Gödel's Incompleteness (extra footage 2) - Numberphile
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- Опубліковано 2 чер 2017
- MAIN VIDEO: • Gödel's Incompleteness...
More links & stuff in full description below ↓↓↓
Extra footage part 1: • Gödel's Incompleteness...
Professor Marcus du Sautoy is Simonyi Professor for the Public Understanding of Science and a Professor of Mathematics at the University of Oxford.
Professor du Sautoy's book as mentioned...
In the US it is called The Great Unknown - amzn.to/2sfkWpb
In the UK it is called What We Cannot Know - amzn.to/2r5yztp
More of his books: amzn.to/2ryEk4r
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumberphile
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I like that guy. I hope we'll see him again in future videos!
He was well-spoken, has a neat accent (to my ear), and has a lot of non-domain-specific knowledge. This was really great to listen to.
Marcus du Sautoy has done a few documentarys for the BBC about maths which are really great if you want to see more of him. His series "The Code" is excellent.
he is awesome
Jakey C, thanks for the hint, I'll check it out!
Yes!
enough said
Q) Why do you need to make so may 'extra footage' videos on Gödel?
A) Any sufficiently long series of videos on Gödel must also be incomplete!
Wanted to like but saw that the number of likes was at 42, and just found that too meaningful
@@lostvarius someone's ruined it; the answer has been lost, back to incompleteness.
@@lostvarius I was the 99th like.
Proof: Any new attempt to discuss Kurt Godel's effect on the world introduces his concept to either new people or in a new manner. This thus changes or increases the effect in a way that can't be known until after the video is made and released. Thus making it impossible for it to cover everything.
Well, right now it's 262 likes, which is a nice semiprime of 2*131. I feel at least a little better even though it isn't 42.
Prof. du Sautoy's shirt really resonates with the topic they are discussing...
同意
Exactly!
You should do a thing discussing how the incompleteness theorem relates to the halting problem, and how the whole Turing machine thing was used in Turing's proof.
They actually cover that excellently on the Computerphile page.
@@frankbrody239 Link?
Thanks for including all the extra footage, there's so much more than the main upload. It's great to see prof MDS on your channel, I remember his RS Xmas lecture was superb.
This reminds me of the best expression I have come across for the inherent limit of human knowledge: "bounded rationality".
When I saw there was extra footage I hoped it was going to be an hour long. Please do more with Markus on this topic, it's absolutely fascinating. Thanks for the video!
That was a great series of videos. Thank you!
I want more of this guy!
Great series on Godel - and besides doing such a great job of explaining
More videos from him please!!! :)
Waiting for Extra Footage 2 to go live was the closest I've had to a cliffhanger is some time. _What's he gonna say? I don't know. Exactly! Gah!_
Can you still prove or find Godellian concepts in an axiomatic system that doesn't have primes? Are there different ways of encoding logical statements?
What if we cannot find the answer to the question "Are there questions about the universe which we cannot find the answer to?"
SmileyMPV then there are (at least one)
I'd say then there are none. The question about questions about the universe is not itself a question about the universe.
I would argue that questions about the universe are themselves part of our universe, thus asking a question about said questions falls within asking questions about the universe
Nathan Minsk
So in your mind, any question within the universe is a question about the universe?
Does your universe contain itself?
We can, because Goedel proved it.
I'm not sure Marcus does this on purpose, but I feel that he is seriously misrepresenting Godel's theorem. It does NOT prove that certain truths are unknowable. It only shows that there is not one formal system (i.e. one algorithm), that can prove all true statements.
Godel's own theorem is a case in point: for every formal system a Godel statement can be formulated that is unprovable from within that system, yet conscious humans can understand that the statement is true. So it is a knowable true statement, yet one that cannot be proved with an algorithm.
Marcus kind of brushes over this, and writes this down to our understanding being another formal system. However, it is also conceivable that human understanding simply does not equate to a formal system.
The link to this video in part 1 of the extra footage is missing
First time when UA-cam algorithm saved me.
