# Math Has a Fatal Flaw

Not everything that is true can be proven. This discovery transformed infinity, changed the course of a world war and led to the modern computer. This video is sponsored by Brilliant. The first 200 people to sign up via brilliant.org/veritasium get 20% off a yearly subscription.

Special thanks to Prof. Asaf Karagila for consultation on set theory and specific rewrites, to Prof. Alex Kontorovich for reviews of earlier drafts, Prof. Toby ‘Qubit’ Cubitt for the help with the spectral gap, to Henry Reich for the helpful feedback and comments on the video.

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References:

Dunham, W. (2013, July). A Note on the Origin of the Twin Prime Conjecture. In Notices of the International Congress of Chinese Mathematicians (Vol. 1, No. 1, pp. 63-65). International Press of Boston. - ve42.co/Dunham2013

Conway, J. (1970). The game of life. Scientific American, 223(4), 4. - ve42.co/Conway1970

Churchill, A., Biderman, S., Herrick, A. (2019). Magic: The Gathering is Turing Complete. ArXiv. - ve42.co/Churchill2019

Gaifman, H. (2006). Naming and Diagonalization, from Cantor to Godel to Kleene. Logic Journal of the IGPL, 14(5), 709-728. - ve42.co/Gaifman2006

Lénárt, I. (2010). Gauss, Bolyai, Lobachevsky-in General Education?(Hyperbolic Geometry as Part of the Mathematics Curriculum). In Proceedings of Bridges 2010: Mathematics, Music, Art, Architecture, Culture (pp. 223-230). Tessellations Publishing. - ve42.co/Lnrt2010

Attribution of Poincare’s quote, The Mathematical Intelligencer, vol. 13, no. 1, Winter 1991. - ve42.co/Poincare

Irvine, A. D., & Deutsch, H. (1995). Russell’s paradox. - ve42.co/Irvine1995

Gödel, K. (1992). On formally undecidable propositions of Principia Mathematica and related systems. Courier Corporation. - ve42.co/Godel1931

Russell, B., & Whitehead, A. (1973). Principia Mathematica [PM], vol I, 1910, vol. II, 1912, vol III, 1913, vol. I, 1925, vol II & III, 1927, Paperback Edition to* 56. Cambridge UP. - ve42.co/Russel1910

Gödel, K. (1986). Kurt Gödel: Collected Works: Volume I: Publications 1929-1936 (Vol. 1). Oxford University Press, USA. - ve42.co/Godel1986

Cubitt, T. S., Perez-Garcia, D., & Wolf, M. M. (2015). Undecidability of the spectral gap. Nature, 528(7581), 207-211. - ve42.co/Cubitt2015

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Special thanks to Patreon supporters: Paul Peijzel, Crated Comments, Anna, Mac Malkawi, Michael Schneider, Oleksii Leonov, Jim Osmun, Tyson McDowell, Ludovic Robillard, Jim buckmaster, fanime96, Juan Benet, Ruslan Khroma, Robert Blum, Richard Sundvall, Lee Redden, Vincent, Marinus Kuivenhoven, Alfred Wallace, Arjun Chakroborty, Joar Wandborg, Clayton Greenwell, Pindex, Michael Krugman, Cy 'kkm' K'Nelson, Sam Lutfi, Ron Neal

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Written by Derek Muller, Adam Becker and Jonny Hyman

Animation by Fabio Albertelli, Jakub Misiek, Iván Tello and Jonny Hyman

Math City Animation by Another Angle 3D Visuals (www.anotherangle.ee)

Filmed by Derek Muller and Raquel Nuno

Edited by Derek Muller

Music and SFX by Jonny Hyman Additional Music from Epidemic Sound

Additional video supplied by Getty Images

Thumbnail by Geoff Barrett

Associate Producers: Petr Lebedev and Emily Zhang

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Komentarjev

© 2010-2021 SLthrow Spletni video

Matthew RodgersPred 28 minutami

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MsUncleKevinPred 59 minutami

Thing you cant solve is womens mind. Doesnt matter if u are godel or poincaire. Logic fails.

