Kurt Godel: The World's Most Incredible Mind (Part 1 of 3)
Вставка
- Опубліковано 16 жов 2011
- Kurt Godel: The World's Most Incredible Mind.
"Either mathematics is too big for the human mind or the human mind is more than a machine" ~ Godel
Kurt Godel (1931) proved two important things about any axiomatic system rich enough to include all of number theory.
1) You'll never be able to prove every true result..... you'll never be able to prove every result that is true in your system.
2) Godel also proved that one of the results that you can never prove is the result that says that the system is consistent. More precisely: You cannot prove the consistency of any mathematical system rich enough to include the known theory of numbers.
Hence, any consistent mathematical system that is rich enough to include number theory is inherently incomplete.
Second, one of the propositions whose truth or falsity cannot be proved within the system is precisely the proposition that states that the system is consistent. "
What Godel's proof means, then, is that we can't prove that arithmetic-let alone any more-complicated system-is consistent.
For 2000 years, mathematics has been the model-the subject-that convinces us that certainty is possible. Yet Now there's no certainty anywhere-not even in mathematics.
More...
teachingcompany.12.forumer.com...
Goedel's Ontological Proof.
For those interested in a discussion of Goedel's reasoning for God, then I suggest starting with this heavily annotated work, which I recently stumbled upon.
www.scribd.com/doc/95364925/Goedel-s-God-Proof-Annotated-Version
"It's not that God is subject to the Freedom Proof or the Doubt Proof.
According to Gödel, He's not. But we have to be, or else we are not free. So
our truth game with God turns into something like Feynman had described.
Feynman's Gods, every time physicists think they have the rules of the game
figured out, throw in a new wrinkle. They let people like Feynman make
progress, but if the Feynmans of the world learn too much, physics will stop
being the joy and challenge that it is. The Gods don't let that happen.
Gödel's God has to be very careful about how he lets our universe unfold.
If the world becomes totally controllable and comprehensible, we'll be God.
God does not object to that. In fact, according to Gödel, that is our destiny.
But it is also the end of us as free human beings. And human freedom is an
essential part of the beauty of God's universe."
~ page 251 - Наука та технологія
I love maths, and I think that everybody should learn math continually all their lives.
It is very satisfying.
I'm absolutely sure that not everybody should learn math continually all their lives. Who'd grow the food and who'd pick up the garbage if everybody was obsessed with math? For most people math is mental torture and not satisfying at all.
dlwatib
They could fake it.
I like it very much but its to hard for me.
Never was very good at math anything past basic shop- type math. That’s why I’m not an engineer. One thing I learned about all that hard math I couldn’t do. Unless I was and engineer or something like that I would never use it in life
Agreed!
Blind faith in religion has simply been replaced by blind faith in Science.
But i guess the flip side is that the educational system mostly stifles creativity and curiosity, and children are "made" to study rather than them "wanting" to. This state of affairs will never lead them to ask questions which are on the edge, which question the results, which stretch the science beyond the banal examples which are given in the classroom and which lead to incremental learning...
What Godel shows with incompleteness theorem, is that the human mind has a capacity for intuition and creativity, which ultimately lay at the foundations of reasoning. This is in direct opposition to logicistic attitudes, which suggest that the formal axiomization of mathematics could lead to all truths. Godel understood that truth (including logic) relies on the foundations of axioms, not all of which have been discovered, and our infinite.
great stuff. i've been watching too much garbage on youtube lately. i need to get back to this stuff. it's very satisfyingly interesting.
calabiyou Watch "Gödel, Escher & Bach" instead. :)
this is kicking ass and taking names. thanks for posting this video and the great link info like Goedell's ontological. Greatly and gratefully appreciated. life is good.
There's some confusion in the comments about implications of the theorem. I've studied Gödel quite a bit. An analogous finding was Turing's "undecidability", which proved that every program has a problem which it fundamentally cannot solve even if you gave it an infinite amount of time. The two are analogous to everything: if we build some kind of thinking process out of rules, that system will ALWAYS be flawed. I'm attempting to write a book on expanding the logical implications and some more commonsense analogies.
Do you have the book written? Or is still "undecidable"?
COVID got him maybe. Wish we could see his work
All systems are necessarily finite and therefore flawed to that degree.
