Math's Fundamental Flaw

Publicado el 21 may 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, 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
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Comentarios: 45 995

  • There was a brief moment while reading Hofstedter's

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

  • I very strongly wish mathematics was taught in a wider perspective like this video is.

  • What all of this really proves is that eventually everything devolves into philosophy.

  • I’ve been waiting years for a video like this, even fantasized about doing it myself. When I learned this, it struck me as simultaneously profound and accessible - it doesn’t need too much formal mathematics to still be absolutely blown away, and I’ve been dying to share it with others.

  • 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.

  • 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.

  • This is one of the most beautiful things I've ever seen in my life

  • 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!

  • 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.

  • I passed an algorithms class that spent weeks on Turing machines and decidability, but I didn't understand the halting problem until now and it feels like a revelation

  • 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.

  • So basically...

  • 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

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

  • 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 =)

  • 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.

  • 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

  • Teacher: Your math is flawed.

  • I believe that those theories and the explanations followed, would have been absolutely difficult for me to understand yet due to the simplicity of this magnificent work i can claim that at least i understood the general concepts