The paradox at the heart of mathematics: Gödel's Incompleteness Theorem - Marcus du Sautoy
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- Опубліковано 19 лип 2021
- Explore Gödel’s Incompleteness Theorem, a discovery which changed what we know about mathematical proofs and statements.
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Consider the following sentence: “This statement is false.” Is that true? If so, that would make the statement false. But if it’s false, then the statement is true. This sentence creates an unsolvable paradox; if it’s not true and it’s not false- what is it? This question led a logician to a discovery that would change mathematics forever. Marcus du Sautoy digs into Gödel’s Incompleteness Theorem.
Lesson by Marcus du Sautoy, directed by BASA.
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Gödel casually making mathematicians notice that they have wasted 20 years of their lives in an unsolvable problem while being that cute in the video lol
At least they wouldn't spend more
Yeah
Godel was a Mathematician too. I understand that people might be confused about logic. But logic is just a branch of mathematics.
Still , math works , we live in the world full of stuffs that uses maths directly or indirectly
Life is a waste no matter what you do
A long time ago, the mathmaticians lived together in harmony. But everything changed when Gödel published his incompleteness theorem
guys it's been over a year, PLEASE stop replying
Guys it was funny for the last two years, but you can stop replying to this now.
help
Mathematicians have regularly suffered existential crises since the beginning of history: Zeno's Paradox, irrational numbers, Non-Euclidean Geometry, Russell's Paradox, Halting Problem etc. One would think they'd have gotten used to it by now.
@@nHans isn't this video's paradox the same as the halting problem?
@@ELYESSS There are similarities in the proofs, yes, in the sense that in both proofs, you use self-reference to create a paradox. The problems themselves are, of course, quite different and were solved by different people:
• Incompleteness: Gödel
• Halting Problem: Independently by Alonzo Church and Alan Turing.
(I know that Turing's proof uses self-reference; unfortunately I'm not aware how Church solved it.)
@@nHans okay lets say paradoxes have ruined mathematician lives and not even einstein would even safe math
it all changed when the fire nation attacked
"Its Gödels all the way down"
most underrated joke in this entire series.
Surely a reference to Terry Pratchett ...
Came to the comments to say exactly this and was glad to see it was already taken care of.
Came to the comments to say exactly this and was glad to see it was already taken care of. (2)
@@dineshgoswami6237 wat does it mean
@@devilvocano420 brief history of time.. stephen hawking
Imagine if he didn’t manage to prove his Incompleteness Theorems because they turned out to be true but unprovable. Complete mindfuck that would have been
=))))
If they were true but unprovable that would have been proof right there, but that means it would be true and provable which makes it so there is no proof, which would make it true and unprovable
@@Sir-Taco wow
@@Sir-Taco if the only way to prove that some mathematical states are unprovable is itself unprovable, then you can't know if things are unprovable. You wouldn't know it was unprovable. It would just look like it was very hard to solve.
I think that is the case! Incompletene or Inconsistent may be Unprovable truth.
Mathematicians: trying to prove that all equations can only be true or false
Gödel: hippity hoppity your certainty is my property
Lol
sheldon, is that you?
Wrong
@@historicwine1283 w-what's wrong 😟 😅
@@AKHELUS. What Gödel says is that there are statements that cannot be proved, no that they are not true nor false.
"Jim is his own worst enemy, and enemy of my enemy is a friend. But...." - Dwight Schrute
Same Russell Paradox
Barber problem
Unexpectedoffice
Lol
That's not really a paradox, is it?
@@reetjaiswal3950 then what is it?
Godel almost received his Nobel Prize but his theorem was incomplete.
Nice one! But mathematicians don't get a Nobel, they have the Fields Medal.
@@cerendemir9977 John Nash got it.
@@rince7A Yes, in economy
@rince7A The Nobel Memorial prize of economics is a fake Nobel
@@cerendemir9977 Fields Medal was introduced in 1936 and Goedel who lived until 1978 did not receive it.
*"...and* *he* *was* *even* *less* *confident* *that* *Mathematics* *was* *the* *right* *tool* *to* *investigate* *this* *problem."*
This is a big lesson on life. Sometimes you have to go outside the system to identify and solve the problems within it. And you have to have the courage to do so, even if doing so leaves you completely alone, and working completely alone for a long time. The moment you get to strongly suspecting there's a major problem in something that matters a lot to you, you should start looking into it and you should consider that the structure of the system in which the problem lies may be contributing to it. The work of Gödel is a textbook example of this: the structure of Mathematics disguises paradoxes within Axiomatic proofs. So Gödel divested a bit from Mathematics and went outside the field to (a part of Philosophy called) Logic to try to identify (and maybe even solve) these problems. And he ended up revolutionizing his field.
Whether the system is mathematics, or the world-economy, or even your own government, if there are problems you're noticing more and more, you may have to go outside the system to truly understand what's going on. The system may be disguising or even contributing to these problems. And you may be the only one who can solve (or begin to solve) them because you may be the only person who is able to see them.