Mind, if properly cultivated, can intuit the Infinite and realize It eventually.
@@manuelfrn still writing, not dead
@@thenowchurch6419 humans can recognize a loop and intuit that it has no end, but we cannot grasp an uncountable infinity.
The alarming way to clear a crowded room is to yell “Fire!!” However, the safest way to clear such a room is to tell the crowd that you are going to talk about complex problems in math. You will soon be enjoying the empty space.
*Tips fedora
or was that enjoying the empty "set"
Or bring a baby to a lecture
I would not leave the room based on someone's claim alone. I would wait until the proof, e.g. smoke.
@@artoffugue333 smoke on its own isn't a rigorous proof, it's just a hint you can build a conjecture on.
I first heard this presentation as a podcast and was impressed at what a good communicator Mark Colyvan is, and what an interesting talk he gave here. I still think it's one of the best offerings on Godel out there.
Starts out as a bit of a dry treatment for those not intellectually inclined. But for those who can hang in for a few minutes there's a lot to enjoy, and even a few good laughs.
You are right about that....We cannot prove any property of any system inside the system itself. But Godel proved the INconsistency of arithmetic outside of its system
@Dent Niggemeyer: "Uncertainty" is an enormous threat to the established system and those who control it and benefit from it. If the public were to become uncertain, then they would become less vested, and perhaps turn to alternatives, or turn to themselves, or perhaps turn to direct relationship with the spiritual. All of these trajectories disintermediate the current power structure. Hence, Goedel's findings are extremely dangerous to the status quo.
I am amazed that people are still talking about the mind/machine issue. The resolution is simple. An analogy goes like this: Ask this question: "In general, is it possible to trisect an angle?" The knee-jerk mathematical response is: "Of course not! Everybody knows that!" But the better response is: "Of course you can! Just not with a compass and straightedge." The mind/machine issue similarly hinges on the technical definition of the word "algorithm": a finite set of instructions that is guaranteed to produce a correct result in a finite time. Human minds are not an algorithm. Neither do they need to be anything more than a computer to do what they do. A human mind is a gigantic (but finite) non-terminating (except by death) Monte Carlo process, some parts of which run in a deductive (mathematical) mode. It has been possible to build something mind-like for decades now. The trick is to recognize what a mind is by recognizing what it accomplishes, and then noting how it does it. It's like evolution: once you see the trick, there is nothing hard about it. What turns out to be hard is encompassing all of the consequences of the little trick.
The best trick I learned in this life was how to tell shit from shinola. :D
My father know how to trisec the angle but you need a different tool for it.
Nice. I like what you say sir, and I appreciate you saying it.
+Steven Abell
Well, it is possible to trisect an angle using origami. There are some problems that are impossible by ANY finite construction, like, for instance, squaring the circle. So this means the mind/machine issue is solved? Um, ok. So how did you reach the conclusion that the mind is a giant Monte Carlo process? What exactly do you mean by a Monte Carlo process? The Monte Carlo process refers to the generation of random numbers in a computer to solve some problem. So, if the mind is a Monte Carlo simulation, does that mean the human mind is a computer generating random numbers or...
I'm not clear what your point is.
Agreed. I think this opinion is similar to that of AI pioneer Marvin Minsky.
paradox lies at the heart of reality.
Very interesting and worthwhile video.
Very good general explanation of Godel incompleteness theorem and it's implication;
I hoped it was longer :(
Very good Global!! Very good.
We need more solicitation to improve college learning facilities and expand our horizons into the next generation of well wishers.
Thanks for this! Want to do a Comp Sci degree, really enjoy this.
Very interesting !
Thanks
I don't really know how anyone could NOT believe in some higher power after understanding Godel's First Proof, the Incompleteness Theorem. Man simply does not have command over nature, and that is obvious no matter how many elitist academics, media personalities or the like state to the contrary.
good discussion. thanks.
Video grapher should have zoomed out long ago. Great information but film can make your eyes tired.
@prof5string: The sentence "this sentence has five words" is NOT self-referential. It refers to a numbering system that defines the number of words in the sentence, and that numbering system is outside the language of the sentence.
Nevermind... Saw that was answered below. Thanks!
