see the logic behind what you said, how the human system can give rise to these thoughts about a god and how the entire notion of god is essentialy one big fallacy. "Mathematics is the art of quantifying these patterns that we perceive. Mathematics is a man made system, a system that at least attempts to describe everything in the known universe, as well as everything in the imaginable universe. but for the sake of the point i'm trying to make, let's say that mathematics, in one way or another, can quantify, or describe the entire universe as we know it, from the smallest particle to the entirety of the universe, however, in order to prove that the system of mathematics is true, one must go outside the system, but the system already includes the entirety of the universe, including the entirety of human cognition, so how then, do we prove that the mathematical system, and everything included within the system, to be true?'" 1. Math is a man made system intended to express as much true statements about the world as possible, we know however that (a formal system of) math cannot be complete and consistent at the same time. If the system is consistent it means that certain elements are excluded from the system and therefore it is not complete, all derived statements are true but not all true statements can be derived in this system. If the system is complete then all truths and non-thuths should be derivable via this system. This is not possible in a formal system (which is usefull for anything) because the only way you can be sure of consistency and ditinction between truths and non-truths is when you close the system and thereby cause incompleteness. I hope this makes sense. 2. The universe is infinite so it is impossible to have a complete and correct (consistent) understanding (what is and what is not) of the universe (unless maybe you are the universe as a whole). ''so in summary, the human mind sees patterns in everything, literally, and feels the need to quantify these patterns, however it does not stop with patterns we can actually observe, but goes further to try to quantify what we can not yet rationally quantify, which results in an abstract idea, which is often named god' 1. it is true that humans make assumptions for certain patterns etc. For example if you throw something in the air it will fall back on the ground so you will expect such a thing to happen again. I don't think that if I can't observe something it is perse an irritaional quintification, I don't need to go to antarctica so see with my own eyes if gravity works the same over there. (simple example just to demonstrate the idea). 2. ''an abstract idea which is often named god' is similar to 'god of the gaps' in the context of your sentence. 3. Abstract ideas are always based on rational ideas, the uncertainty rate just becomes higher. For example: Is gravity real? 99.9999....% certain. Will it rain tomorrow?: 65% Are there horses on the moon?: 0.0000......1%. Maybe this doesnt get my point across but what I am trying to say is that there is no need to give the unknown a name, we should instead focus on what we do know. I also must say that these statistics are what is real according to humans, to be very correct. Some additional thoulds of mine: -We do not know what we don't know, therefore I dont believe anything without (some) reason, otherwise I would have to consider all imaginary things as a serious thing to investigate. -A human as a system is limited, we are limited to our senses to perceive and to our internal processing. We are also bound to our surroundings. -We cannot get out of our own system. The most important thould here is that humans are as a system incomplete and possibly inconsistent, just like a formal system for mathematics. A system which is limited and not capable of knowing all truths and possibly stating contradictions. Hope this is interesting for you.
first off i just wanna say thanks for the reply, very interesting insights. I think that maybe i was misunderstanding what was meant by an incomplete system, but the way you explained it makes alot of sense, however it raises another question for me, is there even such a thing as a complete system? i can't think of one unless maybe the universe it's self is? idk. also yes i realize i am basically restating the 'god of the gaps' idea from a slightly different perspective, but i'm trying to take it slightly further to explain why the god of the gaps is so prevalent. I would also agree that abstract ideas are based on rational ideas, or at least birthed there, however I think abstract ideas can get out of hand quickly when people treat them as if they were completely rational and factual ideas and fail to make the distinction between factual ideas and abstractions. i know there can be alot of grey area between the two. "-We do not know what we don't know, therefore I dont believe anything without (some) reason, otherwise I would have to consider all imaginary things as a serious thing to investigate. -A human as a system is limited, we are limited to our senses to perceive and to our internal processing. We are also bound to our surroundings. -We cannot get out of our own system." i have often thought the same thing, we cannot study ourselves from a purely objective sense, there will always be an element of subjectivity when we are trying to understand ourselves, which really brings into question, how much do we actually know about ourselves and our knowledge? how much of it is distorted or just plain wrong because of this subjectivity? I think that is also part of what makes it so fascinating trying to understand humanity, that we will probably never be able to completely understand ourselves simply because we are either incomplete or inconsistant or both, making the task so much harder than it would be otherwise! on another side note, if you like philosophy and psychology, and trying to understand humanity, look up Jordan peterson here on youtube, he is constantly dropping bombshells of utterly simple yet profound observations regarding humanity. you won't be bored listening to him loo.
.......................the crystal of DNA vibrates differently than the way a crystal of silicon vibrates...........I believe the modes of vibration are different from one another.
@beefking69er: if reality hinges on "elusive rationalism", and if this reality is "demonstrated thru the sciences", then is science the driver or reporter of reality?
Just remember this, man's formal systems of logic, which are used in an almost Godlike manner to direct all humankind, ultimate rely on FAITH, because there are proofs that we know are true but can NOT be proven by these supposed infallable formal systems of logic. The takeaway is this.... that man is not so powerful as we are told, and that man does not have command over nature as we are lead to believe.
The worst thing that's happened to mankind is the introduction of nought, zero. It's a nonsense, anyway, though it makes arithematic easier. From nought we've leaped headfirst to negative number and it's corollary, negative value. It's only a stone's throw from there to Benatar and beyond; not just collateral damage but necessary damage, doing away with life in the name of progress, to stop suffering, the "we're doing you a favour" mentality. That was impossible to think before we thought 0 was something. Well, I have news for you: zero is still nothing. It's nonsense to say I have no eggs; might as well say I have no dinero or no Martians. Then, to take it further to it's absurdity, can you hear the unborn Martian babies screaming to be born?