Question: What is the cardinality of all the unprovable statements under our axiomatic mathematics, how much larger is it than the cardinality of the set of those statements which may be proved?
Both the set of proveable und unproveable statements are countably infinite.
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 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.
If there are "things beyond science" they are actually a part of science, we just don't know how to incorporate them. Things work the way they work, end of story. If it's transcendental in a whole new mathematics/logic kind of way then that's how it works.
There is no such thing as a "scientific world" and a "mystic/transcendental world", because as soon as one interacts with the other, then that is the way it is. And it works in some way, according to some rules.
Checkmate mystic atheists
GinoTheSinner Well put with a deep insight. I agree. Things are happening in the way they're meant to or supposed to. We tend to define them as scientific if we understand how they work and as mystic or transcendental if we don't have a clue on how they happen. The term 'scientific' or 'mystic' are not something absolute but quite relative and simply for human's convenience to define something.
@@sangyongpark2137
"and as mystic or transcendental if we don't have a clue on how they happen"
Maybe some people but not me :p
I just say "I don't know, I have no clue, neither science yet" and don't call these mystic nor transcendental ;)
You're making too many assumptions about that which might exist outside 'science'. If it is outside this universe, then we can know nigh nothing of it, and certainly never understand it.
Compare your worldview with that of an ant, or a two-dimensional being, you can affect them in ways unimaginable to them.
@@edomeindertsma6669 But then they know that's just how things work. And become part of their understanding.
Also if they completely understand their own realm, they will be able to identify what influence comes from outside of it.
@A K, but they will not be able to explain it, or find the reason.
They can only describe the effects.
Why does it trail off that way?
you're assuming that there are rules by which the universe behaves fundamentally. Does this necessarily have to be true? Is it also possible that, at a very small scale things just behave very randomly and all the things we think we know about the world are not true but give correct results with a high probability?
How was Heisenberg's Uncertainty Principle not brought up in this conversation?
Because it follows form the axioms ;)
Its about the mathematical fact, that a wave cannot be determined in both time and frequency domain
Despite *appearing* similar in nature, the two are quite different. Gödel deals with the fact that we *cannot* prove whether everything that is true is true. The Uncertainty Principle *states* that our information will always be incomplete.
Again, Goedel's theorem deals specifically with the Principa and related systems. Isn't it rather fatuous to extend that to anything else?
I think Gödel's theorems have the same power in stronger systems(loosely, more axioms), but less power in weaker systems (less axioms). Examples of weaker systems are Peano arithmetic, true arithmetic, first-order logic.
Great stuff lately, Brady ! How come everyone talks so much about Goldbach these days?
Because the Baader-Meinhof Phenomenon, that's why.
I doubt it's that because it was referenced like 5 times in the last few weeks here, and it's not that I've first encountered Goldbach on this channel, I've been aware of that mathematical question for quite some time now.
Just like there were tons of Riemann hypothesis videos before.
Gaps going to infinity got me going :D
Gödel Extra II: Gödel Harder
why did i just get a notification for a video 2 days ago
wait it was uploaded today but all the comments are 1 or 2 days ago? what is this youtube
it is impossible to prove the nonexistence of anything, because you can always add properties making the thing undetectable in more and more ways.
I thought Pauli already covered the same thing in quantum mechanics. But maybe I am mixing up my analogies😁
Empirical knowledge is qualitatively different from abstract knowledge in the sense that it cannot be understood objectively. It doesn't mean science can't be right, but that it could always be wrong.
Now as for the philosophical consequences. even seemingly "a priori" truths about nature (such as our perception of time and space) are empirically derived as even movement in space-time is a form of interaction. To say there exists some form of transcendental ideality in which through cognition nature can be perceived is to redefine existence. It's not an ontological statement, but rather a semantical one. The absence of "a priori" truths about nature implies only empirically derivable ones are ontologically (and scientifically) meaningful. While there could be technical or even theoretical impediments to what types of knowledge can be understood empirically, this does dissolve the most abstract part of the "unobtainable truths" conundrum.