A power set has a property {a,b} = {b,a}
but, where {a,b} *= {b,a} the set is defined by a linear algorithm
No one said set theory is inconsistent. It is not a fabulous question. Being unable to prove consistency does not demonstrate that something is in fact inconsistent. The only way to do that is to find an inconsistency and that has not been done.
The movement, pacing back and forth, induces nausea.
A few seconds in and I'm already getting seasick. Dude stop moving back and forth! 🤣
He paces back and forth with a rhythmic tempo. The acoustics change in this distance. It sounds like he's being recorded with a slight flanger.
@Pooja Deshpande Set theory is not inconsistent.
In order for a system to be inconsistent, it must be the case that both a formula of the system and that very formula's negation can be proven within the system (I.E. you can prove some formula P and you can also prove NOT P).
Set theory is a consistent system. It is impossible to prove both a formula of set theory and that very formula's negation within set theory.
Godel's incompleteness theorem does not demonstrate that set theory is inconsistent (this cannot be demonstrated, because set theory is consistent). Godel proved that set theory is incomplete. A system is incomplete if there is some formula of the system which is true, but cannot be proven within the system. Godel's theorem demonstrates that there is at least one formula of set theory which is true, but cannot be proven according to the deductive rules of set theory (it follows that there are actually infinitely many formulae of set theory which cannot be proven).
Now, to answer your question: you asked why set theory is still taught in schools even though it is inconsistent. If set theory were inconsistent, it would hardly make sense to teach it at all. For example, if basic arithmetic were inconsistent (it is, in fact, not), then we would be able to prove both that 1 + 1 = 2 and that 1 + 1 =/= 2. In an inconsistent system, you can prove anything, no matter how crazy-sounding! So teaching it would be a breeze because every formula you demonstrate can be proven. The only downside would be that you could come in the next day and teach the exact opposite of what you had taught the previous day without breaking the rules of the system. That is why it is important to teach consistent systems.
However, there is a much more interesting question of why it is that we still teach incomplete systems (like set theory). I find this question much more open to discussion, as there are many different arguments in favor of teaching incomplete systems. One point to be made is that systems which are both complete and consistent are often not considered "interesting" as fields of study. For example, first order predicate logic is a system which is both consistent and complete, but the complexity of provable statements within first order logic comes nowhere near the complexity of some of the results provable within set theory.
Another argument one could make in favor of teaching incomplete systems is to appeal to the results provable within the system themselves. Much of our discoveries in mathematics rest on set theory as a foundation for demonstrating our results, and giving up set theory might also mean giving up on those discoveries.
Again, my answer to this second question is much more speculative than the first, but I hope I was able to clear up the confusion about the consistency of set theory and explain why we still teach set theory in schools even after the discovery of Godel's incompleteness theorem.
I always thought of the Russel paradox as an oscillating paradox.
because it's a good approximation that works in enough cases to be useful. that's the answer for all theories - we'll probably never find a perfectly consistent system that perfectly describes the real world, partly because of the limits of language, our brains, logic, etc.
delightful!
Just to be clear, Russell's letter to Frege was written in 1902. Zermelo's work on set theory was published in 1904-1908.
You must get ahead of inconsistencies and find involvement with true conjecture a future which satisfies a dream yet unattainable in present circumstances of inner desertion.
This guy is a very gifted lecturer. What is the setting of his presentation? Thanks.
What is the highest level of math that you have used in your everyday lives? Me? In the days before supermarkets listed the price/volume, I would do a simple ratio to determine if the economy size was really a better deal than the other. I weigh stuff to make sure that I am not being cheated, too.
I'm not quite sure why I shared this...
I once when working in retail used the integration of a disc to prove the length of a coil of wire was accurate...now I don't work retail anymore...
I do that same thing for best pricing at the market! I'm glad I'm not alone on that
@Nukutawiti: And that makes arithmetic INCOMPLETE. Exactly what Goedel had said.
Hi- who is the speaker here? I'm sorry if it is mentioned somewhere, I just couldn't see it myself... Thanks for putting it up.
Incredible videos. Who is the speaker?
Is mathematics merely the natural language of physics, or is it also indispensable at the core of its conceptual development, as well? In other words, is the conceptual foundation of physics ultimately mathematical in nature?