NYTLYFPred uro

This channel is sponsored by illuminati.

Daniel DwikyPred uro

Finished watching this and i just realized that mathematics and physics really are complicated

Ern de ChePred 4 urami

By design, science only disproves, consequently autoreference is only scientifically useful when it disproves. Illusory truth is a contradiction in terms. Finite minds cannot process infinity in any true form, insanity is the demonstrated result of relentless attempts in spite. Formalist is a euphemism. 0 doesn't exist. An infinite set is a contradiction in terms. g is irrelevant. No one can stand on the shoulders of giants if midgets are permitted to anklebite them down, let alone encouraged. Believing a lie defiles your mind, and the more obvious the lie the greater the defilement.

Barely RicePred 6 urami

yo does anyone remember that tile game from a dong like 10 years ago

Snigdha SarkarPred 7 urami

Can we just appreciate the animation quality and hard work he put?!!

Krispyking24Pred 7 urami

i want to watch the conways game of life in conways game of life

Michael ShawPred 7 urami

Moobs

robert ottoPred 7 urami

Brain just melted 😂 smart guy!👍

Kazimierz KrólPred 7 urami

About Cantor's Diagonalization Proof (I am not a mathematician, I have read some Wikipedia articles, but couldn't understand them fully, so what I say here might be bollocks): If we create another real number by adding one to next digit in all the real numbers, then I see two problems: 1. Here is the contradiction without using natural numbers at all: earlier we assumed the list was complete, and contained every real number between 0 and 1, so if we can create another one, the assumption was wrong. 2. We can do the same operation on the natural numbers on the left: add one to next digit in all following numbers (padding all with zeros on the left), and look at this: we have got another natural number, different from all previous ones. This will be our newly created index for the created real number. Isn't this a proof the number set sizes are equal? Of course this suffers from the same contradiction as in #1, but I wonder, did no one think about this simple thing before?

Tom SvobodaPred 5 urami

@Kazimierz Król well, the fact that naturals have finitely many digits and decimals have infinitely many digits implies that one set is larger than another, but not trivially. it needs to be proved, which is exactly what the diagonal argument does. for example the set of all rational numbers has the same size as the set of natural numbers, and yet most rational numbers have infinitely many digits when represented by decimals.

Kazimierz KrólPred 5 urami

Thanks. So can't the conception that there is more real numbers between 0 and 1 than natural numbers be derived just from the fact that the number of digits must be finite for naturals, while can be infinite for reals? The diagonal proof does not seem necessary at all.

Tom SvobodaPred 5 urami

1. it's a proof by contradiction. assume the reals can be listed => derive contradiction => the assumption that the reals can be listed was wrong (which we wanted to prove in the first place) 2. the diagonal argument fails for the natural numbers because it produces an infinite string of digits which is not a natural number. natural numbers have only finitely many digits.

Suzanne FreedmanPred 8 urami

I bet Hilbert was looking over at Gödel thinking like... “You really going to build your entire career around saying I’m wrong? That’s rude, what’d I ever do to you?”

Suzanne FreedmanPred 8 urami

And now I’m wondering if Hilbert DID do something to Gödel to spark that amount of determination 😂

Peter SteelePred 9 urami

The thing about mathematics is that numbers and the values they represent always get bigger & bigger much like ego... until they don't... almost as if self aware, they change their values.

JeremiasPred 10 urami

i feel like the barber reference isnt that accurate, if he cant shave him self and the barber cant shave him. he would be exiled or go to jail? not welcome in that"set". No?

