Sir Roger Penrose explaining Godel's incompleteness theorems.
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- เผยแพร่เมื่อ 24 พ.ย. 2024
- "What is Godel's incompleteness theorems?" By Roger Penrose.
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I don't own any part of this video. It's just only for educational purposes.
Thank you.
Sir Roger Penrose is a genius
Creativity comes from the ancient Greek root Xreo, which means a few things: a) to open up, explicate, unfold b) to mix.
This is interesting as we actually do have a modern definition of creativity which states that creativity is:
a) The ability to do many things with one method or tool
b) The ability to do the same thing using many different methods or tools.
For instance, a musician might compose a variety of different melodies which follow the same Rhythmic pattern, or create many different rhythmic patterns to perform the same scale of c major.
Creativity would also be the combination of two things to generate a new thing, thus it can be emergent.
Interpretations of quantum mechanics are creative, as the observations are the same, but people have generated different ways of interpreting the theory.
In martial arts, the ability to carry out different techniques that lead to different positions but from the same starting situation, permits one to choose a particular path that best suits the specific scenario. For instance, let's say two men attack, one behind the other. You block the punch and deliver a front thrust tick, sending the first man into the second man, knocking them both over. However, let's say person 2 attacks from your left just after person 1 attacks you from in front of you. In this situation, a more fitting response might be to parry person 1's punch and throw them into person 2. Knowing two or more defenses to the same initial attack gives you more facility to defend yourself.
Knowing multiple Rhythmic patterns for the same melody means you can perform the same tune in many different ways, so the audience experience is different each time. Many hit records evolved from earlier songs or modifications of other songs.
Creativity's purpose is to generate different utility.
Evolution in that sense is creative, because sexual reproduction mixes two genomes to generate a new genetic pattern.
Evaluation is the tool you have to use to determine what variations are suitable for any given situation.
Thus, in chess, creativity is generating lots of different pathways from the given position. The right choice might be determined by calculating material or positional advantages/disadvantages resulting from a particular set of moves/countermoves.
Therefore, creativity and evaluation are best combined to generate choice. In order to evaluate, there needs to be understanding of what constitutes advantage. In chess, material advantages may not play a role, if you can hem in your opponents pieces, effectively preventing their involvement. However, with only a few pieces and the right moves, it might be possible to coordinate and overwhelm an opponents position in fewer moves than they are capable of mounting a defence within. Thus, to succeed in chess, you need to be able to evaluate three factors: Material resources (how well you will do out of a particular line of moves in an exchange of pieces) , quality of the position (whether your pieces are more or less active and coordinated than your opponents) , and time (how quickly you can attack or defend in a situation).
"Create" comes from Latin "creare," meaning "to cause to grow", from Proto Indo-European _*ker-._
@@dougmarkhamWell, I have a PhD in linguistics and studied Proto Indo-European for many years, so I'm not the amateur you suppose. First, you contradict yourself by saying that "create" doesn't derive from Latin, and then saying right away that it derives from Latin "creo." Second, with the exception of loan words, Latin does not "derive" from Greek in any way. Both derive from PIE, a language that has been extremely well reconstructed for many years to great success, and enterprise that brought linguistics for the first time into the domain of the natural sciences. To call it ""mostly guesswork" can only mean you know nothing of the theory. Third, I know of no cases in which a single script has made everything change. In fact, quite the opposite tends to happen: the discovery of Hittite, for example, provided direct evidence for the existence of hypothesized laryngeal sounds in PIE, which no other languages had preserved. You should learn something about linguistics; clearly you are very interested in it. I can recommend many books to you on the historical method, which has been successfully applied to many language families.
@@dougmarkham Do you regularly reject input from people who have something to teach you and are willing to help you learn? It's not a good habit.
@@erichgroat838 This is very good insight, but nobody speaks Latin anymore. So when people make the contemporary argument that machines or AI can't be creative, the definition of creativity should be derived from modern usage. I believe "creativity" is to be able to create - basically what the original commenter said. To create novel things. Based on this, some AIs can indeed be said to have creativity.
Can you imagine taking secondary courses from Dirac and Bondi all at the same time.
The key point to Godels Incompleteness Theorem is this. There is only a countably infinite number of proofs, but as Cantor proved, the set of number theoretical functions is uncountable, therefore he constructs a true number theoretic function which cannot be proved.