Godel and Feynman's gods reminds me of Q from Star Trek. I can't seem to find any connections though. Does anyone know if the Q is an exploration of this idea of Godel's or Feynman's gods?
I'm not an expert in any sense. But from the description i have seen here. Incompleteness is an observation if anything. It doesn't need to have a mathematical framework justifying it.
It's not set theory, but Arithmetic that Godel examined, and he showed not that it is inconsistent, but that it is consistent only if incomplete.
I have 1 question. If Godel proved that we cannot guarantee that the mathematical axiom's are consistent, doesn't that ironically undermine his own proof? He used a framework to prove that there is no framework wich is fale safe, or at least we can't determine it.
The moment he placed Darwin in the ranks of the greatest thinkers, I knew this guy doesn't have any idea about what Darwin REALLY BELIEVED AND WHAT HE PROVED.
Consistency allows some things to be true and others false. Inconsistency makes everything and its opposite true. You really have to expect the answer to the consistency question to be “yes”.
In an inconsistent world, you can answer the Consistency Question and any other question “yes”. In a consistent world, “yes” is the obvious answer to the Consistency Question.
So that’s two choices. One is “yes” and the other is “yes”. Gödel"s Proof showed the answer to the Consistency Question was “no”.
It's only inconsistent if you're too loose with your definition of "set". That problem was eventually solved.
The concept of cardinality (or "size"- two sets are the same size if their elements can be put in 1-1 correspondence) is, for finite sets, the underlying concept that the "counting numbers" 1,2,3... are based on, so sets are indispensable in math.
Yes, my point was that the inconsistencies have been resolved.
I mentioned it because the lecturer mentioned some of Godel's personal beliefs.
"A word is the skin of a living idea". We are talking about mathematical certainty here, not human certainty. The importance of the proof is not that when you measure the length of a building's shadow and its angle with the ground that you should distrust the height you calculate. The importance is that a software developer who is writing a software program that determines the correctness of other software programs can give up the impossible and create a video game instead.
(con't).....I realized that there were as many branches of math as there are types of literature & in fact maths is literature of numbers, relationships, etc. So the equivalence is that maths & literature begin as one, then diverge developing into their recognizable forms based on how people decide to develop their characters.
oh Mark, still waiting for Godel.
I'm tempted to say that "this sentence has five words" is a true self referential statement but i dont really know.
"For 2000 years, mathematics has been the model-the subject-that convinces us that certainty is possible. Yet Now there's no certainty anywhere-not even in mathematics."
Are you certain?
@MagisterPridgen: yes, the "anchors" are outside the mathematical system
the mind which rules the dream is not within the dream, the brain waves are not part of that system. Holographic 3d universe is ruled from 2d reality.
@Pooja Deshpande: What a fabulous question!
What's preposterous is that children are taught these tools without being told of their limitations, especially when compared to the human mind or the real world. But I guess if you want to create an elite supersystem you must convince all of the people within that system that science is a god that its subjects must religiously follow. And when science takes over, humanity is marginalized, as numbers, charts & algorthms drive all decision making.
@tattoconga: I added a link and some notes at the bottom of the video description. Thanks for asking.
ZFC set theory is not inconsistent, but the standard set theory at the time was. Modern mathematicians use a set of axioms to avoid paradoxes.
Just waiting for the door to open. Counting is involuntary and leads to confusion in a formal sense of course.
Everyone holds all truths of life in their heart, me and many others have found all truths needed. I feel blessed. God bless you all on earth!
Can someone correct me if I am wrong; according to the first incompleteness theorem in every axiomatic system related to arithmetic there will be statements that are true but not provable within that system, second incompleteness theorem is special case of the first one - each axiomatic system related to arithmetic cannot prove its consistency.
If true, what is relation between first and second theorem?
You're account of gödels incompletness theorems is roughly correct, depending on what you mean with "related to arithmetic" and "provable". The english wikipedia article on the theorem is probably an ok read.
It is wrong though, to say that the "second incompleteness theorem is special case of the first one". I see no connection between them. It is however very hard to explain that, as the whole conext of argument is extremely technical and it feels impossible to give an appropiate account without laying out the formalisms.
A few things I can say though:
- beware that the 1. and 2 . theorem do not exactly speak about the same sets of axiom systems.