@ziggityfriggity: My issue is not with mathematical consistency and completeness, My Issue is that academia, media, government institutions and corporations try to sell science to the public as the new god, and that's simply not the case. Math & Science, and therefore computer networks, have severe limitations that should be understood.
To bad the guy toward the end had exposed his unconvincing wishful thinking and bias. He is afraid to see that the emperor has no clothes. Again and again, sweeping under the rug the original Godel's interpretation and fact that likes of Godel, Cantor, Penrose or Chaitin were/are all Platonists, and not formalists. Kant has already showed us the epistemological limits of reasoning (in "Critique of pure Reason") and solipsistic A Priori for true consistency of Godelian argument.
One line of poetry can crack mathematics apart, based as maths is on 2-D abstraction with contradiction of the excluded middle. It's logically flawed, in other words. Pick your own favourite line before you write 2 + 2 = 4 large on a sheet of A4 and fold it in half. "Season of mist and mellow fruitfulness." The point is noone can tell all the associations, personal and general, this has for you. It's not reducible to a Holographic Principle on the boundary. At the same time you can't weigh a thought; you can test for energy use, etc., and cut the brain open. You won't find the thought, though, in the dead meat. That implies mind is not meat, as our current bunch of academics want to prove. It's surprising that string theorists can believe in 11 dimensions but reduce us to one or two. And let's not go from pure maths- and that's only Mind- to the land of fairy, Multiverses. Next we'll be introducing the Mandela Effect and parallel worlds, if you follow the logic. Didn't the news in 1985 say Demi Roussos was dead; and now I find he died in 2015. What's your Mandela Effect moment? Godel had a logical proof of God, by the way, which I can write if anyone's interested, though the formal language won't come out in a comment section.
There is something wrong with reality. For every particle there is an anti-particle. Again, for every physical law, the opposite is also true. In such a situation, in my mind it is futile talking about "set of all sets", it simply is an undefined quantity like the concept of "infinity". A theory of everything will need an algorithm with infinite "axioms", or like a book with infinite pages, you can safely set aside. For every other sub-set like a physical system, consists of finite axiom algorithm that enables us to not only know about it but also use it with certainty, that does not fail us.
but what does it mean to say "This sentence IS provable" or "This statement is TRUE" ? How can a statement be true or false if it contains no information?
that is a bit wrong of a way to look at it. godel's theory is a part of mathematics. as far as i understand, you can either prove all theorems(but then some proofs would be wrong, in the sense this is a lie is wrong) or you can avoid all such statements and have correct proofs(but then you would not have all the theorems, some theorems you would never be able to prove). it requires a radical rethinking(for me at least) of how to approach mathematics.
yeah, Gödel would have disagreed. But I find the proposition, that humans are somehow special and are "above" any formal systems much more elitist. Why should that be the case? I treat everyone like a "carbon-based machine" - that's why I'm vegetarian. How and why does it devalue your or anyone's life if I DON'T believe in something special/magical about minds? I don't think it does, on the contrary, I think it's quite humbling...
"this sentence is false" does not really need to be pondered over for hundreds of years.. it is simply an incomplete proposition. it doesn't relate to maths cos maths doesn't care about syntax.
+shenmueso You do realize that that sentence is the dumbed-down "fake" version of the core of Godel's Theorem? Godel wrote a book about his Theorems. If you disprove him, you can win a Nobel. Go for it, stud.
shenmueso The real version is a book consisting mostly of math and formal logic. The liar paradox is a rough analogy which Godel can be said to have drawn his inspiration from. It's useful tool for explaining the general idea to laymen without using numbers. Most of popular science media is like that: Full of dumbed-down "fake" versions of the actual thing, used to give laymen a rough idea. For example, electrons having a "spin", or string theory being a bunch of "tiny vibrating strings" making violin sounds which create the universe. There's no point laughing at how stupid all that sounds, because they were intentionally made to be stupid enough for people on the street.
Mlai00 i understand the concept of metaphor. tbh i don't remember why i typed that, i'd have to watch the vid again to know. but i was probably being epically insightful. ;)
shenmueso As long as you're not another one of those science deniers who troll science vids in order to make dumb comments about evolution or "just a theory", we're good.
None are known yet but there are theorems known to be of Godel type, that is proven to be unprovable from the standard axioms of maths. The most famous is the continuum hypothesis: that states that there is a set strictly larger than the set of natural integers but also strictly smaller than the set of real numbers. Well this is undecidable in ZFC...
@@DeusVivus Does Godel's Incompleteness Theorem implies that 2 + 2 can be 5??? i.e. You cannot prove that 2 + 2 = 4? Is there something like that as an implication of this theorem?
@@oneshot2028 No no, Godel theorem says that no matter what axioms you choose for Maths, either they are inconsistent or there are statements that do not follow from them. 2 plus 2 is 4 alright in ZFC this is proven and so far ZFC or otq variant with the axiom of choice have all shown to be consistent, that is that there is no logical inconsistency following from them. But at the same time there are questions which cannot be decided from these axioms like the one I mentioned. So Godel said that either our Maths is inconsistent or we will never know everything, even in principle
@@oneshot2028 sorry it is a typo I meant "its variants" as several systems of axioms have been proposed depending on what additional axiom you impose. Typically it is the axiom of choice added to the ZF axioms to give ZFC. To be honest the axioms are all pretty natural so it is good that they are not inconsistent but there will always be some truths of nature that are beyond them or beyond any set of axioms you choose as your basis for maths
Hmmm so is it safe to conclude that mathematics is not the language of "God" I dont mean this religious per sé ... just more like Einstein put it.Was there ever an attempt to create an alternative mathemetical system ? Or is there allready such, which is superior to the current one ? Just seriously wondering about such. Infinity seems to be a natural phenomena which can perhaps be holistically explained. I mean if all is included in itself. But again just wondering.