Formal logic professor. The amount of math students I come across who come to philosophy to solve problems in other fields is shocking. Philosophers have known about this for a while. But what's really interesting about Gödel is the theorem breaks math specifically but not formal logic. One thing about logic that really sets it apart from math is every rule in logic is provable. You're not allowed to use rules in logic unless you can prove them. Which is why his proof proves formal logic to be complete but like mathematics to be incomplete.
Another thing people don't typically know about logic is this whole notion of "true or false" is called the law of the excluded middle. Now mathematics is completely built in this but logic has systems like fuzzy logic which is typically the logic used for AI doesn't operate on this law. In fuzzy logic you can actually have degrees of truth. So you can have 0.7 full and 0.3 empty both be true vs the 100% full and not 100% full.
Philosophy gets a lot of shade but you can't do logic without understanding why the rules are the way they are formal logic inherently and automatically transcends this very problem to begin with. Which a lot of us logicians speculate why he identified as a logician and not a mathematician lmao.
We need more people doing logic. We're kinda an extinct species.
I'm finding as my career goes on as a professor is the higher someone in math goes the more philosophical they get lmao. The number of math students in my classes have vastly increased. Which has brought me a great deal of joy. Not only do I get to shader they belief in math with this proof but I also get the joy to too using logic to shader their belief in science as well because of how inductive logic is inherently works. Inductive logic will never be able to prove anything because inductive logic can't be valid. And unless something is valid it can't be certain. Science is a guess at best. Inductive logic is like medium level logic. Deductive logic is where certainty lies not inductive logic.
Deductive logic relies on inductively proved premises, invariably. So deductive logic in practice does not give you certainty.
@@psychvision101 deductive logic gives you guaranteed conclusions if premises are true.
Math is also deductive, without empirical content. They are a priori true. Its not like you need to go and check every empirical instance of two couples to see if it adds up to four, like you have to make sure you are not wrong about that sum. You only have to check your thinking. There is no possibility you are wrong and one of these days it will somehow count up to five. Furthermore, each fresh experience of two couples adding up to four in the real world does not further confirm its truth like the way scientific claims are.
So deductive logic does give you certainty. Math is an example.
Can you explain how logic is any different from mathematics when it comes to self-referential pitfalls? Because, for instance, if I say that "this statement is false;" how would logic cope with that? Or the liar paradox? Or any number of those logic paradoxes.
I enjoyed your post. I have a few questions I hope you can find the time to answer. Let me begin (and I am happy to prove my argument but only if you are interesting in hearing it) with this; Quine’s liar paradox is pure nonsense, meaningless and that Goedell admired it and Quine I find very disconcerting. If I am right about this, the conclusion to his scheme was that statement, “this statement is true but has no proof”. Well, he who makes such a claim about the statement being true would have to know how and why it was true and of course, that “is” the proof. So the contradiction we were promised Goedell did “not” introduce seems to in fact to be there. Also, I thought he claimed a hole in formal logic as well. Am I wrong in that?
Also, as for inductive logic, was that what Hume and the empiricists claimed was the impediment to cause and effect? If so, consider; I was not accusing you of appealing to authority. I was speaking editorially. You have been very instructive in your responses. Anyway, I do believe Hume was guilty of that of which I accused him in this way; he surrenders to certain conventions for were he to not, he could not even formulate his propositions. Consider….
1. He agrees by implication at least, that entities can be and are distinct. A square is not a circle, a billiard ball on the billiard table moving toward another is not rock and also that second ball, etc. He even distinguishes the billiard ball from any other ball such as a beach ball such as a baseball or beach ball or whatever they had in those times. This is an acknowledgement of the objective nature of the objects he employs in his theory.
2. He by definition as a consequence of 1. above, surrenders to the understanding that the distinctiveness of these objects is because of their characteristics, the ball is round and not square, etc. His empiricist notions are irrelevant here for he admits that the square, regardless of any distortion of its true nature by our subjective minds, appears as such to all of us, always (and as wholly distinct from the circle).
3. That he surrenders to the 1. and 2. above, he also as a consequence, admits to the understanding that motion is not and cannot be one of those characteristics of the ball (in this case) moving. Motion is not tangible as is the ball and thus is apart from it though effecting it, etc.