- the first incompletness theorem does not exactly give you "statements that are true but not provable", instead it gives you statements whose truth cannot be determined, within an appropiate axiomatic system. it is not clear wether these statements are true or not in an intuitive sense, for all I know, it could be that alle the statements whose truth cannot be determined are false. (the concept of truth has to be treated very carefully here, as the only way we can communicate truth are axiomatic systems, so its weird to say that something is true even though all the seemingly good axiomatic systems do not tell us it is true)
- the second gives you statement where one cannot infer that it is true. that is different from saying you cannot infer its truth value, because "one cannot infer that it is true" still entails the possibility of you infering that the statement is wrong. "you cannot infer its truth value" does not leave that possibility open. the first theorem gives you such statements.
- the wikipedia article says that the proof of the second theorem can be obtained by formalizing the proof of the first one. I don't know about that. We didn't do it that way in my class, never read about that either, not even in philosophy books. seems like a cumbersome way of doing things.
I'm not sure what you mean. Completeness has to do more with formulas in a system which are also theorems in that system. So a system is considered incomplete if there is a formula in a system that cannot be proven. I haven't taken enough number theory, though I would imagine it wouldn't be trivial to attempt to prove this rigorously enough for mathematicians. If there is something I'm missing, I would like to know.
"For something to be either true of false it cannot be self referential, it must refer to something outside itself."
Really? "This sentence has five words" is self-referential and true.
@QuantumBunk: To be more specific, the "problem" is that the public does not understand the limits of Mathmetics. Further, I believe that these limits are deliberately hidden from the general population, so that science can be sold to the masses as its new god.
No mention of Euler?
@Nukutawiti:Right, one has to rely upon an EXTERNAL system to prove consistence of the target system. That's incompleteness. If the mathematical system were complete, then it would be able to prove it's own consistency. But it can't.
I like this Godel fellow, I think we could hang out.
@dekippiesip: I too, am struggling with this same question. If anyone has anything to add here, I'd appreciate.
Thanks for giving such an enthusiastic talk about Gödel. Maybe you know my booklet (written together with Casti) "Gödel: A Life of Logic" (Casti+DePauli, Perseus Books, Cambridge MA, 2000.) Also: "Wahrheit und Beweisbarkeit" htp+bv (Hölder-Pichler-Tempski + Bundes-Verlag, Wien 2002 Volume 1 and 2). Or: "EUROPOLIS5 Kurt Gödel, ein Mathematischer Mythos" NOVUM_Verlag, Horitschon, Austria 2003). There exists also a film with the same title, produced by the ORF= Austrian Television network, 1986, you can buy from the ORF-shop (But the film is also in German).
We must write rules for god to follow,,im not kidding, it will cure all illnesses and it will cure god and when we have written all the rules for god god can finnaly become pure good and can heal us and we all can live a perfect life etc. Thats all we need to do. Thx me on a later time in the universe.
The reason why not many good books are written is that people that know stuff don't know how to write.
Set theory has not been shown to be inconsistent. It is taught because it is useful.
The speaker makes an error by saying "set theory was shown to be inconsistent". What he means to say that the Naive Set Theory of Cantor where sets could be described without types or classes or additional axioms restricting what defines a set. In this case, he is evoking Russell's paradox, namely "is the set R which is the set of all sets which are not members of themselves contained in itself?"
The part where he jokes that philosophers have been somewhat silly in asking questions like in the liar paradox for millenia, the same reasoning can apply to the question whether we are not computers!
4:22 "Doctoral dissertation: completeness of first order logic"
The set theory everyone is talking about here is *equivalent* to fist order logic. Which Godel proved was COMPLETE. Not incomplete, not inconsistent; Consistent and Complete.
Please do not forget, that people like Einsten and Gödel lived in times, when being an atheist was not an option. If anyone would have taken the position of an atheist, he wouldn't have the chance to study or the chance for a job. At this time, churches have been overcrowded on sundays. If you're not there taking part in praying silly crap, you'd have a very good excuse or become an unadapted outlaw. Like in the bible belt today.
................................................shared again................................!!!
No we are NOT, and Godel helps to show this. John von Neumann took Godel Numbering and used it to help create binary numbering systems, which can be "gamed" to create a Complete Formal System where there is none, via a computer controlled virtual "reality".
