I'm reading Goldstein's book about Gödel now. Thanks for posting this wondeeful lecture.

Having just graduated in philosophy, not with this particular university, I found Rebecca's talk fascinating. I thought it was great infact, because any philosophical talk ought to inform you of a position or some form of knowledge while allowing you to come to terms with the proposals brought forward. The allowance for you as a viewer to question parts of this talk, to me, represents a key aspect of philosophy; , that is to allow ideas to develop and flourish in your own mind and at your own pace. Those of you viewing this for the first time who are, wisely, embarking on a philosophy course, should come back after each year and see how the information sits. You may understand it better or you may not. Either way, you ought to be able to challenge some of the positions outlined in this talk. I thoroughly enjoyed it. Thank you for posting this.

@naimulhaq9626

7 жыл бұрын

Godel's theorems stems from the 'liar paradox', intending to find the truth value of a statement when the opposite of the proposition is incorporated within the proposition. How do you assimilate the concept of malevolence with benevolence? Like quantum uncertainty, due to the dual nature of reality.

philosophy is a wondering subject, but very interesting. i am learning so much.

@huyenmai223

10 жыл бұрын

Good morning from your favorite spade...

It's been 7 minutes into the video and I am beginning to wonder when something interesting begins. I see from the comments that it doesn't look too good.

The thing that is often forgotten in these discussions is Godel's completeness theorem: It implies that if we can prove the Godel sentence but a machine can not, then the only correct explanation is that we made more assumptions than the machine did. What was that additional assumption? Well, it was that the axioms are consistent. This observation is what lead to Godel's 2nd incompleteness theorem. How do we know the axioms are consistent? Simple: we added it to our list of assumptions (often without realizing that we did that).

I am reading it right now.

The interesting thing is to contemplate the possibility of inconsistent but complete formal systems. A good example of this is the game of chess. It's symbols, syntax, and rules of deduction can be set up formally, and making use of them we can prove a contradiction: namely white wins and white does not win. Both of these statements are theorems of the system and the process of proof is the playing of the game.

@ianhruday9584

8 жыл бұрын

+Burton Voorhees As a chess player, I find that both intriguing and unsettling... Yah, really disturbing. But is chess complicated enough for the GIT to apply? I thought she said it only applied to formal systems complicated enough to support arithmetical propositions.

@RadicalSolver

7 жыл бұрын

+Burton Voorhees, can you give a reference or refences for your claims? For example, Where can I find an explicit description of the symbols, syntax, and rules of deduction as a formal first-order predicate logic system? Can you provide a reference that shows that from a fixed set S of axioms in this language of which you speak, one can prove that "white wins and white does not win"?

@RadicalSolver

7 жыл бұрын

+Burton Voorhees In first order predicate logic, the logic system in which Goedel's Theorems are formulated, every inconsistent theory is complete. This is because of the simple fact that from a contradiction, every sentence is provable. Recall that a theory S is complete provided that for every sentence p in the language in which S is formulated, either p is provable from S or the negation of p is provable from S. In the case that some contradiction q is provable from S, we can write the proof of q, and then add to that proof the proof of p, no matter what is the sentence p. Thus in this case, from S, BOTH p and its negation are provable, so of course, at least one of them is provable.

how to use mathematics and mathematical in every sentence ...

Who gave her the title of professor?

I would like anyone to venture a comment on Godel's theorems and philosophy of mind. What other possible implications might exist in human thought and experience ?

@JamesJoyce12

4 жыл бұрын

there is no relation between Godel's ideas and anything except mathematical systems - a bunch on math-challenged philosophers have tried to hijack it to say we can never prove a consistent moral system or a language system - it plainly does not have anything to do with those undertakings

@juanf.gonzalez9986

2 жыл бұрын

@@JamesJoyce12 Wouldn't Gödel's theorems apply to any formal system as well, besides mathematics?

Good G-d, I can't believe Nagel educated this person.

Now, That's Paradoxical!

So much over-confidence in the comment section about knowing what gödels theories actually potray, even though among their piers there is most definitely still debate upon what it actually means.

