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.

As an engineer, numerical methods saved my sanity.

You’re a “mathematician” but still can’t figure out the golden rule? Stop it.

But don't just all of those unsolvable problems have the same thing in common: they somehow reference themselves. So by being able to prove that all unprovable problems are somehow selfreferencing you would have a prove for everything

A "working mathematician" in the sense that you are paid to do sums? You experience fear when you cannot what you call " solve a problem"? Of exactly what are you afraid?

@Ern de Che Proven to whom and to what standard?

Fun fact: Einstein and Godel were close friends. Einstein once said later in life that he kept going to the Institute for Advanced Study (where they both had a position) just to go on walks with Godel. Godel once found a solution Einstein's field equations that he presented to Einstein as a birthday present. There's also a funny story where Godel applied for US citizenship, but his paranoia led him to conclude that the US constitution is inconsistent and allows for a dictator to take power. He then tried to present his discovery during his citizenship test, but the judge, a friend of Einstein, thankfully cut Godel off.

@Gerrie van Boven why you bring this up here is beyond me and completely idiotic. So far so bad. Specially considering the previous one. I'm not a US citizen nore do I live in the US. I have the distance to have a non biased opinion, still a strong one based on facts , and facts only. I do agree, however, that Biden is too old for the job. Senile is going much much too far howevervand shows a complete lack of objectivity on your part.

I would of liked to have heard from Godel how the U.S. Constitution would allow for a dictator...

@Saaber Shoyeb pulling out is a good thing but it was a rout. Trump was pulling out of many areas in a much more balanced manner. Although as a Muslim, I am glad that Sharia is now law in Afghanistan.

Democracy as a concept comes with it's optional self-destruct button. If you want to make sure that button is never pressed, you would have to commit undemocratic actions. In Turkey, what I mentioned was somewhat used to justify the 1960 coup that overthrew a democratically elected administration and manufactured a new democratic system. It was accepted as common narrative that a coup can be justified using the argument of the self-destruct button. Now it is believed to be unjustifiable. Which makes more sense. Wether you like the outcome or not, if you truly support democracy, you must allow the system to self-destruct democratically if the people wish to do so.

Amazing story!

I can't get over how good this video is.. I'm 3/4 into my Introduction to set theory course as a math major and I have watched this video before the semester and rewatched it now and it just explains so many advanced concepts (that aren't even in my course) so good it is incredible

This is a great video because most people who aren't deep into math or studying at university never get these problems under their minds. These problems and issues with artificial intelligence programming in computer science are what initially led me into philosophy.

But as a 14 yo, I think way too much of this that it s destroying me. It distracts me from my real life because of all the random deep thinking moments. Idk if that's a good thing 😕

@Jordan Eisenman I'd be interested in more details regarding your areas of interest.

And philosophy led me to math and data science. Weird how they wind together.

@Kevin Austin No and no, learn mathematics instead of armchair philosophy.

@Kevin Austin Do you think Mathematics is more like a science, more like a language, or more like a logical system? Do you think Mathematics is one thing or a plurality of things?

I'm glad I was an engineer. I learnt to use advanced mathematics to build things but I never had to worry about this stuff thank goodness. I think it's kept me sane.

YKW‘s glad he‘s an engineer aswell..but Holland

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

@Happy 747 you

@Happy 747 you u

@Kathan Shah Uuuuuuyu

@Kathan Shah 7

@Pineapple you know uiuiiiiu8iiiiii88iii88iii7i UI is 7uu

This is the best video I have seen in a long time. You litterly blew my head so many times, the video is amazingly made and it is so interesting even for a teen like me who doesn't like school math too much and doesn't know math, school should teach us stuff like this because students will actually have interest in classes

