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Math's Fundamental Flaw

  • Published on May 21, 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.
    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
    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
    Executive Producer: Derek Muller
    Writers: Adam Becker, Jonny Hyman, Derek Muller
    Animators: Fabio Albertelli, Jakub Misiek, Iván 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

Comments • 46 161

  • Tyler Jane Bronson
    Tyler Jane Bronson Year ago +13002

    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!

    • Ken Fryer
      Ken Fryer 3 days ago

      @Alex Hetherington no... the game of life used as a computer is very inefficient. Your saw how slow it was to run the game of life on just a few pixels. It would grind to a halt after just a few factors.
      But theoretically it could, but practically not

    • julio
      julio 4 days ago

      Me too

    • sam
      sam 6 days ago

      @Alex Hetherington If we had a big enough board, could we simulate a human brain? After all, your brain is just a bunch of dead things coming together to make an alive thing, could we do that here?

    • Elliot Janzen
      Elliot Janzen 10 days ago

      Game numbers moment

  • Steve Sether
    Steve Sether Month ago +839

    I very strongly wish mathematics was taught in a wider perspective like this video is.
    We teach mathematics as if it's a world onto itself, disconnected from everything. In reality, it's highly connected to history, philosophy, and nearly everything.

    • Luceat Lux
      Luceat Lux 4 hours ago

      @Steve Sether Take a quick fun little test, if you're inclined enough :)
      You're correct. It essentially means nothing that it's supposed to mean (with the function stacks and all that garbage). I find that it IS useful incidentally and as a blunt instrument to determine an individual's basic intrinsic perspective and modality. For instance, it describes how an individual answers the test questions; how one emotionally CHOOSES to engage the task. So, it just reveals a fuzzy cursory outline.
      Funnily enough, most individuals who I ask ,y last comment's question to are, no lie, INTPs...
      See? Incidental utility that can connect with intuition once in a while.
      I'm the, so-called, INFP (the one at the bottom of the list, as far as what type one would freely choose to be, given the option... "The dreamer..." They might as well go ahead and include in the description, "Losers who live in their own heads." Careers that best fit this type, according to the, "Experts?":: "Artists... Writers... hobos... tramps... etc. One site only listed, "Artist," having listed ~10 suitable careers for each of the other types. The have the lowest life-happiness scores, by far, needless to say. lol
      Anyway, for some reason, I tend to regard INTPs opinions highly. There seems to be some sort of ego-divorced side to themselves that's they're able to access. They seem... more able to think outside of themselves and play with perspectives. There is a somewhat comical approach to stuff with them that I connect with.
      I don't really have much incidental data on how this type generally regards me. But I seem to empathize with their general emotional perspectives and approaches the most.
      Thanks for chatting with me!

    • Steve Sether
      Steve Sether 5 hours ago

      @Luceat Lux From what I've read, myers briggs isn't really repeatable or valid beyond introvert/extrovert.
      I recall taking it 35 years ago. I might have been INTP, but I don't remember. I'd be curious if it's shifted over the past 35 years.

    • Luceat Lux
      Luceat Lux 6 hours ago

      @Steve Sether i have the exact same regard for it. i was just curious about your thought process.
      If you feel like further engagement and aren't bored or burdened:
      Though i don't subscribe to such designations, do you happen to know the classification of your personality type (Myers Briggs, etc)?

    • Steve Sether
      Steve Sether 8 hours ago

      @Curtis Euga They take more of that approach in many European countries where you start specializing more early on.
      I have some mixed feelings about that. The downside is that if you take a direction early on, it's hard to reverse course and go another way.

    • Steve Sether
      Steve Sether 8 hours ago

      @Luceat Lux Sorry, I just clicked in the wrong place when opening up the rest of your comment, and reversed it. I didn't downvote you. I generally ignore a lot of the upvote/downvote stuff. I've become less of a voter over the years.

  • Linuxdirk
    Linuxdirk 28 days ago +647

    So Gödel basically said “The next sentence is wrong. The previous sentence is true.” but in a super complex and complicated way.

    • John Doe
      John Doe 16 hours ago

      @Keen Observer Well, tell me when you've found a proper answer because as far as I see we're making things way too complex, more than it has to be. The point we struggle understand might just be that because it just is. Take existence for example, people always say that time is infinite but doesn't realize that while infinity doesn't have an end, it might just not have a start. The idea is that paradoxes are repeatable answers. They might be our "neither", we're trying to make something of it but what if it just is? Advanced mathematics won't get you very far from where you are now. You are free to take my word for it or go mess around and find out, that'd be a time waste. A time wasted on only defining infinity which'll be a small contribution to giving our existence a reason. Be warned though, you may just end in a downwards spiral, and once you realize and acknowledge it, it may just be too late for you.

    • Keen Observer
      Keen Observer 17 hours ago

      @John Doe And to go back to your idea of right and wrong, I believe "neither" might be a viable answer as there are some things are are neither right nor wrong. For example, if I ask the 2 questions: do we live in a simulation? Do we live in the real material world? What if the answer to both is "yes and no"? i.e We cannot live without simultaneously being in a simulation and in real space, and one cannot exist without the other, hence the answer "yes and no".
      I may be completely wrong or illogical but I am sensing that we may be lacking this sort of perspective in maths.

    • Keen Observer
      Keen Observer 18 hours ago

      @John Doe In Physics there are sometimes multiple solutions for a set of equations that are mathematically correct however we only take the ones that make sense from a Physics perspective and discard the rest because we know maths can lead to nonsensical answers. I think that's one way of getting answers. It might just be that maths as we invented it to this day is full of inherent paradoxical flaws and that it needs to be re-invented to avoid those flaws.
      We probably also have to re-define the concept of infinity at it's core or re-think how it applies to maths. It seems to me (but then again I have not studied enough pure maths to make such claims) that we don't properly understand concepts like infinity and even zero from a purely philosophical standpoint, which then leads to a lot of weird conclusions (such as paradoxes) and ideas further along the line in maths.
      The same way Physics has been continuously re-invented (and still is being re-invented to this day), maths probably has a long way to go.
      And then philosophically speaking there is this possibility that it is impossible to know some things for certains (kind of like this video was saying, but in a more general sense not limited to arithmatic).
      For example, if the universe is cyclical, it expands and then eventually contracts and creates another big bang, and everything repeats itself, it would be impossible for us to know as we are limited by the information of OUR big bang, not being able to see the information from the previous big bang that happened before. In quantum mechanics we are also starting to see the limitations of observation at the smallest scale.
      PS: All of this is making me want to study advanced maths.

    • John Doe
      John Doe Day ago

      @Keen Observer "I am just speculating." So am I. I agree with you. But how do we solve this? From my viewpoint this is explained in the thumbnail. "Incomplete" technically defines infinity as both really doesn't have an end but the part they overlap is infinity doesn't have an end but incompleteness has a designated one. But my defense is that if infinity is incomplete, then incompleteness may be infinite, therefore being tied to our subject; I have a feeling there really isn't an answer just an infinite spiral of thoughts, questions and equations. Whenever we get close to the end another ring is added. Point is, yeah we need answers but what if there is an infinite amount of possibilities that may or may not result in an infinite amount of answers or questions and those may just give way for infinite amount of possibilities again. A weird loophole. What could we even propose to overcome this?

    • Keen Observer
      Keen Observer Day ago

      @John Doe I am just speculating. But we seem to create a lot of paradoxes with "right" and "wrong". And recent advances in Quantum Mechanics suggest the possibility of super-imposed states. So maybe the idea of "Neutral" or "right and wrong" is something to consider. It isn't an absurd concept either. When you look at the number zero, it is neither positive nor negative, a "neutral" number.
      When you think of the ice point of water being 0 degrees celcius. There is a temperature where the water is not liquid and not frozen but really in-between.
      I think part of the issues we encounter in Math might be due to our black and white reasoning, but of course I am just speculating.

