Math's Fundamental Flaw
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- čas přidán 21. 05. 2021
- 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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Executive Producer: Derek Muller
Writers: Adam Becker, Jonny Hyman, Derek Muller
Animators: Fabio Albertelli, Jakub Misiek, Ivy Tello, Jonny Hyman
SFX & Music: Jonny Hyman
Camerapeople: Derek Muller, Raquel Nuno
Editors: Derek Muller
Producers: Petr Lebedev, Emily Zhang
Additional video supplied by Getty Images
Thumbnail by Geoff Barrett
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Ironic that Godel's death was the result of a self-referential paradox: he died in order to not die
This comment deserves more likes
Underrated
Woah...
You nailed this comment
This comment is just too good
So basically...
Can math prove itself?
No.
But math can prove that math can't prove itself.
hahahahha good one
well... you can't prove the rule using a rule because the rule is universal and immutable
Yesn’t
"math can't prove itself" to the power of -1
I was asking myself the exact same question
I love how the set theorists answer to self reference was "I changed the definition so that doesnt count."
It's like a kid on a playground saying they weren't playing when someone else tags them.
@dominicbonogofski i dont feel like thats a valid analogy, theres nothing wrong with going youre right this is a flaw and trying to adjust the rules to fix it. Maybe its just the problem with analogies is that they can also be unproveable though so its also a contradiction based on perspective 😯
@@EonsEternity I was just implying that it had the same energy behind it.
this brings up a question: what if the turing machine's answer to haltability was to simply make a new rule: the turing machine cannot accept itself as an input. that would remove the proof against haltability. so does that mean mathematics could be decidable as long as it doesn't self-reference? or does this prove that set theorists were in denial? if neither, then what makes set theory different from mathematics in that in can exclude self-reference and still be useful, while mathematics/turing machines cannot?
@@smarchar@dragonsaige I had that thought as well, but then that would eliminate self-reference, which is very useful in answering a lot of questions correctly. At least, that's what my logic led to. I'm just a software engineer with a passion for maths. I could be entirely wrong.
Everytime people get into the weeds with math like this i feel like im just listening to philosophy with a different label.
Philosopher ask a question,Phisicists Turn questions into math
Indeed. Just remember that numbers aren’t real. I mean that in the sense that they are always tied to an object or idea. You can’t go out and find a 7 in nature, you especially can’t find a negative seven.
thats because they are philosophers, They are natural philosophers.
PhD student here. Math is applied philosophy. You cannot have one without the other.
Exactly, the foundation of mathematical proofs came from the Greek philosophers.
Godel's friends: "No one's trying to kill you Godel"
Godel: "You can't prove that!"
He actually refused to eat any food not prepared by his wife. Unfortunately she was hospitalized, and couldn't prepare food for him, causing him to starve to death.
@@nbjornestol he couldn't prepare his own food?
@@lavabeard5939 He was a mathematician (logician) after all.
@@segmentsAndCurves
Does that excuse a man from being able to provide for... himself?
@@kindlin It doesn’t excuse, but it explains why he didn’t prepare his own food.
If you're a mathematician and you are labelled a "corrupter of the youth", you are doing something very right.
nerd burns
@Linus Fu Yet neither can prove nor unprove logical paradoxes. The same way no one figured out why we can pin point an electron's vector and position separately at the expense of the other, and never both.
I watched this video when its title was still "There's a Hole at the Bottom of Math".
@@Aereto wait whaaaaaat
Exactly. If one proposes a theory or statement that pushes all of our minds to think hard enough, regardless if it's wrong or not, overall it's something right.
Why didn't they just have three people stand beside John Conway after he died?
This is the best math joke I’ve heard in a month
Oohhhhhhhhhh
Give me the reference please 😂
@@1stlullaby484 In Conway's Game of Life, 3 living cells around a dead cell make the dead cell alive again so, the joke is that they could have resurrected Conway by having the people (cells) surround him (the dead cell).
Wouldn't work. They'd have to stand there for a whole generation and everyone's bound to use the restroom at some point 😅
So this is how these things are connected to each other. In my CS degree we had to study about almost every one of these topics (at least a very little of every topic) and they seemed very disconnected and apart of each other. Discrete mathematics, Automata, Set theory, proofs... etc. This video connects dots. An actual tear fell from my eye at the end of video. Thanks for making these amazing videos.
Me too brother, me too....
Proofs are horrible, man
😢
Fr its so awesome seeing everything ive learned over the years recontextualized into a cohesive story of cause and effect, i wish my professors told me about this connection before 😂
Weil fell off not tear
"How about you just hire another barber?" Said the engineer
And you only need two barbers to break the paradox. They can shave each other; the rules never said that wasn't allowed.
Engineering student here, my first thought as well
2 barbers 1 town
@@jeffirwin7862 IYKYK
@@theknightwhosayn1 only the barber can shave anyone, that was one of the rules
I don't know why but I love the idea of mathematicians gathered in a room yelling and hurling insults at one another
"You are proof that one can actuality have a value of zero!"