Yes, it could still be that nature is governed by a set of rules which does lead to undecidable truths, but then again, it could always be that nature is not consistent (whatever that could mean) in the first place.
But why is it okay to shrug it off undecidable empirical truths in science? Well, because science only worries about truths which can be apprehensible empirically (not to be mistaken with "heuristically") in the first place. Undecidable empirical truths are all over the place. Was the universe created two seconds ago? Is there a 5th, 6th, 7th fundamental force unbeknown to physicists and by some theoretical thingamajigs, impossibly discernible from known ones? Which interpretation of quantum mechanics is correct? Etc. Occam's razor, falsifiability and other criteria don't always/ever answer these "undecidable" questions, nor do they dismiss them as "unscientific", but instead "shrug them off" as things we shouldn't worry about, at least while there are no known tools to tackle them. Like a "currently not worth worrying, but we might get back to it later" sticker on an idea.
But isn't the nature of logic apriorical ?
That is correct!
What I stated is simply that cognizance is not purely rational/aprioristic. And I state that space-time is not just some sort of background in which phenomena can be perceived, but is also something just as phenomenologically perceived (and in that regard, not much different from events which happen in it). This is also true for knowledge of the self, since self-conscience arises from the perception of the physical self by the manifestation of the self. There is no "a priori" knowledge of nature.
I look at it in the precise way:
- natural science or more generally empiricism uses induction & abduction
- mathematics uses deduction
Thanks for correction!
Empiricism is not necessarily inductive (for example, it could also be abductive).
Felipe Hindi we as humans fall to see that existence, universe do not require any level of understanding. They do not even hint at that. However, the clincher is that such setup gives birth to matter organized in such way IT self-assigns the licence of proof seeking. Hence humans and the quest for proof. I intentionally skip the word "truth" since it is irreparably anthropocentric.
Time did not exist ! Is this statement true or false?
Can an all-powerful being create a bolder that he can't lift?
something like that
Depends on your all-powerful definition.
1) can do anything
2) can do anything over everything that exists
Choose one
Depends on what you mean by can’t. Unable to or not permitted to⁉️
So how would one discover that the statement in question doesnt have a proof but its true in practice/real life if one has one of those nasty true but unprovable statements?
Exhaustive search.
i think this has little bit more debate just over language then it is over reality
Man the universe is sooo fucked up.
I love it.
It's fundamentally untrue that the statement of theology is that there is an unknowable truth. It's the claim of knowing an unknowable truth.
I'm interested on the question: on what basis may we say that something is 'true' in a system, while nevertheless being insusceptible of proof *within* that system? What justifies the truth claim here--what perspective (knowledge) from outside the system? Intuition? Consensus? If the proposition in question is properly basic, it cannot be deduced from a(ny) larger system, and can only be added as an axiom. So is it 'true' only in the sense that we make other axioms 'true'--i.e., denominated as rules in the game? This seems like a rather weak definition of truth viz-a-viz Godel's theorem.
A bit piece of the puzzle here is theory and metatheory. In mathematical logic, a theory is a collection of axioms. And the theorems of the theory are those axioms which can be proven. Now, in examining a particular theory and the theorems which can be proven from that theory, you might adopt a metatheory - a different set of axioms which you assume to be true. In this case, let's suppose you assume the axioms of set theory are true, and you are studying the theory of the natural numbers, in particular. Let's say the Peano axioms. Things you can prove from the Peano axioms alone are theorems of the Peano axioms. And if your goal is to study the Peano axioms, you might be able to prove things about the Peano axioms or about the natural numbers using the axioms of set theory. These would be considered metatheorems since we're using the metatheory as opposed to the actual theory we're studying.
In the case of the Gödel sentence, we find a sentence which is independent of the Peano axioms but, nonetheless, is a metatheorem. It can't be proven true from the Peano axioms themselves, but can be proven true from a broader set of axioms which encompasses the natural numbers as well. This is something pretty special about the Peano axioms. Gödel's theorems apply to any formal system capable of encoding the arithmetic of the natural numbers, so Gödel's theorems also apply to the metatheory. And there are sentences which the metatheory itself cannot prove nor disprove. However, unless we have a metametatheory from which we can prove or disprove such a sentence, we cannot conclude "truth" or "falsity" of such a statement.