  • EdgyShooter
    EdgyShooter 28 days ago +145

    Meanwhile in physics:
    "Can you prove this statement is true?"
    "I'm just going to assume it is and continue from there"

    • Dale Johnson
      Dale Johnson 12 days ago +5

      A lot of why we do what we do in modern quantum theory (physics, mechanics, gravity, etc...) is entirely based on knowing these principles as well. Once you establish that there are some things you may never be able to prove, you can assume that if your model is in fact flawed, you will be able to prove that it is flawed eventually with enough evidence and research or computational power, or the correct real world simulation that answers the question, just as everything in this video was more or less shown conclusively (except for the things which conclusively couldn't be, because yay uncertainty principle). If your assumptions are in fact correct, it should actually be easy to prove they are, even if you don't know WHY they are. There are actually numerous technologies which we know work, but have no idea why, and the same goes for systems within the human body and specifically the nervous system in particular. Some of the imagery you'll see or otherwise experience mentally, while on psychedelic drugs like mushrooms, LSD, DMT, and even dissociatives like ketamine and phencyclidine, match up with the kind of fractal geometry you'll see when you feed certain known mathematical patterns into a computer visualization system. On some level our own brains may in fact be Turing Complete computing systems. I suspect as we go further and further with the research into neural networks, and simultaneously try to properly understand the method of functioning behind the biological computer we all use to think, which simultaneously gives us our sense of self, and the ability for meta-consciousness, the ability to be conscious of one's own consciousness. You can dive off the deep end into theory all night on that one and at the end you'll be even more confused than you were when you dived in to begin with, what with everything you learned, but someday somebody is going to figure it out, and completely revolutionize the world yet again. After the ascension of quantum computing, that will most likely be the next major computer revolution, assuming they don't happen simultaneously in some ultimate singularity event.

    • Bug Dracula
      Bug Dracula 24 days ago +4

      I mean that is how the halting problem works

  • seniorgir
    seniorgir Month ago +418

    The more I learn about Turing the more amazing I realise his brain truly was. Ever since I watched The Imitation Game I've been fascinated with Turing, and honestly the fact that he was driven to suicide makes me feel disgusted at the waste of a revolutionary once in a generation brain. Imagine how far science could have come if he lived longer.

    • Jyrek Lordinthrain
      Jyrek Lordinthrain 19 hours ago

      @PatCam I disagree, all the people responding to Ben are trying to engage him in a conversation, to respectfully argue their point. Ben is actually behaving in a way I don't really understand, he keeps saying that he wants to end the conversation but then keeps responding. So far I think everyone is being respectful, confrontational, but respectful.
      I also disagree with Ben Botts. I respect his right to engage or disengage with this discussion but if he keeps responding then he can't really expect people not to do the same just because he asks them not to.

    • PatCam
      PatCam Day ago

      @Tamaz G I think you just need to chill, I don’t agree with Ben botts but at least they’re being respectful of others having a different view point. They’ve stated they won’t change their mind but you can’t accept that, because you don’t want to believe someone will always think you’re wrong.

    • Ben Botts
      Ben Botts 4 days ago +1

      @Tamaz G People really need to dial back their savior complex on trying to persuade me with their own belief system in regards to "Choice" versus "No Choice".
      Have a good night.

    • Tamaz G
      Tamaz G 4 days ago

      @Ben Botts “believe differently” isn’t the right way to describe this, as the topic at hand isn’t subjective nor up to interpretation. This is just denial in the face of a fact

  • DoomMantia
    DoomMantia Year ago +4298

    This is one of the best videos on this channel ever. My brain hurts a little, but I thoroughly enjoyed the experience.

    • George Tsitsiani
      George Tsitsiani 4 months ago +2

      Took words right out of my mouth.

    • Tako Au
      Tako Au 4 months ago +2

      I can’t imagine this 30-minutes video covers one of my major course about finite-state and Turing automatons in college. Natural language, primitive recursive functions and state machines are always my favourite topic!

    • DoomMantia
      DoomMantia 4 months ago +1

      @Peter Codner Way to be needlessly pedantic.

    • Peter Codner
      Peter Codner 4 months ago

      With what organ do you experience the "pain"(hurting) of your brain?
      Can a mirror reflect itself?
      It is axiomatic that it cannot.

  • Mr Gïdds
    Mr Gïdds 5 days ago +4

    Mathematics is amazing because it transcends numbers. The reasoning we find in math can be transferred to any logical problem outside math, and the building blocks of the technology that powers our world is made possible by math. So, my hope is that in the near future we can teach math in a broad perspective like this so that people won't grow to hate it, rather they will grow to appreciate it and use it daily. Math is connected to everything, so it's about time we started treating it that way.

  • Manlio
    Manlio Month ago +168

    The video itself is very beautiful and very well made. From the math perspective however there's some deal of confusion. To be precise, we should make a distinction between provability and decidability. The former is about the possibility to prove something in a given formal system, and this is what Godel's incompleteness theorem is about. The latter is about "do something algorithmically", and this is what Turing's work is about. In the video, we often jump from "we cannot know something" to "there's no algorithm to do something". The two things are very different. In particular, it is true that there is no algorithm to determine whether *an arbitrary* statement is provable or not (that is, there is no algorithm that, given a statement, tells you whether that statement is provable or not). However, this tells nothing on the possibility to prove a specific statement in a specific formal system. In fact, if you fix a specific (consistent) formal system S, then for any given statement there are only 3 possibilities: it is provable in S, its negation is provable in S, neither it nor its negation can be proven in S. For such a statement THERE IS an algorithm that tells you in which of the 3 cases you are. The problem is that you don't know which one it is!
    The topic is much wider than what can be explained in a RU-clip comment. And nobody reads past the first lines anyway, I'd be surprised if someone reads this last line.

    • Manlio
      Manlio 8 days ago

      @Ripple Reader Lol, a bit! Were you?

    • Ripple Reader
      Ripple Reader 9 days ago

      Were you surprised?
      Btw, I'll be surprised if you read this comment.

    • Manlio
      Manlio 10 days ago +1

      ​@Macarena Cabral mmm not really. You probably know this already, but in the P vs NP problem, P stands for "polynomial time" and NP for "non-deterministic polynomial time". So the problem is to determine whether non-deterministic machines are strictly more efficient (can solve a problem faster) than a deterministic one. Notice that here we are only talking about thing that we know we can compute! In fact, every problem that can be solved via a non-deterministic algorithm can also be solved via a deterministic one, it just takes exponentially more time. So there's no non-computability involved here. However, we are still talking about proving the existence/non-existence of an algorithm, so, in a sense, there is mild connection with computability theory. I am not an expert in computational complexity (I work in computability theory), so I'd be happy if someone more expert than me can add more details.
      Now, the statement "P=NP" MAY be independent from ZFC (the most common set of axioms used in math), but, again, this is a very hard open problem, and we have no real clue on whether this is true or not. Somebody believes that, since people have worked so hard without being able to prove it/disprove it, it must be independent from the axioms, but these are just guesses, and the truth is that we don't know (yet).

    • Macarena Cabral
      Macarena Cabral 10 days ago

      @Manlio hi, sorry if this is a dumb question, but is this then related to the P versus NP problem, or am I tripping balls?

    • Manlio
      Manlio 15 days ago +1

      @Dathaniel The former! The actual algorithm is probably disappointing: if we conventionally decide that 0 means "independent from the axioms", 1 means "provable", and 2 means "the negation is provable", then it is enough to consider three algorithms (from 0 to 2), each one doing nothing but saying its number. Now, for any statement, either it is independent, provable, or its negation is provable. So there is an algorithm that answer your question, but knowing which one it is is as hard as answering the original question.

  • Renegade Vile
    Renegade Vile 16 days ago +27

    To show how important Turing is to compute science, I have never heard of someone studying a degree in Computer Science and not seeing the concepts of Turing Completeness in their math classes. Unless you work in specific fields, it's unlikely you will actively use any of that knowledge, but it's still very important to know.

