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

Math can prove that math cannot prove

brilliant

So I was right, my math teachers were all totally off base demanding that I prove my answer. Show your work? Yeah, show it to my butt!

Another proof that we live in a simulation.

i like my brain is melting💨

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.

@schroecat1 You are right on that. The irony is that many utilitarian CS stuff they teach is already old barely used tech now, yet they ignore eternally useful info such as the themes you mentioned.

@Joseph Anglada You're not wrong, but they're often missing even the fundamentals of that practical foundation. Not only are they graduating students who don't understand Turing machines and the fundamental math, they also don't understand IP addressing and how protocol stacks work. It's been a bugbear of mine for years now.

Computer Science classes today are more utilitarian and practical than formal. It is sad, but to be honest, there is so much we could cover in CS but we cannot because little time.

teaching Turing machines is a complete waste of time for the vast majority of people who just want to put food on the table

@Azerty And it's only a property of purely theoretical computers as physical computers don't have infinite memory.

Serious material and serious work making, hell yeah man, over the years your content is just getting more and more detailed, in depth, fantastically written and beautifully illustrated. Good work my man!!

I suspect that for many people, making this video might be considered a lifetime achievement. But for Derek, just one more brick in his incredible, historic castle of outstanding teaching.

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

@JureDoon underrated comment.

Sorta... I'd say it's actually applicable to everything, but not connected unless we force it to be. The strength and weakness of Mathematics is that it's not reality (thus it can be isolated from uncontrolled external influences - a very desirable trait), though it can be used to help understand reality. However, any mathematics that fail the reality test is a failure of the mathematics and our understanding of reality, not a problem with reality itself.

I second this

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.

@DSUM Is that a problem of language or a problem of mathematics? Maybe its a problem of the English language. Is mathematics (just) a language. One of many?

Try to disprove Gödel Incompleteness theorem 💀

4:32 Excuse me! There are 0 natural numbers between 0 and 1 and natural numbers start from 1.

Im a former mathematician (is there such a thing? lol, do we ever stop?). And the answer to your question is you, I and others are not working hard enough. There is a major flaw in the Godel system. It itself rests on some logical fallacies, so the proof resulting is nul as far as I'm concerned. (Obviously Im no longer in any university, and this being a mere comment might do little for actual math debating, but still...its my only contact permited in this field). The biggest issue is that many "schools of thought" have some false statement that goes by unnoticed. Much like the twin prime conjecture that I knew was solved. I do know that primes separated by 15 or some such number was solved. But thats just a footnote here. Whenever I see a proof relying on prime factorization there is already a hefty dose of unknown. How can you create a formal math language while using something we still have not figured out yet? We have no way of predicting the number of primes found in a random integer sequence that is arbitrarily large (ie how many primes are there b etween 1 trillion and 3 trillion, and so on). What I was working on, was in fact that, and I cant go into because for one, I need to draw lol, and two, I am petty I suppose and an taking my work to the grave. But, primes are knowable, predictable and consistent, if starting off correctly. (This is my unproven as of yet assumption. The results so far are scary as far as where they appear, in rather, how I have not found anything yet where they do not appear based on my work. From atomic structures, energetic orbital geometric shapes, to quantum particle trajectories in various mediums). Going back, the issue with proofs is the data given. Not at the level of axiom, but if those axioms apply to a system. Euclid's axioms are just that, but with the additional added info of operating in a Euclidean space. Two parallel lines may intersect in a non E space. There is additional info missing with the Godel system, or some restriction. I dont know the author of this channel and what his field is, but I have a hunch its more along CS or perhaps some physics. But he clearly seems convinced of certain statements, which may not necessarily be true. Turing machines may have a paradox, but a paradox is only such, if given data that is was either not designed to fail, if the system ommited deliberately or not said data or scenario, or maliciously setting up the system to fail, as in giving it data it could not process, for which it was not meant to, or a legitimate failure point was found. (I use the term malicious, not in the day to day use. I hope its clear). Sorry for the long comment. Its still something I miss, such conversations, but alas, my coworkers were...malicious. Was fired while learning how to walk again after a spinal injury.