@@viacheslav7870 lmao
@@viacheslav7870 I'd rather listen to the first 10,000 digits of Pi than some irrational numble like you
*crowd commotion intensifies*
Hello! How are you all? If anyone needs someone to listen, someone to talk to, or a friend. I am here to talk, listen, and be a friend. I hope you all are safe and well. Know that you are amazing and have rights as a human. I am very sorry for anything that seems bad that may have happened in your life. I want you to know that you are incredible and are capable of wonders. What matters is your inside, not your exterior. Love yourself and cherish yourself. Words cannot explain how astonishing you are. You deserve care, love, and happiness, don't let anything make you feel otherwise. Please have appropriate action for anything that you know is wrong. Anything that seems bad or wrong in your life right now will get better. Please don't do what is wrong, fighting back and harming others will not solve the problem. Please understand that and do the good thing. It will one day come back to you. The people in the world are so much more than what we know about them, not everyone opens up about the beautiful things and acts they have witnessed, not all those amazing doings are acknowledged. Please understand that and know that. If you feel like no one cares about you, know that I care about you. Keep your head up high and never give up! Together, we can be a better community! Stay safe, healthy, happy, kind, understanding, positive and strong!
"You are more irrational than any number I've ever seen!"
I love how tightly intertwined mathematics and philosophy are
Philosophy is everything. Mathematics is based on logic, a branch of philosophy of "reasoning", which forms the groundwork for the formal science which is in this case mathematics.
There are other formal sciences like Computer Science, Statistics etc. Sometimes logic is also included in the list of formal sciences.
Truly one of the greatest mathematics-related video out there on CZcams. I often find myself returning back to this video, and thanks to you, I was inspired to major in engineering. I started loving math; it's such a great language!
It really is and highlights how amazing math is and how useful it is despite its limitations. I think most people would be surprised that our whole modern system of science is built on these grounds, but it works amazingly
Not to disway you, but if you love math, go to math! Not engineering. As much as math is involved in engineering in a lot of ways, in practice it does not.
mom: why did you get a B in math!
me: math has a fatal flaw
B is good
@@cohensmith6100 and A is excellent.
@@ALBINO1D ya but like why get mad abt a b when most mfs fail math
@@cohensmith6100 is your benchmark just to be better than worst, or to be the best?
Learn a lesson from Ash Ketchum.
@@ALBINO1D Hes like over 20 and hangs with 12 yrs old girls ill pass man
Mathematicians: we must prove this equation
Engineers: Eh, it's good enough, we'll just use it
bridge collapses
@@mattstokes3881 and they learn from their mistakes, and makes better bridges
I feel seen
@@cipherxen2 No no no as a civil engineer student u have to prove some math equations to make sure the measurements are right.
So idk what tf are u talking about
Mathematicians: "We must prove this equation is true in all possible scenarios across all possible universes."
Engineers: "Bro, do you even constraints? I only need the equation to be true _on Earth for the next 50 years."_
I learn about the Veritasium by watching this video about 2 years ago, and it turned out to be the most valuable 30 minutes that I spent on the internet. Maths became almost religious to me after watching it, for it is capable of proving its own limitations within its limited system, although my father kinda disagrees with me on the religious part as a professional in maths. But there's a reason for me. It actually reminds me of the philosophical question of "what we are" and "what we are made for" since they are also related to the self-reference paradox.
I have been suffering from depression and anxiety at the time when I first saw this video. I was about to graduate but had no idea of what I am going to do nor what is the meaning of my life. But this video somehow saved me. For no reason, I suddenly feel relaxed after learning about Godel number and the answer and proof to the three questions. I realized that, just like maths, life is not about meaningful or desidablility either, but we may find what we have done meaningful years later. And this is a proven truth. Just like one of the most famous Chinese poems said, "everyone was made with some talent that must be useful".
Anyway. I finally found my own belief after watching this video. And now, after 2 years, I am back to the college for postgraduate degree and for working out my own value of life. Many thanks to Veritasium for the great work. Wish everyone a great life.
Wow! I’m sincerely happy to see that you’ve found your path! Keep it up! …And don’t worry; no religion makes any sense.. if you found something to believe in, charge on!
I've been wondering how numbers relate to life. I'm approaching it from the perspective of psychology, where we often use null hypothesis significance testing and the law of big numbers. We take a group of people, do an experiment, and we check the average.
But our phenomenon of interest is the (average) individual, which is different from the group average! Group-to-individual generalizability cannot be taken for granted. The difficulty lies with multiple realizability: -2 and +2 are the same as 0+0. Except for the standard deviation of course, I guess.
But clearly, taking averages obfuscates things. Why? Because counting things reduces information. When you say a pair of shoes, or even two shoes, you equivocate two non-identical things. And it is the same for people with depression, who may not even have a single symptom in common with each other. One study which checked "depression profiles" of ~3,000 people found that the most commonly occurring profile occurred 1.4% of the time amongst the ~1,000 different depression profiles identified. Yet, if a study is done on depression, imagine how difficult it would be to test a psychotherapy or psychoactive drugs without being able to refer to depression as a single concept
The question is whether mathematics applies to the real world. The answer is obviously still: yes, extremely applicable. Numbers allow us to see patterns in the world, which is an utmost necessary condition for intelligence to work. The way the universe and its objects worked in the past is, at the very least, a really great analogy for the future
But might it be possible to describe accurate and precise truths about the world with numbers? The capacity to abstract is fundamental for us to make to be capable to think and talk about the world -- to talk about "depression", without having to mention all the specific cases in mind which represent that concept. Yet, if the utmost of specificity is desired, would that be possible? It would be interesting to see if quantification, or counting, is valid in its strictest sense. Is there any phenomenon in the world which is identical with another (as opposed to merely equivalent), and of which we would therefore not lose any information if counted? Or might every single thing, in its strictest sense, be different from one another?
This is actually the MOST interesting video I have EVER seen. Every minute was amazingly clear and intriguing.