Jack DanielsPred 10 urami

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Joaquín HinojosaPred 13 urami

My brain hurts

XeL NaGaPred 13 urami

well if you knew what they feed the cattle youd think youd get poissoned too lel worst monocrop used for cattle full of pesticide and gmo. basicly giving the cheapest worst most toxic chemical food possible for cattle. the steak gotta be cheap :D

PoliticsPred 14 urami

Glad there is a mathematical answer to why I can't figure out the truth about Covid!

Adam McKinziePred 14 urami

If the game of life can run the game of life, then it’s possible that the game of life could run the game of life running the game of life and continue to do so on an infinite scale

Sven CroonPred 14 urami

only a mathematical system that has no axioms, can ever be truly complete, consistent and decidable. Axioms are the evil that corrupts everything ;-)

Thomas KlughPred 14 urami

Now my head hurts.

Marshall115Pred 15 urami

give it a few decades-current math is wrong-it will change but still just be humans way of measurement.

neil ungerPred 16 urami

And here I read this as Meth Has A Fatal Flaw.

EliteTrollingGPred 16 urami

5:16. I just wanna ask, why don't mathematicians just agree on what number natural numbers and real numbers should stop at. Like the end of the number line is 10,000,000,000,000,000,000,000,000,000,000 starting today. And the end of real numbers is 0.999999999999999 until it goes to 1. I mean that would ruin all the work we have put into trying to find the end of pi, but my question is, WOULD it help solve problems mathematicians have been trying to solve for centuries, or would it just create more problems? I'm actually curious.

PiotrekPred 17 urami

Hi, i ve got some idea, about part about cantor set theory. If u try to groupe natural numbers and numbers between 0 and 1 like this: for 1, theres 0.1. For 27, theres 0.27. For 29010123, theres 0.29010123. Then every Natural number, will be grouped with exactly one number between 0 and 1, and also if u try doing something like in the video, u just take another natural number matching this one. Does it work? Please lemme know.

PiotrekPred 17 urami

@Brauggi the bold oh, u re right, thank u. I didnt considered irrational numbers. But still, i dont believe this😒, even if the proof showed by cantor is 100% true and logic, my mind cannot take this. :c.

Brauggi the boldPred 17 urami

It does not work. You will only ever hit rational numbers which have a decimal representation that ends in repeating zeros that way. Your mapping will for instance not include ANY irrational number between 0 and 1.

Sumit JuyalPred 18 urami

Wow, self-reference, what a beautiful link, this channel is the perfect blend between epistemological philosophy and mathematics, as someone who is a math graduate I would've loved to be introduced to subject histories like that before mindlessly trying to solve problems and pass the exam. People need to normalize and popularise the idea that mathematics applied philosophy which is just applied inherent logic. Your channel does wonders for us appreciators of the mind but are still rigorous in our understanding of the same.

Dylan HasePred 19 urami

I want to launch my own branch of math! How do you do that

HPred 20 urami

Maybe I don't understand it but wouldn't the diagonal number be on the list since the list goes to infinity and because at one point you will have to roll back 9 to an 8 making it equal to that number in the diagonal the minute you add +1.

Abdullah AhmedPred 20 urami

9:25 need explanation

Azio PrismPred 22 urami

Having 2 apples is impossible. Identical numbers are not possible. What makes numbers and apple differ is the environment around them :9

CroldfishPred 23 urami

1:25 i forgor :skull:

Heaven&HellPred 23 urami

none of this can be proven to be true.. concurrent quantum states prevent this.

KLEISPred dnevom

Math can prove a lot of things but it can't prove life, it is coz of math is itself incomplete by knowledge of human has created, completeness or creation of all things that has life is fundamental things that it is not related by math. So math is not absolute and it is just one of many knowledge to prove something by what creation has created

TheBeastPred dnevom

just learned set theory for my computer science degree really interesting stuff

Bijou SmithPred dnevom

@28:20 so, the undecidability of the spectral gap property amounts to the first proof quantum physics, as we conceive it presently, does not admit reductionism? Have I got that right? That's pretty signifcant for philosophy of science, which has previously generally operated under the paradigm that science (whatever it is) is reducible _in principle_ to base physics, through obviously not always in practice. So either that's a false paradigm or quantum physics is not base.

ayy lmaoPred dnevom

Russell's paradox is a violation of the law of excluded middle. Ergo the law of excluded middle is wrong. Ergo superposition.