U can always try the oldest profession ever 😳

I literally teared up at the very end. Thank you Derek for inspiring millions of people to pursue math, science and engineering!

I'm binging away on your channel's content, and it's absolutely incredible how good your videos have become over the years. Amazing development in quality, delivery, pacing, scripting, and everything around it. Keep up the brilliant work!

This really makes me feel like Perspective's and approach to what scale you are focusing around is very important. All over the universe there's multiple scaled levels of properties of systems all over the Universe.

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

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!

In the 70s I developed an ehanced version of this, with rules that allowed a "live" unit to have unique characteristics (just like real life) -- that is, some could be more predatory, or need more resources, etc. It's really just infinitely variable to represent whatever sort of competative system.

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

Me too

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

Game numbers moment

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

Never had the appreciation for math as much as I do now. This video must be watched by every engineer/natural science student

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

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

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

762 pages just to prove 1+1=2? Did no one teach these blokes finger counting in childhood.😂😂😂

Can't agree more.

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!

I LOVE combinatorial game thoery. The idea of things like star numbers and even weirder values like on , off, up, down, onoff, oof, high, low, hot, cold, etc. and the way they are defined in our system of math is absolutely amazing to me. There's a really good video on it called "HACKENBUSH: a window to a new world of math" by Owen Maitzen. It only has 100k views, and it's his only video on that channel, but it's extremely good, and it's a nice place to start with combinatorial game theory.

Hello. I just wanted to tell you that this is my top 1 video on YouTube. During that year I have seen it like 6 times and every time I find something new and fascinating. Just wanted to thank you for your work.

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 😇

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

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

@brien maybe you becoq

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

At 17:56, I think it would be good to note that one could write down something like the expression "==", which obviously doesn't mean anything, but which still has a Godel number. Specifically, 2^5 * 3^5, or 7,776. So some of the Godel numbers correspond to meaningless expressions/statements.

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.

Thanks to content creators like you who make the time spent here absolutely worthwhile!

My incredibly-intelligent 11yo son introduced me to this channel. Your content consistently teaches me new things. Love your work.

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

There is no video on the entirety that I have rewatched more times than this one. I yearn for the next time Derek revisits pure math. (excluding music vids and vids for school that you obvs have to watch many many times)

This melted my brain. I'm more inclined to believe we're living in the matrix than ever

I love you, I have been looking for simplified explaination about all of these for more than a year. I am very glad to find it here. I am eager to see your future videos on the np-complete problems, or maybe Simulation hypothesis which could be the continue of this video. Another thing, I have heard a lot from mathmaticians that "it has been proved that, this is unprovable ". As I have seen, Godel just proved that, not necesserly every true statements are provable. But how anyone can prove that something cannot be proved ?

I have seen this video many times, and I absolutely love it. Great Job Derek!❤🧡💛👍👍👍👍

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 🤣

Please do more math videos they are extremely fascinating!

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!

OMG! Derek, thank you for so many awesome videos. I love them, especially the ones about math!

I'm just here to say I love this channel. Keep up the good work !

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.

About the halting problem: 1. The way that h+ is defined seems to be it must get h's answer BEFORE it runs the second part. 2. Since h is assumed always worked, if h receives itself as input, it will always print out "Halt" 3. If h+ received itself as input, h is the one that actually processing the input FIRST. 4. For h to get h+'s result it has to run h first and get the answer, after that, the second part will execute the opposite action 5. But since there's no input for the second h(the h inside the input h+), and by (1) and (4) hence h+ cannot be simulated by h+ This is my understanding of this problem, my conclusion is that this program(h+ that receives h+ as input) is not able to run properly because the lack of the input for h, not by results contradiction But I think because this is a video presentation, and the original paper is probably way more rigorous

If P=NP is figured out can it be used to test a group or category of math problems to find out if they fall within the "provable" category? I suppose that depends on how you define the properties of those kinds of problems that fall within the category your testing... still id love to hear if someone may be able to explain if there is any connection between the P=NP problem and whether it is either effected by Incompleteness or whether it might help delineate what types of problems which may fall within or without the "scope" of incompleteness? Its probably obvious that I'm not a mathematician so everything i just said above could be absurdly wrong haha but obviously I'm still curious!