  • BradyPostma
    BradyPostma Month ago +120

    I rewatched this again. This is one of the best educational videos ever. Not just on this channel, not just on this site. One of the best in this world.
    Profound but conversational,it makes connections between a dozen aspects of our society and describes the fabric of logic itself,the setting in which our thinking occurs.
    What an accomplishment!

    • BradyPostma
      BradyPostma 7 days ago

      @Hjertrud Fiddlecock If you go away knowing that there will always be true facts that cannot be proven, then you carry with you one of the greatest realizations in human history. That's not moronic. That's brilliant!

    • Hjertrud Fiddlecock
      Hjertrud Fiddlecock 8 days ago

      @BradyPostma cool. I've watched it twice now and i think I'm just gonna accept that I'm a moron and move on to greener pastures

    • BradyPostma
      BradyPostma 9 days ago

      @Hjertrud Fiddlecock I think I understood by the second watch-through. It's complicated, but it doesn't make you prove it. It just tells you the premise that was proved.

    • Hjertrud Fiddlecock
      Hjertrud Fiddlecock 9 days ago

      @BradyPostma sooo... are you less or more confused after 10 times than after your first watch?

    • BradyPostma
      BradyPostma 12 days ago +2

      @David Klausa My words indicated that I watched it at least three times. The unspoken reality is that it's more like 10 times.

  • Judy Petree
    Judy Petree Year ago +619

    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.

    • 2 minute vape reviews
      2 minute vape reviews 20 days ago +1

      I'm 104, male. I'm grateful I watched this video

    • Peter Codner
      Peter Codner 4 months ago

      "Education" is a rather vague portmanteau word into which any number of sins and evils can be crammed, just as useless information is rammed down the throats of small beings who would rather play or do some useful work, but No, they must be "educated" whatever educated means, but let us just call it bullied.

    • Peter Codner
      Peter Codner 4 months ago

      "Reference" is a noun in pure English, not a verb; one can no more reference than one can parent or debut- except in that dialect of pure English that is American.
      If the salt has lost its savour, wherewithal shall it be salted?

    • blue sewage
      blue sewage 9 months ago

      @capratchet this is honestly might be the most beautiful way I've seen the edutube community described and encouraged yet. cant wait to share a classroom with everyone else too.

  • tophercruzio
    tophercruzio 6 days ago +2

    This is one of the best videos on youtube that I've seen. It's an all-encompassing summary of the nature of mathematics and logic

  • Pablo Sampaio
    Pablo Sampaio Month ago +63

    Para mim, como professor de Matemática Discreta e Teoria da Computação em cursos superiores de Computação, este vídeo é simplesmente apaixonante! Pela quantidade de assuntos profundos dessas disciplinas que ele apresenta de forma tão intuitiva e pelas informações históricas que eu mesmo não conhecia em tantos detalhes (e que Derek apresenta de forma legal, como um tipo de romance histórico). Vídeo obrigatório para quem é da área de Computação!

    • Anderson Marinho
      Anderson Marinho 16 days ago

      Professor o problema que nenhum professor fela da pota leva isso pra sala de aula. Brasil eu te amo! Um dia seremos mais

  • Shreyas Padhye
    Shreyas Padhye Month ago +82

    Not everything that's true can be proven. Incredible video. Every adult who is even remotely connected to science, math or philosophy needs to watch this.

    • Jacob Lee
      Jacob Lee 9 days ago

      @Ron J Applied math people and engineers don’t necessarily learn Godel. Most engineers I know didn’t. Of course many do now thanks to this video

    • Ron J
      Ron J 20 days ago

      @pyropulse Its obvious if you have an education in math or philosophy. Thanks

    • pyropulse
      pyropulse 22 days ago

      @Ron J It is an obvious statement? How wrong you are, since first order logic is both complete and consistent. Is that also an obvious statement to you?
      Goedel's incompleteness theorems only apply to axiom systems that are powerful enough to express first-order arithmetic.

    • pyropulse
      pyropulse 22 days ago

      wrong; you can prove any statement you want, given the right axiomatic framework.
      It is just that within any given framework, you cannot prove all statements within it unless you want it to have a contradiction

    • Victor Dantas
      Victor Dantas 22 days ago

      @Ron J you'd be surprised...

  • Ishaan Pandey
    Ishaan Pandey 10 days ago +6

    This is one of the most beautiful things I've ever seen in my life
    and it's hard to digest the fact that we may never know the ending to that "life game" and similar conjectures which keeps on going on...
    Thank you Derek for providing us such wonderful content every time 😇

  • DSUM
    DSUM 10 months ago +7677

    As a working mathematician, the scariest part of incompleteness is that when I can't solve a problem, I don't know if the problem I'm working on is just really hard... or if it's actually impossible.

    • Meow Mix
      Meow Mix 12 days ago

      Sounds like my dilemma with creating financial market algos lol

      ERNEST PATRIOT 12 days ago

      E = 0²

    • daniel gautreau
      daniel gautreau 13 days ago +1

      About 1970 a problem (for the readers) in Amer. Math. Monthly was actually undecidable in ZFC: Does the sigma-algebra generated by {AxB: A\subset R, B\subset R} contain every subset of R^2? (R is the reals.)

    • Blije Bij
      Blije Bij 15 days ago

      @AdultHumanFemale That, you can also experience as fascinating.

  • hey ho
    hey ho Month ago +26

    Amazing Video... there was a time when i understood this better... now I'm still not sure I get it =) To me this is very roughly a formalized and airtight version of the paradox: "If there was a machine that could answer everything, you could ask it to phrase a question it can not answer. If it just tells you "that doesn't exists" it didn't really answer. If it phrased that question it wouldn't be a machine that can answer everything anymore. So in a way there can not be a machine that can answer everything." Any logical system, complete enough to ask a hole into it's own completness, can't be complete. Yet, it needs that capability to be complete. I think their fight boils down to a weird human mentality, where some people are intereseted in math because they consider it to a path to "perfect order and truth" while others, like me, are fascinated by it, because of its riddles and the way it lets you glimpse into the paradoxical and chaotic. I like questions more than answers =)

  • Soyuno Forinfo
    Soyuno Forinfo 3 days ago

    A truly brilliant video. The most fundamental theorems of meta-mathematics and computability, described clearly and beautifully in half an hour. Absolutely outstanding.

  • GrandMaster Pritt
    GrandMaster Pritt 9 days ago

    I learned about some of this stuff in my CS class Data Structures and Algorithms, but you actually made it interesting! This was cool to look back on after taking that class, it helped me gain some appreciation. So, thank you for that

  • Jamie Crookes
    Jamie Crookes 16 days ago +12

    I always hated maths lessons at school because I couldn't wrap my head around them, but i've always been fascinated by things like this due to my work with computer. Only found this channel today.
    Keep up the great work.

  • Mackinstyle
    Mackinstyle Year ago +3028

    If you're a mathematician and you are labelled a "corrupter of the youth", you are doing something very right.

    • august pettersson
      august pettersson 6 days ago

      @brien maybe you becoq

    • No Name
      No Name 17 days ago +1

      Socrates agrees with this statement

    • Ethan Newland
      Ethan Newland 23 days ago

      @Brien831 Just want to say your explanation of Cantor's proof is really solid. Especially compared to other people in this comment section that have never studied in a related area and as such when they hear about his proof dont fully grasp it.

    • Victor Rabinovich
      Victor Rabinovich 26 days ago

      I suck at math and this is awesome!

    • Tea Formula
      Tea Formula Month ago

      Not always

  • alcatel😊😊😊😀☺

    When a disagreement arises between an infinite number of mathematicians there is a non-zero probability that approximately 3.14 of them will form a circle and start throwing pi at each other.

    • Bug Dracula
      Bug Dracula 12 days ago

      @alcatel😊😊😊😀☺ ah, understood

    • alcatel😊😊😊😀☺
      alcatel😊😊😊😀☺ 12 days ago +1

      @Bug Dracula When a whole person drops out of all their math courses after the first semester they are .14 of a mathematician. The other .84 of them becomes a psychology major working at Starbucks.