This stuff is usually taught in courses like real analysis, discrete math or logic in universities, unfortunately not with all these cool animations :(. The course itself doesn't have so many prerequisites so you could potentially teach it to whoever knows what a limit or an integral is, but it's so difficult to understand certain topics and theorems that if you tried to actually teach these things to every high school student 99% of them would stop paying attention. And you can't explain some concepts non rigorously because it would be a mess to understand correctly what is what. For example you can take the part where he talks about "countable sets". But what does "countable" actually mean? It could look obvious but it's not. If you don't explain it in a formal and rigorous way, you wouldn't understand why Q is countable but R is not. This one was probably the easiest part (since countable is still a concept that you could grasp without formalism), but that's pretty much it. I would recommend to find the beauty in things that you are currently studying. Ask questions to your teacher if you don't understand why something is like that, be curious. You will eventually reach this part of math and it will be great if you will still have this curiosity! Or you can go to YouTube channels like "the bright side of mathematics" (probably the best one that I found so far) and learn those topics right now. If you like the way he explains things and cool animations I highly suggest 3Blue1Brown, another great yt channel!

Literally literally literally, you know? Literally, literally! xD

I literally couldn’t agree more

The lineage Cantor->Gödel->Turing has one additional ramification: Rice's Theorem (1951). Wikipedia does not do it justice, failing to give a layman explanation of it. It is a very important theorem which generalizes Turing's answer to the Halting Problem. It predicted why UML would fail, why code verification software is snake oil, why some programming languages, contrary to marketing will remain hopelessly slow... Very useful indeed. Ignoring it has wasted billions.

@Sam ahahahah.

Code verification is not snake oil, because they do not verify arbitrary powerful languages and programs but specific ones and they check certain properties. I would discuss this with an expert before making such claims. Theory and need in practice are not equivalent.

@Jimin Park, cannot agree more with you. The extreme promises of code verification tools are the issue. The problem is already in the name.

You're ignoring a key point: a program doesn't have to be perfect (complete) to be *useful* . Rust's lifetime analyser is an example; while it can't *always* prove that a Rust code is memory-safe (hence some theoretically safe codes will not compile without using unsafe blocks), it nevertheless can gaurantee, for some but useful cases, that such code don't contain a bug related to memory-safety. Even this incomplete code analysis tool turned out to be very useful and practical for removing a major class of bugs. It's similar for type systems, code verification tools, and description languages. Rice's theorem does tell that a perfect solution doesn't exist, but it doesn't tell at all that a useful solution does not exist.

@Vincent Lextrait I guarantee that Google employees use sequence diagrams. Maybe they're not officially using all of the UML standards, but there's only so many ways to communicate software design. What's your alternative? Do you just draw simpler diagrams and say they're not UML?

I have heard about Alan Turing or the "Turing Machine" at least 500 times in my life. And, this was the FIRST TIME anyone EVER mentioned how his life ended. The guy saves countless lives, and because he isn't attracted to females, they destroy him?

Yeah, I was pretty neutral on gay rights, but this was a major window of reconsideration. It was a really big tragedy what happened to him.

Britain in the 50s looks as barbarious as old civilizations.

@Art Donovan About him yes, but even I've noticed that untill the last decade and a bit, any story involving him rather than focusing on his life, generally omitted the parts that this video and The Imitation Game handled well, I only really knew as early as I did as I was told by others in the LGBT communit. I've found most documentaries, even British ones from 15 years ago plus and still some YouTubers now, generally don't mention it unless they're LGBT them selves.

@Rehak Mate whats wrong with you

@Art Donovan Yep, never heard of his demise. But, I don't know why that's hard to believe. For, whilst he had a impact on a lot of things, he was pretty consistently a secondary figure. If you do research on WW2, Computers, Free Will, Transistors, or Code Breaking, Turing will definitely be mentioned, but it's not like his LIFE STORY is relevant to those topics. His contribution/education/background is mentioned, and that's basically it. His personal life isn't congruent with those topics.

This was very enjoyable to watch! I wish everyone would find this sort of thinking invaluable and necessary to better understand ourselves and our world. Thank you for fostering critical thought!

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!

@Henry 1 ppöppåäppäääp

Åååå

And it's all running inside a game of life!

@asdfasdfasdf1 check fundy's minecraft video about that, that is possible now

Wait until you see the game of life running inside the game of life running inside the game of life.