Teacher: Your math is flawed.
Student: No, math itself is flawed.
dank meme
lmfao
I’m gonna go to my math teacher and be like “math is incomplete and inconsistent,” and she’s gonna say no it is and then I will now more about math than her and I will be so happy
@@Scipio_Africanuss ahahaha bro let me know what she says 😂
@@getonthecrossanddontlookba5004 I assure you math and time are constructs of man, not God.
"1+1=2"
"The above proposition is occasionally useful."
What's 3x+1?
Or y3X+1 it is impossible to get an answer it's like pi
@@Jayess-c lol dude they literally made a vid about that, it’s that where you got it from
@@kam9910 what are you referring to?
@@Jayess-c if you were trying to pose it as your own equation you made up, I’m not sure rlly, I’m just 11 lol
Sorry, I'm a little late! 😂 The part near the end, where the game of life was running in the game of life? Wow. That was beautiful. It gave me chills. I'm severely deficient in math, at about a 6th grade level (6th grade in the 80s, not the madness they teach now, I'm so lost 😂) but the abstract concepts, the beauty and art has always fascinated me. Thank you for this video, and all the others on this channel. The way you explain it makes it understandable even to a math dummy like me. ❤
There's a program called 'Golly' that includes a bunch of pre-built simulations like that. Worth checking out, and it runs very fast on even slow hardware!
I've known these topics for years and also watched many videos, I have to say this one is so beautifully done, all those thoughtful illustrations
I love you
Ah yes, the iconic half way point of the video where I stop comprehending a single thing said
that feeling
Read Douglas Hoffstadter and comprehend even less. In an entertaining way ;)
If we had had videos like this in high school, I wouldn't have come out of math class convinced that 2 + 2 = CAT . . .
It is a proof which proves that not everything that is true can be proven after all
¯\_(ツ)_/¯
it's zone out time
Me: *failing my math class*
Veritasium: “they could be something like the twin prime conjecture”
Me: go on...
Lmfao
tbh the conjecture itself is pretty elementary to understand.
@@tejasdeepsingh456 ditto
@@wildanimus2559 Charizard
@@SoumilSahu what is gobbledygook? - In particle physics, a lepton is an elementary particle of half-integer spin (spin 1⁄2) that does not undergo strong interactions. Two main classes of leptons exist: charged leptons (also known as the electron-like leptons or muons), and neutral leptons (better known as neutrinos). (fûr′mē-ŏn′, fĕr′-) Any of a class of particles having a spin that is half an odd integer and obeying the exclusion principle, by which no more than one identical particle may occupy the same quantum state.
I truly believe that this was the most spectular video I've ever seen on this channel! Congratulations to you and your team, Derek!
Oh, well done sir. Your closing line here very nearly sent a chill up my spine. Thank you for another well-spent half hour.
"There will always be true statements that cannot be proven." Oh yeah? Prove it.
....He proved it.
Brains!
Plp are to smart
you mean like Epstein not killing himself
Proving something is impossible is also a proof
Dis gave my brain a new wrinkle
I honestly and wholeheartedly believe this is the best video this platform had ever seen. I share it often with my students.
Derek, Amazing work and Thanks.
This is top 10 videos of all time. Literally the best explanation ever of one of the most interesting scientific (and existential) concepts.
7:49 - 'corrupter of the youth' haha
"Hey kids come here, you want to learn about some illicit infinities"
wanna learn how to divide by zero?
Noooooooo
lmaoo
illicit infinities are creations of the universe, just like ourselves.
Socrates back from the dead
I'm a PhD in computer science. This is a full-on Discrete Mathematics intro course. This is amazing.
I never saw much of this in DiMa... most of this I picked up somewhere along the line and often in the actual CS introductory courses or while trying to understand more basic concepts using YT. Only to be distracted by that one video on the side called "The halting problem" or some such and getting curious. :D
Right on! A semester of DM in one video.
@@Kirmeins Yeah, thing is that DM is so vast that it is really easy to set up a course that doesn't touch on any of this material. The DM course I took was like this... introduction to game theory, a little combinatorics and cryptography, coin weighing problems, stuff like that. But I think the important thing is the ability to get students interested in the material, and then they go looking for other courses that cover it.
I agree, this is also the key for appreciating the role of AI/ML theory. And randomised algorithms.
I had this in my theoretical CS module more than the discrete maths one and while I hated the exams and the assignments, I thoroughly enjoyed getting my mind blown by such a profound topic. I've never thought that we actually would go into deeply philosophical questions about the fundamentals of logical systems, truths and math itself while studying computer science. And how it all connects to computers in the end. Brilliant video, it creates this amazing feeling of profound enlightenment I had when I first encountered this topic and I hope it reaches as many people and blow people's minds just like it had mine.
This channel teaches the basics so easily. When explaining something such as complex numbers, they go into the most basic foundations, akin to explaining an organism from the level of quarks and gluons as opposed to the conventional educational system which just tells properties outright. Brilliant chose an awesome channel to sponsor
Wow, this is great content, Veritasium! Sometimes your videos just transcend the brilliant educational films they always are, and become pure art.
Gödel was also first to ask P vs NP question and he asked it in the letter to John von Neuman.
Those dudes had some world changing conversations.
Was waiting for P = NP after The Halting Problem. Maybe next time.
Nice
Veritasium needs a video on P vs NP! Would be amazing.
meanwhile me to my friend: Do you think dogs know theyre adorable?
