Math's Fundamental Flaw

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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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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.

▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀

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

▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀

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

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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!

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

@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.

@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).

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

That is astonishingly incorrect.

When you have two or more things that contradict each other, you need to go find which is correct and which isn't.

What if logic was not bimodal and there existed a 3rd state "not true, not wrong"? In that case your two sentences would be consistent with each other, no?

Wrong, not false. A sentence can be wrong while being correct, if you interpret "wrong" as "immoral."

@Mario Carneiro That's often how it is in proving seemingly simple concepts, most of the proof is in the setup, and making the setup robust enough that the proof is actually general and does not require some specific properties of the setup without realizing it.

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.

@Steve Sether trust me, as great as it is to know this stuff, it isn’t useful to know for a layman. Heck, even engineers barely understand set theory, and only do so in areas where it is useful.

Have you ever hear of the Golden ratio! They been looking in all the wrong places. But creation, there can only be one way!

Anything that is related to time is related to history. However, Mathematical principles, concepts or equations were never at the whim of time. Equations involving numbers are not a function of time and therefore history cannot affect mathematics.

@Craig Kallenbach I beg to differ about the public sector teachers. I've been through the public school systems, and generally most of my teachers were good. There's a few exceptions, but there always are. We used to call one of my HS history teachers "The Movie Channel" because that's all he did, show movies he recorded from cable TV. At the same time, I've had extremely good science teachers in my HS. The private isn't much different. I've had teachers in private schools that are just as bad or worse than in the public system.

This falls to the quality of your instructor, which in the public sector is atrocious. but to be fair, it also falls to the wonder of the student, which is even worse than the former.

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

@Luke Nasti math is fundamental, we just invented the written language

@Luke Nasti Math is an artificial perception of the Fundamental nature of the universe. A comfortable way to think of it would be as follows : Is colour real ? Not really. They are generated by our brains when specific wavelengths fall on our eyes but ultimately they don’t exist, all light (with some variation in parameters describing them ) are essentially the same. There are even Wavelengths are eyes can’t detect. So ultimately maths like colour is the human perception of something fundamental that (probably) can’t be fully experienced.

@Luke Nasti I despise solipsism, and do not believe that humans "hallucinate reality." Reality existed before we did, and will continue long after we cease.

Devolves it does.

@Doc S.A.T. Sorry but 1+1 doesn't always makes 2, even if you do not aree much with Popper, I do not in fact, I'd read his works a couple more times. 1+1=2 if and only if 1+1=2, a tauthology (spelling!? ) that doesn't add anything to knowledge. Mathematics means "that which is learnt" so it most probably invented by us and not discovered. So I'd be very careful to both extend it to study the world outside our mind and to believe it is "perfect" and consistent. Maths works, sometime. P.S.: 1L of water + 1L of ethyl alcohol doesn't equate 2L of liquids.

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.

There should be a study on how closely tied mathematics are with psychology and individuality. There are rules to math but not so much as there isn't a unique perspective derived from basic mathematical concepts. I sort of see math as a language, one that can even carry a bias, and be unique to one's logic, even a slang terminology if you will.

next time, do it when u feel like doing it. and do it how u would want it done.

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.

Sounds like my dilemma with creating financial market algos lol

E = 0²

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.)

@AdultHumanFemale That, you can also experience as fascinating.

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.

@Ben Botts this entire conversation / argument / etc is kinda ironic given it is beneath a video explaining how certain axioms are unproveable yet also may be true, and how along the way to trying, great insights can be made. I for one can say from my perspective as a gay woman that I didn't choose to be gay. Can I ever prove it to you? Hell no, your mind is already made up. But please, do humour me and read on. It was a choice between accepting the clear signals my body was sending and having some chance of a life where I did not loathe myself vs a lifetime of misery, gaslighting and self-hatred. I contemplated suicide a few times. Never went through with it out of a fear of death even greater than my self-hatred at the time. When stated in those terms... It's not much of a bloody choice now is it mate? Going around saying being gay is a choice ignores the fundamental unknowns of the human brain and the methods by which it functions. If you disagree, I would like to invite you to try and choose not to breathe. Go on. See how far you get before your body *forces you to comply.* That is the limitation of your ideal of 'choice'. It is fundamentally limited by factors you choose to ignore just because it makes you feel secure and safe in your comfortable little bubble of existence. I know this won't pop your bubble. You'll just respond with 'let's agree to disagree' or something like that. I'm posting this anyway. Deal with it.