And so the work begins, and in all likelihood it's going to be heavy. And as you work, you may have to endure a lot of push-back and isolation before you can make a big change happen. There's almost always consequences for people trying to fix the problems of the world. You should do it anyway. It's only through honestly representing your truth in the face of the falsehoods of your era that you and the world will know peace.
brother, thank you for this comment.., I appreciate it... I
@@naimejb7921 you’re welcome ✌️
@Anonymous :)
Thanks
@Israel Gawiseb you’re welcome! Thank you for your kind words.
Isaac Newton dancing for Gödel is now ingrained in my mind.
It's Gödels all the way down!
Oh lmao
@@foxgaming76yt24 no you do not
@@hieuminh9164 Huh
That was funny
heisenberg's uncertainty principle: *here comes trouble...*
gödel's incompleteness theorem: *...and make it double!*
Imagine a hero mathematician comes out of nowhere and just solves them like that russian mathemician dude who solved an equation which was very difficult and just went back into living his normal life as a regular dude
To protect the world from assertations
To confuse the people of every nation
To denounce the evils of truth and false
To extend all measurements containing faults
Werner
Kurt
Team Uncertainty put error bars on the speed of light
Surrender now or your certitude will be out of sight
Meowth, is that right?
That's a definite A for effort. Was not expecting this on the comments. Thank you
@@mrsugar7528 Grigori Perelman on Poincare conjecture?
@@JohnathanLeeSprite That guy looks like a hippie but takes life seriously. But he seemed like an idealist. He should have just taken the Fields Medal.
Mathematicians : *represent numbers by alphabets.*
Godel: *turns alphabets back to numbers.*
Math students: "Is this the power of a god?"
Gaara from Naruto. I understood it. Nice one
More like he turned it into holders of information that could be represented in language
Isaacus Neuutonus.
/*Jehova Sanctus Unus*/
No the power of godel
gödel was a mathematician himself
Dude took "Math is just numbers" to a whole new level
LOL, Godel, Noether, and Hilbert are drawn so adorable
@@DyslexicMitochondria Hey bro I watch ur videos. Love your channeI
Ohhh it was Hilbert! I was thinking Russell.
Who is LOL?
Definitely one of my favorite videos animation wise
@ʜᴏɴᴇʏᴘɪᴇ bruh i was just kidding 😆
"Breaking math" is the most badass thing a person could ever achieve and you can't change my mind
Can you prove that your mind cannot be changed?
@George Khoory agreed, he just "broke" the concept of "math is this because of this, period"
@George Khoory it is not 100% impossible to violate physics and you can't prove otherwise.
@George Khoory imo just like math or any other things, physic is just knowledge and physic laws are just agreement that people agree upon based on what have been known, and because of the fact that there are things that we dont know that we dont know, physic laws maybe true today but not so true in the future. That why i thing it is definitely possible to violate a 'current' physic law
How about breaking the jokers mind to where he snuffs it? 🚨🚨🚨
I remember when I was in thrid gradr my math books had written in the cover "Maths make sense". And as a kid that hated math, I spent time trying to figure out any mistake in it, something that didnt make sense. I actually did it a few times, buuut it was actually just me making mistakes, not maths. Well, glad to see one guy did my childhood quest
But it still makes sense
not only do i appreciate the concise synopsis of the theorem, i also appreciate getting to see godel dancing around in glee
You guys should make a video about me titled: "the man math broke"
that would be pretty unremarkable seeing that since every highschooler can be called that
@@manuelmathew848 not every :)
@@spoopyscaryskelebones3846 goddamit
@@manuelmathew848 more than 50% then 😂
Underrated comment
Teacher: Why didn't you show your complete solution?!?
Me: well ma'am, according to the Incompleteness Theorem....
Teacher : but that solution was proved to be proveable , i know since *I DID BY SOlVING IT*.
Teacher: "So you've chosen death 💀"
@@LolwutLol2000 They are an definition , not a statement.
Teacher: I'm well aware of Godel. Too bad your oversimplified Ted-Ed video didn't teach you enough.
@@-Subtle- you may have a brain but... I HAVE A GUN
I love how it’s Hilbert (we must know, we will know) who’s walking down the cliff at 4:09
Truly wonderful detail, that you used Hilbert as the character, desperatly tries to fix the towers. since he dreamed the most about a complete system of axioms! wonderfull
“The man who broke math”
Me when the calculator says *syntax error*:
*look what he needs to mimic a fraction of my power*
+
Lol
My math teachers always hated my questions. Ignored them with nervous laughter or acted bothered or said something along the lines of "that's a whole other discussion." Yeah. They were afraid of me.
@Franklin Roe how to make baby? I want baby
@Franklin Roe why was six scared of seven?
Me: can barely do fractions
Gödel: *breaks the entire field of mathematics spine over his knee*
He Bautista Bombed it on a table!