    • SuperGamer5
      SuperGamer5 22 days ago

      4 is a much better approximation.

    • LOG!C
      LOG!C 22 days ago

      @Bug Dracula uhh 1/7.14 if a person duh

    • John Daniels
      John Daniels 22 days ago

      infinite number * non-zero probability = it has happened, it is happening, it will happen again

  • Sylvie KS
    Sylvie KS 6 days ago

    This video gave me chills 💙
    Thus, I've decided to describe the paradox of my own mind for you.
    I absolutely love mathematics, especially such abstract problems.
    However, I bearly understand the simplest things at the beginning, and the more complicated it gets, the less I can comprehend. I know, that no matter how much I'll try, I'll never understand the complicated parts. I know, because I tried, and it was a total waste of time.
    So why am I interested in this so much?
    It seems my whole life is nonsensical, I'm smart enough to know, how awesome such topics are, but too mentally limited, to understand the exact things, that are exciting and interesting for me?
    I live in constant disappointment, regret and no happiness in my life.
    Why the fu*ck do I even exist?
    I wrote this comment, because I have a feeling, it's somehow corresponding to the content of the video....
    Or not.
    I don't know.
    I'm too dumb.

  • Rad Dead II
    Rad Dead II Month ago +36

    I had this moment of epiphany when the game of life simulated itself… I legit felt like I just travelled through dimensions at that moment with this sense of enlightenment and clarity… I just regained my motivation to pursue aerospace and astrophysics again

    • Burned Pizza Crust
      Burned Pizza Crust 17 hours ago

      @pyropulse how is that obvious? can u explain?

    • pyropulse
      pyropulse 22 days ago

      This really isn't that amazing..... it is an obvious consequence of the rules that govern it
      What is amazing is that someone actually pulled it off by actually doing it; but the fact it could be done was obvious from pure logic

    • sophia ouchari
      sophia ouchari 29 days ago +8

      that’s nice to hear :)) hope you pursue your dream and achieve great things man

  • Mary Hart
    Mary Hart 8 days ago +1

    Thank you for this video. You have given me a new appreciation for mathematics. Math has always frustrated me but I love how this makes it almost alive.

  • Michael H
    Michael H 7 months ago +4011

    Teacher: Your math is flawed.
    Student: No, math itself is flawed.

    • pyropulse
      pyropulse 22 days ago

      this doesn't show math is flawed, but I'm glad people still like making jokes of them being so dumb and trying to excuse it with super dumb jokes they think are clever

    • clash o clan
      clash o clan Month ago

      You missed the point of the video kiddo

    • Janis Thompson
      Janis Thompson Month ago +1

      My son tried that line in calculus, disputing his teacher. Was not spoken to, by the teacher, the rest of the year. Hes44 and just fine ❤️🇨🇦❤️

    • A Czech Man Going His Own Way
      A Czech Man Going His Own Way 2 months ago +1

      Bro, the school is about following and repeating what the teacher says. Not about discovering the ultimate truth (or about convincing/converting the average teacher).

    • Peter Codner
      Peter Codner 4 months ago

      Depending on how you define " flawed".
      Is a mirror not a mirror simply and only because it cannot reflect itself?
      It is axiomatic that a mirror cannot reflect itself. If axioms were not a priori they would not be axioms.

  • Ivan Bakin
    Ivan Bakin Day ago

    This was a great and entertaining video, but I highly recommend Dr. Ron Eglash’s TedTalk on African Fractal Thinking to provide a greater depth to the preservation of the concepts of infinity and self reference. Even binary has strong roots from sand divination that came from Africa as part of the use of fractals in African societies.

  • Madhur Garg
    Madhur Garg 12 days ago +2

    This incompleteness theorem completely changed my perspective towards mathematics. You are doing a great work.🙌

  • Nishad Dixit
    Nishad Dixit 18 days ago +1

    I wish I could understand all this.
    And it is scary for me that this has been easily understood by so many people.

  • Joe_509
    Joe_509 Month ago +2

    I’ve been subscribed for a long time and I watch all of your videos, I absolutely love your content, you have truly made a mark on my life and the way I perceive space, time, and just the world around me. Please never stop making content like this.

  • Matthew Ao
    Matthew Ao Year ago +2203

    Can we just appreciate how well animated and produced this video is? God, so much effort.

    • Ward Fadel
      Ward Fadel 4 months ago

      @Peter Codner just to tell her that incompleteness theory is agreed everywhere and it is a breakthrough and no way to compare it with the electricity video of this channel which oversimplified some aspects of the experiment although it was a nice one.

    • Peter Codner
      Peter Codner 4 months ago

      @Ward Fadel So, or therefore, what?

    • Peter Codner
      Peter Codner 4 months ago +1

      Far simpler clearer and quicker to advance the axiom that a mirror cannot reflect itself.

    • John Wicked
      John Wicked 4 months ago

      @Fred Esch nice 👍

    • xodz
      xodz 4 months ago

      The chart scene looks lile Flash MX discontinued

  • KimTGF
    KimTGF Day ago

    Didn't know humans were so experienced in living with existential crises. The crises just kept changing when we got accustomed to the previous one.

  • Iso bOOl
    Iso bOOl 5 days ago +2

    Math is the most compressed simulation of the universe. Since the universe is infinite and not a perpetuum mobile it is completely reasonable that math deviates from the outdated human assumption that there might not be perpetuum mobiles when we take parts of the universe into focus - but not the universe itself. Therefore this hole never will get patched without creating another one.
    It's like imagining the borders of the universe - only that a border by definition consists of at least two sides - so what's on the outside if this border if not even more of that sweet little universe.

  • peter mitchell
    peter mitchell 27 days ago +1

    Thank you Derek for this amazing video. This is why maths and engineering are so intriguing to me. Simply brilliant.

  • Shein Phyo
    Shein Phyo 20 days ago +2

    I don't mind long video, @Veritasium. The videos here are the one I can watch in one sit without knowing how long the time has passed. Keep up the great work!

  • Niklas
    Niklas Year ago +1066

    This is basically my whole computer science studies in 34 minutes.

    • Eric McKinley
      Eric McKinley 3 days ago

      @kotzpenner cs degree is a waste of time anyway

    • Zoran Matijević
      Zoran Matijević 11 months ago

      @Brandon puntin One of the first things general science methodology and logic professor explained to us.

    • kotzpenner
      kotzpenner Year ago

      @a set theory and all that is what I struggle with the most

    • a
      a Year ago

      @kotzpenner what math are you in rn? Just got my CS degree so maybe I can give some insight

    • kotzpenner
      kotzpenner Year ago

      Well, I'm studying CS and the math is hard af to the point of considering dropping out. (And it's only 2 courses out of 3 years)
      Like the whole video allover I was thinking "what's the point" like 80% of the time lol.
      And I studied that stuff for months and have another exam in 2 months again because I botched it the first time.

  • Proskillz
    Proskillz 17 days ago

    This is why I love math. One of my favorite sayings is that math is the language of the universe.

  • Proskillz
    Proskillz 17 days ago

    This is why I love math. One of my favorite sayings is that math is the language of the universe.

  • NIHCAS com
    NIHCAS com 6 days ago +1

    Although I dont understand some part of this video. But eventually I thought this is really great video for scientific persons and the editing of video was also nice.

  • Raul Giron
    Raul Giron 2 months ago +3

    You explain it so well that I was able to follow along the whole video until I realized I had no idea what I had just watched.

  • Amecha
    Amecha 11 months ago +9863

    So basically...
    Can math prove itself?
    But math can prove that math can't prove itself.