How are you so good at explaining things? You are just too good!

This is one of the best videos I’ve ever seen. I’ve probably watched it 20 times but it never gets old

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.

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

Love these high quality informative videos

Some of the most brilliant minds of all time met such tragic fates. It's unfortunate. Hopefully we as a species can learn to be kinder to people who come forward with ideas that seem strange and counterintuitive at the time they are introduced.

You basically managed put all of Logicomix in one video, well done!

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).

7:05 the answer of the limit is -π/2

I joined Brilliant before watching this video. I think I'm in love - this is so much more interesting than the best stories I've seen. Synchronicity - maybe it's just the right time... If you're reading this, Brilliant is a spot on way to spend your everything.

Yes it's true: the only concept I understood in this video was Dwight and his enemy of my enemy is my friend thing, nonetheless I have found this video fascinating. Thank you @veritasium

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.

It will eventually shut itself down

what if i never shut my computer?

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.

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

Your graphics are excellent. I’d love to be able tell a story so visually complete.

Absolutely fantastic video! Great work, my friend! 🤝🤓

I've watched this video thrice already, pretty sure I'll have to do it again to prevent myself from feeling dumb! Brilliantly made, without a doubt!

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

Engineer, knowing does not require proof, when experience is sufficient.

@_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.

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.

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!

It's incredible revelation that there are more. decimals between 1 and 0 than the natural numbers between 0 and infinity. This fact can have far reaching implications

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

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.

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

This easily should be YouTube’s #1 most watched video ever. Super cool! Everyone should watch this.

Meanwhile Turing invented his side-kick "computer" to support/solve his main subject .. What kind of geniuses we had!

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...

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

Magnificent work!, you reminded me my student times :)

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

i learned about math logic at first year of my junior high and i thought it was easy, until i saw this video 17 years later

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.

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

I am watching mathematics!!! Never thought I'd do that! Awesome job man

Very well made video. Great job.

Continue to be among the best videos out there!

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 😂😂😂😂😂😂😂😂

As a Brit, I hear many people say that they're ashamed of X thing in our history, but what our government did to Turing and other men like him is top of my list. Turing in particular because the loss of his mind probably held back our technological progress by a decade or two. Think of the contributions he could have made during the digital revolution...

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...

I like your simulation take.

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

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

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

@kotzpenner cs degree is a waste of time anyway

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

@a 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.

Fun Fact : All the thinkers were native German or English speakers. I think, it came from educational quality in these countries, nowadays it has globalized, but they were the first, leading mankind into modern age, although they came from very conservative societies, but they became the most progressive of all societies because of the power of thoughts and technical inventions.

"Mathematicians did not like it one bit." So even mathematicians don't like extra homework. Now that new foundations were being discovered

I rate this as the most informative video I ever watched on youtube

I was having difficulties sleeping, thanks to this fantastic documentary I am now able to sleep at my will.

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

You could have just used the ontological argument that there is never "two" of a thing. The only place in the universe where there exists "two" of a thing is in our imagination.

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

Best video on the subject. The visual aides help so much.

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

For me, the biggest takeaway of the whole thing is this: how amazingly smart must Gödel have been to come up with that proof? Obviously, every other Mathematician mentioned here is also incredibly, incomprehensibly smart, but with the other mentioned proofs, I can kind of reconstruct how one might have arrived there. But with the incompleteness theorem, I just cannot fathom how one might come up with it. The guy must have been able to just straight see the matrix.

he was one of the best mathematicians, that's for sure

@linkinlinkinlinkin Some of are born with vast knowledge, for it is not even close to the first life one has lived.

@Spirit None I'm not a maths guy, I'm sure the proof the rigorous and I wouldn't be able to do it... but the concept itself seems very simple unlike, say, hyperbolic space time :p -- more impressive to me is that they've found real world applications of the incompleteness theorem, and it's not just a pure maths logic loophole.