The conclusion I've reached is that Atheism is: Admitting you don't YET understand God.
Religon is: Pretending you NOW understand God.
(...something along those lines..)
Couldn't you just create an axiom that says "All unprovable statements are true"?
No, because if you add this axiom, and then you try to prove unprovable statement, you proved it with the axiom, therefore, it is not unprovable, and you cant prove it with the axiom, so once again, you create a loop that messes with your system and makes in inconsistent
@@NeedForMadnessSVK "All statements unprovable by the other axioms are true"?
But not all unprovable statements are true
@William White In the main video, he showed that any mathematical statement that can't be proved is necessarily true. That obviously doesn't apply to statements about physical reality
Ah ha. My comment last video about _The_Ultimate_Proof_of_Creation_ by Dr.Jason Lisle was more on topic than i had known. I hope he shall read it.
Human self-awareness is something science can't know.
Dr. Marcus, wouldn't you say though that while you feel that the constructs of religion about God mustn't be right, there is this to consider. Suppose that God is that 'Goedel formula' (let us hazard to be this impious provisionally) which is Goedelian in even the uber-consistent system that might come to be known as the 'Theory of everyTHING' : i.e. a reality (truth) that cannot be disproven (or proven) in terms of THINGS - the terms of the Theory. It would of course mean that the Theory is only of provable - i.e. controllable and measurable - things and so not even a theory of intangibles like beauty. Now if it may be supposed that God (with whatever actual true attributes) may be such a Goedel truth - meaning not amenable to proof 'from below' , this does not preclude that God can (and well may by his nature) cause men to know the truth by self-revelation. The analogy I have in mind (though it limps) is the 1st Goedel Theorem itself, impressing itself on Goedel in an 'aha' or eureka moment. Imagine how daunted he'd have been if that Theorem turned out to be one of the Goedel sentences itself, not only in the axiomatic system of math, but in the next level of coding, and the next .. and the next .. etc. Then we'd only have the Goedel Conjectures, and 'secularist' or determinedly skeptic mathematicians (what I called [consistency-bound]) would treat them as laughable. What serendipity it was, that Goedel's Theorem is not a Goedel sentence in the first levels of predicate logic !!
Well, maybe we have not yet devised the right "KILLER" axiom!
There is no sense to say something is true if it iso not proved. True is what you can prove to b true. if no, so, it may not be true or it may also not be false. The entenses are not divided in true or false. There is a third kind: not true and not false. Any way, there is no conjecture in mathematics, people should be read Wittenstein.
If science, the process, requires agnosticism, to operate as an observer of least bias, then atheism has a deliberately indiscriminate bias against emotional beliefs that shouldn't be necessary, if the process of science is kept separate from what is generally socio-political. Except when it's the subject of study?
Atheism is almost always just a type of agnosticism. It's a lack of belief in gods, not an absolute belief that there are no gods. I agree that atheism isn't necessary in science and shouldn't be necessary anywhere in a perfect world, but unfortunately the world we live in is still full of religiously motivated privilege, ignorance, and destruction.
Right, so I watched all three and didn't see it addressed so I'll just bring it up here...
Seems to me that Gödel's theorem is based on a number of conjectures, some of which are in need of more detailed analysis.
1. That there are an infinite number of possible mathematical propositions. It's obvious that there are things that are outside there realm of mathematics, so the set of all things mathematical is definitely bounded. It could still be both bounded and infinite, but that is not easily proven.
2. That the combination of mathematical symbols and the combination of mathematical propositions (which are two different things) are both somehow analogous to multiplication. Why multiplication? Why not addition, or exponentiation, or even some other mechanism not found in mathematics at all?
3. That every consistent mathematical system must have a finite number of axioms. This is the basic assumption of the proof, and it does seem intuitively logical, but still it can't be automatically assumed.