I have a feeling this guy started bullshitting at the end to give humanity hope and a possibility of complete understanding which is sad because that is definitely not one of the beliefs goedel had about his work.
we are paralel, computer at to, random, exzist in computers. we are from atoms, phisics computers are, what is computer what are you, you are the same.
@MrIuliux: I believe Goedel would DISAGREE with you on your assessment of the human brain. From all of Goedel's interest in Metaphysics, I believe he thought there were more to us and our brains than simply a "network of decision cells". In fact, I find this viewpoint quite elitist, as it supports elite goals to convince the public that they are merely Carbon-based machines, and can be treated as such.
@myLocalixations: Thank God! For if man knew all, then man could control all. If man controls all, then man becomes a god. And God won't allow that to happen, because once some men become gods, all other men lose their freedom.
I have been fascinated by Gödel’s incompleteness theorems for years, ever since I was a graduate student in mathematics many years ago. This is one of my favorite lectures on the subject. Well done!
really enjoy this chap's lecture style. accessible, uninflated, folksy and concise- his explanation of the seats to people in the lecture hall as a way of explaining that you don't have to count everything to arrive at accurate numbers is a great example of how different teachers offer different examples, some easier to understand than others. he, thankfully is very helpful and not obscure.
Excellent. It is not just that Godl found a quirky solution to GR, but because there are so few analytic GR solutions it also shows the enormous skill of Godl.
Excellent three video set on Godel. It is worth noting that Godel made significant contributions to relativistic cosmology, circa 1949. Einstein and Morgenstern took Godel to be sworn in as a U.S. citizen. He was brought before a Federal judge to be sworn in. Godel corrected the judge on several minor matters. The judge then stated that the troubles in Europe with Hitler and Nazism could not happen here in the U.S. Godel had studied the Constitution and Bill of Rights, etc., etc., and started to correct the judge stating that it could easily happen here. Einstein, however, tapped Godel's leg and cut Godel off before he got denied citizenship. Many would love to have heard Godel's comments on how a dictator could take over the U.S.
Best explanation of Gödel I have heard so far in a short and concise manner.
An excellent post. Goedel's ideas on relativity (popular book) are in "A World without time". One interpretation is that time as we usually view it, doesn't exist -- there is no time travel, since time doesn't flow. Of course, we are still *barely* 100 years into the theory of relativity, so much remains to be seen.
Also, the whole "I am lie-ing" problem was (to some extent) dealt with by one of Russell's students Spenser Brown in his book "Laws of Form", but by the time he wrote it, Russell's mind had pretty much been broken (by his own admission) from the years of intense work (and prison for protesting WW I ). But, he said that the work was better than the theory of "types" and "classes" that they had come up with to attempt to fix the original paradox.
Brown's work postulates that self-referential statements and such systems involve something like "imaginary" numbers except as a sort of logical equivalent in terms of proofs, sets, etc.
Onward into the 21st century. Poincare's theorem is solved, on to the next --- ;)
-r.
Rion Breffeny i must agree that the conception of time is purely ideal
Wonderful presentation. I loved all of it. Being an engineering major with a huge interest in math, I've always wanted an overview of the incompleteness theorems, but found them quite inaccessible in formal mathematics. Now with an intuitive grasp, I think I can read the mathematics behind them, though I have quite a few questions about them. Huge thanks to the uploader!
Great lecture!
Enjoyed these three - Very thought provoking vids.
Science is simply a ‘method’ which functions in a common sense/rationalistic view of the world. I take that only through a commitment to rationalism, can we "demonstrate" science.
Quoting Ed Feser (paraphrasing E.A.Burtt):
“the thinker who claims to eschew philosophy in favor of science is tempted “to make a metaphysics out of his method,” trying to define reality as what his preferred techniques can measure, rather than letting reality dictate what techniques are appropriate for studying it.”
The pattern to all these logical paradoxes is that they are syntactically correct, but semantically they imply nonsense decision procedures. In each case the decision procedure boils down to an infinite recursion with no base case. Infinite recursions are never computable without a base case. It's like trying to navigate space with a black hole in it. You never get to the other side. You always get sucked into a bottomless well.
But we are exploring space.
you'll never get thru with arguments of computability to mathematicians and logicians that are idealist instead of materialist. it literally took until alan turing, a materialist mathematicial, to actually start the computer revolution.
And he clearly disagreed with Gödel's idealism: he suggested in his reply that Gödel's solutions of the cosmological equations might have "to be excluded on physical grounds".
First time I heard of Godel's arguments for time travel very cool!
To be precise, it applies to any system, that contains natural numbers. ZFC (Zermelo-Fraenkel-Set Theory) for example is nowadays usually used to define sets, and within it I can model arithmetic. So the incompleteness theorem holds for ZFC. It is then possible to prove arithmetic consistent within ZFC - but you can't proof, that ZFC itself is consistent, so... yeah. ^^ I love mathematics as well, especially logic, and in fact, Gödel is the reason why I am a mathematics student ;-)
on youtube we get to grade professors' lectures: this guy gets an A++
Logic proving logic? No.
Faith proving logic? Yes.
I was simply pointing out that reason is priori empirical science. You can’t base reality off empiricism alone. Godel demonstrates that nothing is ultimately certain, and that everything hinges on an elusive rationalism, that of which cannot be proven empirically, but which is seemingly and undoubtedly demonstrated through the sciences themselves.
Because it has been proven to be complete ;-) Keep in mind, that being complete for a logical calculus is something different than for a theory. A theory is complete, if for every sentence phi, either phi itself or its negation follow from the theory. A calculus is complete, if, for every theory, every sentence that actually DOES follow from the theory (so, that holds in every model of the theory) is PROVABLE from that theory in this calculus...
1. the books "Gödel's Proof" and "Gödel, Escher, Bach" contain proofs.