@Shaun Smith You continue to drag your opinion along. End the discussion, please. You have your side and I have mine. I believe we have choice and you believe we do not. Good night.

@Ben Botts not sure if human behavior and intrinsic parts of personality and identity are matters of belief but alright bro

@Shaun Smith Reference my earlier response(s). I believe 1 thing you obviously, another. This isn't a discussion to continue.

@Ben Botts "one who makes an active choice would willingly do so if it would grant them true freedom of self" but that statement shows that we make choices that reflect our "self" ie aspects of our personality. Let's say I like vanilla ice cream but hate chocolate ice cream. Did I choose to like vanilla and hate chocolate? No, not really, because these things derive from how my brain interacts with my taste buds, which is something sub conscious. What I do get to choose is wether I eat chocolate or vanilla ice cream. But in the vast majority of circumstances, why would I eat chocolate ice cream if I hate it?

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 😇

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!

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

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!

@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.

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

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

So you watched it 3 times?

So basically... Can math prove itself? No. But math can prove that math can't prove itself.

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

Lol

It's curious that such people cannot approach more public forms of controversy in the same comprehensive manner. The most controversial subjects contain more substantial evidence to provide an indication, yet no one wants to take the responsibility and stick their neck out . . as an example, the notion of global warming aka climate change . . I'll only say -- objectivity flies out the window even for some of the most intelligent people and they lose the ability to check various sources, including independent sources . . We bow to a process of consensus building . . Normalization rather than conspiracy is how influence and corruption create and take precedence, so that awareness is hostage to the needs for anything other than ourselves to define what is real and acceptable -- the need for any form of consensus to rule us unaware, lest we ever accept responsibility.

Which unlocks stupidity

Yes it can prove it can't prove itself. By default if it can't prove itself then it doesn't need to prove it can't prove itself because it's yet to be proven. Math is based on human perception it's accurate in specific and closed systems but is much harder the more "noise" or variables are added. Theoretically you could form an equation to predict everything but in order to do that you would have to have knowledge surpassing what's already in existence hence impossible. But exact precision isn't required to be correct enough of the time for progress to result.

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

@reabsorb The same word can have different meanings depending on how you use it. For example, I could say, "My alarm went off, so then I turned it off." and in that sentence alone, the word "off" meant working and resting. In Greg's statement, the word cover has two meanings. The first being synonymous with the phrase "going through" and the second use being synonymous with the word "concealed". So in your hypothetical scenario, you're going through topics pertaining to what a teacher covers, and that information must be un-concealed for you to learn anything. Basically, human language isn't that precise, and that paraphrase was just a play on words utilizing that fact. There was nothing "dumb" about it, you just used the definition of cover as an absolute, which you're not entirely at fault for since human language depends on our perceptions of different sounds/symbols linking to the same ideas.

@Gregory Zak this is a dumb quote. what if i am in search of learning about the things that a teacher covers? then it IS about what they cover. and if i learn more about the UNcovered , thats fine too.

It’s not about what material the teacher covers, but what they uncover Paraphrasing Walter Lewin

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.

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 RSloft comment. And nobody reads past the first lines anyway, I'd be surprised if someone reads this last line.

@Ripple Reader Lol, a bit! Were you?

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

@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).

@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?

@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.

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

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

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

You missed the point of the video kiddo

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 ❤️🇨🇦❤️

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).

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.

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.

@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

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

@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.

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

@Ron J you'd be surprised...

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

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.

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

"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.

"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?

@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.

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

Very well done, sir. Great presentation of ideas that I was vaguely aware of, but I had no real understanding of. Thank you.

I just came across your videos today, beginning with "The Big Misconception About Electricity". That one answered a question that I had debated my college roommate about 65 years ago. Never heard a good answer about it until today. Brilliant -- all of them so far.

I would recommend you to watch the response video to that by ElectroBoom!

He was kind of wrong in that video about electricity. There are several responces to it on youtube.

This video is really brilliant. I have seen it multiple times, and each time it brings me something new. Thank you Derek!