Kim Tae HwanPred dnevom

The voice is good though I'd never understand the contents...

AfqwaPred dnevom

This really murders the idea that math is some kind of divine tongue bestowed upon us by the gods. Murders it in its crib by smothering it with a pillow. Why does math work . . . _uhhhh sometimes it has empirically useful results._

Giap ChinPred dnevom

"This is the game of life, running on the game of life." Then proceeds to slow zoom out. Wow my mind literally was blown.

Andrew C. MummPred dnevom

Enter quantum entanglement... a proof can be true and false at the same time until observed. And when observed, the universe splits into different realities... :D

AscotPred dnevom

Mind Blown

Andrew C. MummPred dnevom

The game of life animations at the start are awesome - does anyone know how they were made?

Gerard LigondePred dnevom

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enthusiasticGeekPred dnevom

25:57 and it vanishes in a puff of logic

hieu dangPred dnevom

somehow i heard "godel" as "good old" for almost the entire video

Babe RootPred dnevom

The language of God...the Creator/Designer, and mind of ALL information. The language of God...Mathematics. Amazing...isn't it? ☀️

AndrePred dnevom

_"The language of God...the Creator/Designer,"_ There is no god and no creator.

ayy lmaoPred dnevom

More like word salad

flobbiePred dnevom

Yeah, but it is neither a flaw, nor is it fatal.

AndrePred 18 urami

@flobbie _"It just is that way. This does not make math any less useful to me."_ If there is even one contradiction it is. But we have never found any. It is just that there is no proof.

flobbiePred 19 urami

@Andre, i don't see that this makes it flawed. It just is that way. This does not make math any less useful to me.

AndrePred 22 urami

@flobbie _" i don't understand. There are no flaws in math."_ You can't prove that a formal system is free of contradictions inside that system itself (for those systems that fullfill all conditions of Gödels theorems). That is a problem, because if there is a contradiction (even a contradiction we do not know today) then you can prove anything. Math is useless then. You maybe can prove this in some other system, but then how do you know that this system is free of contradictions? _"Your axiomatic system may be flawed, as it is contradictory. "_ And for many, many of those systems you cannot prove that it's not. _". But there is no flaw in the general way of how somone is supposed to process language."_ A "language" that contains a proof for A and not A is not really useful. _"Do some math and you will see there is no flaw."_ You can't prove this by examples. Do you even know what math is?

flobbiePred 22 urami

@Andre, i don't understand. There are no flaws in math. What is that even supposed to mean. Your axiomatic system may be flawed, as it is contradictory. Or your proof may be flawed as it contains errors. But there is no flaw in the general way of how somone is supposed to process language. Do some math and you will see there is no flaw.

AndrePred dnevom

Not being able to prove consistency is something I would call a "flaw".

Robin CollinsPred dnevom

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darth biernotPred dnevom

My brain nearly exploded while watching this! People always say: "Math is logic" I personally am more comfortable with language. I speak German natively, English and French fluently but when it comes to a simple mathematic equation with a variable in it my brain goes: "system just crashed due to missing math.dll" 😆

MrBizaaroPred dnevom

Excellent Video ! Feels like one of your best

Jeffrey WiegleyPred dnevom

Holy crud!!!! I *finally* understand the diagonal proof after 30 years... Prof. Leonard Adelman (The 'A' in RSA) used in Gödel's Incompleteness theorem in Second Order Logic class and I blew that on the final exam. Oh... I can die happy now.