@@DavidLiMusic yeah because there isn’t enough n/np out there
“1+1=2
The above proposition is occasionally useful “
I need this on a poster for my classroom 😂😂😂
😂
“1+1=2
The above proposition is occasionally useful “ It's also racist. smh
So trueee
@@sdgathman "I proudly and loudly misunderstand things"
@@ccgarciab sounds like you weren't aware that math and logic are constructs of whiteness which inherently oppress people of color
This is the only CZcams subscription channel you need... Can't get enough. Thank you so much.
Thank you so much for you video. The soft and friendly presentation helped me to hang in there until the end. You articulated a lot of what I suspected-that at the heart of the way math is practiced is the flawed belief that if we just work it out until the end, we'll have a pretty, perfect answer. I think it's why I feel so hostile overall to Newton and his systems, despite him being undeniably brilliant. I never thought I'd say this EVER, but your video makes me want to learn more and maybe even DO and practice some math myself! Thanks again!
As a mathematician I haven't seen a more elegent presentation of these concepts,especially Godel's theorem. Amazing job thank you.
I just don't understand where equation g came from. Why would it have been a contradiction to prove g, just because it said "this can't be proven"? If one had proven it anyways, Gödel's statement would have been wrong, yes,but what of it? Why did he write "this can't be proven"? Purposefully trying to MAKE a paradox by setting contradicting rules and then saying "See? Major problem, math incomplete." doesn't make any sense to me. If things naturally contradict, isn't it the axiom's fault? Shouldn't we just rethink the basics?
@@WritersMoment well if he didnt do that contradiction then we wouldnt know the completeness of math
@@WritersMoment I didn't watch the video, so I don't know how they explained it, quite possibly very incorrect. However the point of the 2nd Gödel incompleteness theorem is if your axioms fulfill a bunch of desirable attributes (such as being able to prove all true statements about the natural numbers), then you can encode its own consistency. Those are known as Gödel sentences. As the axiom system can not prove that, it's therefore not complete if it's consistent. It's possible for an axiom system to not have arithmetic, but be complete and consistent, have arithmetic, be complete but not consistent or be consistent, have arithmetic but not be complete.
So it's not possible to rethink the basics to get all desirable quantities. Math is not flawed tho, since having arithmetics and a consistent axiom system is possible and absolutely sufficient for everything that mathematicians do.
@@henningbreede6428 Wait, do you always comment in comment sections of videos you haven't actually seen?
@@WritersMoment No, this is the sole exception. I clicked on the youtube video because it was recommended and after reading the comments I'm not very motivated to watch it either. It doesn't seem to do a good job at addressing common misconceptions.
OVER HALF AN HOUR OF CONTENT, youtube> TV any day
and here I am, watching this on TV ;)
WOAH WOAH WOAH!!! Let me get this perfectly straight: You comment something that is completely unrelated to the fact that I have two HAZARDOUSLY HOT girlfriends? Considering that I am the unprettiest CZcamsr worldwide, it is really incredible. Yet you did not mention it at all. I am VERY disappointed, dear te
@@AxxLAfriku Are you OK?
@@AxxLAfriku what
@AxxL Some types of madness are beyond the limit of infinity
This is a fantastic video and the best explanation of Godel's incompleteness theorem. 👍 👍 for showing how it relates to the halting problem, too.
I wish I had access to your videos back when I was in college trying to study engineering. I find them truly inspirational.
I admire the fact that you take concepts and bring it to life. Taking us on an adventurous journey making it more fun. You really brought my intrust in science back I had took a break but coming back here after a year feels great. Great work! Nice explanation on how math is incomplete and inconsistent and how a turing machine program with binary code makes it different.
"19th century mathematicians HATE this one weird trick!"
Haha when will those ads stop being a thing? Gödel would have known
Funny thing is Henry Pointcare seems to be a formalist at heart, as he claimed “later generations would have recovered from the disease” - meaning maths is Complete, Consistent and Decidable.. since he was sure that there would be a system that could with certainty disprove Canter 😏🤷♂️
Oh, look. A meme.
You won't believe what Kurt Gödel looks like at age 115!
Comment of the year.
Seeing the game of life running inside the game of life gave me goosebumps. Had to pause for a minute to digest that. Just beautiful!
Where?
Just like the human dimension...
@@RAMBO14001 It's simulations all the way down ....
So wait... if the camera kept zooming out on the game, it would continuously be simulating itself?
Same feeling 🤩
i literally clapped in a room I was alone..For the great men on whose shoulders we have build the modern scientific and technological world... Kudos to those scientific minds mentioned in this video...
Man you are overreacting.
You clapped, but while you were alone?... Does that mean you just clapped ursel-
What happened to Turing was a tragedy, then and now.
While i am an American i love Military History, specifically WWII. So i am familiar with Turing and the work he did during the War. (Not so much his other work which is fascinating) It has always bothered me how he specifically was treated after the war and it was good to see him get his Roaly Pardon in 2013.
The moment he showed the game of life running inside the game of life, I was totally blown away. Such a mind bending topic to contemplate.
I felt like i was going to start crying!
What game? Can you mention time
@@pushparahi5681 around 30:00
I wrote an implementation of Game of Life as an A level project on a Commodore PET. I had to use machine code as BASIC was too slow. I got a bad grade compared to others in the class who wrote simple stock entry systems, as the teacher didn't understand what I was trying to do.
@@albanana683 That sounds great! If only this video was available back then, then the teacher would have definitely given you the best grade. The game of life is awesome.