Gödel is the bane to maths’s batman
It's more like giving it the ability to turn invisible
50 years of attempts, beginning with the work of Gottlob Frege and culminating in Principia Mathematica and Hilbert's formalism, to find a set of axioms sufficient for all mathematics just gets thrown under the bus when Godel finds his Theorem
You lost me bro.
A math professor I had said this, "No arbitrary system of rules can explain itself without external input"
Alfred Tarski.
now let me introduce myself. I, the man who's broken by math
😂😂😂
Nerds watching this:This man needs some milk
:( Both you and me..
😂😂
“Someday a real rain will come and wash all the certainty off the streets “-Gödel
I also like his other quote:
"The accumulated filth of all their certainty will foam up about their waists and all the physicists and mathematicians will look up and shout 'Save us!' and I'll whisper 'No.'" -Gödel
Certainly.
Wow!! Profound!!
"He didn't say that." - A. Einstein
@@mrrodriguezHLP "I guess it comes down to a simple choice,really,get busy proving or get busy assuming"-Gödel
It's good that you make it clear how Gödel’s self-referencing sentence is only interesting because it is stated in a language inside the system! But we also need to determine if there’s a possible flaw in translation from the verbal sentence "this statement is false" to the mathematical translation…
And finally we need also determine whether an axiom kan be self-confirming or self-denying without creating a systemic paradox or placing itself in an order of axioms, not part of the class of all common axioms….
I'm sure the mathematicians thought about that.
ANd it failed, which is why they called it a theorem.
The art and sound design of this video is especially satisfying.
For those who don't quite understand what "It's Godel all the way down" means.
The phrase "It's turtles all the way down" comes from an anecdote told in the opening to Stephen Hawking's A Brief History of Time(Edit: although it is not the anecdote's or the saying's real origin):
A well-known scientist (some say it was Bertrand Russell) once gave a public lecture on astronomy. He described how the earth orbits around the sun and how the sun, in turn, orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a little old lady at the back of the room got up and said: "What you have told us is rubbish. The world is really a flat plate supported on the back of a giant tortoise." The scientist gave a superior smile before replying, "What is the tortoise standing on?" "You're very clever, young man, very clever", said the old lady. "But it's turtles all the way down!"
The phrase is used to describe any system that appears to have dependencies that never end. In the anecdote old lady said that the Earth is flat and is supported on the back of a turtle, but that creates the fact that the turtle needs something to stand on, so the lady says that "It's turtles all the way down" which means that every turtle stands on another turtle which stands on another turtle and so on.
For another example, imagine accountability in a (hypothetical) police department. The citizens are policed by police, the police are policed by internal affairs, which might lead to the formation of an "internal internal affairs" to police internal affairs. Someone might describe this system of policing as "turtles all the way down", meaning that the system of policing never ends.
So "It's Godels all the way down" means that even if someone tries to make unprovebly true statements new axioms it would create new unprovebly true statements and if someone tries to make them axioms there will be new unprovebly true statements and so on, like said in the video.
I do hope it wasn't long enough for you to get bored, but detailed enough so that you now get what's going on. Have a great day.
Edit courtesy of @silver6054 : In the form of "rocks all the way down", the saying dates to at least 1838, when it was printed in an unsigned anecdote in the New-York Mirror. A version of the saying in its "turtle" form appeared in an 1854 transcript of remarks by preacher Joseph Frederick Berg addressed to Joseph Barker:
My opponent's reasoning reminds me of the heathen, who, being asked on what the world stood, replied, "On a tortoise." But on what does the tortoise stand? "On another tortoise." With Mr. Barker, too, there are tortoises all the way down. (Vehement and vociferous applause.)
- "Second Evening: Remarks of Rev. Dr. Berg"
So, I suppose Stephen Hawking was just the more known person to popularize the saying. Thanks to @silver6054, again, for the correction.
Well you'd like to know i generally don't read this much in a comment like you said "get bored" but I did find your explanation so intriguing... that look! I am even leaving a comment to notify you😂. It was good btw!
Thanks for this explanation. I'm no mathematician (far from it) but very intrigued by them, and I found this video quite distrubing and interesting at the same time. Your comment just completed with a great metaphor the theory of unprovable axioms I wasn't sure to get properly. Plus the fact that's still very modern problem (flat earthers and so on).
Thanks a lot a lot a lot (and so on ... :D )
Kudos to your patience.
Ty! You da real mvp
Men thankyou thankyou thankyou thankyou
After this video, only one statement comes to mind: "I understand nothing" -Michael Scott
Who knew Michael Scott was a fan of Socrates all along?
What is there to understand if there's nothing in the first place 😂😂
@@nada__ that is actually a really good example!
Its basically schrodingers cat,
You cant prove its dead or alive until youve opened the box
well they made a whole video about math without showing the equation they were talking about. no one could understand what is not shown
I love how Ted teachers me in ways that lead me to understand subjects & concepts I struggled to learn in school
yes I also really struggled with Godel's incompleteness theorem at school
This is such an excellent intuitive explanation of a complex idea!