2. Roughly: Take the Sentence G="This sentence is not provable". If the system is consistent, G must be true. Therefore, if I could prove the consistency, it follows, that G is true, therefore I would have proven G. However that would mean, that G is wrong and my system is therefore inconsistent - contradiction.
3. It could theoretically exist - most mathematicians assume, ZFC (and Peano Arithemtic specifically) is consistent
I think it has a groundbreaking implication outside mathematics, and that is it may just show that we can never ever find a so called physical theory of everything. A framework that can derive all physical truths abiut the universe we live in.
"Autoinfanticide" hehehe. Totally stealing that one
Computers don't have faith. Our faith on the axiom of choice made us accomplish very interesting results like Hahn-Banach theorem, Banach-Alaoglu theorem, Baire category theorem, open mapping theorem, the closed graph theorem...
Godel's solutions of Einstein's equation could serve _limitatively._ as an example of fictional supernatural spacetimes to which the AdS in AdS/CFT might have to be compared. In Headline form: "AdS/CFT holography a Godel-encoding of Godel's own closed-wordlines solutions to Einstein's Equations?"
Or, less ambitiously: "AdS/CFT holography as an analogue to the closed worldlines in Godel's disk-shaped models of Einstein's equations"
Given Godel's proof that 'it is impossible to demonstrate, within a mathematical theory, that this very theory is non-contradictory,' Alain Badiou argues that:
a reasonable ethic of mathematics is to not wish to force this point; to accept that a mathematical truth is never complete. But this reasonable thing is difficult to maintain. As can be seen with scientism, or with totalitarianism, there is always a desire for the omnipotence of the True.
philpapers . org / archive / BABCPO.1.pdf
Mathematicians are underrated only in the minds of mathematicians. Everyone else recognizes that the "problems" of mathematics are almost totally of the mathematicians' own making. They are the penultimate in ivory tower academicians, philosophers being the ultimate.
+dlwatib
And yet this beepy flashy whirry thingamajig you are typing on and communicating through, is built entirely on mathematics.
>>> As a follow up to my recent posts on (Goedel’s incompleteness theorem) the architecture of materiality and that of the realm of abstraction, the two structurally linked, which prohibit the formulation of conceptual contradictions, I present the following for critique.
After watching several video presentations of Geodel’s incompleteness theorems 1 and 2, as presented in each I have been able to find, it was made clear that he admired Quine’s liar’s paradox to a measure which inspired him to formulate a means of translating mathematical statements into a system reflective of the structure of formal semantics, essentially a language by which he could intentionally introduce self-referencing (for some unfathomable reason). Given that it is claimed that this introduces paradoxical conditions into the foundations of mathematics, his theorems can only be considered as suspect, a corruption of mathematic’s logical structure. The self-reference is born of a conceptual contradiction, that which I have previously shown to be impossible within the bounds of material reality and the system of logic reflective of it. To demonstrate again, below is a previous critique of Quine’s liars paradox.
Quine’s liar’s paradox is in the form of the statement, “this statement is false”. Apparently, he was so impacted by this that he claimed it to be a crisis of thought. It is a crisis of nothing, but perhaps only of the diminishment of his reputation. “This statement is false” is a fraud for several reasons. The first is that the term “statement” as employed, which is the subject, a noun, is merely a placeholder, an empty vessel, a term without meaning, perhaps a definition of a set of which there are no members. It refers to no previous utterance for were that the case, there would be no paradox. No information was conveyed which could be judged as true or false. It can be neither. The statement commands that its consideration be as such, if true, it is false, but if false, it is true, but again, if true, it is false, etc. The object of the statement, its falsity, cannot at once be both true and false which the consideration of the paradox demands, nor can it at once be the cause and the effect of the paradoxical function. This then breaks the law of logic, that of non-contradiction.
Neither the structure of materiality, the means of the “process of existence”, nor that of the realm of abstraction which is its direct reflection, permits such corruption of language or thought. An entity cannot be at once here and there. Likewise, one cannot claim that he can formulate a position by the appeal to truths, that denies truth, i.e., the employment of terms and concepts in a statement which in its very expression, they are denied. It is like saying “I think I am not thinking” and expecting that it could ever be true. How is it that such piffle could be offered as a proof of that possible by such a man as Quine, purportedly of such genius? How could it then be embraced by another such as Goedel to be employed in the foundational structure of his discipline, corrupting the assumptions and discoveries of the previous centuries? Something is very wrong. If I am, I would appreciate being shown how and where.
All such paradoxes are easily shown to be sophistry, their resolutions obvious in most cases. What then are we left to conclude? To deliberately introduce the self-reference into mathematics to demonstrate by its inclusion that somehow reality will permit such conceptual contradictions is a grave indictment of Goedel. Consider;
As mentioned above, that he might introduce the self-reference into mathematics, he generated a kind of formal semantics, as shown in most lectures and videos, which ultimately translated numbers and mathematical symbols into language, producing the statement, “this statement cannot be proved”, it being paradoxical in that in mathematics, all statements which are true have a proof and a false statement has none. Thus if true, that it cannot be proved, then it has a proof, but if false, there can be no proof, but if true it cannot be proved, etc., thus the paradox. If then this language could be created by the method of Goedel numbers (no need to go into this here), it logically and by definition could be “reverse engineered” back to the mathematical formulae from which it was derived. Thus, if logic can be shown to have been defied in this means of the introduction of the self-reference into mathematics via this “language” then should not these original mathematical formulae retain the effect of the contradiction of this self-reference? It is claimed that this is not the case, for the structure of mathematics does not permit such, which was the impetus for the development and employment of this language in the first place. I would venture then that the entire exercise has absolutely no purpose, no meaning and no effect. It is stated in all the lectures I have seen that these (original) mathematical formulae had to be translated into a semantic structure that the self-reference could be introduced at all. If then it could not be expressed in mathematical terms alone and if it is found when translated into semantic structures to be false, does that not make clear the deception? If Quine’s liar’s paradox can so easily be shown to be sophistry, how is Goedel’s scheme any less so? If the conceptual contradiction created by Goedel’s statement “this statement has no proof” is so exposed, no less a defiance of logic than Quine’s liar’s paradox then how can all that rests upon it not be considered suspect, i.e., completeness, consistency, decidability, etc.?