I can't help thinking that the structure of this lecture is wrong if we don't get the actual theorems until half an hour in. For people who didn't already know what the GIT are, the preamble would have been much so much waffle.

I wonder what are some examples or what is the proof of saying, there are certain true statements without proof. Or there are certain true propositions that are unprovable. Like example from godel

@aleksandarnedeljkovic8104

3 жыл бұрын

From Godel I don't know. I think "0 exists" is example of such statements

Anyone from Economics who is seriously in confusion with our positivism ? Anyone thinking empirical enquiry maybe foundational for our subject?

i think she did a good job approaching this from a different than math or compsci standpoint. shes cute too.

0:30:20 "A system in which you can actually express the truth of arithmetic". That's wrong; she means a system which proves a sufficient portion of arithmetic, for instance Peano arithmetic. Also, it's not true that no such formal system can prove its own consistency: some inconsistent systems can. This is lack of rigor but comparing an inconsistent system with a jury formed by 12 nazis is ridiculous. The talk is also misleading on a number of philosophical issues.

@agimasoschandir

8 жыл бұрын

+Laureano Luna {Also, it's not true that no such formal system can prove its own consistency: some inconsistent systems can.} It sounds like you are saying an inconsistent system can prove it's own consistency. Peano arithmetic cannot format a proof of its own consistency within itself.

@LaureanoLuna

8 жыл бұрын

Agimaso Schandir Inconsistent systems of arithmetic do prove their own consistency. PA doesn't because it's consistent. Remember that inconsistent systems prove all sentences in their language.

@aleksandarnedeljkovic8104

3 жыл бұрын

Wtf are you guys talking about

I learned more about Godel's theorem from Raymond Smullyan's books than watching this lecture.

@mrssrm5053

5 жыл бұрын

Maybe you should have listened to it too.

@lauralaura2293

4 жыл бұрын

The dissertation is pretty weak...

@jamespower5165

2 жыл бұрын

Well obviously. Those books get into the heavy stuff very quickly. This is a gentler approach

Dear Professor Goldstein, Do you contend that there are absolute truths? If not, then why at about 25:35 do you say that the incompleteness phenomena establish ("prove mathematically") absolute limitations on what we can prove? I ask because as I understand it, a large number of people -philosophers, lay ("armchair") philosophers, etc, seem to contend that there is no such thing as a statement that is absolutely true. (If I've misunderstood, please help me clear up my misunderstanding.)

@WriteRightMathNation

8 жыл бұрын

+Karl Ruprecht Kroenen Good question.

@ashimov1970

8 жыл бұрын

The God is the only absolute Thruth

@joshkerr

7 жыл бұрын

The absolute limitation she is referring to is the fact that a system can't be proven consistent from within. So unless we go outside the system, we can't ever know if it is consistent or not. There will always be truths that we just can't reach. For example, how do you get outside the known Universe? If you can't, then you can't prove its consistency.

@RadicalSolver

7 жыл бұрын

The theorem of Goedel does not state unequivocally that a system cannot be proven consistent from within, Josh Kerr. That is a nontechnical description of it for lay people to start getting a grasp of the ideas behind Goedel's results. To be careful about the statement of Goedel's Second Incompleteness Theorem for the purposes of understanding Dr. Goldstein's comments about Goedel's Theorem, one must state it this way, or equivalently: "If S is a system of nonlogical axioms consisting of sentences from first order predicate logic, in which the first order axioms of Peano Arithmetic can be interpreted, then the following hold: (a) The claim that S is consistent can be formulated in the language of the system S; and (b) if S is consistent, then no sentence in the language of the system S that states that S is consistent can be proved from the axioms listed in the system S." Now let S be a consistent system of nonlogical axioms consisting of sentences from first order predicate logic, in which the first order axioms of Peano Arithmetic cannot be interpreted. Goedel's Undecidability Theorems say absolutely nothing about whether there is a proof from the axioms listed in S of the consistency of the system S. Now, the question to Dr. Goldstein is not really about the theorems of Goedel. Please note that it was asked if she contends that there are absolute truths. Also, please note that the questioner did not say that they disagree with her. But they asked for clarification, because they have been under the impression that it is ("Philosophically"?) unfashionable to to believe such a thing.