I can't help but feel like this guy's motivation for developing this was nothing more than spite.
No. The Cantor diagonal problem is another demonstration of the same thing. In some ways it's easier to understand, depending on how one learns.
true lol. ig spite is one of the primary motivation huh
Soo.. Does this mean I can write this in my upcoming math test?
if the guy above the Yve named JiminTae was i, we would form the word "guy" with our names
@@UserName-mf9db now that's another level of observation 😎😂
Socrates: *How did this guy not get poisoned?*
Gödel: Oh wait...
Underrated comment
@@Athlin I’m pretty sure Godel had a severe fear of being poisoned. He only trusted his wife to prepare him food and refused to eat once she died until he died of malnutrition. Something along those lines
@@maximumoverdrive2676 Wow that actually a pretty terrible way to die. Was really someone after him have there been aptempt at his life or did he become paranoid?
@@maximumoverdrive2676 he is just paranoid, his wife was not dead beside, she just has to go to hospital for six months because of stroke and he hadn't eaten much the whole time. When she came back, he was 30kg. She brought him to hospital immediately, but unfortunately, some week later...
@@franxx941 He became paranoid after his close friend Moritz Schlick was murdered.
I feel this title is misleading: Godel didn’t “break” math any more than Ben Franklin “invented” electricity. Godel simply discovered a limitation that had always existed (which is still an incredible achievement btw since he had to construct the proof for that, as the video explains). Frankly, I find that to be far more disturbing: it means that one of the fundamental tools we use to understand the universe was inherently flawed from the outset.
The title has to be designed to attract people to click on the video. The more people they can attract to watch, the more people they end up teaching. And if their goal is to teach, then clickbaity titles will enhance that goal
Is the math flawed... or does it just accurately reflect the inherent uncertainty and incompleteness of reality?
It's funny because it basically means that whatever we are "discovering" could be just an approximation or totally wrong. Similar to the nonsense predictions of the standard model, despite some accurate predictions.
i think what people have to realize is that any tool created by imperfect humans is necessarily gonna be imperfect. people look at science and math as infallible, be-all, end-all solutions, but they're not. now, this isn't to say they're worthless and/or they're not the best tools we've got, but acknowledging that our tools are imperfect is ok (and necessary)
ua-cam.com/video/voWzbIE6ZRs/v-deo.html
Not only love the topic but also the animation and the music 💜
“It’s Godels all the way down” BRILLIANT
Yeah, I loved that too. There's a pompous mathematician friend of mine who I can't wait to use it on.
Physics: Turtles.
Coding: Hand Grenades.
i'm smooth brain, could someone please explain this to me? i've heard of "turtles all the way down" but i have no clue what it means
@@flowercities The phrase comes from an anecdote told in the opening to Stephen Hawking's A Brief History of Time:
A well-known scientist (some say it was Bertrand Russell) once gave a public lecture on astronomy. He described how the earth orbits around the sun and how the sun, in turn, orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a little old lady at the back of the room got up and said: "What you have told us is rubbish. The world is really a flat plate supported on the back of a giant tortoise." The scientist gave a superior smile before replying, "What is the tortoise standing on?" "You're very clever, young man, very clever", said the old lady. "But it's turtles all the way down!"
The phrase is used to describe any system that appears to have dependencies that never end. In the anecdote old lady said that the Earth is flat and is supported on the back of a turtle, but that creates one fact: the turtle needs something to stand on, so the lady says that "It's turtles all the way down" which means that every turtle stands on another turtle that stands on another turtle and so on.
For another example, imagine accountability in a (hypothetical) police department. The citizens are policed by police, the police are policed by internal affairs, which might lead to the formation of an "internal internal affairs" to police internal affairs. Someone might describe this system of policing as "turtles all the way down", meaning that the system of policing never ends.
So "It's Godels all the way down" means that even if someone tries to make unprovebly true statements new axioms it would create new unprovebly true statements and if someone tries to make them axioms there will be new unprovebly true statements and that's a never-ending cycle, like said in the video.
I do hope that wasn't long enough for you to get bored and give up, but detailed enough so that you now get what's going on. Have a great day.
@@DGHeina very good explanation, thanks
Veritasium did an amazing job explaining Godel's Incompleteness Theorem. I highly recommend for everyone if they want a more in depth video of the theorem.
Yes, Veritasium did a much more thorough and in-depth exploration of Godel
Yeah I saw that! I was eating and I almost choked lol 😂 I was so shocked lol
Veritasium misunderstood Gödel’s theorem-specifically the part where it states that certain *true* statements cannot be proved. Veritasium ignored the 'true' part, and went on to claim that any currently unproven statement like the Riemann Hypothesis might be unproveable due to Gödel’s theorem.