That was good. I had the good fortune to live in a communal house with a math student doing his PhD. I decided it helped to be really weird and a bit crazy to be a math type. His PhD, he told me, was to be about mathematical applications to the forest industry, very useful where we live

In retrospect, I just love how the loop was closed, or rather elevated in a loop synthesis. Math is a derivative of Philosophy via humanity's curiosity and inquisitive propensity, whose main aim is for the betterment of mankind and society. Then the imperfection of the derived evolved to achieve the aim of the source, furthered by the same inquisitive propensity by its recipients.

Maths is just amazing, unlike what we learn in school/college as it’s not taught in a interesting way :|

I remembered my Symbolic logic (Russellian logic to fuzzy logic and so on) course from college to my graduate degree. Damn I didn't know the philosophy of logic and philosophy of mathematics behind such course. It makes me want to tackle this areas even if I am not good at it. (Coming from someone who practice continental philosophy). Thank you for sharing this.

Can you do please the Gettier problem, the problem of Criterion, the problem of moral luck, the problem of evil, the omnigod problem, the mind and body problem, the problem that we cannot know the external world but we only know our experience (Immanuel Kant's Noumenon and Phenomenon problem), determinism and freewill (although this has been solved by soft determinism but it still begs the questions that leads to the problem of criterion) these are the topics that are unanswerable in philosophy right now. I am not sure if some of them has been answered but when I taught these problems to my students they were always left hanging hahaha.

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

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.

Teacher: prove it

@Baconstrip so you are saying it’s actually more complex than what math can actually determine therefore defies math? Interesting

@Aaron Durham What? Reproduction doesn't defy math... All the atoms and energy used for reproduction are acquired from external sources, whether that be food, water, sunlight... 1+1=3 is an atrocious oversimplification of a much more complex phenomenon that exists well within the boundaries of math and physics as we know them.

@minnowpanda > Maths seems to be an intrinsic co-emergent expression of lifes potential for semiotic expressions of mentality, one of universal nature's enabling meta-logical principles of being.

This has become my favourite channel on Youtube, so thank you very much for uploading this great content. The fact it has 15M view at time of writing gives me hope for humankind.

I cant stop watching this. I actually wanted to sleep. Never ever would i read anything about mathematics, but damn Veri makes it so interesting and easy to understand that it makes me feel good to be able to understand University level of Mathematics.

This video is truly a masterpiece. Great job!

"Alan Turing" was a great man but his death was unfair. He should have left the country because it did not deserve him. Imagine all the things he could have discovered, all the lives he could saved, all the things he could have done but he had to die of humiliation. All because some people thought that they had the right to meddle in other people's sexual preferences. I don't care what other people say about this opinion because any arguments that you give will mean you are undeserving of any device that came from his discovery and innovation.

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

Definitely more out there to give you headaches if you enjoy them

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.

Hi Derek, As a Hungarian mathematician, I feel somewhat sad because of you missing to refer to Janos Bolyai as one founder of the hyperbolical geometry.  Though we know Gauss worked on the subject, he did not publish his results unlike Bolyai or Lobachevsky. Therefore,I would not deem him as significant contributor to hyperbolic geometry. By the way, the stoy where and how Bolyai worked on his theory and how he published it - in short, it became an Appendix (we Hungarian mathematicians respect this word) to a math book written by his father, for which he spent one year of his income - and what was the afterlife of the theory, how it reached the French Academy, would deserve a video on its own, I would say. Best regards, Tamas

I'm not a mathematician, but this is exactly what I was about to say! Thanks for beating me to it (and putting it much better than I could have). Still an amazing video overall, though. :)

You have been noticed Hungarian mathematician

This was awesome! The hardest class I took as an undergrad at Harvard was a philosophy class (yes, philosophy) that basically covered all of this.

I went to Harvard and took that class and it was easy af

Yeah you went to Harvard, good for you

The beauty of it is that math prefectly represents its imperfections.

The definition of ‘perfection’ is the colloquial one in this discussion. A problem without a tangible solution is perfect in that one’s understanding of the same agrees with the fact;; ergo, a good man knows his limitations.

You plainly have no idea what "perfect" means; it means finished or accomplished, not without flaw- whatever you take to be a flaw. First define your terms.