    • Nikolay Antipin
      Nikolay Antipin 2 days ago

      Depends do you consider logic part of math or not :)

    • sam
      sam 6 days ago

      Someone should tell my geometry teacher this, I had to sit there and learn about proofs for nothing

    • Izzaya
      Izzaya 8 days ago

      How dare you sir

    • Apurv Speaks
      Apurv Speaks 13 days ago

      That's like child giving excuses why they can't but not why they can

  • Best China Info
    Best China Info Month ago +6

    I'm not into 'maths', at all, but this video, and explanations therein, are very watchable - good job!

  • Christians Making Music

    I com back to this video from time to time. This is something that many who use math never think about. If the model works and produces consistent results, why bother. But it is a real issue and very profound suggesting that there are limits in all our endeavors. I’m not sure it is right to say this is a flaw in mathematics. It simply is what it is, and there is no objective standard by which we could contend that this is a “flaw”.

  • Alex Korinth
    Alex Korinth 4 days ago +1

    This kind of reminds me of when we discovered irrational numbers. Every time we try to iterate a new part of the item, it ends differently. It becomes no longer predictable if every iteration into infinity is different from the last with no pattern.

  • Andrew
    Andrew 25 days ago +5

    I work in theoretical computer science and love this video because it so closely relates mathematics and computability! The first time I learned about the theory of computation was an eye-opening moment for me and a small introduction to incompleteness. Whenever someone asks me what I do and what my field is, I tell them that the most famous guy, the guy who really started the field I'm in is Alan Turing.
    A nice way of explaining modern day computers is that they are equivalent to TM's.
    Great video!!

    • Andrew
      Andrew 21 day ago

      @Haytham Hammud Well I'm planning on going into academia, but there are some jobs in industry where theory is very important, mostly in research. It really depends on the niche you are in!
      A quick example: a friend of mine works for a subsidiary of a big company that produces chips and he does research on optimizing the building and manufacturing processes of these chips.
      But it is true that there are less jobs in theory than in most other parts of CS! I would definitely describe myself as a computer scientist/mathematician.

    • Haytham Hammud
      Haytham Hammud 22 days ago

      What jobs are there in theoretical computer science ? Basically I’m from the same field but under the headline of math

  • Kyriakos Mousias
    Kyriakos Mousias 7 months ago +1563

    As a mathematician I haven't seen a more elegent presentation of these concepts,especially Godel's theorem. Amazing job thank you.

    • Aisha
      Aisha 6 months ago

      Any tips on becoming good at math as a high schooler?☹️

    • william mabon
      william mabon 6 months ago

      @Dayton Robar What's naturally good? Opinions are endless.

    • Dayton Robar
      Dayton Robar 6 months ago

      Presentation is everything for people that are not naturally good at math.

    • Michael Salisbury
      Michael Salisbury 6 months ago

      This is the perfect medium for this stuff.

    • william mabon
      william mabon 6 months ago

      Godel like Cantor did not see that change is a subset of Infinity. Change allows for a contradiction to operate as a constant in a stream of logic that changes an identity within a mathematical extremity. This fact do not make math incomplete. It simply allow for the growth of change which is actually an expansion of a set's identity given that any contradiction must contain elements of identity to the set in question.
      Any contradiction is based on finding a counter or opposite identity with like elements thereby making the contradiction a mirror set or a set turned in the opposite direction.
      Example: the elements of the negative number set do not contain any positive numbers within it but positive numbers do exist. Both sets have like elements within a larger set of change. Each of these sets have an equal number of elements that oppose the direction of the other yet both sets share the identity of likeness of size and division of spatial order.
      Here we have an order creating a disorder of self. A contradiction or simply an expansion of its spatial self.

  • qwertyuiop 1tiop
    qwertyuiop 1tiop 17 days ago

    Hello, I want you to know that you are a saviour to my final year undergraduate maths history grade. Our lecturer didn't write notes, gave us a ridiculous reading list of 20 very dense maths books, each over 1000 pages, and didn't record the lectures, all for an exam that is 50% of 1/8th of the final year that is weighted at 60% (so in total 3.75% of my degree). We are expected to understand the full history of maths from prehistory to know and also understand all the different areas of maths philosophy. This video gave me a fundamental understanding and allowed me to exit the anti-philosophy-learning stance I had taken.

  • ziisch
    ziisch Month ago

    How you take the origins of computer science, content of several lecture and just put it in a concise youtube video is amazing!

  • gerkey
    gerkey Month ago

    This should be a part of curriculum for undergrad math students everywhere. Such an amazing work!

  • Krombopulos Lincler
    Krombopulos Lincler 6 days ago

    Our life revolves around numbers. One of the revolutions of history was being able to do boolean algebra with numbers. We could create our own simulated worlds with enough computing power and we probably are already living in one.

  • Arthur Barbosa Câmara
    Arthur Barbosa Câmara Year ago +3664

    I'm a PhD in computer science. This is a full-on Discrete Mathematics intro course. This is amazing.

      FAST N BULBOUS Month ago

      My dad's best friend at Cambridge university was Dave Masser. Do any of you know that guy? Formulated the abc conjecture..

    • Kevin Rinkanya
      Kevin Rinkanya Month ago

      Some poor kids are about to be forced to watch this

    • Joseph Smith
      Joseph Smith Month ago

      I have a basic math knowledge but do to videos like this I understand some theory

    • Zarth Sajuuk
      Zarth Sajuuk Month ago

      There is a fundamental flaw in the real vs natural numbers challenge.
      The way Veritasum is presenting it - is a trickery.
      It is presented as if natural numbers N are being opposed to the real numbers with the length of N, which is wrong. Obviously, the natural numbers from 1 to 100 will have fewer combinations than a real number with 100 digits in length. But that is wrong comparison. The correct one is comparing natural numbers with K (infinity) number of digits in it vs real numbers with K digits in it.
      So, if this task is presented properly without tricking the viewers into substituting of the natutal number count with the real number length, then it will be obvious that this task has a valid conclusion (see below).
      In other words, lets say the "infinity" (or "lim") is N, and assume it's 2-digit value (K=2). That means on the natural number side you have 100 possible values between 0 and 99. On the real side you have got "random" non-repeating values between 0.00 and 0.99.
      Please note, the trick in the video lures you into an impression that you would have more digits in the real numbers row, e.g. you could use 0.991 value, but it is wrong because of the premise that you have reached the N (in the natural numbers) and that is the "infinity". Otherwise you could say "well, whatewer is the last natural number, I will add 1 in front of it and I will get a new unused natural number". But the idea is - you have reached the limit.
      But this means, you are supposed to reach the same limit in real numbers that will tell you that there is no more digits to continue your real numbers.
      Therefore, we are playing in the same field and the limit is the same.
      So if we go back to our 2-digit "lim" for natural 0 to 99 where you have 100 variants or real of the same lim between 0.00 and 0.99. Now you can try applying "adding 1 to the digit" in the real row.
      What do you get?
      And the answer is - you get it duplicated. Or you have to violate the limit.
      So, the conclusion is - there are as many natural numbers between 0 and 1 as there are real ones.
      The importsnt understanding is that natural number 1 and real 0.1 are in fact: 0000...infinity...0001 and 0.1000...infinity...0000
      And if their length is the same - they have have the same number of combinations.
      Thanks for reading this if you reached this line :)

    • Bazm0
      Bazm0 Month ago

      Whats a PhD in computer science? Isnt that called a geek 🤣

  • Isaiah Inman
    Isaiah Inman 14 days ago

    i used to loath math, now after college its become one of my favorite subjects. lol wish math was taught like this

  • SuperDeadRooster
    SuperDeadRooster Month ago +2

    One of the greatest videos on RU-clip. Congrats, brilliant.

  • Victor Rabinovich
    Victor Rabinovich 26 days ago

    What a window into the history of the 20th century, thank you!

  • Garrett Bellinghausen

    I just coded my very own game of life for an assignment for computer architecture for my CS degree and wow did it give me insight to ideas discussed in this topic

  • jonasba276
    jonasba276 Year ago +460

    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.