But that's wrong. It has been known since Euclid's times that if you start with a finite number of axioms (or axiom schemas), there are always statements that cannot be proved using those axioms. What's new in Gödel's theorem is that even 'true' statements cannot be proved.
Most people have difficulty understanding the concept of "true but unprovable." Is the Riemann Hypothesis true but unprovable? Or, like Fermat's Last Theorem, is it just a matter of time before somebody proves it?
Veritasium unfortunately created some confusion in that matter. Some students came up to me and said that the Riemann Hypothesis cannot be proved because of Gödel's theorem, and referred to Veritasium's video as the source.
Don't misunderstand me, I love Veritasium's videos in general. But Derek frequently gives in to hyperbole and click-bait titles, possibly because he depends on UA-cam ad revenue for a living. So he called Gödel’s theorem a 'fatal' flaw in mathematics. Well, that 'fatal' flaw has existed in mathematics since the beginning of time, but mathematics is still alive and going strong. Forget being dead, it's not even crippled. 🤣
Agreed. Feel like TED-Ed took the idea after realising they've been a bit too bias politically lately...
@@nHans That's not what interpreted but, as always, it was a long video connecting several ideas
When you can step back a frame reference and look from a more higher level/simpler view the details start to get blurred but new trends emerge on a higher level.
Mom and Dad have a joint bank account. They make 1 and 2 and a combination of 3(sided combination like 70/30 of the way its used. etc.)
Thank you so much for uploading this video. It is helping me get through the pandemic!
4:05 That man with the hat falling down the hill was the great mathematician David Hilbert....who asked 3 most important question about whether math is complete, consistent or decidable. Gödel answered the first question using his Incompleteness Theorem.
thanks i wasn't able to remember his name ^^
The one who raced with Einstein about the equation of TGR?
@@ronharleypantaleon1824 yes, the same Hilbert
Fun fact, It was later proven that math is also undecidable.
What does this mean? It means that there are some equations, algorithms, processes, and changing arrays that we will never know whether or not they come to a conclusions or loop endlessly.
I believe Gödel also proved that mathematics cannot prove its own consistency. In this context, consistency means 2+2=4, always, or that adding two numbers always creates a bigger number. While some basic statements are pretty much a no brainer, the foundations of mathematics cannot be used to prove their own validity or consistency.
So at the very best, mathematics is either:
1). Incomplete, Consistent (but we will never know), and undecidable
2). OR incomplete, inconsistent, and undecidable
@comh33 I believe that Gödel proved in his second incompleteness theorem that the statement "Mathematics is consistent" also falls into the category of statements that can not be proved but is true regardless, just like those self-referencing axioms which cease to be proved by it's axiomatic/logical system.
There's something cathartic about even mathematics, something we both create and discover, having equally mysterious side as discoverries in nature and space.
Agreed, the inherent mystery of the universe, it’s pretty fascinating
I think it shows that mathematics isn't merely invented
I know this is an old video but whoever did the sound design for this video deserves an award for all these sfx
Gödel and Turing are my idols. Turing's Turing machine and Gödel's Gödel number both brought to life my adoration for mathematics and motivation to work in the field. Thx guys.
ua-cam.com/video/voWzbIE6ZRs/v-deo.html
And both came about to prove hilbert wrong
@@stellaleicht4035 I also like the outcome of Hilbert's 10th problem: give an algorithm for finding the solutions of a diophantine equation (the integral solutions of a polynomial equation in multiple variables with integral coefficients).
In 1970 Matiyasevich completed the proof that any computer program (any turing machine) can be encoded as a diophantine equation. Since there is no algorithm for the halting problem, the algorithm that Hilbert asked for does not exist.
ChatGPT can be rewritten as a diophantine equation. Don't try this at home.
"Godel rocked a mohawk in real life". This statement cannot be proved
To which reality are we talking about?
~ A random dimensional hopper
It cannot be proved that the original poster of this comment was referring to the mathematician Godel, and was instead referring to any other person named Godel.
No it's just nonsense
Hilbert: "math is complete"
Godel: "I'm about to end this man's whole career"
It "Godels" all the way down
maths is complete, just it's only internally consistant.
all this is nonsense.
@@calhackit9806 keep telling yourself that :)
@@calhackit9806 mathematics is provably incomplete. That's what Goedels incompleteness theorems tells us. In order to make the claim "maths is complete", you need to disprove Goedel's theorems, rather than just decry them as nonsense
@@epicmarschmallow5049 Yeah but that's too much work I'm not capable of doing, so I would rather call it nonsense on the internet.
"The worst he can say is my formula is wrong"
This person:
I like that in 3:42 the human is Gilbert who believed that math is comple and you can prove every true statement, and now after discovering Gödels incompleteness theorem he suffers the most
Ah here is another video which makes me question everything I've ever studied
Now that's confusion ladies and gentlemen.
I’m not a religious person, but when I first learnt about this and read more about Euclidean axioms, the more I started to believe that there is some higher power. Obviously it would be amazing if we could prove why axioms are true, but something tells me the reason of thing that makes them true is outside the capacity of human understanding.