It may be that what the authors are struggling to convey can be boiled down to: A mirror cannot reflect itself-which is indeed axiomatic.

For a deeper dive into the Foundational Quest of Mathematics, you can read the graphic novel "Logicomix". It tells the story from the perspective of Bertrand Russell and is very well written.

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

@NeoFrontier Technologies you're going on philosophical tangent

You can not prove something which must be comprehended. You can not force comprehension. The individual must choose to comprehend. Life is choices - personality choices which effects intelligence and perception of reality.

@Magnus Kramnik engineer learned from mistake, made a better bridge, and by doing so, found the better math equation that mathematician was trying to "validate", and now math guy wasted a whole bunch of time trying to prove something that you can just experimentally demonstrate is wrong? woah wait what? And now mathematician has new job... "validate" this new equation the engineer found, which, already works demonstrably well in practice. Yeah, well he decided to study math! I mean, we cant let him just.... like, just give him something to do man... feel sorry for him now.

LOL!!! Reminds me of the history of Legacy schlockware AKA bloatware which made Gates & colleagues hyper-rich while addicting the rest of us to ever-more patches, upgrades, hacks, attacks, ransomware, botnets, cyber-war, etc.

@AloysiusVo Of course you do. Unlike in mathematics, though, the proof consists of proper references to the handbook, the standard, and relevant legal documents.

Read a book as a kid in the 80's called Gödel, Escher, Bach, and it really opened my eyes.

Dr. Hofstadter is a real genius.

Wow! Understood very little, but nevertheless appreciated the brilliance of the presentation! It seems to me that mathematics is all about finding patterns within numbers… if I’m correct, then I’m intrigued… what are some of the patterns hidden within the recesses of math? And are there any impressive examples?

@SaiLens yup, you found it, and yup, stuff in nature obey this sequence. its not just random xD

@Charles Trudel It's A Rough Guess, But Is It 13 Followed By 21?

​@Charles Trudel Brilliant! Thank you!

Fibunacci is one such example: 0+1+1+2+3+5+8+? I let you find what follow 8 here... thats one of the weirdest sequence of number ever.

That just blew my mind! 🤯 literally speechless! You sir are a very gifted teacher and a brilliant mind

It certainly provokes a number of rather mindless clichés of the sort beloved of those that struggle with language.

I nearly failed Grade 11 cause I got obsessed with a exponential list of number . That i just couldn't solve an equation for and thats what i worked on every day through every class after school . it consumed me . 🤣 i filled up a pile of note books trying to solve it . Eventually my physics teacher asked to see it and told me to give up cause its impossible to solve and predict.

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

What does you having a PhD in computer science have anything to do with this being an intro course to discrete maths? Also congratulations on completing your PhD

You 'are' a PhD are you? I doubt it, you 'have' a PhD in lying

Intro course? Lol more like PhD level

how do you know when someone has a PhD? they'll tell you.

I had a question if barber can't shave himself than why doesn't give his blade to the 2nd barber in that way they can shave eachother's beard without shaving their own beard by themselves. Explain me please.

My only regret is that I can't like this video separately for each and every time I watch it. This is my 5th or 6th time watching it in its entirety and I'm sad I only get to like it once. Thank you so much for giving us such great content in an easy to understand format.

Just coming back to this, it's the best video on youtube. Will probably watch a 3rd time in a few months. Legitimately a masterpiece of a video and very inspiring

I"d just like to point out that 'there will always be true statements that cannot be proven' (one of the first sentences in the video) is not quite what the incompleteness theorem actually claims. What it says is that in every sufficiently complicated formal system, there exist undecidable arithmetical propositions. That is similar to the difference between saying 'There is somebody, whom nobody knows' and 'For each person, there is somebody whom they don't know'. Undecidability characterizes formal systems, not particular propositions. Also, Godel never claimed that such undecidable propositions are true. They may be true, but we could only know that if we knew that the formal system they belong to is consistent, and we may not know that.

so "there will always be true statements that cannot be proven' ; so or therefore what? Of what syllogism does that form a premise? Is it not axiomatic that a mirror cannot reflect itself?