I realize that I am no equal to Goedel, who himself was admired by Einstein, an intellect greater than that of anyone in the last couple of centuries. However, unless someone can refute my critique and show how Quine’s liar’s paradox and by extension, Goedel’s are actually valid, it’s only logical that the work which rests upon their acceptance be considered as invalid.
@12:00 I've been thinking about this for about 5 years now. The impossibly possible.
@jimbo: Yes, that talk would further differentiate the power of the human "mind" from that of the machine. Perhaps the speaker did not want to further highlight those differences? @5:30 he virtually dismisses these differences and the significance of Goedel's work in illustrating them.
I think one can give similar arguments that a machine cannot create another machine. The only way to do this is the human mind programs the first machine to create the second machine. That is, machines cannot create algorithms. Or in other words, it is impossible to give an algorithm which creates another algorithm. This leads to the fact that "thinking" is not an algorithmic process.
nice talk. thanks for sharing
That is so cool! Thanks. So this argument applies not only to arithmetic but to any system which can encode every mathematical statement? If so that is way more awesome than I thought. I just love mathematics! ^_^
people make mistakes by not thinking enough, it doesn't mean that human thought is axiomatically inconsistent.
+Ian Jackson
Godel's inconsistency/incompleteness stems from the liar paradox, which includes the opposite incorporated into the proposition like the true statement 'I am not a liar'.
Turing using 'self-reference' to overcome incompleteness and prove ingeniously, the halting problem in computers.
I believe Kubrick's "2001" is much more about these blind spots that machines don't have than space travel.
there are provably consistent systems of course, however, they are not powerful enough to even model basic arithmetic. For example take the set {r,p,s} with a relation R, such that R(r,s), R(s,p) and R(p,r) are true. r,p and s stand for rock, paper and scissors and R is the relation that tells you, what beats what. It's obviously consistent, but you can't do a lot of math with it ;-)
Wow, that's mind blowing.
Wow, simply brilliant.
But for that to work, G must be expressable WITHIN the system. Gödel achieved this, by encoding formulas as numbers, requiring arithmetic. He then was able to talk about formulas and proofs just by talking about numbers with certain properties. This makes it possible to say "there is no proof for this sentence" by saying "there is no number c, such that c has the Property P" - and that sentence itself has a certain number assigned to it that can be put into the sentence itself... boom ^^
Present theories are known to have contrary conclusions if taken far enough, so there is no way to get remotely in the range of the idealizations that concern axiomatic logic.
Thanx for uploading.
@packe777: Yes, I must admit that I have been troubled that Goedel is a Platonist. Can you provide more info on "original Goedel interpretation" that you speak of. Thanks.
Really good talk, but I don't think (& it wasn't expected) the speaker took into account the autopoiesis of the individual as a sub-set of it's environment (rhizome) when trying to compare the human mind (second order cybernetics) to that of a computer (first order). That would be an interesting discussion to hear.
@OptimusPanzer: YES ! ! ! !
A unified theory of physics is impossible.
I would hope that people can start to see how dangerous Gödel's finding's were to men with power, especially those attempting to promote science as the new god. Perhaps Gödel's fear of being poisoned by food were not so illogical.
It's not odd that a humanly constructed model that mimics some aspects of human minds seem to be oddly reminiscent of human reasoning. It's about as unsurprising that a lifelike statue is oddly reminiscent of what a human looks like. Anyone who thinks computers think is practicing a modern form of Pygmalionism, doing the same thing children do before they learn that their teddy bears and dolls aren't alive.
Interesting question... I had that same thought. I'm nowhere near versed enough in math or logic to have my head around these ideas in one viewing (or likely 100)- but it seems to me be a fair question from a logical standpoint. I'm sure that this fellow would have an interesting answer.
Well, wouldn't an experimental physical proof be a measure of our physical world, which is a system that is external to theoretical physics? And given that our physical world is not a formal system I don't think Goedel would apply. So yeah, if I understand your comment correctly, I believe your first point is accurate.
I'm relieved to know that when my computer crashes, it's just a 'blindspot'.
+philipm06 Its probably a virus. lol
Thanks. But, I don't understand why any system that can model arithmetic must not be provably consistent. Do you know where can I find Godel's proof? But another question is then, can a system capable of modeling arithmetic be (unprovably) consistent. I am simply asking if such a system could theoretically exist, not if we can find it (because the answer is obviously no to that).
Does Godel's theories apply outside of mathematics?
You could say, a 'Divine Simplicity'; as argued for by St. Thomas Aquinas.
I have a question: has it been proven whether there are any systems that are consistent? I understand that no system can prove that it itself is consistent, but do we know if a consistent set is even logically possible? sorry if this is a stupid question.
@ShopSmartShopSMart1: Are you able to expand on this comment (within TH-cam's posting constraints)? How would one "unify all information". Can you refer us to any reference material, articles, etc.? Thanks.
no, he said there are theories that can't be proven, but not ALL theories are unprovable.
The brain could be computational AND be something which transcends computation, they're not mutually exclusive as this fellow seems to imply. See R. Penrose.
I really dont know.
What cannot be proven formally will need an experimental proof.
But, on what formal basis will they explain and connect the results of thoses experiments with the unified theory...?