This channel is just awesome.
"There are known unkowns, and there are unknown unknowns... Things we don't know that we don't know!!"
May Rumsfeld Rest In Peace.
@@stansantos4733 I believe that this came from Rumsfeld's recollection of the Allegory of the Cave from Plato's Republic (Book VII).
He said it to justify the Iraq war, what he didn't tell people is that the 4th kind is also there, the unknown known which would have prevented the war, looking at what transpired since then we know the answer now, his statement was a very elaborate cope for an excuse to start a war in Iraq
Gödel really went above and beyond to say "The situation here is that the question is badly worded" and he was right.
Well Math is much more complicated than human language
Yes I agree
@@princemachiavelli6570 wait, whos language is it if not human?
«Gödel really went above and beyond to say "The situation here is that the question is badly worded" and he was right.»
----
Imho attention is to be paid to how HE worded it.
The first statement is insubstantial. It’s like just saying “I’m lying”
Great video! As useful as mathematics is, we can only prove what our limited human minds can process. But there is truth outside of our ability to understand.
There must be a handoff between mathematics and philosophy.
Alternative Title: How to give a Mathematician an existential crisis.
Kind of. But we settled to not think about it too often.
I think most mathematicians have accepted by now that we do have no natural right to proofs. Everything we can prove is basically a miracle. And in some way, that makes it even more exciting. Now it's like, "Look, guys! I fought the universe and won!"
@@lonestarr1490 I guess it depends on how fixated the mathematician is on the assumption that maths will help them find Truth.
so gödel have himself an existential crisis, because he was also a mathematician
Glados: This sentence is false.
Wheatley: True, I'll go true
if you are in the danger of robots just close your eyes and shout out :
This has absolutely nothing to do with the video, but I saw a dude with an Aperture Science t-shirt today and hope was reinstalled in me for the future
the sounds are so lovely to my ears
I don’t really understand what it is that Godel did at 1:53. Turning mathematical statements into random numbers is supposed to facilitate what, exactly?
1) He didn't convert mathematical statements into random numbers. He converted them into numbers in an orderly fashion according to a specific scheme that was universal. We all do the same habitually now via our computers, smart phones, etc., with ASCII, Unicode, binary, etc., for numbers and also image and video formats for photos (jpeg, png, etc.) and video (mp4, avi). All of these, at their very basis are just numbers. Everything in every computer essentially does the same thing that Godel did.
2) Having converted everything to numbers, he described how to them put those numbers into specific functions (which also could be described as numbers). We do the same. We feed a .docx word document (which is really just a complicated number) into a word processor (Microsoft Word, which itself is just a complicated number) and we can manipulate it. Same with images and video. We take an mp4 (a complicated number) and feed it into VLC (a video program, also just a number) or UA-cam (a complicated number we get online) and watch a video.
3) Godel did the same, but he fed his numbers into a specific function, which was a function that purported to be able to determine whether a given function was provable. He was able to show that a specific number that he could construct using his numbering scheme would have a numeric representation that corresponded to the mathematical statement that that statement itself was unprovable. The details of that are highly technical, just as the details of how Microsoft Word works, or UA-cam works, are also highly technical. I've greatly simplified things here, but I hope the analogy of how we now convert everything into numbers for computer purposes helps to illustrate how and why he did what he did.
If you want a more detailed explanation, search for Veritasium Godel, and watch Veritasium's video,which is titled "Math's Fundamental Flaw"
Mathematicians: You either die a hero or live long enough to see yourself become the villain.
Godel: Yes
That made me laugh way too hard 😂😂😂
I see what you're doing there
@@robinbruce7838 can u prove??😂😂
Fun fact (actually pretty sad): Goedel went insane. He thought everyone except his wife wanted to poison him. When she ^went to hospital for longer/died (I dunno what it was) he starved himself to death. 😊
Me: "Breaks math"
Teacher: That's cheating.
Me: *dies*
Gray: "we don't need Math where we're going"
If you could mathematically break math, your teacher would impress (if only they're good at math, ofc)
Me: Prove it!
When things were at their very worst:
2 Suns, Cross in the sky, 2 comets will collide = don`t be afraid - repent, accept Lord`s Hand of Mercy.
Scientists will say it was a global illusion.
Beaware - Jesus will never walk in flesh again.
After WW3 - rise of the “ man of peace“ from the East = Antichrist - the most powerful, popular, charismatic and influential leader of all time. Many miracles will be attributed to him. He will imitate Jesus in every conceivable way.
Don`t trust „pope“ Francis = the False Prophet
- will seem to rise from the dead
- will unite all Christian Churches and all Religions as one.
One World Religion = the seat of the Antichrist.
Benedict XVI is the last true pope - will be accused of a crime of which he is totally innocent.