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

Socrates agrees with this statement

@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.

I suck at math and this is awesome!

Not always

This video is CRT SEL grooming!

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

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.

I mean that is how the halting problem works

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!!

@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.

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

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!

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.

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!

Game numbers moment

@Alex Hetherington I don't think we have any reason to believe that it's an infinitely nested game of life, I'm not sure if that's possible. All that's been established here is that by setting the correct parameters within the game of life, you can create the GOL within the GOL. Assuming you can set the same parameters within the GOL you created within the GOL, you should be able to great another GOL on top of the GOL you simulated. If that's true, then you hypothetically could create an arbitrary number of nesting levels. Actually, if my understanding is correct, then if you can set the parameters within the simulated GOL, you should be able to create an infinite recursion. But that's a big assumption. It may be, that because of the way it's constructed, the simulated GOL has limited possible starting starts, which might preclude it from simulating another level. I don't know enough about this topic to give an informed answer.

How is that amazing

@Jan Arne Wirths how. how that is just you. what are you trying to say.

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.

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”.

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

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

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

😂😂

@Matthew N AMEN!

@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.

@Veritasium

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

Thinking back, I remember responding to a friend as a kid that kept telling me everything was possible : If everything is possible, Is it possible to find something impossible? Everyone thought that once right?

@A Czech Man Going His Own Way What reality and whose imagination are we talking about? I mean any such logic is pointless as it only leads to more questions - all without an unequivocal answer. So, like the snake eating it's own tail, it ends at the beginning.... and the other way round :)

Do you mean possible as in possible in reality or as in possible to imagine? Not sure if that question is legitimate, though (or a false dichotomy).

Can God create a stone so heavy that even he can’t lift it? 😉

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.

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

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.

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

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

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

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).

🤣🤣🤣

@Nobody Knows 😂😂😂😂😂😂😂😂

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

This was just great. I won't pretend to understand how the Godel number logic but it was fun stretching my brain. I'll have to do some more research on that one. Wow, some of us are really clever aren't we? Most of us, not so much, but some of us just are amazing.

Besides the great concepts, fantastic animation/illustration 👌

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!

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

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

Took words right out of my mouth.

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!

@Peter Codner Way to be needlessly pedantic.

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

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

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

One of the greatest videos on RSloft. Congrats, brilliant.

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

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

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

Some poor kids are about to be forced to watch this

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

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 :)

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

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

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

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.

As a scientist, at the late stages of your Bachelor's or in the beginning stages of your Master's program, you become familiar with two concepts. *Deterministic* , and *Probabilistic* . The more you study Physics (And hence math), the more you realize that inadvertently, you are moving _away_ from deterministic and more _towards_ Probabilistic. If your luck is absolutely and utterly rotten and you go bellow the _Nano scale_ , in certain branches of electronics, telecommunications, photonics and Physics, you drift into the Quantum realm where deterministic ceases to exist, and it all becomes probabilistic. Then you realize it's foolhardy to always look for *_certainty_* . There are some things that you can't prove with certainty, but have been demonstrated to be true. To turn into a more poetic avenue than math and physics, love is such an example. You will never be able to prove you love someone else, you can only demonstrate it.

You are just rambling. None of what you said has any connection to Gödel's incompleteness theorem whatsoever. It just made you feel emotions, and those emotions were felt by you in the stuff you just listed, and so you naively related the two, which started your rambling, and then ended on 'love,' for some reason. I literally just defined 'prove of love' to 'ability to demonstrate it.' Therefore, you can prove your love for someone.

The schrodinger equation is 100% deterministic; it determines exactly how the wave function evolves over time..... also why they hell are you doing a masters program? I went from physics to theoretical physics phd program; if I want a masters, I can do an 'en route.' Why waste the time tho?

"Beings"- that is to say real things -are fundamentally acts and not static "clear and distinct" ideas. Therefore there is always to some degree something or other we do not know about them no matter how much we actually experience them. Thomas Aquinas identified be-ing and beings as fundamentally act/acts hundreds of years ago. This was how he was able to conclude that who/what we call "God" can only be "actus purus" (Pure Act) and thus not able to be known directly by a human mind, but only by indirect inference. So the next time a earnest Christian tells you he can "irrefutably prove" that "God exists", not only can he not, but also he's using the wrong language. "God" doesn't "ex-ist" because Pure Act does not "ex-" from any other being. Rather He/She/It simply IS. Which brings us back to the beginning of this comment: "Is/To Be/Being" is an act...and to the point when human languages get really clumsy. :)

This hit very hard mate, thanks for your comment.

wow

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

@_Nines pi = root (g)

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

yes, the more practical

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.