What is the name of the lecturer?
@Bloke Poppy: My point in the earlier post is to illustrate that a system where the public believes in God-over-men creates more freedom for the people living in that system. When God is disbelieved, men can fill that role and exert godly powers over the public, resulting in massive suppression.
What can I say but be smart and suave.
I doubt the correctness of the subtitles
The labyrinth finally destroyed Goedel as it did Newton, Weierstrass, Cantor, Frege, Ramanujan, von Neumann et al. They used bad infinities & not the transfinite fractions of Non-Cantorian set theory. I painted a large triptych called The Madness of Mathematics.
Who is the speaker?
If anyone has any recommendations similar please recommend!
I suggest readers look into Godel's Ontological proof on why God exists. Ultimate, it is what keeps humanity free... free from other men who purport to be gods. It's a beautiful system, but the public is not supposed to understand this, because it would render those seeking "godship" powerless in everyones eyes.
@HebaruSan but what if a = infinity?
I am thinking that that is either inconsistent or incomplete...
Jim Overbeck
[A]: The Copenhagen Interpretation implies that stasis aka equilibrium [reached after a while] results in the cat being simultaneously alive & dead [B]: an observer sees the cat as either alive or dead > this poses the question when exactly SUPERIMPOSITION - quantum superposing - ends & "reality" resolves into one possibility or other > I suggest 'resolution' rests in 2 impossibles > Einstein said to Svhroedinger that unstable gunpowder superposes both exploded & unexploded states [B]a: mathematically, this involves linear solutions for Schroedinger's equation + the EPR Paradox considerations of "reality" - hence, I responded [cont].
I responded by having transfinite fractions negate tertium non datur & equated it with the Fourier-Bolzano [= Grandi] series 1 - 1 + 1 - 1 + ..... [D]: Physicists and mathematical theorists cannot go beyond this impasse, without the codes into deific realms. Cantor's Paradise is a redoubt of hell - hence, Cantor's madness - an insane substitute for Paradise Accessed. The CAT state of quantum physics negates [divine] identity & tertium exclusi> mortal life = immortal death & the tertium is IMMORTAL LIFE. s madness - an insane substitute for Paradise Accessed.
I think SUPERPOSING is really the superimposition of mortality on to immortality = 2 diametrically opposed conditions with 1 APPEARANCE in time, giving rise to an invalid conjunction [= alive & dead at the same time], this simultaneity having a reversal point I named aleph- p / q [= the 1st transfinite cardinal fraction], which destabilizes death [= mortality] & flips it into immortality. The infinities of death are Cantor's aleph-null & 2 to the aleph-null, whereas the infinity I found is immortal - hence, I negated Schroedinger's equations & probabilistic mechanics by taking this cat into the Ineffable, Immaculate and Infinite beyond, owned by God Almighty.
@QuantumBunk: One could say paradoxes are "stupid & fake" if they are KNOWN and ACKNOWLEDGED. One of the biggest problems with mathemetatics, science and computer technology is that they are being sold to the public as techniques and systems that are infallable. The general population does not understand the weaknesses and faults inherent in these system, and therefore they place them on a pedastal that is undeserved.
What about when Nature seems to conform to mathematical laws? I am not advocating a sort of Platonic mathematical ideality, but I still believe that mathematics is something objective and independent of the human mind. I am not altogether sure I understand my current conception of what mathematics -is-.
hahahah his comment on Godel's death... priceless!
who is the lecturer?
This sentence is true.
...some infinities are bigger than others...
I am a man. Self-referential, but not "stupid and fake". The problem with those self-referential paradoxes is that the object of those sentences are not complete enough to ascribe "truth" or "falseness" to. "This sentence" is neither true nor false, by itself. It does not assert or deny anything. Same thing for "time", or "space", or "apple", or "man", or anything like that.
What Is The Point Of His Marcus Elieser Bloch ?
To say "biggest problems w/ math, science & computer tech...." is erroneous because that's like saying, "....the biggest problem w/ literature is....." No. There's no problem w/ literature or maths. The 'problem' isn't a 'problem.' People have a tool, they just don't know what they want to apply it to. That's 'the problem.' They want it to answer a question, they just don't know they don't have a question.
read "godel, escher, bach - an eternal golden braid" - you might enjoy it