"Many events, including ecological upheavals, wars, the schism in My Church on Earth, the dictatorships in each of your nations - bound as one, at its very core - will all take place at the same time."
1 November 2012
The Book of Truth
i really loved the sound effects too..!!!
So this is what has plagued me for 5 years. Thanks for helping me confirm it wasn't just me, TED.
Me trying to explain to my maths teacher why I didn’t do my homework:
I'm a mathematician. The animation of the video was very cool. However, many concepts put in this video are difficult for a layman (including many mathematicians). This theorem does not deny or refute that 'a chair exists', he argues about some existing indeterminations when trying to 'prove' that 'a chair exists' within a very specific context. Basically, the foundations of the mathematical thought consist of axioms.
Axioms is what is 'pure faith', but not a 'blind faith'. An almost religious essential of mathematics. For they are 'things' that you cannot prove exist by definition. But if you assume that they exist and are true, everything you build from them makes sense and is consistent. Mathematics is beautifully built on top of axioms. A specific part in which one seeks to prove this consistency in a specific context is about Gödel's Incompleteness Theorem argues.
This question is addressed mainly regarding the philosophers of mathematics who question the consistency of demonstration methods. And maybe you ask yourself, why are people worried about this? Well, stop to think about computers. How does when entering your bank password, what makes the computer 'validate' that the number 6 you typed is actually 6, instead of 9?
what exactly is "mathematician" ?
I understand the context , but the phrase
"I am a mathematician" , is that a job ?
what do mathematicians do on a daily basis?
@@tanishqarora2647 they do research and teach university classes usually
@@tanishqarora2647 They eat math as breakfast. xD
Strikes me as something that's only really a problem if you're on the side of the fence that claims that mathematics is discovered. If you take the view that mathematics is invented, then resting on unprovable axioms isn't really something that will shake your confidence too much. Seems a bit like building a house: If you put one brick on top of the other the right way, you'll end up with a building. This is true as long as bricks are what we mean when we say the word. We don't need to prove how they are what they are if all we care about is building a house.
Wow! He was 5 year earlier than Alan Turing solved the Halting Problem. Which actually is the same kind idea. That's awesome!
Beautifully animated!
The dislikes may be by those who understand why this simplified version can be argued to be erroneous.
That's precisely correct it was an oversimplified explanation and most people misunderstood it.
The time it took to write this comment could have been used to create and post a copy-pasta of relevant URL links, search engine terms, and a brief message. With the assumption, of course, that spreading knowledge is your mission as opposed to self-admiration.
*This copypasta was created on 7-4-2007. Please reuse.*
@@guymanperson1 brilliant
thank youuu Ted Ed! it was one of the best projects i've worked on, and im glad people are enjoying our animations :> !!!
Your art is so good!
@@lenl8004 thank you! :>
Your art is beautiful ❤
‘We now have a true equation of mathematics that asserts it cannot be proved.’
Wow! Absolutely mind-boggling, but mega-interesting. And of course for the mathematical community groundbreaking.
He translated it and doubted math but he didn't doubt his translation. Truly a "mastermind"
He translated it Because he doubted the certainties of it. His translation only pointed out the flaws. Go back to the video and watch it again because you didn't get it. And also, in a way you are right, but this is next next level, if you know what I mean
You too spotted on it.
p.s. if you there, read my other answer to the "next level" guy that answered you.
@@vitalismed «He translated it Because he doubted the certainties of it. His translation only pointed out the flaws. Go back to the video and watch it again because you didn't get it. And also, in a way you are right, but this is next next level, if you know what I mean»
----
It is you that should go back and think about "what is a translation" (what has to satisfy etc etc). A hint: transliteration is not a synonym for translation.
Save me eventual further parroting lessons on next levels.
@Volty De Qua woah man chill, you’re tensing so hard
@@vitalismed «woah man chill, you’re tensing so hard»
----
Can't. I feel like an inflamed chord when I smell parroting dressed as lecturing.
Anyway your previous comment was more than tensed.
I gave you a hint - follow it, or ignore me and continue with your recital.
my dude hilbert at the end was badly injured yet still happy to celebrate gödel's achievements.
I chuckled at that bit
In short: self reference with negation ruins everything
There are more than that
@@Noname-67 see halting problem
I see no problem simplifying your shortcoming
If a liar lies, does he say the truth?
@@fragileomniscience7647 no, he's lying, hence he is not saying the truth
I love how Hilbert is shown without reference expect with his iconic hat and him trying to formalize a system but running into trouble.
They should have added Bertrand Russell.
Principia Mathematica is dense that it needs some recognition.
Love how Godel rocks that Mohawk, sort of like a punk rock mathematician
this channel educates us in a way that's so visually and mentally pleasing , i hope current educational systems would do something similar !
The animator did an extremely good job with this video.