4. He would know analytically that if motion cannot be a characteristic of the ball’s physicality, as per the above, then he knows, also analytically that it has to have been imparted to the ball by the force of some other entity of which that motion could not have been a part of either, it having been imparted to that entity by the force of another also.
5. If that is the case, that motion is imparted by the force of an entity upon another entity, the one ball to the other then there is no denying cause and effect and that the second ball will in fact move if struck by the first.
I leave out a great deal of detailed argument here to avoid writing another book, but I do think it clear. Hume was very wrong and did in fact “appeal to truths to formulate a position that denied the existence of truth”.
What do you think?
I was a Computer Engineering major for my bachelor's degree. I remember taking a philosophy of logic class as an elective thinking that it would be an easy A. (I admit I was not exactly a good role-model with my study habits, especially while I was still a teenager. But for those that do not know, "Computer Engineering" as a major was a combination of Computer Science (programming) and Electrical Engineering (focusing on digital circuits and chip design etc.) Since all digital computer circuits are in fact logic circuits and by that point I lived boolean algebra for years, I figured I would breeze through this 100-level philosophy class.)
Instead, I was blessed with a teacher as good as you sound. While the boolean algebra computation itself was second nature to me that point, she introduced me to many of the meanings behind the math that engineering teachers tend to ignore with their "shut up and calculate" attitude. Furthermore, similar to your post, through ideas like Gödel she introduced me to areas that I never dreamed were applicable had I just lived in my ECE bubble. I am very glad I took that class. (And I'm also glad professors like you are still out there inspiring ignorant math students to think outside their bubbles...thank you!)
6:07 One of the most profound things a human can learn.
I am a big fan of yours mr penrose I will meet you one day
Así es la inteligencia está por más haya de lo mecánico
could anyone critique my notions here?...............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 prohibits for 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 place holder, 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. 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 is 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 its development and employment 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 not equally 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.
Thank you V M
it is like this i think: consider infinity, the concept itself is a unending stretch, but hypothetically, if we had very less memory power that prohibits more than 100 counting, that is we cannot possibly hold more than 100 variables of concept, then whatever beyond that is infinite. for us this is false because we can hold lot of variable using zero alone we can extend the concept to infinite, even if the whole universe is turned into a paper and we write and finish a number we still have lot more to write, so i think Kurt self-reference is in a essence explaining the force that make us to feel, to exist itself could be a reason for our mathematical concept, imagine if we had 8 fingers or 18 fingers would we consider base 10 as easy to count or base 18? as there is many-world theory that says we have lot of alternate universes, there could be lot of mathematical concepts greatly varying from ours, so from their axiom our's would be false, from our's their's would be. i think so...
@@bennywins2930 You English is a bit tricky, obviously a second language for you so forgive me if I mistake your meaning. So….there is no infinity in materiality. None. Even as a concept it cannot be paired or joined with any other which is of materiality for to exist within materiality, all entities must be finite, delineable and quantifiable. There are no exceptions. Also consider…that within any quantifiable instant in time in material reality, there can be no “complete” infinity, Dr. Penrose notwithstanding. All we can do if we are to remain true to the understanding of materiality, is consider the concept of infinity as a really long progression, but still quantifiable at any place along that progression. As a concept, it is beyond our reach in an attempt to pair it without material concepts or entities. Consider, it is said that a “line segment” is composed of infinite points for in science and math, points have no scope or dimension. But this is conceptually problematic. IF we consider that a line segment, 1” let’s say, exists, it does so by beginning somewhere in the entirety of space and ends a quantifiable 1” away. This is measurable, quantifiable. Given this, each end by definition must be designated by a point beyond which there is no other (or the line segment cannot be defined as such to begin with). IF there is then a point beyond which there is no other at each end, again by definition there must be an aspect which is quantifiable or also again, the line segment cannot be defined. Which way do you want it? Can’t have it both ways.
Consider a companion theory that a line segment, let’s say of 1”can be subdivided infinitly, e.g., halved and those halves halved, ad infinitum. This too is problematic for when we have the line segment the first time, the problem articulated in the above remains and each half would have to have its ends designated by points beyond which there are no others. Additionally, the two ends at the place of that first dissection would be designated by points both beyond which there were no others BUT also, by definition, adjacent. This would require quantifiable factors. The conclusion to be drawn is that in actuality, the line segments cannot be composed of infinite points. Now this may throw a wrench in the mathematics and physics associated with these two concepts, but it only means that some “cheating” is taking place to allow certain conclusions to be drawn. That does NOT change the fact that logic and language are being violated and nothing about them is revealed to be “insufficient”.