"Later generations will regard set theory a disease", "No one shall expel us from the paradise that Cantor has created"
Those dudes felt *really* strong about abstract maths back then.
It did remember 'God don't play dices' from Einstein.
you might want to read mathematicians debates nowadays... nothing has changed
Later generations are just making tiktok videos.
it's not at all surprising that they had strong feelings. they were literally debating how reality works. not just physical reality, but abstract reality too.
Pythagoras beat them at their game though
This is so beautiful!
Thanks for being one of the people that helped me truly discover mathematics.
I grew up hating math, but thanks to mathematicians, physicist, computer scientists and programmers here on youtube i have grown to really love and appreciate the subject.
great work. definetly have to rewatch this more than once to understand, but it's a great video.
Seeing that "game of life" running inside "game of life" gave me goosebumps .... inception seems like child's play infront of it.
The dislikes to this video are from people who are watching it sitting/standing upside down.
Agreed. It's like watching videos comparing the scale of astronomical objects.
It's like watching videos of Minecraft made inside Minecraft.
Which several people have done, apparently.
And I thought, well if Windows exists inside Windows due to virtualization, and you could even run deeper layers, than it doesn't surprise me, that math's followed the same logic... A paradox that is working, by self referencing itself...Which gave birth to computers...
I didn't get that bit, I thought the game of life was essentially a set of rules, so what does that mean to see those rules running on those rules?
I physically exclaimed "OH DEAR GOD" and my wife heard me from the other room and yelled "oh no, what's wrong??"
It's okay, she knew what I was watching and I just shouted back "MORE MATH" and she knew what was up.
I'm 75, female; I am grateful that I have had enough education to have at least heard of the people you reference. Awed that you explained it all so well that I could not stop listening. Lastly, so proud to have lived this era from beginning to undecidable end.
I get my education from youtune videos:)
@@carealoo744 Self education is better than forced education!
Have a good day!
@@kebekbutcher well said
So awesome to have people of all ages getting so much from these videos. I’m 38 and make, and have watched Ve videos for what feels like a decade.
I hope you live long and healthy 🙏❤️👍
To me this is one of the most influential videos I've ever seen on CZcams. I think this video should be a prerequisite for children to watch in education. Why does this discovery not disturb more people?!
The quality of this documentary is astonishing. I wish I had access to such materials as a kid, actually I felt like a kid again for over 30 minutes
For me, the biggest takeaway of the whole thing is this: how amazingly smart must Gödel have been to come up with that proof? Obviously, every other Mathematician mentioned here is also incredibly, incomprehensibly smart, but with the other mentioned proofs, I can kind of reconstruct how one might have arrived there. But with the incompleteness theorem, I just cannot fathom how one might come up with it. The guy must have been able to just straight see the matrix.
He was only 25
@Markman Dave Thanks a lot bud I have been in an argument with my brother and we clearly defend different ideas, I should probably do the research on it my self. Though it’s not completely about math only partly. Where arguing about videogame strategy, where when it comes to math we usually agree, but if it comes down to different strategies were clearly on different start ups we are clearly on different opinions.
It's cods wallop. Has anyone commenting here ever studied probability? In all scientific claims you must provide the figures to back your claim. That is not done here.
@@ValMartinIreland Are you referring to the video? Or the comment? Or someone’s reply to the comment?
@Markman Dave While it's true that groups limit one's freedom, they also expand it. If there are no other people one would be one's own input and output. This would mean we'd never get other information besides the ones already existing in one's individual system. Thus we'd be systems of stagnation. On the contrary, the more people we listen to, the more information we can get. Especially if the others have a different point of view. Thus we have a lot of contradictory Information we can work with. Or in other words: "We have an abstract horizon". With this we not only have the chance to solve the contradiction, but also a synthesis.
Whether a group is beneficial or obstructing for an individual, is based on the structure of the group and the level of self-confidence of the individual. Being self-confident means to stand your ground, but also being able to reflect on the critique. Only then you can find the most differentiated solution for your time. 0nly then you can build up on the horizon of your critics and convince them.
I have to admit, seeing 'the game of life' running 'the game of life' was impressive. That's mind blowing.
Yeah, out of the whole video that part blew my mind more than anything else.
Wait. If the game of life can run its self, then the game of life will run its self that will run itself that will run its self...
(edit)
...and so on.
@@uttie3408 I actually think it would be worth the effort to build one more iteration on top of the two. Perhaps I'm being unreasonable.
@@uttie3408 I dont get it is the game of life something that can run itself infinitely. It's just confusing tbh.
@@HassanAhmed-rf9xr you can write a computer program that simulates every computer component (that is what is called emulation), and you can make this emulated computer run windows with the same program running in it. this is the same thing: every next level of emulation requires large amount of setup, and takes a very long time to execute.
but a turning complete system is not difficult to simulate: all you truly need is a way to do if-then and store a state, everything else (operating systems, games, hardware drivers, is just built on top of having a set of instructions in the memory modifying the memory and choosing between 2 option based on the memory)
Amazing video. I'm very impressed about the quality of this content; through and deep, yet approachable. Congratulations.
Tone is sick dude - keep it up 👍🏻
Mathematicians: “I used the math to destroy the math”
Math is not destroyed. It is a science. It improves itself.
@@hisxmark i do fear the day human is no longer able to wrap our brain around maths, that we might hit a "wall", if we have not already had.
@@caber1487 that is not how it works
0 is like infinity and infinity and is like 0?