...
Don't get me wrong. At heart I agree that (probably) there should be mathematical propositions that can be both true and unprovable. At least, that's what my intuition says. The definition of mathematical consistency only implies that you shouldn't be able to take a false statement and prove it true. The converse of that doesn't automatically follow. But this particular proof seems incomplete to me. Or at least, this presentation of it as I've never actually seen the original in full.
In the first video, the axioms are primes and the resulting statements are divisible by its primes/axioms. He intentionally simplified why primes/multiplication was the mechanic used, but the key is that those statements are now "mathematical".
@@jacobshirley3457 CONGRATULATIONS!!! You've successfully demonstrated that you watched the video!
You have not, however, addressed nor answered even a single one of my points.
At this point it's been months since I've commented anyway, and I'd have to watch the whole thing again to be able to cohesively discuss the proof properly. And honestly, I'm just not about to do that so there's no need for you to bother anyway
@@SumNutOnU2b Oh, so you were asking why the whole prime system was used in the first place instead of another? Instead of asking why addition with primes wasn't used?
godels theorems do not help the theists in any way. they just allow them to abuse the logical fallacy called "argument from ignorance"
Exactly. Mathematicians who know better never say Goldbach conjecture is unprovable just because they couldn't find a way to either prove or disprove it. It's only after proving a certain thing is unprovable can we say it's true but unprovable. What theists do is just line up some words which sounds like logical when compared with Godel's incompleteness theorems to justify their fairy tales. One can also come up with such an argument about unicorns too. That doesn't justify unicorns' existence.
We as muslims say Allah is outside his creation hence, time any natural laws hence he doesn't have to fall under maths and science.
Chapther 112
Say: He is Allah, the One and Only;
Allah, the Eternal, Absolute;
He begetteth not, nor is He begotten;
And there is none like unto Him.
Argument of Ignorance, not really saying this only saying you will never fully logicly know everything him. What I say is he is the only neccesary excistence, for everything else is contingent. And you cant have an full contingent set in maths.
life is meaningless!
Does he also define himself as a Marxist? I would love this.
læl
Maybe you should consider declaring your self an agnostic theist.
" Marxist Theologian "....!! WTF ?!!
Shirsho Mukherjee nothing strange about that really.
jsammelin are you a Marxist Theologian too ?
Studying theology doesn't make one a deist. It's not so baffling that there are "dialetical-materialists" theologians.
Okay....withdrawn
There's the Liberation Theology, which was an attempt to bring some socialist thinking into catholisism. It was important in Latin America at some point when the church was competing against atheist marxism.
check psytrance for a journey to the unknown ;)
Gödel had a proof of God's existence, wonder why it was not mentioned.
Because it's nonsense.
IUIUI ontological proof?
That's for Theologyphile :P
I highly doubt it was any better than what apologists peddle today.
Try using this "proof" with someone who knows his stuff.
There's a reason why you don't see apologists use this "proof", and it's not because none of them has ever heard about it.
A marxist theologian ?He sounds like a lunch bucket.
Love this guy too... but ugh!.. Marxist theologian?.. can't think of anything less cool, at least from an atheist-libertarian mindset, lol
This takes it too far I think. This theorem talks about mathematical proof. There are tons of mathematical statements that can be said to be true in a scientific sense (NP complete algorithms cannot be done in polynomial time, the Riemann Hypothesis) but have not been proven mathematically. All this theorem would say is that you can't prove certain statements in physics in a mathematical sense, but since we can't prove *any* statements in physics from a mathematical sense (which is different from using mathematical proofs with regards to physics equations) it's moot.
Marxism so cool oh look everyones starving
God is not unknown. For most beliefs God is clearly defined by his interactions with individuals. The majority of the people in the world have a personal relationship with God, it's only the weird minority that seem to lack that capability.
God is often incomprehensible, but that's different from being unknowable.
weird minority? really? what makes people that don't believe in something unprovable weird? please explain that to me.