Yes, one of the particularly mind bending things about a Godel sentence is that you can add the axiom that it is *false* to your formal system, and still get a consistent system. Normally people (e.g. Roger Penrose) consider it obvious that these statements are true, but you don't necessarily create problems if you take them to be false. Another way to phrase Godel's theorem would be... "there's always a remaining degree of freedom in your formal system; another statement or its negation that you can add as an axiom without causing inconsistencies".

Have given this lots of thought over the years. Godel's Incompleteness Theorem seem to imply that, in principle, you can't know everything... Somewhat disturbing to a mathematician. :-)

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

Extremely well done and an excellent explanation of difficult material Poor Frege and Wittgenstein - lately they have been getting left out.

I come to this channel for answers and most of the time I leave with more questions and these questions get stuck in my head all day. They are the type of questions I could have never came across in my life and everything would have gone smoothly and I wouldn’t be having this headache yet I here I am.

Lucky you. Blessed are those that find themselves with more questions than answers although another put it as "Blessed are those that hunger and thirst after righteousness."

This is such a beautiful idea. Literally makes me emotional

I wish I had you as a teacher in high school. Would’ve actually paid attention.

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.

I remember going through all this at Uni studying Computer Science.

This is an incredible watch, well done!

I just learned why I hated maths so much at school, my teachers were so uninterested in maths. This guy is a brilliant maths teacher!

If this was how school is taught, I would enjoy it so much more

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

@Linus Fu H Hhhh ⭕ ⭕ নমস্কার

"Right" meaning likeable or desirable? Maybe but for whose purposes?

tell that to my uncle the pedo mathematician

Hahah ya

@Umar Ahmed Still doesn't make sense to me. Infinity for me is an abstract concept. If something is countable it is not infinite by nature. Infinity is that unreachable number to which no addition, multiplication by a number greater than 1 and subtraction can be performed unto. Or when it is the denominator, the result is a perfect 0. It's just an abstract mathematical limit. Once you reach it, everything is "frozen" or "locked" mathematically. Cantor proof is nonsense. And no Hilbert thought experiment doesn't work because you can't perform a subtraction to the last room once it's occupied. Infinity - 1 doesn't exist because infinity is disconnected from the countable series. So you can't reach it through infinite addition. Such a set {0,1,2,3.....infinity} cannot exist.

One of the best YouTube videos ever. Wish I could have seen the extended cut!

May I ask, what software do you use for creating infographics like this?

I love the concept and discussion of infinity.

An absolutely superb video, as always, but I have to take exception to the title. To describe something as flawed means you must start with a preconception of what something should be, and then find that it falls short against that preconception. To describe the flaw as fatal you must be assuming that the thing has a ‘purpose’ which its flaws prevent it carrying out. Surely we have learnt from Hilbert (and we learnt sooo much from Hilbert), and the experience you describe so well, that mathematics is just what it is. We can discover things about it, but if we have created preconceptions of what it should be like then that’s our problem - it doesn’t detract from the beauty. And if we have intentions of utility then that’s our problem as well (thank you, Hardy). Mathematics can never be flawed. But our preconceptions of what mathematics ‘should’ be often are.

The term for which you may be struggling - and are spot on about preconceptions is "hidden premise." It is a preconception or assumption that borders on religion, which can be defined as any set of related unquestioned assumptions presumptions, preconceptions or norms.

The title is click bait, Veritasium jas a video on click bait you might wsnt to watch lol

Can you elaborate on who this Hardy is and what his contribution to the conversation was that you referenced here? I would be very curious to learn more.

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!

@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.

@sdfsdgsdfsdf23423423 Yep. Glad I'm not alone judging along similar lines.

@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).

same. exept for the card parts i didn't tried to understand because i remember i understood the puzzle already and do not need to do it again lol (heahach))

@sdfsdgsdfsdf23423423 You make some interesting points; can you elaborate further on your last statement about the technology world flowering in spite of Godel? This video made it seem like his studies inspired Turing and the development of computing systems.