@@hisxmark race and IQ proves many things regardless of what you want to believe lots of anti science ppl here. odd.
I got literal chills at the "It's the Game of Life... running on the Game of Life" part
Like our own life.
Like a matrix
But can it run Doom?
the game of life can be run on a game of life inside a game of life tho
@@Legobuild123 xkcd 505
When you put it like incompleteness it sounds negative, but I think that the fact that mathematics is essentially endless is incredibly hopeful, when viewed as a human activity.
I mean when you view it that way then a theorem can absolutely be complete and consistent and decidable.
A system that assigns true to every statement in inherently complete and consistent and decidable.
Even a system that assigns the value to be an oracle that is possibly obtained tomorrow is consistent and complete and decidable - all undecidable problems are simply deferred.
All Gödel proves is the symbols not, and, or, sets and whatever logic used to manipulate those are incomplete.
@@Zenovarse Well, not quite. Basic logical systems of propositions are complete. But once it's got the machinery to encode number theory then yeah.
@@MindForgedManacle WDYM? If you write your logical system as propositions in your logical system, Gödel shows either it will not be complete or not consistent?
@@MindForgedManacle a kind of a trivial system that works is a system that has 1 statement, corresponding to the system validity is true. But in a non trivial system like the ones we use it is not the case?
Bro, the flames you are lighting in my heart with these videos... Thank you so much
Can we just appreciate how well animated and produced this video is? God, so much effort.
everyother youtuber: animates their ideas to make it easier for the viewer
vertasium: climes mountain with no context for a nice backround, spends hours making 3 words with a line through them and custom prints an entire set of cards just to express an idea, just to name a few.
Yeah, but that's irrelevant really - I read all this in a book already. It's the information that matters, not how nice it's presented.
@@unripetomato4312 He has a big team around him. Its not a one man show.
hey I recognize you from ut eng
@@sunnyjim1355 Uhhh... no, actually no. You, me, and a lot of other people may find it easy to understand written, objective, and scientific language, but many others don't. Some people understand artistic, subjective language easier, some others understand abstract languages easier (like the way sounds and colors relate, and "talk" to each other, like people who know how to use colors to tell a story, or people who write melodies, etc.). So probably a lot of people have a hard time with the math and stuff, and to help them have as fun as we have in this beautiful world of math, people (like veritasium) adapt the math to a more visual, artistic language.
Your lack of empathy for people's different necessities helps no one, showing off you read books helps no one, belittling other people hard work helps no one. When you understand that reading books is just one of the many valid ways of acquiring information, and it doesn't make you "cooler" or "smarter", you'll definitely cringe looking back.
:)
I suspect that for many people, making this video might be considered a lifetime achievement. But for Derek, just one more brick in his incredible, historic castle of outstanding teaching.
Yeah fr
look closer at the bricks composing the castle what are the bricks composed of.
Derek: "... But for me, it was Thursday..."
I suspect that you are one of his groupies.
If only he is my math teacher or history teachers
thats byt far the best way anyone can ever conclude a study session. Like you professor
mathematics is so beautiful. i loved the way you were trying to explain the godel's number. having never heard of it before, i found it even more exciting.i am planning to persue a career in mathematics. would surely come back to this video once i have actually studied about it.
When he showed "It's the Game of Life... running on the Game of Life" it literally blew my mind.
Can someone explain that better? It was cool but I think I don't fully comprehend what is happening
@@ritwikism he put an input in the game of life that it's output, instead of random patterns, was the game of life itself.
@@ritwikism they basically built a computer on the game of life that runs the game of life
@@ritwikism Since the Game of Life is Turing complete, that means you can essentially program anything with the Game of Life. At 29:50 they zoomed out to show how someone had programmed the Game of Life inside of the Game of Life.
The idea is somewhat similar to simulating a computer on a computer, like a macbook running a virtual machine of that same type of macbook.
@@cookiecan10 hence going back to Derek's first answer: Life. If life is turing complete (which it must be), there must be a way to fully simulate itself
Engineers be like:
"Does is work tho?"
"Well yes, but if you look closel..."
"Then yes"
Word
Two principles I follow. KISS and IIWIAS
KISS = Keep It Simple Stupid
IIWIAS = If it works it ain't stupid
@@amanawolf9166 that's enough for me, let's leave the puzzles for those who can bother
Mathematicians: Start crying uncontrollably*
@@thephantommarauder7748 Don’t look at the way we do trig.
Bro, I swear I sleep hearing your videos like hundreds of times. Your voice is so calm and one-pitched it helps me sleep every time.
I watched this in high school in 2020, and I did not understand a thing and thought it was really uninteresting.
Now im watching this again in college, and while Im taking my first discrete Math class(Berkeley's CS70), I now understand and find everything interesting!
same here
a single CS semester course of discrete math + another foundational course on proofs, logic, sets in my math minor and boom everything in this video is familiar and understandable
This is one of the best videos on this channel ever. My brain hurts a little, but I thoroughly enjoyed the experience.
Hard agree. This is your best work. The animations, from the cartoons, to the 2D graphics, to the 3D models, were spectacular, and you and the folks that produced them deserve a huge amount of credit.
This and the one equation will change your life vid
I wanted to say the same thing. This video is giving me a dopamine hit like none other. So well researched and presented. I love how he's connecting all these concepts and theorems across math, computer science, and history. What an amazing journey through time!