The "human mind is not a machine" is not the conclusion. It only implies that the human mind -might- not be a deterministic Turing machine. My only criticism. Otherwise great lecture.
Godel's theorem being applicable or not doesn't say anything about existence of a theorem. If it's applicable to a given physics theorem say string theory for example then it says that string theory can't prove it's self. But people can still write down string theory and scratch their heads about it.
Thanks for the answer!
Can't experimental proofs of physical theories kick physics out of Godel theorem applicability range?
Or considering physics laws and experimental results as axioms means that Godel theorem applies to physics?
Proof by experiment in physics is a form of induction. Proof in mathematics has to be a logical deductive proof with axioms only. Proof by induction in physics is never true proof in the mathematical sense. An example would be newton's mechanics to be disproved by Einsteins relativity. But logically Einsteins theory could be wrong and it only represents our understanding of physical world through observation. It is not the same as logical truth.
I was not able to understand the thinking that produced the paradox (that may not be the best word to use) in "This sentence is true." or "...false", because those 4 words are not a sentence, they don't convey any information. Just 4 words strung together.
Also had difficulty with Cantors' uncountable numbers, the tortoise and the hare race where some kind of concept of infinity emerges.
What is the sum of 1 infinity and 2 infinities? Or 1/2 infinity + 3/4 infinity? 😉
great! thanks for the vide
Does Gödel theorem implies that a grand unified theory of physics is impossible?
Try search a talk by Hawking, named "Gödel and the end of physics"
Maybe the problem is that BiPolar Logic does not apply in reality. Perhaps, Spectral Logic (some call it Fuzzy Logic) will yield a more realistic set of Math Systems which better model "reality".
Hmm so I've been thinking about Godel's incompleteness theorem, and i am not a mathematician or physicist, but I think that taking his theorem a little further, we can explain why it is human nature to believe in an omnipotent god. I am not saying this theorem proves or disproves the existence of said god. I believe that at the core of human cognition, is simply pattern recognition, we see patterns in everything, we even see complete randomness as a form of a pattern, a lack of a pattern if you will. Mathematics is the art of quantifying these patterns that we perceive. Mathematics is a man made system, a system that at least attempts to describe everything in the known universe, as well as everything in the imaginable universe. but for the sake of the point i'm trying to make, let's say that mathematics, in one way or another, can quantify, or describe the entire universe as we know it, from the smallest particle to the entirety of the universe, however, in order to prove that the system of mathematics is true, one must go outside the system, but the system already includes the entirety of the universe, including the entirety of human cognition, so how then, do we prove that the mathematical system, and everything included within the system, to be true? We cannot actually prove the system that we are a part of, which is where an omnipotent god comes in. we imagine something that is infinitely powerful, that is essentially infinity raised to an infinite power, raised to an infinite power to infinity, and that something is what we label as god, however i believe that "god" is simply a placeholder for the system which proves the entirety of our currently known, quantifiable system. so in summary, the human mind sees patterns in everything, literally, and feels the need to quantify these patterns, however it does not stop with patterns we can actually observe, but goes further to try to quantify what we can not yet rationally quantify, which results in an abstract idea, which is often named god. so then by that logic, once we discover the system that explains everything, including our entire known system, then the need to ascribe the label of god to anything, will vanish? So then is string theory the theory that will replace god? all this is simply ideas rolling around in my head and I realize that my logic may be somewhat fallacious, however I'd love legitimate feedback, either positive or negative on everything i just said.
I find your thoughts on god interesting. If I am not mistaking you are basically saying that god is 'the unknown'. I prefer to keep the unknown for what it is and dont replace it by some sort of entity but I can see the logic behind what you said, how the human system can give rise to these thoughts about a god and how the entire notion of god is essentialy one big fallacy.
"Mathematics is the art of quantifying these patterns that we perceive. Mathematics is a man made system, a system that at least attempts to describe everything in the known universe, as well as everything in the imaginable universe. but for the sake of the point i'm trying to make, let's say that mathematics, in one way or another, can quantify, or describe the entire universe as we know it, from the smallest particle to the entirety of the universe, however, in order to prove that the system of mathematics is true, one must go outside the system, but the system already includes the entirety of the universe, including the entirety of human cognition, so how then, do we prove that the mathematical system, and everything included within the system, to be true?'"
1. Math is a man made system intended to express as much true statements about the world as possible, we know however that (a formal system of) math cannot be complete and consistent at the same time. If the system is consistent it means that certain elements are excluded from the system and therefore it is not complete, all derived statements are true but not all true statements can be derived in this system. If the system is complete then all truths and non-thuths should be derivable via this system. This is not possible in a formal system (which is usefull for anything) because the only way you can be sure of consistency and ditinction between truths and non-truths is when you close the system and thereby cause incompleteness. I hope this makes sense.
2. The universe is infinite so it is impossible to have a complete and correct (consistent) understanding (what is and what is not) of the universe (unless maybe you are the universe as a whole).
''so in summary, the human mind sees patterns in everything, literally, and feels the need to quantify these patterns, however it does not stop with patterns we can actually observe, but goes further to try to quantify what we can not yet rationally quantify, which results in an abstract idea, which is often named god'
1. it is true that humans make assumptions for certain patterns etc. For example if you throw something in the air it will fall back on the ground so you will expect such a thing to happen again. I don't think that if I can't observe something it is perse an irritaional quintification, I don't need to go to antarctica so see with my own eyes if gravity works the same over there. (simple example just to demonstrate the idea).
2. ''an abstract idea which is often named god' is similar to 'god of the gaps' in the context of your sentence.
3. Abstract ideas are always based on rational ideas, the uncertainty rate just becomes higher. For example: Is gravity real? 99.9999....% certain. Will it rain tomorrow?: 65% Are there horses on the moon?: 0.0000......1%. Maybe this doesnt get my point across but what I am trying to say is that there is no need to give the unknown a name, we should instead focus on what we do know. I also must say that these statistics are what is real according to humans, to be very correct.