I am not a scientist, but I'm a science enthusiast, and I've always thought about math and physics as closed, incomplete systems that couldn't fully explain everything. It would be like a goldfish trying to fully explain his water bowl while being inside of it. It would have only a partial visual of the bowl while only us humans watching from the outside could see the whole. At the same time physics seems unable to explain the universe since we are inside of it and we are limited by intrinsic factors such as the speed of light which creates edges of the visibile universe. Tell me what do you think!

That's a really interesting proposition Riccardo, one I have often thought about too. I also think that our physical brain is a limiting factor too.

@Peter Codner Whoah man no need to get semantic. A scientist is just one that participates in science. Science (As a verb) simply being a process of creating or applying information.

Whose science of what? You use the words but have no idea what you mean by them or seek to convey when you use them. Surely anyone that has any direct immediate personal experience or knowledge of anything is a scientist is he not?The word science coming from the latin infinitive sciere to know, and its first person singular scio -I know, thus science is no more than a species of knowing or knowledge which can only be direct immediate personal experience, as in you know that you have a headache. If science be not knowledge of sorts, what the devil is it exactly?Self evidently all know ledge must be the knowledge of something in particular by someone in particular, or by " science" do you mean de jolly clever secrets of de jolly clever men in de white coats?

As someone who has studied both math and propositional logic at a high level (as well as some computer science), this video is truly a treat.

I’m relived. I now know why my math problems in school were seldom correct, they were unknowable….. (at least to me)🤣

This was an amazingly interesting topic and that says something because I am terrible at mathematical concepts.

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.

wow - good one -- though it is difficult to grasp it all. I think about these things in relation to language and law - always incomplete! And the Turing ending always makes me sad.

I wish I had all this knowledge on youtube available to me when I was a kid.... might have influenced me on a different path... who knows now

Well, let's say that science endeavors to explain everything (the "theory of everything", the "one-inch equation" etc). This video suggests that you'll never get there, and not only that , you'll never know for sure whether or not you'll ever get there. BUT, you learn all kinds of things that transform your understanding of the existence, and therefore the Endeavor is totally worthwhile. Like that old adage, life is not about the destination (the destination, of course, is death), but about the journey (everything you experience along the way).

This is such high quality content keep it up

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

Fascinating, I wrote poetry as a youngster, and eventually arrived at a similar solution to exactitude. The symbols I developed differed of course, but were extremely accurate in their logic and detail. A much older young man loved my poetry, and discouraged me as much as he could, as he could no longer read them. It was just another avenue of expression to me. Kids can be unbounded by formalism. I was.

It takes two different components to create a continuum, one physical, the other electrical. This way they work together in order to complete the one continuum.

I was really studying Sets in my coaching and now realise everything taught to everyone is wrong, reality is not taught..

Dude, you're a freaking inspiration

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

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.

@Baalaaxa You cant expect anyone to live up with their standard, especially when they are in doubt.

This kept coming up in my suggestions, and at first I didn't watch it because I already read Godel, Escher, Bach 40 years ago. But anyway, every time I saw "Math has a fatal flaw", it made me think, "Space has a terible power".

"I don't like math" Veritassium showing that there is much more mathematics out what you call math, and that you actually like that.

This video makes my brain hurt, and I am literally the closes thing you can get to proving that human brains are computers in and of themselves.

I have always loved science but despised math. I took to math fine, understand it easily, but find more annoyance from it than anything unless I am using it directly as a tool in my scientific works and learning. I also find myself to be philosophical in thought. Which is another reason I do not find pleasure in math. For I hate the idea of definate. That takes away the beauty of imagination. Which is the biggest angst I hold against math. It, in how I have been taught so far, is definite. Now, I started watching this video with an expectation of pain from the examples of maths claimed definity. But was amazed at what was shown. And of course thankful to learn of this. I never once expected math hating self to say this, but... that math was interesting, thought provoking, pleasurable. How very interesting. Wonderful.

Me: *failing my math class* Veritasium: “they could be something like the twin prime conjecture” Me: go on...

@jorgepeterbarton What do you mean? That infinitesimals are being represented by other numbers? Actually, to prove 0.9999999 = 1, you need calculus. And limits aren't actual arithmetic operations. All that the 0.99999 = 1 proof shows is the limit of 0.9999999 is 1. It may be close but not 100%.