Yeaaa it makes me feel cool
@@mfadhilal-fatih1427 XD
Seeing the game of life being carried out in the game of life was a really impactful moment in this video
FACTS, i don't know how to explain it but that was mind blowing
I actually cried. I'm not sure what came over me.
You can actually find files with game of life running on game of life that is in turn ran in the program. So its game of life all the way down.
I was reading about it 2-3 months ago so I my self made some patterns.... But then it because headache..... And not after watching this video I got to know why it was a headache....
So the game of life can run the game of life but that game of life can run another game of life but is the original game of life running on another game of life?
Blowing our minds. I love to watch your videos, because I always have something to learn.
Fantastic, one of the best video I have never seen. Well done.
Simple Maths: Everything is certain.
Complex Maths: *Nothing is real.*
very accurate hahahaha
Complex Maths: Everything has an imaginary component
This whole Episode has just thrown me into a very weird existential crisis:/
...or rather:
Complex Maths: *Everything you can imagine exists. Not everything you can imagine is Real.*
@@glircom Im(religion) = God
Veritasium: “Math has a fatal flaw”
Me: So that’s why I failed my math test
your math test failed you
Bruh
@@enveloreal True, It denied the possible that your answer is concrete and relevant
@@enveloreal You are not taking a math test, but rather the math test is taking you
"Math has a fatal flaw" I believe this was my repeated assertion for the entirety of my school years... ; D
The game of life creating and destroying as it keeps iterating is one of the most incredible examples of the beauty of mathematics shown visually. Seeing all of the different patterns it creates is so cool
I read a book that introduced Gödel's theorem and didn't quite understand how mathematical statements were supposed to be expressed as numbers, but this video worked wonderfully on that.
I didn’t get even a sixty percent of the math in the video but I’m grateful for those amazing people who thought about these things and still do.
I mean the whole video was about paradoxes and contradictions
This is about the heart of mathematics, the most abstract thing that humans do. 40% is pretty good for a first try.
@@shoam2103 Definitely. Mathematical foundations are not for the faint of heart, but when presented in this way it can become quite accessible. You don't need to be able to DO the math here in order to appreciate it or even talk about it. I haven't watched very much of this channel, and I'm going to have to remedy that.
This Is the stuff I study everyday... I still don't get it. Math's are hard, but they wouldn't be so beautiful I they weren't
This is fine. It's a really abstract topic that most people will never have a reason to understand to begin with. But understanding the gist of it gives you this amazing feeling of having found out something so profound and fundamental about the world that is mathematics and any system our mind creates by applying logic. It's beautiful, awe-inspiring and depressing at the same time
Hilbert: I proved everything
Goudel: I am about to end this man's whole career
lolololololololol
Actually no, Hilbert didn't proved everything, he created a system of proofs, a formal way to prove everything in mathematics and every other field.
On the other way, Gödel didn't want to disproof all mathematics, he proved that not ALL mathematical statement can be proven, that is, there will be always some true statement that we will not be able to prove, but still there will be mathematical statements that CAN be proven, till this day we prove new and old mathematical laws, the problem is we can't know which statement can be proved or not, we might not find the answer right now and say that it is unprovable and 500 years later someone prove it, it is just undecidable, that's the point of Gödel's study.
@@GabrielLima-gh2we ikr
Godel: Can you prove yourself tho?
@@edwardhuang5885 Descartes: Yes
Excellent video/audio presentation and explanation.
It is important to note that in the "incompleteness theorem" that even though it is possible to create a "well formed" statement within the context of a specific axiom system, that may not be provable within that axiom system, it does not preclude proving or disproving the statement in a "larger" axiom system .
Some problems formulated in the context of geometry may not be provable with axioms of geometry, but may be provable in the context of algebraic geometry.
"I'm right"
"Okay, prove it."
"I can't"
Prove that you can't
Trust me
@@monkestronk1227 Prove that you can prove that you can't prove
Is that you Al Gore????
Said every YT comment ever.
Godel really woke up and said "i'm gonna ruin this man's whole career".
He failed... I mean... I've heard Hilbert's name propably a thousand times but this video felt like the first time I heard about Gödel. And I'm from Germany... :D
@@Kirmeins its a joke my dude
That's probably because you're from Germany and Hillbert was german. I'm from Austria and I heared about Gödel way earlier and lot more than Hillbert.
Gödel could have nuked the basis of formal logic, for all I understand.
@@Kirmeins even if you're only studying mathematics, you won't really hear Gödel's name until final year of undergraduate study at the earliest, unless you do an awful lot of advanced reading. Hilbert, on the other hand, has his name scattered throughout undergraduate and graduate topics in mathematics and physics. Gödel's work is considered fairly esoteric and difficult, while Hilbert is more spread out. So it's not surprising that you wouldn't have heard of Gödel if you haven't touched on foundations of maths.
Thank you for having a channel that contributes something positive to the online community. It is very rare to find a channel like this compared to all the other noise online.
Did you get an advanced degree in mathematics at some point?
Thanks Derek, I *LOVE* your videos, and this one was no different!
Godel : *refused to eat any food in order to not die*
Master Oogway : "One often meets his destiny on the road he takes to avoid it..."
I just came from a Kong-Fu Panda video LOL What a coincidence...
@@afiffarhati4580 wow 🤩 😂
One can learn so much from movies that were intentionally made for kids.
fun fact: did u know that more people die from pugs than from sharks!!!?? i will post regular videos like this so make sure to subscribe!! btw i'm a kid
@@THINKPATH wow rlly
There was a brief moment while reading Hofstedter's *Gödel, Escher, Bach* where I felt I truly understood the concepts... This video brought me right back to that feeling! Very well written, presented, and produced! BRAVO!