Watch a rocket take off into the realm of genius.

@RadicalSolver

7 жыл бұрын

What do you mean?

She could have shown Lauren Bacall some acting tips in the movie "The Big Sleep".

As an engineer, I've always seen Godel's theorem much as I do Zeno's paradox. Eventually the arrow does, indeed, hit the target. In spite of this theoretical limitation, mathematics has provided an amazing space in which practical things can be accomplished. Does anyone know of a practical example where Godel's theorem has provided an example in the real world? Is this just current philosophy's "how many angels can dance on the head of a pin" debate? "In theory, communism works."

@lerakmahendu1231

9 жыл бұрын

Jeff Griffith I would say it's better seen as a limitation on the practical things you can do with mathematics and logic. For instance, there are algorithms that can derive a lot of mathematical theorems given the right formulation (see the history of LISP). However, because of Godel, no one would attempt to write a program that could derive ALL theorems, because there will always be truths that have no proofs (aren't derivable) in the current system. That doesn't mean you don't write theorem provers, but you DO consider how to wring as much out of them as you can. Another example (sort of): Roger Penrose uses Godel to argue that AI can't produce machine intelligence/consciousness. I believe it goes something like this: No algorithm can be coded to derive the Incompleteness Theorem from first principles, because it's not computable (there is a proof of this if you re-interpret Incompleteness with Turing's work). HOWEVER, the human brain CAN. Therefore, something more is going on in our heads than mere computing. If he's reasoning correctly, it's a fairly stark limitation on what you can do with AI. Another (this one is cheap, because it exploits metaphorical rather than precise language): Grand Unified Theories of physics (if they really exists) are sometimes loosely called 'theories of everything' (physicists themselves don't like this term -- it's meant for newspaper reporters). Godel's result prevents ANY theory to be about 'everything.'

@chrisofnottingham

8 жыл бұрын

+Jeff Griffith The point with Zeno's Paradox tho is that it is clearly wrong, because things do move, hence the paradox. Whereas GIT is still thought to be correct. I don't think it is possible to have a concrete example because we can't tell the difference been true things we haven't proved yet and true thing that are unprovable. We call them all conjectures.

@JRobbySh

6 жыл бұрын

Well, it does provide a limit to the world view of the Enlightenment.

@Tadesan

6 жыл бұрын

Derp factor five. Zeno's paradox is relevant to understanding the existence of a limit. I can tell that your understanding of the word limit in it's mathematical context is incorrect. Therefore what you are trying to say sounds interesting but is fundamentally vacuuous. I don't think calculus can shed light on axiomatic set theory in terms of Godel's Incompleteness Theorem.

@Tadesan

6 жыл бұрын

And to simply answer your question: No. Nobody knows of a thing that is demonstrably true yet absolutely unprovable and that you can go visit in your Chevy. Sorry.

So not a great talk, but seriously she doesn't deserve the crap she's getting in the comments. Let me tell you a secret. Intelligence and the ability to give a good talk... Not the same thing. These two things can and often do come apart. Secondly, understanding of a topic and depth of analysis in a public lecture for undergrads... Shocker, also not the same thing. Unless anyone here has read her technical work, pleas refrain from commenting on it.

@neutralcriticism4017

7 жыл бұрын

Ian Hruday There is no 'technical work' of her. She has some fictions though.

Excellent lecture. Excuse me while I go be mind fucked.

It seems by sciences own words and testimonies, quite a few important people in regards to science have befallen within some very harsh conditions and ends, without being rude, physically and mentally, while during and what could be considered as under the very sciences umbrella. If science cannot look after science, then science is not worth it. If science intends deliberately to jeopardize itself, then science is not worth it. I say this because without godel there is no general relativity, which nonetheless there still isn't, but that much that is, without godel there is not. Respect.

Alas, I can never again say I went to a great party. From now on I will have to tell people last night was shit, like every night, & be branded a cynical depressive :(

@agimasoschandir

8 жыл бұрын

+Ono Jin Host your own.

