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.

@Scott It implies both ways, we don't know if it is solvable or not. No point pondering if it could be solved in the future with far superior intelligence and better mathematical tools because WE DON'T KNOW. We simply cannot see the future so it is a waste of time to be thinking about it. The only way to know a problem is solvable is after solving it, but there is no way of knowing if a problem is solvable without solving it. Even chess, even though it is a "solvable" game, is extraordinarily complex. Even the most powerful engines and AI currently are not even close to solving chess, the number of possibilities being only one factor.

@Scott Chess being solved by brute force is one of the most ridiculous things I have heard, especially when you mention it would be solved in the next decade or so (even if you give a problem an infinite amount of time to be solved by brute force, it won't always be, depending on the problem). I don't think you comprehend the number of possibilities involved in chess.

Sounds like my dilemma with creating financial market algos lol

E = 0²

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

@Marcus Nelson I went to different types some bog standard public US, private, free school,college and at home in both Michigan, England and long Island New York. Before going to university. I find what kind of education you get depends on the parents background and education. The more manual labour the environment the less value will be placed on learning. Observation, logic and the 4 principles of the scientific method will stand you well in navigating people.

@Pedro I claim no such thing. I only claim math can be taught in a more holistic way that attaches it to other areas of knowledge.

@Luceat Lux I have also heard that MBTI is nonsense. I do find it useful and largely agree with the result and description that comes along with it, for me. Many of my friends throughout my life have ended up being the same type. I get on best with NTs I think.

@Number 6. which type of school did you go to and in which country?

@Steve Sether that is you deciding without any empirical evidence that all the viewers understand the content and the constructs. :) Using statistics we can easily break down which portion of the viewers actually understand all of the concepts etc…

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

Imagine the cow is a sphere

@Dale Johnson actually learned a lot from that thanks

@Michael Asta as long as you go to the trouble of proving or disproving your assumptions in an independent system the method is not flawed.

Don’t forget to ignore air resistance and friction.

Physicists rarely question whether or not they *should* assume such and such to be true in the first place. Einstein at least had an axiomatic criteria (least number of assumptions in the simplest possible form which frames the most general kinds of problems). Even then, his long talks with Gödel likely helped him to come to terms with the fact that axiomatic (assumptions-based) mathematics which has prevailed for 3000 years is fundamentally flawed.

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

godel was explaining psychohistory aka cliodynamics, the prediction of the future by using the past if we can predict what the past will be we predict the future but we cant predict the future only the past and we can't predict the future in the past only the past in the future... so its highly complex to say the least...the calculations are infinite indexes of historical databases of everything that can ever been catalogued into a tiny little quantum dot and to have our tiny little peon brain wrap itself around that is highly egotistical for our mental comprehensive capacity at our stage in development within the cosmos... we are just the smallest fragments that make up a seed that will one day cover the universe...

@Keen Observer  The problem is that no matter how we define it, it will always have flaws. There’s an excellent comment above in this thread that explains it more thoroughly, but essentially Godel proved that any system strong enough to define basic arithmetic is either incomplete or incorrect.

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

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

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

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

@Ben Botts ,may I recommend you ,,Man’s search for meaning “ by Viktor Frankl? I think you will like the philosophy he teaches. Now about the topic at hand, let me just say that rejecting your own sexuality is no easy feat. I do believe that the analogy @Outback Catgirl made is not entirely appropriate though. Breathing is the most basic need of (almost) all living beings on earth while sexual orientation might be only present in humans. While it may be caused by a difference in the behavior of the limbic system as many studies have proven, this does not imply that it is unchangeable. The behavior of every individual plays a big role in the development of the brain. And I think that here is where the real ,,choice” could in theory apply. And make no mistake, this choice is more akin to ,,I choose to live my whole life as a monk” then ,,I choose cereal for breakfast” And even then, one might never be able to radically change their likes and dislikes, just controlling the amount of control said ,,likes and dislikes” exert on their behavior. Thanks for reading my ramblings -An aspiring neurologist

@Riya Choosing one's truth in life comes with good/bad outcomes from those you consider family, friends, et al. I believe this is a choice. Being gay, straight, or non-associative (whatever the correct term is for that 3rd option) ... I believe those are all choices. If we cannot claim agency of our own lives and what defines us then we are subject to simply being rolled some cosmic dice in a game of universally bound craps where we have no ownership of ourselves. As before I have stated I believe this is a choice and not something we are inherently born with/into ... others do not agree with me & that is okay. I'm not vying for anyone to agree with me nor am I hoping for someone to shift my perspective or change my mind. I appreciate the feedback, even if I respectfully decline your belief; I am thankful to read others' thoughts/input on the topic. Some folks want to railroad me or paint me to be some ignorant clown. Which is awkward, to say the least. Have a good night.