Matthew MeansPred dnevom

Alan Turing a story is my favorite depiction of humanity. A single human was enough to determine the difference between a world with and without Nazi Germany thru his contributions which would have otherwise plummeted us into a darker world with less insigh, yet in response to his glorious contribution to humanity, he was treated as a plague for being gay and made the other on premise of his differences that hurt no one. Humanity will respond to it's very saving with its own doom.

why breadPred dnevom

This video and his “how a infinite hotel ran out of rooms” video match up I just thought about it

Peter ShmainPred dnevom

Cantor's diagnolization proof is incorrect because when you use that method to think of a new number, it must then also be assigned to a new index which is just 1 more than the previous number therefore disproving infinite inequality. However, that's not to say that his ideas are incorrect. It's just that this proof doesn't completely work in this way

AndrePred 4 urami

@Peter Shmain _"given that the list is static, then you should not be able to create a new number from the decimal side either, "_ No. Cantor has shown that you can do this. That is the whole idea behind it. _"thus defeating the point if being static,"_ No. _"because then logic stands to reason "_ What "logic"? _"So if you can take a decimal and add 1 to each digit, why can't you take the index number and add 1 to it?"_ All indexes are taken. The list is static.

Tom SvobodaPred 5 urami

@Peter Shmain the list is not an equation, you're not "doing" anything to it. it's an assignment of a decimal to every natural number (which can be pictured as an infinite list indexed by the natural numbers). for any natural number n there's a given decimal d_n. one such assignment would be d_n = 1/n, which in the "infinite list" representation looks like d_1 = 1/1 = 1.0000... d_2 = 1/2 = 0.5000... d_3 = 1/3 = 0.3333... ... this is just an example. clearly a lot of decimals are missed, for example 2/3 never appears. the diagonal argument proves that we can find a missing number for any such list, therefore it's impossible to come up with a formula for d_n which would account for all decimals.

Peter ShmainPred 5 urami

@Tom Svoboda given that the list is static, then you should not be able to create a new number from the decimal side either, thus defeating the point if being static, because then logic stands to reason that what you do to one side of an equation, you do to the other side, correct? So if you can take a decimal and add 1 to each digit, why can't you take the index number and add 1 to it? Maybe im missing something here?

Tom SvobodaPred 5 urami

there is no new index. the list is not being generated, it's static and completed from the moment you consider it. all the indices (the natural numbers) are already taken.

AndrePred dnevom

_"Cantor's diagnolization proof is incorrect"_ No. _"when you use that method to think of a new number, it must then also be assigned to a new index"_ No. _"t 1 more than the previous number "_ What "previous number"? _"However, that's not to say that his ideas are incorrect."_ The idea and the proof is correct. _" It's just that this proof doesn't completely work in this way"_ It does.

Michael BorisowPred dnevom

I don't know how many of all 8 million of yall are understanding this, but I'm gonna have to pause and look up stuff from this vid another ∞times before I understand anything

Aymane SghiarPred dnevom

Best video I've watched in a looooooooong time.

Anaya BarataPred dnevom

So is the fact that not all true things can be proven also unprovable?

AndrePred dnevom

It is the opposite.

Robin HodsonPred dnevom

But this depends upon the assumption that recursion and logic contradictions disprove systems. That's not necessarily universal, otherwise we wouldn't be capable of comprehending them. Saying "This is incomprehensible," an apparent paradox, is actually comprehensible, and thus not not invalid.

costaranPred dnevom

MEGALIKE 👍

James CoreyPred dnevom

Thank you for this waste of time.

Michael MAnvillePred dnevom

More like insufficient computational ability to prove true, but can never be prove false.

Релёкс84Pred dnevom

Absolutely nothing to do with "computational ability"

Boysfifa010Pred dnevom

What does it say about me that i read the title as "Meth has a fatal flaw" ??

Furqan SiddiquiPred dnevom

That's the reason I love watching this channel. It forces me to "THINK"

James WhitePred dnevom

I don't know if there is truth to be found studying mathematics, but there is much beauty to behold.