Some additional thoulds of mine:
-We do not know what we don't know, therefore I dont believe anything without (some) reason, otherwise I would have to consider all imaginary things as a serious thing to investigate.
-A human as a system is limited, we are limited to our senses to perceive and to our internal processing. We are also bound to our surroundings.
-We cannot get out of our own system.
The most important thould here is that humans are as a system incomplete and possibly inconsistent, just like a formal system for mathematics. A system which is limited and not capable of knowing all truths and possibly stating contradictions.
Hope this is interesting for you.
first off i just wanna say thanks for the reply, very interesting insights. I think that maybe i was misunderstanding what was meant by an incomplete system, but the way you explained it makes alot of sense, however it raises another question for me, is there even such a thing as a complete system? i can't think of one unless maybe the universe it's self is? idk. also yes i realize i am basically restating the 'god of the gaps' idea from a slightly different perspective, but i'm trying to take it slightly further to explain why the god of the gaps is so prevalent.
I would also agree that abstract ideas are based on rational ideas, or at least birthed there, however I think abstract ideas can get out of hand quickly when people treat them as if they were completely rational and factual ideas and fail to make the distinction between factual ideas and abstractions. i know there can be alot of grey area between the two.
"-We do not know what we don't know, therefore I dont believe anything without (some) reason, otherwise I would have to consider all imaginary things as a serious thing to investigate.
-A human as a system is limited, we are limited to our senses to perceive and to our internal processing. We are also bound to our surroundings.
-We cannot get out of our own system."
i have often thought the same thing, we cannot study ourselves from a purely objective sense, there will always be an element of subjectivity when we are trying to understand ourselves, which really brings into question, how much do we actually know about ourselves and our knowledge? how much of it is distorted or just plain wrong because of this subjectivity? I think that is also part of what makes it so fascinating trying to understand humanity, that we will probably never be able to completely understand ourselves simply because we are either incomplete or inconsistant or both, making the task so much harder than it would be otherwise!
on another side note, if you like philosophy and psychology, and trying to understand humanity, look up Jordan peterson here on youtube, he is constantly dropping bombshells of utterly simple yet profound observations regarding humanity. you won't be bored listening to him loo.
.......................the crystal of DNA vibrates differently than the way a crystal of silicon vibrates...........I believe the modes of vibration are different from one another.
isn't the difference between a computer and a human mind the fact that a mind is housed in a body and a computer is housed in a box?
@Grak70: Thank God for this Irony, for it is what will ultimately keep us free.
@beefking69er: if reality hinges on "elusive rationalism", and if this reality is "demonstrated thru the sciences", then is science the driver or reporter of reality?
Just remember this, man's formal systems of logic, which are used in an almost Godlike manner to direct all humankind, ultimate rely on FAITH, because there are proofs that we know are true but can NOT be proven by these supposed infallable formal systems of logic. The takeaway is this.... that man is not so powerful as we are told, and that man does not have command over nature as we are lead to believe.
The worst thing that's happened to mankind is the introduction of nought, zero. It's a nonsense, anyway, though it makes arithematic easier. From nought we've leaped headfirst to negative number and it's corollary, negative value. It's only a stone's throw from there to Benatar and beyond; not just collateral damage but necessary damage, doing away with life in the name of progress, to stop suffering, the "we're doing you a favour" mentality. That was impossible to think before we thought 0 was something. Well, I have news for you: zero is still nothing. It's nonsense to say I have no eggs; might as well say I have no dinero or no Martians. Then, to take it further to it's absurdity, can you hear the unborn Martian babies screaming to be born?
I think,You are meaning some things are proved ,i.e. simply at it.
@ziggityfriggity: My issue is not with mathematical consistency and completeness, My Issue is that academia, media, government institutions and corporations try to sell science to the public as the new god, and that's simply not the case. Math & Science, and therefore computer networks, have severe limitations that should be understood.
Time travel without incorporating Entropy? which is incomplete pun intended
To bad the guy toward the end had exposed his unconvincing wishful thinking and bias. He is afraid to see that the emperor has no clothes. Again and again, sweeping under the rug the original Godel's interpretation and fact that likes of Godel, Cantor, Penrose or Chaitin were/are all Platonists, and not formalists. Kant has already showed us the epistemological limits of reasoning (in "Critique of pure Reason") and solipsistic A Priori for true consistency of Godelian argument.
how do you know if you can't see them?
What's the name of the teacher, pls?
great. thanks.
One line of poetry can crack mathematics apart, based as maths is on 2-D abstraction with contradiction of the excluded middle.
It's logically flawed, in other words. Pick your own favourite line before you write 2 + 2 = 4 large on a sheet of A4 and fold it in half.
"Season of mist and mellow fruitfulness."
The point is noone can tell all the associations, personal and general, this has for you. It's not reducible to a Holographic Principle on the boundary. At the same time you can't weigh a thought; you can test for energy use, etc., and cut the brain open. You won't find the thought, though, in the dead meat. That implies mind is not meat, as our current bunch of academics want to prove.
It's surprising that string theorists can believe in 11 dimensions but reduce us to one or two. And let's not go from pure maths- and that's only Mind- to the land of fairy, Multiverses. Next we'll be introducing the Mandela Effect and parallel worlds, if you follow the logic. Didn't the news in 1985 say Demi Roussos was dead; and now I find he died in 2015. What's your Mandela Effect moment?
Godel had a logical proof of God, by the way, which I can write if anyone's interested, though the formal language won't come out in a comment section.
One cannot know the knower of the known.