@Shambhav Gautam infinitessimals. No you can prove 0.000....1=0 easily enough, hence never actually writing the 1 after the infinite process of zeros is a really quick way to look at it An infinitessimal is simply unwritable its a problem. Imo they are still iffy though, perhaps 0.3333...=1/3 is missing an infinitessimal, but assuming it did equal a third then we could probably say 0.9999 does equal 1. Despite 0.9999... being 1-0.0000....1 and this being a proof it doesnt seem to actually apply as such if we subtacted zero all along. Well the proof is usually more like take 9/10 of that number, its 0.9, then bring up to 10/9 so its either 0.9999... or 1 same thing. Still cant help something arbitrary happening- if no i finitessimals are involved then maybe something can be made that DOES express one? Anyway infinitessimals exist and yet they use a different framework. I know some disagree with the recurring 9s as for me dont take my word fr it in the slightest, someone please correct me.

😂👌

@Poorvi Singh 😝🤪

@Tony Suda insane

What happens if you take an infinity that runs clock-wise and then put it up against another infinity running counter clock-wise. Then run more infinities clock-wise and counter clock-wise (every other one) until you get a perfect sphere of them?

Amazing. They should give prizes for this caliber of videos.

This has huge implications for A. I. and the concept of infallibility of science, technology

Videos like this make me fall in love with math again... thanks 😊

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

great video. i think i have to watch this a few more times to understand as much as im able to.

The Game of Life bit in this blew my mind open to the idea that every biological system is just a system of systems of systems, all the way down. Society is our attempt at organizing the next tier of life, and if we intend to be successful, we need to seek answers within our own microbiology. If cells are like individuals and tissues are like industries, what does that make DNA?

My mind is blown. I am experiencing a paradox of confusion and awe of beauty at the same time. Just wonderful.

great video indeed, and amazing subjects in all the videos.

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.

@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 learned about this from the book, "Godel, Escher, Bach" by Douglas Hofstadter, back when I was doing my Joint Honours Math and Computer Science degree.

I simple solution that stuck out to me from set theory on, would be to simply eliminate self reference. Instead of saying that there are no sets of sets, just say a set can't contain itself. (I am not an expert in this field, so please explain why if I am wrong)

@Noel Lundström So my idea does exist! Thank you very much for explaining that to me.

@Hyrum Leishman your idea is called the well foundedness axiom and it is an axiom of the modern formulation of set theory (Zermelo-Fraenkel axioms). It just wouldn't make a difference if you added it to naive set theory (the set theory where the set of all sets exists) since adding another axiom to an already broken theory wouldn't really change anything. A contradiction exists here already so adding another axiom won't really change anything.

@Noel Lundström But does the set of all sets contain itself? Also, please explain why this doesn't solve the paradox, because I would really like to know if my idea wouldn't work.

That doesn't eliminate the paradox, in naive set theory the set of all sets has to exist.

There is another really weird infinity numbers property. That always blow me away - Galileo's paradox

This is brilliantly explained! These problems are what got me interested in set thwory and the axiom of choice! I was lucky enough to run into these when I studied The Banach-Tarski paradox (an interesting topic for a video, if you have not covered it already)! This both shows the brilliance and power of humankind as well as its darkness and self-destruction. Two brilliant minds lost early from something preventable. Turing's case is particularly frustrating and ridiculous. The man literaly saved hia fellow countrymen lives as well as other countless lives. He shoukd be considered a hero, at least for Britain. But no, they drove him to kill himself over the most stupid thing ever, being gay. I don't care it was another time, shiw me where is the logic and humanity in that. How in the world is him being gay in his personal life the government's problem. It is not like he was indoctrinating children or somethin (even then he did not deserve it). Truth is we did not deserve Alan Turing. Britain certainly did not deserve him. And so many others have not had their stories told or their potencial shown. In his very short life he changed the world forever...imagine what he cluld have done if he had not died at the government's hands. And what Goedel could have further acomplished with proper mental health treatment. Deeply saddening.