"This statement..." cannot be used in itself, because predicates can absolutely never be applied on themselves. Also, the statement is not yet complete, so you can't apply any statement on it. It needs to be completed first and then you can use it to define anything. I've said the same thing about sets of sets - the set of all sets, that do not contain themselves as an element, doesn't contain itself, because sets can only contain elements, that have already been defined before-hand. And if you're creating this set of all sets, that don't contain themselves, you shouldn't add it to itself, because you're still in the process of creating it (and the same goes for if you change it afterwards). There're no paradoxes, just get your definitions straight.
I have understood nothing but it's a very interesting matter 😉
A few thoughts from someone with a strong interest in this area:
Mathematical statements don't always have to be either true or false - it's common to assume they are, but can be useful to reject that assumption. For a simple example, statements about variables can be thought of as potentially true or false, but not necessarily either one.
Another example is axioms. These don't have to be "undeniable" - they're more like conditions specifying the sort of situation we're currently interested in. If these axioms apply to a situation, and we use these deductive rules, then this result follows. In particular circumstances an axiom might be true, or false, or indeterminate.
Going in the other direction, a statement might be provable but false, e.g. if the axioms are inconsistent. It might seem silly to use inconsistent axioms, but Gödel in effect proved that any list of axioms sufficient for ordinary arithmetic is potentially inconsistent: it can only be proved consistent if we add in extra axioms - and then proving this new list consistent would require extra axioms on top of those, and so on.
One reason these ideas are important: when a statement can't be proven using particular axioms, it can often be thought of as being false for some models of those axioms. If you think it should actually be true, that's because you're implicitly assuming extra axioms that you weren't stating... and there's no way to list all the axioms you'd want without also including some that you don't want.
On the other hand, only a tiny fraction of these "philosophically significant" axioms are needed for the vast majority of results used in science, technology and engineering. That's why most of the people interested in this stuff are logicians or philosophers.
Yet that is why we have SPECIFIC TERMS for those: predicates (depending on variables) aren't statements, axioms aren't statements. ;-) Although intuitively i'd include "we don't know yet" in the neither true nor false category, but i feel neither you nor the video are considering those.
As for the "provable but false" i think you're misinterpreting something. If you have a proof, then the conclusion is by definition true. Even though its negation would ALSO be true (because the system is inconsistent/contradictory), that doesn't mean the non-negation wasn't true.
By the way, something feels off about your explanation of the "potentially inconsistent" arithmetic... Those phrases seem to be more about completeness than about consistency. 🤔
3:43 love the David Hilbert reference with "We must know, we will know!"
Haha
I feel bad for Hilbert.
@@segmentsAndCurves i heard that he died before the discovery of Godel theorum
@@damnguen1726 nope: Hilbert died in 1943 while Gödel published his paper in 1931 a darn full dozen years of misery for ma mann Hilbert
@@WolfgangGalilei tks, for Hilbert that is depressing
for some reason this video made me so happy
Thank you very much!
0:22
It's
Tralse
I had to do a study on godel. He's a legend. Dude made Einstein doubt his relativity theory. And then was so convinced someone was going to poison his food that his wife was the only person to make him food. When his wife went into the hospital he starved to death
As a kid, I always wondered, if I told someone "I'm a liar", would they believe that sentence, or would they consider it a lie.
Your video is really cool and amazing! I want to be good at math like you!
Ahh! This is always the case, but what a good mini lecture with such lovely art to back it up! I am always amazed!
That was extremely interesting lol. I happened to pass by it while scrolling for something to watch. The video played with no sound, but had subtitles. So I sat and watched the whole thing while reading along to what was being said. Thanks for the video.
The key point is representability; which made me search for proofs in some grad textbooks on logic (because it's not there in the 1931 paper ! ). Although judging how strong system P is, it's kinda obvious (and if it's false then theorem 7 tells us that there are arithmetic relations not representable).
Perhaps going along why relations 1-46 makes sense is the tedious part.
The sounds in this one are so satisfying
Smarter than me, that's for sure.
sussy baka
sometime you don't need proof
Whenever something is unprovable true I’m just gonna call it tralse
Why not
Falue
Frue
All meaningful statements are provable. Godel is simply saying that the process of producing stronger mathematical theories goes on forever.
Then prove this statement is false by it being tralse.
@Hand Grabbing Fruits Dammit you’re right
That accordian made the not ever knowing feel a whole lot better ..
I'll be honest, I feel off the train when godel turned mathematical statements into numeric strings. How does one do that?
@Scott Doherty sounds like it only goes one way though. Regardless, I appreciate your reply.
I love this style of animation! I just really like the design if the characters.
Random Mathematician guy: damn I love how everything in math can be proven using a set of basic axioms
Godel: yeah about that...
Yea just going to calmly destroy 50 years of work at trying to find that everything in math is provable with sets of axioms
i wouldn't say that gödel destroyed mathematics, in fact he actually expanded mathematics
Love the animations here :)
A math professor was teaching informal logic. He called it logic in its undershorts. Logic is also used in philosophy. Philosophy is the bridge that connects science with the arts.