There is something wrong with reality. For every particle there is an anti-particle. Again, for every physical law, the opposite is also true. In such a situation, in my mind it is futile talking about "set of all sets", it simply is an undefined quantity like the concept of "infinity". A theory of everything will need an algorithm with infinite "axioms", or like a book with infinite pages, you can safely set aside. For every other sub-set like a physical system, consists of finite axiom algorithm that enables us to not only know about it but also use it with certainty, that does not fail us.
but what does it mean to say "This sentence IS provable" or "This statement is TRUE" ? How can a statement be true or false if it contains no information?
that is a bit wrong of a way to look at it. godel's theory is a part of mathematics. as far as i understand, you can either prove all theorems(but then some proofs would be wrong, in the sense this is a lie is wrong) or you can avoid all such statements and have correct proofs(but then you would not have all the theorems, some theorems you would never be able to prove).
it requires a radical rethinking(for me at least) of how to approach mathematics.
The lecturer certainly enjoys using and re-using the term "blind spots."
Great, now you gave me a headache.
@mrc560: Only applies to formal systems of logic, of which math is one.
ddstar: No, he proved it.
yeah, Gödel would have disagreed. But I find the proposition, that humans are somehow special and are "above" any formal systems much more elitist. Why should that be the case? I treat everyone like a "carbon-based machine" - that's why I'm vegetarian. How and why does it devalue your or anyone's life if I DON'T believe in something special/magical about minds? I don't think it does, on the contrary, I think it's quite humbling...
"this sentence is false" does not really need to be pondered over for hundreds of years.. it is simply an incomplete proposition. it doesn't relate to maths cos maths doesn't care about syntax.
+shenmueso
You do realize that that sentence is the dumbed-down "fake" version of the core of Godel's Theorem?
Godel wrote a book about his Theorems. If you disprove him, you can win a Nobel. Go for it, stud.
no i didn't know that, so whats the real version of the core of his theorem?
shenmueso
The real version is a book consisting mostly of math and formal logic. The liar paradox is a rough analogy which Godel can be said to have drawn his inspiration from. It's useful tool for explaining the general idea to laymen without using numbers.
Most of popular science media is like that: Full of dumbed-down "fake" versions of the actual thing, used to give laymen a rough idea. For example, electrons having a "spin", or string theory being a bunch of "tiny vibrating strings" making violin sounds which create the universe.
There's no point laughing at how stupid all that sounds, because they were intentionally made to be stupid enough for people on the street.
Mlai00 i understand the concept of metaphor. tbh i don't remember why i typed that, i'd have to watch the vid again to know. but i was probably being epically insightful. ;)
shenmueso
As long as you're not another one of those science deniers who troll science vids in order to make dumb comments about evolution or "just a theory", we're good.
Can somebody give an example of "inconsistency of mathematics"??
None are known yet but there are theorems known to be of Godel type, that is proven to be unprovable from the standard axioms of maths. The most famous is the continuum hypothesis: that states that there is a set strictly larger than the set of natural integers but also strictly smaller than the set of real numbers. Well this is undecidable in ZFC...
@@DeusVivus Does Godel's Incompleteness Theorem implies that 2 + 2 can be 5??? i.e. You cannot prove that 2 + 2 = 4? Is there something like that as an implication of this theorem?
@@oneshot2028 No no, Godel theorem says that no matter what axioms you choose for Maths, either they are inconsistent or there are statements that do not follow from them. 2 plus 2 is 4 alright in ZFC this is proven and so far ZFC or otq variant with the axiom of choice have all shown to be consistent, that is that there is no logical inconsistency following from them. But at the same time there are questions which cannot be decided from these axioms like the one I mentioned. So Godel said that either our Maths is inconsistent or we will never know everything, even in principle
@@DeusVivus What does "otq variant" mean?
@@oneshot2028 sorry it is a typo I meant "its variants" as several systems of axioms have been proposed depending on what additional axiom you impose. Typically it is the axiom of choice added to the ZF axioms to give ZFC. To be honest the axioms are all pretty natural so it is good that they are not inconsistent but there will always be some truths of nature that are beyond them or beyond any set of axioms you choose as your basis for maths
@najtofni: I agree with this statement. This is my understanding as well.
Hmmm so is it safe to conclude that mathematics is not the language of "God"
I dont mean this religious per sé ... just more like Einstein put it.Was there ever an attempt to create an alternative mathemetical system ? Or is there allready such, which is superior to the current one ? Just seriously wondering about such. Infinity seems to be a natural phenomena which can perhaps be holistically explained. I mean if all is included in itself. But again just wondering.
I have a feeling this guy started bullshitting at the end to give humanity hope and a possibility of complete understanding which is sad because that is definitely not one of the beliefs goedel had about his work.
@najtofni: "I agree that globalbeehive has a nonsensical stance on science"
So, where am I going wrong?
I wonder if you see the irony of that statement. I'm guessing not.
So according to Godel, Godel's theory can't be proven.
we are paralel, computer at to, random, exzist in computers. we are from atoms, phisics computers are, what is computer what are you, you are the same.
@MrIuliux: I believe Goedel would DISAGREE with you on your assessment of the human brain. From all of Goedel's interest in Metaphysics, I believe he thought there were more to us and our brains than simply a "network of decision cells". In fact, I find this viewpoint quite elitist, as it supports elite goals to convince the public that they are merely Carbon-based machines, and can be treated as such.
@myLocalixations: Thank God! For if man knew all, then man could control all. If man controls all, then man becomes a god. And God won't allow that to happen, because once some men become gods, all other men lose their freedom.
Thanks...you gave me a headache
hm... killing of your own young vigorous grandpa? what would be effect of such"scientific" murder to human stupidity: profit or decline of?
Look its richard buckland's brother..
Your girlfriend has many blind-spots in her logic and reasoning. Don't try to understand her, don't try to fix her, just live with her!
To understand Godel, just try to understand how benevolent God created Devil.