    • AR
      AR 3 months ago

      @Peter Codner Horrible attempt at sounding smart. Did you feel bad this video didn't including anything about your field of expertise which you could flex. Sit down buddy, you've successfully cringed me out.

    • Peter Codner
      Peter Codner 4 months ago

      @jonasba276 Nice etymology: Middle English (in the sense ‘stupid’): from Old French, from Latin nescius ‘ignorant’, from nescire ‘not know’. Other early senses included ‘coy, reserved’, giving rise to ‘fastidious, scrupulous’: this led both to the sense ‘fine, subtle’ (regarded by some as the ‘correct’ sense), and to the main current senses.

    • jonasba276
      jonasba276 4 months ago

      ​@Peter Codner Damn you sound like a nice guy

    • Peter Codner
      Peter Codner 4 months ago

      Apparently you decided to skip English , in the pure form of which there are no verbs to major or to minor and thus no gerunds thereof since they are adjectives. Of course there is nothing to prevent you from inventing your own exclusive-to-you language save perhaps that you will be its only speaker.

    • Joseph Shinn
      Joseph Shinn Year ago

      You like conflating. Well that sums up this whole video. Have fun!

  • Neom
    Neom 19 days ago

    Ok, this was my second video of Veratisium I ever have watched...the first was about the impossible measuring of light in one way...combined with this here i would say, "that's brain fxxxing". But that's not the point, the point is, I am loving it! And the game in game declaration is another milestone to the point that we are simulated. An other theorie that is also not to be checkable...

  • Brad Clark
    Brad Clark 2 months ago +9

    Wow! This is by far the best I’ve ever heard/seen this explained. I did my best to read D.H.‘s “Gödel, Escher, Bach” and the actual book Gödel wrote and have tried to wrap my head around it for so long. This helped tremendously!

    • daniel gautreau
      daniel gautreau 13 days ago

      Don't bother with D.H. Try "Godel's Proof" by Nagy & Newman.

  • Isaac John Padilla
    Isaac John Padilla 20 days ago +1

    This was one of the craziest videos I've ever had the attention span to actually sit through!
    I'm not going to lie I was definitely lost halfway through I had to watch it two or three times.

  • Michael Epp
    Michael Epp Month ago +1

    Even if I watched this an infinite number of times, my brain would always reach its elastic limit.

  • Magnus Kramnik
    Magnus Kramnik Year ago +4497

    Mathematicians: we must prove this equation
    Engineers: Eh, it's good enough, we'll just use it

    • howard baxter
      howard baxter 9 days ago

      @_Nines pi = root (g)

    • garak55
      garak55 2 months ago

      Physicists be like : "fools to the left of me, jokers to right, here I am : stuck in the middle with you"

    • Mattical
      Mattical 2 months ago

      yes, the more practical

    • Jarred Mosdal
      Jarred Mosdal 2 months ago

      Mathematicians: we must prove this equation.
      Engineers: Eh, it's good enough, we'll just use it.
      Lawyers: the evidence is inadmissible. But Godel's numbered cards are a gold mine. I'll add Bates numbering to each and consult until the funds available are exhausted.

    • Evan Pederson
      Evan Pederson 2 months ago

      @Jacob Lee lots of empirically derived formulas in fluids and heat transfer too.

  • The Cosmic Melodies

    What happens to the game of life when you wrap the "infinite" starting grid around a sphere? Might need another shape other than squares to make the grid. Might need to tweak the rules a bit too. Is this interesting for cryptography? An extremely simple set of rules that produces extremely complex results, and being enclosed on itself, a modular cyclical behavior. Something about 3D cardioids ... ? Apples and such.

  • Jon R
    Jon R 2 months ago

    In a world full of self promotion you are sharing knowledge and have such a great chill vibe. Thanks for your service.

  • EdgyShooter
    EdgyShooter 28 days ago

    Wasn't Cantor's diagonal argument based on a binary number where when you created your diagonal each digit was complimentary (ie, if it was a 1 originally it was changed to zero)?

  • Bartley Troyan
    Bartley Troyan Month ago

    I tried to scrape off the dark spot between the zeroes that appeared at 6:50 and faded at 7:12.
    Otherwise, this part of the video contains the clearest explanation of Cantor's diagonalization argument that I've encountered.

  • Peter Jerde
    Peter Jerde Year ago +1848

    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!

    • daniel gautreau
      daniel gautreau 13 days ago

      @Rob Inson I agree. If you didn't already understand Godel's work, Hofstader's book would just confuse you.

    • Victorel Petrovich
      Victorel Petrovich 6 months ago

      @Leah C Checkout Babbage, and others, which were develloping computers regardless of Godel and their math plays. Turing was just one of the many who dabbled into computing.

    • Victorel Petrovich
      Victorel Petrovich 6 months ago

      @Jonathon Meyer There would have been much more progress in math if Hilbert turned out to be right about all 3 questions. Computers would have been made anyway, don't worry. (read about Babbage and others).

  • KEMNS Art
    KEMNS Art 27 days ago

    This video is just amazing. This is the third time I watched it in a few months and I never get bored thinking about it.

  • milad aghajohari
    milad aghajohari Month ago

    you nailed the explanation. I really enjoyed it. Always wondering what the godel proof would look like.

  • brainswashedthisways

    Besides the great concepts, fantastic animation/illustration 👌

  • JY L
    JY L Month ago +1

    Really massive and complex for me to understand, though I've already watched and somewhat understood this topic.

  • Bruno Ribeiro
    Bruno Ribeiro Year ago +268

    This feels like the start of a new era for Veritasium. The production value is off the charts! And the topic is just beautiful. Congrats Derek and team S2

    • Fractal
      Fractal Year ago

      @Jeremy Raymond the idea of "purpose" is weird to invoke here..it would be better off to state you rather just dont enjoy it much. it is as useless as much of the things he says in his other videos depending on how restrictive you wanna be. there coild definitely be a better spending of time.

    • Misha Finadorin
      Misha Finadorin Year ago +1

      @Jeremy Raymond A statement about the entire system that you use to describe 'the useful stuff' is pretty useful itself.

    • Ben Thorpe
      Ben Thorpe Year ago

      @klobe9 I don't understand why he didn't reshoot that segment lol, was like 10 seconds long

    • RationallySkeptical
      RationallySkeptical Year ago

      No, he's been at this level for quite a while now.

    • Jeremy Raymond
      Jeremy Raymond Year ago

      See I don't like it. Theoretical mathematical truths that aren't provable and/or practical are just neat to hear and that's it.
      It doesn't have a purpose. That's what I don't like. You can make up a story about a hotel with infinite rooms or tell me that there's an infinite number of twin prime numbers but do something with it. Show me an example on why it's worth knowing.
      This is what I'm talking about at 20:59. It's a paradox. Those can be neat. What did you do with the information though? What *can* you do with that information?
      I would MUCH rather Veritasium cover content like where he went into public and asked people things like, "Why does the earth rotate?" or "Why do two objects fall at the same speed?" but that's just my personal preference. I'm happy so many people like where his channel is going. I wish I was one of those people.

  • Ken Fryer
    Ken Fryer 3 days ago

    I've never been so great with maths. So this makes my head ache a little. Maths is really a beautiful human invention. I'm more an artist and creative type so my mind doesnt grasp it so easily.
    Seems so exciting and fascinating and is behind everything. Just always got turned off of it at school.
    Wish they taught the thrill and excitement of maths and science at school rather than the boring dull side of it

  • Menthols
    Menthols 14 days ago +1

    This was an amazing video, really got me thinking. Thank you.

  • Ryan Schlegel
    Ryan Schlegel Month ago +5

    Although it appears unthinkable, if the models were allowed to go on forever the premise of life would eventually expire. As it is for numerology it would be correct in saying that each scenario is true. Probably, if prime numbers get rarer and rarer over time there will be vast expanses where there are no prime numbers. The probability of one will decrease and decrease. The question would then become how long you would be willing to wait until one appears. If the probability of one manifesting is longer than the amount of time left in the universe, then reasonably the relative nature would suggest that it is insignificant and therefore both answers would be correct. There is an end to prime numbers but technically there is no end. -you're welcome.