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

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

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

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."

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

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

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 RSloft, 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?

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

@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!

@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?

“Corrupter of youth” 😂

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

Oh Reginald.... I DISAGREE

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...

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.

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

This was informative to a degree.........thanks for the video man!!

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

@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.

@Ward Fadel So, or therefore, what?

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

@Fred Esch nice 👍

The chart scene looks lile Flash MX discontinued

@veritasium, what do you think about constructor theory in regards to incompleteness and in general constructor theory on its own?

Sad that Gottlob Frege wasn't mentioned, he set the foundation for the formalist logic actually. But nevertheless a good video, thanks Veritasium for your efforts!

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

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

Masterpiece of a video

You really do be popping up in strange places.

@Dr. Michael J. Stefano jeez calm down with the caps

@P. Chakraborty he is just expressing what he thinks about a video, no need to be so critical

@THINK PATH Please Stop promoting your own channel in the comments

Hey its the robocraft man

"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.

@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.

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

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.

@alcatel😊😊😊😀☺ ah, understood

@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.

4 is a much better approximation.

@Bug Dracula uhh 1/7.14 if a person duh

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

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👍

Wow man this is some of your best work yet Tom top notch!

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

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

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

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

This is the perfect medium for this stuff.

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.

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

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.

this is really poetic, beautiful

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.

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

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

Godel's friends: "No one's trying to kill you Godel" Godel: "You can't prove that!"

Haha

The irony that he was a living Godel's theorem. He died, so that he wouldn't die, but the statement 'he wouldn't die' is false as he did die. But he died because he didn't want to die. Damn.

lmao

@Lavabeard No, that would make him become his own wife

@Mooney Makes think he just stated a fact. The guy died from something else he would have said that too.

Very nice video; however, I am not sure it is entirely true to say Math is "flawed" by any means. We, humans, wish to have "certain answers", "clear decisions" and a "sense of security through determinism", even eternal happiness after life (if exists). Yet, nature and existence (whether from a divine source or from itself) do not need to comply with our wishes. Lack of "compliance" does not make "Quantum Acts of Universe" or Mathematical rules (as far as we know them) "flawed". It could be our understanding of those could be flawed, or even if we have a complete understanding, the fact that existence is not deterministic from our point of view, seems, hardly "flawed" to me. The best we can do is to put our best efforts into furthering the science, "decide" as many problems as possible (for peaceful ones), and put a smile on people's minds and hearts, if possible, while they are in this world (not wait until afterlife) :). As a final note, as a cryptographer, there are huge gains from the highest level of entropy, sometimes, pure randomness gives the simplest proof of security. So, random uniform distribution is not that bad after all ;)

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

Amazing and informative. Thank you.

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.

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".

Continue to be among the best videos out there!

Imagine being an actual genius, making the modern computer possible and shortening a war by 2-4 years, and _still_ not getting a gay pass from the stupid government

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

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

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

"How about you just hire another barber?" Said the engineer

He said "Godth"

Ahaha!

That's a manager answer😊

@Smitology a barber can barber a barber while barbering a barber

@thx epsilon thank you really appreciate it. Makes sense to me 🙏🏾😊

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.

Incompleteness is not a flaw! It is evidence of creativity: you need outside intelligence to create a true statement in an algebraic system. Without creativity, what good would mathematics really be?

This is how you tell a story! Bravo!!!

Sometimes the answer you find for a question is an answer to a question you never asked

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.

@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.

@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.

@Peter Codner Damn you sound like a nice guy

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.

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

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

Thank you for giving me a headache because my underdeveloped brain took too much info in at once even if explained extremely well XD

Our understanding of math may be incomplete but not math itself. It's waiting to be discovered / invented.

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? 😀

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

I came

Poolooopooooooooooooooooooooooo

Yes

It was a really impactful moment of my life in general. Due to the music probably.