Its such a good book
@Peter Jerde I just noticed your name. Almost same like mine, funny, don't you think? :)
Same, but for his other book I Am A Strange Loop. In honesty, I have a feel for what the Godel proof is about, but there’s no chance I’d deduce through its formal proof.
We could call this "Gödel, Hilbert, Turing"
@@ilovecomputers I felt much the same until I read David Berlinski's _The Advent of the Algorithm_. I highly recommend it; it makes the subject matter very approachable, and is a super engaging read.
Only thing I regret right now is not having watched this video sooner. I kept putting it off for a while, often for the fear of not understanding Godel's stuff.
Thankfully, Veritasium has done such an amazing job of explaining. This is probably the best video I have watched in my entire life so far.
I guess it didn’t occur to Gödel to just cook his own food
This is one of those videos where I know what he's talking about... But I also dont know what he's talking about.
Its unprovable lol
Ahhh yes quite the contradiction
Now prove it
I know what you are saying... but I don't know what you are saying!
+Cheesy Boi Basically there are several mathematical proofs that mathematicians made. The bulk of these mathematical proofs is setting up an entirely new, imaginary system of math, or numbers or letters etc. In the end, it turns out that none of these systems can ever resolve the following statement:
This statement is false.
Because of this, any system of mathematics or language that we know how to create will always have unsolvable problems.
@ 23:42 he says about the Turing Machine "...although this sounds simple..." ..um, No 😬
I can’t decide if I’m smarter or dumber after watching this.
The smarter you are, the less you know
@@cjc722 The smarter you are, the more you know. But the smarter you are, you know that you know less.
@@YavNe This is at the core of the Dunning-Krugereffect.
You are both, that's the lesson 😉
Both; it's a paradox.
His last line just gave me a brain shock. I have never experienced something like that before. My body literally shook for a second.
"Led directly to the invention of the device you're watching this on"
Agreeing with other praising commenters, I agree this video introduces and explores some of the uncertainties in math and computer science, but it concluded with an optimistic light. Thanks! Another great one from Veritasium!
Gödel: *Thanos Voice* "I used the math to destroy the math."
You deserve more likes!
My absolute favorite mathematics professor, whom interestingly was 'merely' teaching precalculus at a 2yr community college began his course with various proofs which illuminated apparent flaws in mathematics. I
😂😂😂
except the destruction didn't destroy it.
@@voicelikemanywaters1017 Bruh, did you just started another paradox
What a conceptualy beautiful video illuminating a wonderful part in the aesthetics of thinking.
One of Veritasium’s best for sure.
I finally watched this after just ignoring it on recommended for a while, and it was glorious.
Same. It's been hanging there for weeks until I found the precise amount of free time in a day that I could devote to watching the video. Glad I did though.
i was doing an experiment: clicking on my recommendations and, don't paying attention to the video, but scrolling all the way down to the last video on the list of that one... I did it, 6 or 7 times and end up on this video, that called my attention. And like you, I am glad to did found it.
Same here😅😅... thought I would never watch it😅...loved it 🤗
Woah it’s crazy that we waited for the exact same time to watch this. After just ignoring it, knowing we would watch it eventually because it’s interesting 🤔
Mmhmm
I wasn't expecting to get goosebumps from this, but that game of life running a game of life.........
Can I introduce you to the Simulation Hypothesis? ;)
Yeah, that was pretty damn cool.
Oh my god I got it too for real, wasn't expecting to find this in the comments!
Minecraft running Minecraft
Yeah...
19:56
I love how this is just a slightly different variation of the Liar's Paradox or the Russel's Paradox.
And Godel knew this! The Epimenides paradox was the idea that motivated it all. Godel went through all the trouble of constructing his system because he knew that he needed to work that paradox in somehow-though a little bending and twisting was needed in order to force mathematics to refer to itself. :)
Excellent ancillary to Hofstadter’s “Gödel, Escher, Bach”
Thank you!
I thought so too!
My brain initially melted with the infinite hotel rooms and now it's leaking from both of my ears
I hear you...
I failed maths at 14 years old and never got any other education.
My brain sublimated in a cloud of pink fog that came out my nose
@@alex0589 You should try again. The key for leaning is not give up. It's hard and tedious, but the feeling of understanding something is indescribable.
@@alex0589 You didn't fail maths, maths failed you. It's an epidemic in curriculum worldwide; math is perhaps the most consistently mis-taught subject. Like Neto Fransisco above, I encourage you to give it another go. If you're willing, I can recommend the CZcams channels ViHart and 3Blue1Brown, along with the book _Burn Math Class (And Reinvent Mathematics for Yourself)_ by Jason Wilkes.
Understanding math is not nearly as hard as school has led you to believe. With the right teachers, it's the single most intuitive subject you can learn.
trying upgrade your brain to infinite brain it will be easy to understand
As someone who majors in mathematics while minoring in computer science, this video is absolutely awesome. I've learned about a lot of these things in isolation, but this really connects them all.
Ditto
Math glue?
If you want more of this story, I recommend the graphic novel Logicomox
Thanks for confirming this is solid. (I’m not quite awake; need to re-watch when I am! For what I expect will be a more spine-tingly coolness, like when I understood the RSA algorithm. )
The book “Gödel, Escher, Bach” - This remind me to read it!
Hey, I'm the literal opposite of you! (Majored in CS, minored in Maths.)