Naïve paradoxes are frauds, all of them. When used to validate concepts of mathematical paradoxes purportedly to show that there are limits to our logic or language, they too become suspect because material reality does not permit conceptual contradictions, period.
What do you think?
@@jamestagge3429 i am not a mathematician, just a enthusiast and from India. my way of looking at mathematics would be, it is like music, the medium our consciousness could mimic and relate and even connect with, mathematics is something like that too, but it is entirely abstract, thou it also gives us precise value to our reality at the elementary level, it is as if we found a algorithm that the nature of reality itself is mimicking, Einstein without ever taking a telescope speculated there could be mass of star so heavy it collapses itself into a singularity, but then again we should remember we are not apart of materials or the forces that binds the material. as Einstein's theory e=mc2, for energy to become mass there comes forces to make a exchange of energy that creates mass, then atom and electron form chemical bonds to form molecular structure, and this structure is always subjected to gravity, which is not a fore in itself but a result of mass and space' interaction, and also to the nuclear force and the magnetic force, which in result forced a molecular form(DNA) to live and evolve, to eat itself and to convert the reality into its forms(children), but we(humans) are the only kind to not convert the reality into us, but into the thoughts of us. planes, submarines, and satellites, rockets. we don't work for what is there, we work for what it could be like. and all these things using mathematics. without diverting our mind into anthropic principles we could state one thing, mathematics works logically and connects with reality, we cannot create contradictions, but we can have uncertainty, chaos, randomity the close cousins of contradictions. using our formula to predict planetary motion we should be able to tell exactly when and where the earth would be revolving at a remote future, but the three-body problem brings a tremendous confusion. it is not like we cannot but we are not having the system of order without entropy. unlike mathematics which as it increases in unit will hold lot more inputs without confusion, but this can be our incompetence. there is the imaginary plane in mathematics which is not exactly as the left to real numbers, so instead of using this one line of numbers if we could use the complex numbers as how we use real numbers without using another idea to derive it, maybe we would make sense of it, but as i said, the math has mimicking property, it also has leakage of shortcuts and unproved theories, undefined axioms which we can view as contradiction or chaos, uncertainty as we all are formed from uncertain quantum mechanism, the paradoxes are in a way telling us to accept the holes in it.
@@bennywins2930 Thanks for the response. I love this stuff, the study and the debate of it. I however do not believe or accept that there can be ANY material or conceptual contradictions in reality or the realm of abstraction, respectively. There are no actual paradoxes. I believe I can refute them all though some are so subtle they take a bit of time. Theories such as Goedell’s incompleteness theorems are difficult for me because in order to critique the logic one must be able to understand the math by which he arrived at his conclusions. In this case, I am lost. However, after numerous debates lasting weeks with physics graduate students, I am correct in my assumptions with regard to the corruption of the sciences by the piffle of some of those like Penrose, yes, truly a brilliant man but suspect for his embrace of the nonsense of the existence of infinity within reality. As for paradoxes, its all nonsense.
Syntax versus semantics...
So the vertasium video claiming there’s a “hole in the bottom of math” is wrong?
No . The hole is much bigger he says , or higher dimensional if you will. It's the hole of (non)understanding which cannot be filled with logical computation alone.
I haven't seen that Veritasium video, but to me that does not seem to be an embellishment... Godel did in fact show that every formal system with rules capable of referencing itself can be posed a question that will be "true" but cannot be proven to be true by that same system. You need an outside verifier to confirm the validity of the statement. Which is still, I think, an unsettling revelation.
On the surface this would mean that there are realities about life - which we might even be able to see - that cannot be formally proven by math using math alone.
It depends on what you mean by "Math". ZFC Set Theory may be inconsistent. This only bothers logicians. ZCF Set Theory is also incomplete, it may contain statements that can not be proven true or false.
Im the one who taught him that
I doubt it.
@@SadSocks trust me
Yeah he’s smart but does he have a Nobel Prize??
We’ll never be able to prove it within our system of proofs, but I understand that he does
Yes. He does now!! So there...