@Victorel Petrovich To ask what proof exactly is, one should ask what's to be proven. So let's ask if we could proof that there is truth. Truth in this context means there is one assertion which is true no matter the assumptions we take. This might be the assertion: "I exist." Let's define existence as being subject and/or object of something(s). With this definition one could argue, that because "I perceive (Thoughts, Senses), therefore I exist." But everything else could be something like a Matrix or an Inception. I can neither verify nor falsify whether I'm an object of someone else's data or whether I am my own sequented output and input. And if the second is true I can't prove whether other input, but myself exists. So basically: "Everything can be, nothing has to be". So we have to start setting Axioms from here on out. So proof can only take place in a certain field of axioms. So for example I could set the Axiom that free will exists. From there I can say: "If I exist and I have free will, that implies someone/thing else exists." Because otherwise I'd be my own sequenced input and output I'd be a predetermined System → free will doesn't exist. The freedom of humankind is only possible through incompletion. We aren't bound by truth, because we can't prove what's true without setting Axioms. This means we humans constantly create different contradictions, depending on how we - as individuals - set our axioms. As many problems as this causes, it's also the backbone of human creativity and innovation. As Hegel might say: "Every contradiction is part of a synthesis."

@Spirit None Take the alternative that Godel's statement G is true, but unprovable. However, the conclusion was arrived at that it's _true_: the chain of reasoning is a proof actually, thus it's provable. What have we used to prove it? axioms that were not part of the system under investigation? Because otherwise, it shows that it contradicts Godel's conclusion that G would be unprovable. And if so, then must be false. Also, the author already explained that if we consider G false, then G implies it's actually true. Thus, we have a statement that cannot be true nor false. To me, this indicates it's not a statement at all. (It's just a sentence, a meaningless sentence, that cannot have a truth value). So, this problem has to be more carefully investigated. Exactly what rules of reasoning are to be considered as part of the system (of math). What exactly is a proof.

One of my favourite Veritasium videos❤️

Excellent video---super fun to watch, thanks.

i always come back to this video after every few months to see whether i have developed enough to understand everything in it completely 100%. nit there yet.

This is a great channel. I love it!

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

@Honourable Doctor Edwin Moriarty írc it

Bet his magic act was fire.

@Kode Esser hmmmm. Synonymous with energy, matter and guess what? Consciousness... infinity also, very few can grasp the concept. No beginning, no end, it just is. I call it infinity+1 as most people just cannot let go of the need to quantify.

@John Doe I agree wholeheartedly. As a curious minded, gifted level IQ person. I only mention this to give myself some credibility.. as you guys are thinking along the same lines as me. Infinity yes... exactly they way I see it. It just IS. Concoiousness is infinity as I see it. Time is infinity although really does not exist imo... there is only the now. Mathematicians, theoretic physists etc.. most seem to suffer from this reliance on measurement and then reductionist reasoning. Even quantum physists.. that understand the 3rd state of unknown probability.. in particular all believe consciousness is nothing other than matter and energy within the brain. Ask yourself this.. I am conscious. This is undeniably true (to you). How can anything beyond that.. ie my senses, thoughts, emotions etc be PROVEN to be real. Ie brain in a vet thought experiment. Aka simulation theory. I don't personally believe either of those hypothesis however the fact that science has matured to the level where everything can be measured with extreme resolution and particles can be smashed together at ever increasing velocities.. all we have ever discovered is more matter/energy/charge/force based sub atomic particles. What I am getting at is that consciousness perhaps is entirely abstract from the measurable and observable universe. IMO the universe has not discovered a way to observe itself... the universe exists because of the observer. They are interdependent. The maths stated above, to do with incompleteness vs what cannot be proven or disproven.. the paradoxes involved. I cannot help but notice the synonimity here.

The whole of western knowledge is like this. Like a child's blabbering mm Ancient Indian philosophy is the only way to truth.

I'm taking a class in abstract and linear algebra, and we heard of this before.

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

I understood the video, what scare me the most Is that some people can write math demonstration of this concepts and these are basic things. Humanity Is Amazing.

Please make a video on category theory... Can't wrap my head around it so I'm hoping that a youtube channel like you might make it simpler

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".