Jacob AlexanderPred dnevom

Godel hurts my damn head. How would you come up with that.

Prototype 81Pred dnevom

To say things like "always" or "we will never know" is a fallacy. This is similar to clickbait. Nobody can day for certain what can be or cant be possible in the future. That is a simple fundamental of life. So... why is this guy using terms that are incorrect?... What else is he invorrect about. Why trust this guy on anything when hes obviuosly romancing the structure.. Lame.

Mr BluePred dnevom

1 + 2 = 4

János TÓTHPred 2 dnevi

Lobachevsky and Bolyai, Gauss is at most the third.

kamahll goodarzPred 2 dnevi

Perhaps maths really is the language of reality in so much as they are both paradoxical

Solar PlexusPred 2 dnevi

I knew it.. Remember "Computer Code Discovered In Superstring Equations" ?

Galina ZwerleinPred 2 dnevi

We don't know what we don't know.

Reilly 25Pred 2 dnevi

Can you not? Math is hard already stop giving them ideas on making it harder I'm still studying just do it after I graduate ty

infinite definePred 2 dnevi

I don't have to know any of this. I'm richer than yall could ever be. But I only know arithmetic. Chile imma count my money now. Also I don't actually have physical money anymore

Adam KeeleyPred 2 dnevi

There is no preferable side of the equal sign.

GleichtrittPred 2 dnevi

Well now you have it, I am suddenly interested in math....

Fat AlPred 2 dnevi

Hrhr he said "googleplex" like in google... SMART!

PieterPred 2 dnevi

At 17:17, where do the prime numbers come from? 2, 3 and 5. Is it just because those are the first three prime numbers, and the equation holds three elements, or is it something else?

Spider-Jonah- ManPred 2 dnevi

It’s boring

Noriaki KakyoinPred 2 dnevi

Somehow I understood what was going on and it facinated me even though I could never explain it myself

Jesse ThomasPred 2 dnevi

Trying to use math to prove math is like trying to use light to prove light. As you would say, this is a self reference. The light needs the dark to be proven. Math is part ‘logos,’ the logical foundation of all that exists. It’s part of the source of all things.

Sam RePred 2 dnevi

Godel, Escher, Bach

Frank HarcourtPred 2 dnevi

This guy is a total moron and he has no idea why. Everything he says is crap. The universe does not work the way he says it does.

James SonkePred 2 dnevi

The Barber Paradox is not a paradox. Just because something is illegal doesn't mean it's impossible to do. Just because it's illegal for the Barber to shave someone who shaves themselves doesn't change the fact that he is probably going to shave himself It's just an imperfect law. And just because each set supposedly cannot contain itself doesn't change the fact that R (itself) was made up. It exists as an idea. And ideas are infinite

Zora MarslinkPred 2 dnevi

Infinites sort of have different sizes though, so it's not exactly obvious.

Grilled FlatbreadPred 2 dnevi

This is really similar to not knowing whether you are dreaming, alive, or a simulacrum

John WoodallPred 2 dnevi

Math is God. -True, yet unprovable -Consistent as far as we know -Unable to be fully understood

Hedgehog3342Pred 2 dnevi

Math is honestly quite confusing at times. Definitely not my strong area.

Adobe ReviewsPred 2 dnevi

The sad salmon ipsilaterally untidy because yugoslavian genotypically switch underneath a untidy edward. boring, versed milkshake

Philip BerthiaumePred 2 dnevi

I'll stick with 2 + 2 for now, thx....

Jesus ShuttlesworthPred 2 dnevi

Godel grief

Teflon musk Pred 2 dnevi

What a good ole number

KZisNBKosplayPred 2 dnevi

Math is just a Riddle.

Seth Solomon [Student]Pred 2 dnevi

My life is a lie

Tea DrinkerPred 2 dnevi

The set that contains everything can't contain nothing