@Ben Botts but why? it is not a choice, you wouldn't choose to be gay if it could lead to your family disowning you, being ostrasiced by the whole society around you, loosing everyone you love and literally being killed in some countries. you don't willingly choose that. it isn't a thing to agree/disagree on

@Ben Botts Odd way to answer the question, but I feel like the logic is more like: There is a group of people outside your door. One person knocks at a time, and as this happens a couple times each person is asking the same question. If you were to give the group an answer then the group of people will leave, including the countless ones who haven't gone up to knock yet. That saves you time and keeps you from having to keep going to the door and give the same strange response! Most people asking aren't angry, just more confused as we don't understand what you're implying. If you are implying something than we are trying to educate you on the contrary. There is nothing wrong with learning 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!

@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).

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

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

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

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.

not sure if thats a good metric, i agree but id say that example just doesnt really say much thats like saying "to show how important Grug was, his basic teachings of one rock + one rock = two rock to bash other creature over puny head with is still taught in schools today" Turing completeness is just an inherit feature of computers, they wouldn't be possible without that definition and you would not be able to understand computers fully without already knowing that

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

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

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

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

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

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

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

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

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.

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 😇

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

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

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.

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

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

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

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

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.

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

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

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!

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

Math always proves itself. But neither is one who scores a zero in a math exam truly a poor mathematician, nor one who scores a centum in a math exam truly a good mathematician.

@Fomalhaut the rule of math is that it can prove anything doesn't exist, including itself

I bet religious folk can use math to prove math

So if math can prove that math can't prove itself then math is not reliable. Since it's not reliable, we should not trust the proof it makes that it cannot prove itself. A whole other paradox comes

How else would you prove it?

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

@alcatel😊😊😊😀☺ ah, understood

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

4 is a much better approximation.

@Bug Dracula uhh 1/7.14 if a person duh

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

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

@pyropulse how is that obvious? can u explain?

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

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

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

The history of mathematics is absolutely fascinating.

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

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

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

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.

what if i never shut my computer?

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.

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

Took words right out of my mouth.

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

@Peter Codner Way to be needlessly pedantic.

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

Love these high quality informative videos

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".

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

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.

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

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.

Please do more math videos they are extremely fascinating!

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

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.

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.

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

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

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.

Amazing and informative. Thank you.

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

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

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

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.

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

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

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

As someone who majors in mathematics while minoring in computer science, this video is absolutely awesome. I've learned about a lot of these things in isolation, but this really connects them all.

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

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

​@Peter Codner Damn you sound like a nice guy

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

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

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

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.

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

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.

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.

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.

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

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

the completeness bit seems to reflect the"begging the question fallacy". it seems to stem from an implied static state, that the paradox is because of duplicity inherent in the work itself. if the definition is relaxed enough to imply scenarios where the reference palettes from which the word gets its "form", then the static state and the comfort it grants to the mind in its finite form changes. it is this dynamic inference that grants the joke of Jim being his own enemy its hilarity. as it is required for the joke to ignore the circumstances that make Jim his own enemy to be verified, or even true.

This is just out of this world man. A sincere congratulation to everyone that worked on this piece of art. A great way to convey an extremely complex topic. Thank you for publishing this for free. This is the stuff the next great scientific generations will be built on.

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

Continue to be among the best videos out there!

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.

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

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 🤣

I see math as another language, where you have to just agree on certain things in order for it to make sense. If you look too closely at any object, it is made of energy, so in a way it ceases to "exist" in the conventional sense. I see math the same way: if you look too closely things get weird.

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

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

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

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

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

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

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

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

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

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

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

Don't think enough credit was given… the art in this episode is spectacular!

Impressive

credit to who?

This was awesome. However, one thing I suggest you check out is the story of Colossus.

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

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

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

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

Fascinating and brilliant! I pulled Godel Escher Bach off the shelf thanks to this.

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

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

As a computer scientist, this blew my friggin mind

Gödel was also first to ask P vs NP question and he asked it in the letter to John von Neuman. Those dudes had some world changing conversations.

@Anthead The course title was Introduction to the theory of computation, or something like that. I was an engineering major, not maths or computer science. I really only took pretty basic CS for fun. It did turn out to be amazingly useful though (I mostly do bioinformatics these days).

@Travis Collier which course, if I may ask

@Mr Fl0v Depends on how precise you want that description to be. A box of electrical stuff with blinking lights is pretty accurate ;) Seriously, as I've said already, modem computers are based on von Neumann architecture. That basic architecture has been extended and tweaked significantly of course. That "are" stored program electronic computers. It has been quite a while since I took intro theory of computation though. If you want to learn more, great! Do a web search and maybe even look into taking a course. It is cool stuff.

@Travis Collier If computers aren't precisely and exactly Turing machines, then what are they intrinsically? Oh, and please don't just tell me they're an alternative version of "Turing machine", that'd be the idler's way out.

@Andy Polasek Read/skim the Wikipedia pages on von Neumann architecture and Turing machine. Seriously, it is a bit of a pedantic point, but this is really basic stuff. Back in my day (I'm old) it was literally computer science 101 (CS1 at my school) stuff. It's also interesting, at least IMO.

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.

One of my favourite Veritasium videos❤️

30:15 I wasn´t expecting the video to take such a dark turn right after the awesome ''game of life running inside the game of life'' moment

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

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...