@RadicalSolver

7 жыл бұрын

How is this relevant to this discussion?

This is the dangers of philosophers talking about math and logic when they have never specialized in math and logic. Godel at least had an extensive background in both.

@jeremiahwatson1611

4 жыл бұрын

Never?

@aleksandarnedeljkovic8104

3 жыл бұрын

Logic is philosophical discipline.Also mathematical ( formal ) logic is thought in philosophy. It's not so much as not specialising it's not being good at it. You have good doctors , you have bad ones , but they finished same schools , same specialisation.

She needs to learn how to keep her voice from dropping off.

It would have been best if she stayed with presenting her experience at Princeton as it is truthfully. Instead, she took a bite of math she couldn't chew. Now the whole story is ruined with her trying to inject something she doesn't understand. She has failed her chance to deliver us the golden story we would have loved: an outsider's honest glimpse into the historical events played out by the mathematical minds. Honesty would have gone a very long way.

This lady is neither particularly smart, nor does she appear to understand anything substantial about Goedel's theorem. How embarrassingly superficial this lecture is.

@Infinitiely

9 жыл бұрын

Kilgore482 I am a laymen trying to understand the concept. Could you elaborate (if you can remember from 6 months ago) on what she missed?

@ianhruday9584

8 жыл бұрын

+Kilgore482 Seriously, you're judging her intelligence and technical work based on a popular lecture given to undergrads? You have some serious misconceptions about the nature of academia, which indicates that you probably don't understand the GIT as well as you think you do.

@Kilgore482

8 жыл бұрын

+Ian Hruday Hmmm.....with a Yale PhD on Frege and as a chairholder at a research university in symbolic logic, I'll stick by my assertion, my serious misconceptions about the nature of academia notwithstanding. You got that last part right!

@neutralcriticism4017

7 жыл бұрын

Infinitiely It would be an arduous and nit-picky task to point out the things that are missed. If you are really dedicated to understand the concept, you should read real math books on Goedel's theorems, and then you will understand what all these fusses are about. If you would rather read non-technical book(s) just to get the feeling for it, might want to try Goedel-Escher-Bach, but should keep in mind that it doesn't do justice.

@neilanderson891

6 жыл бұрын

Criticism by Kilgore482 (holder of a Yale PhD on Frege and chairholder at a research university in symbolic logic) is correct, although in the spirit of "undecideability" let's say she is either "very smart but unprepared" or "not smart but prepared" or "not smart and not prepared" or "something else entirely".

Wow, she just babels the same thing over and over again.

@agimasoschandir

8 жыл бұрын

+Mahmoud Abdellatif Such as?

This woman should have learned her homework well before giving a lecture ..

Incompleteness theorems have relevance only to the philosophy of maths, (which most mathematicians don't give a fig about, they just get on and do maths). They are interesting ,but this lecture is flabby and unrigorous

@buckstan

10 жыл бұрын

This is the opinion of a machine.

@FrancesdeJasso

10 жыл бұрын

Buck Stanley LMAO!

@RadicalSolver

7 жыл бұрын

+Mark Callaghan, can you give a reference for your claim that Incompleteness Theorems have no relevance outside of maths? Maybe one of them contains a proof of that claim? I'd like to see it.

@markphc99

7 жыл бұрын

Better yet , you tell me of an instance where an incompleteness theorem has been used in another discipline - if you can't , then case closed.

You have to be very careful about how you apply this. His theorems do NOT apply to knowledge. They apply to proof in a specific CONTEXT (formal axiomatic system). Indeed, if you allow the context to vary then ANYTHING can be proven. These types of "proofs" are often used to justify laziness in the form of, "You can't do it so you shouldn't try" type arguments. No Gödel didn't break philosophy. It's still true that *any* claim that is true is provable if you find the context in which the claim makes sense. The giant failure of mathematics is that we have no idea how to find the right context.

@larianton1008

Жыл бұрын

So what is the difference between proof and truth in formal axiomatic system?