  • dubmaster z
    dubmaster z 13 days ago

    Derek, this is really great content. I’m going to coin a term now called “Content Disease“. Meaning = A content provider of any kind using mediums such as RU-clip, with an extreme addiction for the need to constantly produce content. Usually at the risk of devolving themselves from life, family, relationships and social constructs. “
    Is there a mathematical proof in that Derek?

  • Jean le Ronde d'Amelbert
    Jean le Ronde d'Amelbert Year ago +1048

    If there was an Oscar for RU-clip videos, I have absolutely no doubt this would be nominated. Well done sir!

    • Gabriel Carvalho
      Gabriel Carvalho Year ago

      @Jean le Ronde d'Amelbert lol, here's my like sir

    • Jean le Ronde d'Amelbert
      Jean le Ronde d'Amelbert Year ago +1

      @Gabriel Carvalho you can like it now :)

    • Jean le Ronde d'Amelbert
      Jean le Ronde d'Amelbert Year ago +3

      @pottyputter05 I commented without much thought but I absolutely agree. Some (emphasis on some) of the content on RU-clip is absolutely on par with Oscar nominated films, especially some of the lower budget ones

    • Simon B
      Simon B Year ago +2

      So we have the rewind or whatever it is but we don’t have YT oscars? Ricky we need you

    • Gabriel Carvalho
      Gabriel Carvalho Year ago +4

      I was going to like your comment, but it says 404...

  • Diego Arriaga
    Diego Arriaga 29 days ago

    As a computer scientist, this blew my friggin mind

  • yash
    yash 28 days ago

    I don't know a lot of maths but I can prove that these 34 minutes of mathematics have taught me more than 13 years of schooling

  • Alfin Dpr
    Alfin Dpr 20 days ago

    I wish i have this kind of explanation 30years ago. But its never late for clear explanation of fundamental law

  • Rum
    Rum Month ago +1

    I may not be able to grasp everything in this video but it feel so fun to watch it.

  • Chor Ian
    Chor Ian 10 months ago +3278

    mom: why did you get a B in math!
    me: math has a fatal flaw

    • pyropulse
      pyropulse 22 days ago

      me: Damn, I only got a 99% in advanced mathematics course, must be because math has a fatal flaw

    • Enni Roc
      Enni Roc 2 months ago

      Your mom is scarring you for life with her high expectations.

    • Galactic Tech
      Galactic Tech 6 months ago

      if he gets 92%+ most of the time in math, then a (B) would be below his standards of mostly (A)s and would be considered a "not so great grade". At least this is how I understand it as well as my parents(unless I missed a day for whatever reason or didn't understand the concept, then they would understand why).

    • happy gucci
      happy gucci 9 months ago


    • Dr Ghost
      Dr Ghost 9 months ago

      @Nobody Knows 😂😂😂😂😂😂😂😂

  • Nathan Brader
    Nathan Brader 20 days ago +1

    I would disprove Gödel's incompleteness theorems but by Gödel's incompleteness theorems I can't be sure my logic is consistent.

  • mknomad5
    mknomad5 Month ago +1

    Probably the third time I've watched this. Thank you for all your work.

  • Janice Hadley
    Janice Hadley Month ago

    I have to yet again admire the great thinkers, philosophers, etc.

  • Matt Smith
    Matt Smith 23 days ago

    Then there's my favorite type of number: the reccit reverse exclusion clause, a number that can only by defined as being anything other than itself. Brought to us by the great philosopher Douglas Adams who gave an example of of arrival time for a party at a restaurant is therefore the one time in which it's impossible for anyone to arrive.

  • stephan Roche
    stephan Roche Year ago +53970

    Ironic that Godel's death was the result of a self-referential paradox: he died in order to not die

    • Aayush Srivastava
      Aayush Srivastava 7 months ago


    • Niranjan
      Niranjan 7 months ago

      @Matthew N AMEN!

    • Shayer S. Utsho
      Shayer S. Utsho 7 months ago

      @TheUnspeakableHorror Yes. He used self-reference for the benefit of research, while the same self-reference brought about his demise - it's the starkest contrast there can be. It's an irony.

    • Zaraspe, Bong Jr. G
      Zaraspe, Bong Jr. G 9 months ago


    • ctrl
      ctrl 9 months ago

      Great, now I have to clean my brains off the ceiling.

  • Tom Oakhill
    Tom Oakhill 12 days ago +2

    At 20:40 he states the Gödel Incompleteness Theorem the way I was taught it 35 years ago: Any system of axioms sufficient to describe arithmetic will either be able to prove false statements or will not be able to prove true statements, where "prove" means "to decide they are true." There is a corollary in computer engineering: all electric digital logic circuits, complex enough to do arithmetic, will have unused states they can arrive at from which they cannot return. In other words, every computer will need to be shut down now and then.

  • Kevin Bean
    Kevin Bean Month ago

    I liked this video! Since I liked it I also clicked Like so, like squared L(2).
    A lot of material that could become an entire quarter in college math ( perhaps also middle school ). Poor Turing, he was before his time, and did not respect his Era current laws and legal situations, not to mention being a bit paranoid. But those issues did not prevent him from the miracles he helped invent and promote. Perhaps we owe him a bit of slack? A posthumous pass? 😀

  • Shaurya kansal
    Shaurya kansal Month ago

    I am so happy that I started watching veritasium again 😂 idk what happened in the middle just lost it
    this is so inspiring

  • Parniyan I
    Parniyan I Month ago

    And I'm supposed to plan my whole Year 2 and 3 DigiTech units, lessons, assessment, transdisciplinary themes, etc. for the term but I'm watching this coz I LOVE MATHEMATICS!!!!

  • Lemon D
    Lemon D Year ago +1893

    I don't know why but I love the idea of mathematicians gathered in a room yelling and hurling insults at one another

    • MrKotBonifacy
      MrKotBonifacy 5 months ago +1

      @Umar Ahmed Sigh... Yes, SOME of them, SOMETIMES, "once in a blue moon" might have crossed that treshold Also, a duel, although a very confontational act, is not "physical" one (at least not a duel conducted using firearms). "Risky", "harmful" and "deadly" - yes, by all means - but not "physical". Matter of "honour", "dignity" - but NOT a physical confrontation like in a drunken pub brawl.
      Anyway, the first post in this topic was about "mathematicians yelling and hurling insults at each other" (thus "getting emotional", but not "physical"). Others expressed their... doubt's, let's say - "why, scientists are the better breed - educated, cultural and all" - to which I replied "well, they're people too - they have emotions, they can turn nasty, or even spiteful" - and in fact they often do, as it is evident for anyone following "scientists' polemics". There's even that wonderful piece of a fiction story "How the World was Saved" - a "robots' fairy tale" from "The Cyberiad", a book by Polish writer S. Lem:
      _One day Trurl the constructor put together a machine that could create anything starting with n. When it was ready, he tried it out, ordering it to make needles, then nankeens and negligees, which it did, then nail the lot to narghiles filled with nepenthe (...).
      Only then did Trurl invite over his friend Klapaucius the constructor, and introduced him to the machine, praising its extraordinary skill at such length, that Klapaucius grew annoyed and inquired whether he too might not test the machine.
      "Be my guest," said Trurl. "But it has to start with n."
      "N?" said Klapaucius. "All right, let it make Nature."
      The machine whined, and in a trice Trurl's front yard was packed with naturalists. They argued, each publishing heavy volumes, which the others tore to pieces; in the distance one could see flaming pyres, on which martyrs to Nature were sizzling; there was thunder, and strange mushroom-shaped columns of smoke rose up; everyone talked at once, no one listened, and there were all sorts of memoranda, appeals, subpoenas and other documents, while off to the side sat a few old men, feverishly scribbling on scraps of paper.
      "Not bad, eh?" said Trurl with pride. "Nature to a T, admit it!"
      But Klapaucius wasn't satisfied.
      "What, that mob? Surely you're not going to tell me that's Nature?"
      Then give the machine something else," snapped Trurl. "Whatever you like." For a moment Klapaucius was at a loss for what to ask_
      Unfortunately, that piece is a tad on a "lost in translation" side - you see, the original text was in Polish, and Polish term tor "natural science" is "nauka" (which could mean both "learning", "teaching" and "knowledge".
      Which had to be replaced, unfortunatelly, by that rather silly"natural" in translation - but that's not the biggest flaw here.
      In the original text after "Surely you're not going to tell me that's Nature?" came a line, from Klapaucius, "But the Science (= "Nature") is something completely different!"
      To which Trurls' reply was something like:
      "So, you have any better idea? [on what a science is]. Then tell that to the Machine, and it'll make/ create it gladly in no time flat".
      (Slavic languages are "pro-drop" and "null-subject" languages, as bot the pronoun and the subject of the sentence can be easilly deducted/ infered from the grammar of the sentence.)
      To which question/ challenge Klapaucius was lost. (= He didn't know what to say/ answer/ had no better idea whatsoever what "science" is supposed to be.)
      So anyway, because of the "plasticity" of Polish language (and other Slavic languages too), AND a highly "inventive" vocabulary of Lem his works are often next to impossible to translate info languages lacking a "proper grammar" - like, for instance, English).
      But I digress here... Cheers!