Thanks for reminding me of both the dumbest and nerdiest joke I know: Q: What's an anagram of "Banach-Tarski"? A: Banach-Tarski Banach-Tarski

@Lil Yeet theory?...sorry?.... spelling error w next to e...thorry...🙃

@Lil Yeet What?

Thwory

This was brain hurting, I had to go back and forth multiple time to understand things, but i am glad this exists.

Far simple quicker and clearer to pose the question: Can a mirror reflect itself? It is quite simply down that that all those convolutions boil. Nothing more.

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

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

@Gabriel Carvalho you can like it now :)

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

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

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

So I’m here a person with a normal iq and I feel totally out of place in this comment section of total geniuses, hello from the other side my friends .

May I ask something? If the g card is true without proof, isn't that means that it is an axiom? Futhermore, wouldn't any axioms be enough evidence that Mathematics is incomplete?

Bravo. A terrific presentation reminiscent of another ground breaking, terrific presentation of how we got to be where we are… James Burke’s “Connections.” Your analysis went micro and short time span… another proof of the “infinite in-between.”

Is there a "we"? Who experiences it? If it cannot be directly immediately personally experienced, I put it to you that is -by definition, imaginary. Is it not axiomatic that all universals can only possibly be imaginary?

Wow, this video makes me realize how beautiful is mathematics

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

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

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

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

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.

Ditto for my math degree

@8:00 The different size Infinities come from the inherent assumption that you can measure infinite space in many dimensions with a finite number of symbols. Even though you can parse the numbers infinitely in base ten that system crashes when it tries to measure transcendental and algebraic irrational numbers. The writing of number symbols in a line where the placement means something consistent in relationship to other digits was something new really. The placement is consistent which allows for algorithms that rely on relative position and arithmetic which can’t be done with number systems like Roman numerals which is an inconsistent system of numbering. Or a system which cannot have algorithms like long division or adding and subtracting that is the same for all quantity. Anyways the invention of the Indian Arabic number system was really monumental, but the resulting mathematics that we have in modern times is always glitching from the fundamental assumptions that infinities exist in many dimensions and that they can be represented by expansion of the same finite set of symbols. But if I say this is the number pi and then I write the symbol for pi that’s all there is to it. Similarly I can always make a new symbol for any number I come across which can’t be represented as a predictable pattern of digits. But this fatal flaw is really actually a good thing. No useful system of logic and measure etc is consistent and or has no paradoxes. There must be unprovable assumptions made, and in some cases we actually have make obviously false assumptions and then built the house on top of that foundation- like for example the assumption that we can represent all dimensions of numeration with a finite number of symbols and I Infinite parsing of the symbols. It’s similar to how a computer program can get stuck in endless loops etc. But if a mathematician tries to make a consistent system with no paradoxes it is entirely useless. And the fundamental flawedness of math is why it is useful because it can engineer master engineering while also at its limit point to that other level where some of the fundamental flaws or obvious inherent false assumptions were or can be thrown out. And from that level new problems can be solved or new tasks can be completed in the real world. This is why the ocd person really makes a bad engineer or scientist in reality…

i cant even imagine how much maths he have to do to make these videos... my calculator would have been dead too

So interesting. Makes me wonder about probability/ determinism and reconciliation of opposites. Crazy huh?

And at some point, someone knows so much math that they are at trial and error—which is a maths.

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

Yes

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

It is a bit a shame that everything in this video was already quite widely available on Youtube by other people, including Life-in-life. Veritasium adds a great presentation with extraordinary video and visuals. He could add a few more references to the people that made Life-in-life and other modern accomplishments and recognize a bit more that he is presenting other peoples work, but for the rest he is doing a great job.

@Da'Vion Archie this man's on to something....

It’s possible to make computers in minecraft. So we can assume it’s possible to run minecraft within minecraft

This is an excellent description and historical summary of Zermelo-Frankel set theory. Thanks!

so here is my 120 theory, we change the representation of a "whole" from 100% to 120% it's not a new idea by any stretch of the imagination but it certainly makes more sense than 100%. Take it from me, a random guy on YouTube with an anime profile picture

When the new paradox was announced and the music began, one of the funniest moments online, well done. Very well done.

Your explanation to Halting problem is on point. The visualisation is great.