@ThinkTank255

Жыл бұрын

@@larianton1008 Well, it depends on what you mean by "truth". If you mean, by "truth" you mean truth values then these are two completely different things. Often proofs rely on various combinations of true and false variables, in order to prove the final truth value of the desired conclusion is true. Truth about the real world does not exist within formal logical systems. The disconnect is the translation from logic to general knowledge. With every proof simplifications are made in order to construct a model. Those simplifications *always* result in a system that does not accurate reflect reality. Then there are truths that are internal to the formal axiomatic system, which I assume you are referring to. Proof is the process whereby you obtain a truth that is internal to a formal axiomatic system. However, quite often when we write proofs we have a desired conclusion in mind, and due to lack of expressiveness of an axiomatic system it may not be rich enough to capture the desired conclusion (the model of the formal axiomatic system is simply not sufficiently detailed and perhaps overly detailed in abstractions that are irrelevant to obtaining the desired conclusion). Gödel's incompleteness theorem actually fails due to the fact that it relies on the existence of uncomputable functions, which ultimately rely on the Axiom of Choice which is a fundamentally flawed axiom. It was an understandable mistake for early 20th century mathematicians, but later 20th century mathematicians should have corrected the mistake, and yet, they continued on with the same flawed logic.

@sidharthar567

5 ай бұрын

But think of this. In many parts of physics and economics there are places where truth cannot be ascertained. This doesn't mean that mathematics is useless and that formalism is uncanny. However, it could be used as an encouragement to think outside of current modes of thinking. There was a time in physics when physicists thought they knew everything that could be known of. However Einstein blew away the Newtonian determinism. He was able to do it because he imagine a world different to wha Newton imagined. Economics also have problems in which some theories are just not apy for a developing economy compared to developed one. Certain axiomatic theories may cripple down in front of empirical enquiry.

"We know that N is true even though N is unprovable" This is a contradiction in terms. The only way we can know that something is true is by having proven it to be true - if only by pure logic. And it is meaningless to say that something that has been proven is unprovable. As usual, academic philosophy is utter drivel. It's not philosophy at all.

@JebKermantftw

10 жыл бұрын

Uuuhhh that's mathematics. "N is true" means there is a model satisfying N. "N is unprovable" means there is no derivation of N from the axioms. Take a course, maybe?

@KevinSolway

10 жыл бұрын

ivanverano1 "Uuuhhh that's mathematics. "N is true" means there is a model satisfying N." The mathematical definition of "true" is nonsense. A thing is either true or false. A thing can't be both true and false at the same time. For example, it wouldn't be "true" that God exists if I am a theist who brings blind faith to the table, but false that God exists if I am an atheist who brings reason to the table. That God exists is either true or false.

@JebKermantftw

10 жыл бұрын

Kevin Solway long story short, you have no idea what the "mathematical definition of truth" is, and you have failed to present any argument as to why you think truth and provability are the same thing.

@KevinSolway

10 жыл бұрын

***** Just because a thing is called "logic" doesn't mean that it is logic. For example, "modern logics" don't necessarily have anything to do with logic, in the same way that "feminist logic" doesn't necessarily have anything to do with logic.

@Bombtrack411

9 жыл бұрын

Kevin Solway Take "ivanverano1's" word for it, and actually study this rather than make assertions like "The mathematical definition of "true" is nonsense." What justification do you have for that? You clearly don't understand what you're talking about.

This is way too awkward to watch. She might be a little out of her depth on the subject matter. Is she sure it's "mathematical"?

@agimasoschandir

8 жыл бұрын

+modvs1 Why are not you sure?

@modvs1

8 жыл бұрын

+Agimaso Schandir She does a good job of hiding her age. Perhaps she's equally adept at hiding other things too.

@agimasoschandir

8 жыл бұрын

+modvs1 Perhaps. Do you know she is?

If any f you are interested in Philosophy, go to "A History of Philosophy", by Bertrand Russell. But, be prepared to be disappointed in Philosophers. It seems like a history of rubbish. Why" the science is lousy. but the Humanism is debatable. If you need something simple as regards Philosophy, try, "Zen and the Art of Motorcycle Maintenance."