    • Umar Ahmed
      Umar Ahmed 5 months ago +1

      @MrKotBonifacy minus getting physical?! Galois died in a duel at 21. And wasn't Pythagoras rumored to have killed someone for proving that there are irrational numbers?

    • Luka
      Luka 8 months ago +3

      “Corrupter of youth” 😂

    • Grevoron
      Grevoron 8 months ago +3

      the mic drops could've been the hottest known to mankind

    • J K
      J K 8 months ago +1

      Oh Reginald.... I DISAGREE

  • GrimJerr
    GrimJerr 20 days ago

    Godel has been my math hero ever since I started studying the book "Godel, Esher, Bach, the Eternal Golden Braid"

  • Best Me Subliminals and YouTube

    31:37 wow... Almost all geniuses who changed this world, in a way, lived a tragic life. Almost all, Tesla, Turing, Godel.. This just proves the duality of this universe, to know is to not know, to live is to not live. The reader of these will or will not probably know this, but yeah, such is life, an infinity. Idek what horrors I'll spawn from this, but I'm sure some have had this thoughts before.. I'm some guy here on earth and this world is short, I've lived a beautiful life and I will live another beautiful life. Life is one full of the human spectrum, and one full of the infinity of infinity. Life is as it is, meaning we're not here for anything else than to experience it.. Just live it as it is designed, it will or will not matter anyways. Since the universe will inevitably die thus your life and achievements won't mean anything, but in the very moment it will mean everything

    • sketchiiiful
      sketchiiiful 24 days ago

      Its not only that every genius has lived a tragic life, it is that everyone lives a tragic life. Its just that the lives of geniuses and notable people are recorded.

    • eggi
      eggi 24 days ago

      this is really poetic, beautiful

  • Kyla Evelyn
    Kyla Evelyn 28 days ago

    Thank you for mentioning Turing's tragic end... Whenever people talk about how brilliant he was or his contributions during and after WW2, they often leave out the injustice that was done to him simply because he was gay. It's sad to think about how much the world lost out on his brilliance due to this ignorance, and I think it's important to remember so we don't repeat these tragic mistakes.

  • Tom Oakhill
    Tom Oakhill 12 days ago +1

    It is absolutely true the Alan Turing is considered the most important thinker about what computers are capable of. BUT... His designs had nothing at all to do with modern computer circuits. His computers, their circuit designs, were kept a tightly held State Secret by the United Kingdom until the mid-1960's. The UK only declassified them because computers of much greater power had been widely commercially available for years. These computers were based on the work of John Mauchly and J. Presper Eckert who designed, and built, a fully programmable digital computer with internal storage of data and intermediate results from 1943-46. It could decide what sequence of instructions to perform next based on the intermediate results. They then designed a _second_ computer which stored it instructions in the same memory as the data. NO OTHER computer did this: not any by Turing, which had knobs on the front that you turned to program the machine. Every computer to this day names the internal circuit blocks the same way that Mauchly and Eckert named them. In fact Mauchly and Eckert gave these machines the name "computer": ENIAC was Electronic Numerical Integrator and _Computer_ and EDVAC was Electronic Discrete Variable Automatic _Computer_ As Varitasium says, "computer was a job title for women" and EDVAC was an "automatic computer" which automatically did the job of these women. Turning's was called "an electromechanical machine" and was named "Bombe".

  • Xavier Bergeron
    Xavier Bergeron Year ago +1399

    Seeing the game of life being carried out in the game of life was a really impactful moment in this video

  • Saltysenpai
    Saltysenpai Month ago +7

    Cantor's diagonalisation proof, cheats by offering an impossible scenario.
    Think about it for the example to be true we must have a comple list with every real and natural number, let's say for the sake of argument that storing something infinite is even possible. Then it asks us to perform yet another impossible task in the diagonalization test of adding +1 and moving to the next number and so on and proceeeds to explain to us that when we are done the number resulting would be different than any of the listed numbers. But for us to be done in the first place the list has to end and if it end and we extract a number all we have done is make a really long list and made a number that would be on that list had the list continued.
    Here is another example so you can picture what im tryinig to say amke the same naturals and real numbers list but stop at the first natural number, now apply the diagonalization test
    by adding one to the first number (let's say again that you can be done with such a task ), what you are left with is a number that would be on that list had you continued to make it.
    What im trying to say it's that the Example plays with our minds limited capacity for understanding what infinite really means.
    Feel free to tell me if I missed anything

    • Epic Marschmallow
      Epic Marschmallow 19 days ago

      You're very (obviously) wrong. It's also pretty arrogant to assume you know better than the entire mathematics community

    • pyropulse
      pyropulse 22 days ago +3

      you are wrong because you introduced a further, unneeded assumption, which is that we need a complete list for the 'proof' to be valid
      We are working in logic, not actual computation.....
      If you were right, then integral calculus wouldn't work, and yet it does. We never 'carried out the infinite sums."

  • Naterkix
    Naterkix 29 days ago +1

    "Words is hard sometimes" is a phase I use and have used on me when something has been said if it doesn't quite come out right. I think for math we could use "numbers is crazy" but you'd prolly hafta re-title "math" to just "numbers is crazy" or something.

    • Naterkix
      Naterkix 21 day ago

      @pyropulse .......Yes, that was the joke. If somebody says something weird and you correct them with a completely correct statement, that's not really a joke.
      Also, something something capitalization and punctuation.

    • pyropulse
      pyropulse 22 days ago

      when a word is plural, you use 'are,' not 'is,' so yes, words clearly ARE hard for you

  • Yuri the YouTuber
    Yuri the YouTuber Month ago

    26:00 perhaps im missing something, but i cant see how this is a paradox, yes it gives the opposite output from what the initial h+ does, but thats what h+ was programmed to do, so in my mind this isnt necessarily an issue with h+ being flawed, but rather how it was programmed being flawed. Im not an expert in math btw, but I am taking programming courses so im quite familiar with programming stuff that ends up not working for whatever reason. But that reason usually boils down to, "It's not the machine that made the mistake, it just did exactly what you told it to," which seems to be the case here. h+ is forced into an infinite loop if it halts, which makes 'h' do as it's programmed and loop as per the program code used. However, because its part of h+, the output ends up being a halt, because that's what the machine 'h' is part of is programmed to do. And of course the inverse would be true as well. Please do correct me though, i would love to learn why im wrong and expand my knowledge👍

  • Quantos Games
    Quantos Games 7 hours ago

    Honestly the best ending you could've made