Godel’s second incompleteness basically says: “completeness (all true statements are provable), consistency (only true statements are provable), and arithmetic-pick two”
A correction. Fermat's last theorem was not just for third powers, that had been known for a long time and for quite high exponents. Wiles' achievement was to prove it for literally all positive integer exponents.
❤🎉❤❤🎉🎉 Truly truly i say to you all Jesus is the only one who can save you from eternal death. If you just put all your trust in Him, you will find eternal life. But, you may be ashamed by the World as He was. But don't worry, because the Kingdom of Heaven is at hand, and it's up to you to choose this world or That / Heaven or Hell. I say these things for it is written: "Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, *teaching them* to observe all that I have commanded you; and behold, I am with you always, even to the end of seasonal". Amen." -Jesus -Matthew 28:19-20 🎉🎉🎉❤❤❤🎉❤
Just a small correction, AFAIK Wiles did not show that FLT follows from Taniyama-Shimura, that had been known for a long time and isn’t that hard. Also proving Taniyama-Shimura was an extremely important result for mathematics, so proving FLT was more of an icing on the cake.
To correct this correction: Taniyama-Shimura WAS in fact proven by Wiles (and one other), so it wasn't "known for a long time", and neither is it not "that hard". It was regarded as a terribly difficult problem.
@@fysher3316The Taniyama-Shimura conjecture was not proven by Wiles. He proved a specific case of it (semistable elliptic curves) that included Fermat's last theorem (there is an amazing video by Aleph 0 on the topic). Using his work from 1995 on that proof a group of mathematicians finally proved the whole conjecture in 2001.
Hey, in 2:38 you used an image of a painter called Richard Hamilton from London. However, the actual mathematician is called Richard Streit Hamilton and lives in Ohio.
10:38 my understanding from taking discrete math years ago is that godels incompleteness theorm wasnt: “any math system has true statements that cannot be proven true and also cant prove that it isn’t inconsistent” but more so “any math system that doesn’t have true statements that can’t be proven true is inconsistent and any consistent math system has true statements that cant be proven true.” like it’s one or the other. A math system can only be useless (inconsistent and unprovable truths), have unprovable truths, or consistent. Is that wrong?
One way to think of it: 1. A system is complete 2. A system is consistent 3. A system is recursively enumerable 4. A system can express basic arithmetic You can only pick 3. A system can be both complete and consistent, say Presburger arithmetic. It is strictly weaker (can’t even express multiplication) than Peano arithmetic, which is subject to Gödel incompleteness. Tarski even devised a complete axiomization of geometry, but it too fails to satisfy the hypothesis of Godel’s incompleteness theorem like Presburger arithmetic. The hypothesis of Godel incompleteness is that it it can express arithmetic such as PA, once it reaches that threshold it can no longer be both complete and consistent. Edit: #3 also makes it so this only applies to first order logic, as second order logic is not recursively enumerable.
One correction: "algebraic groups" are a concept from algebraic geometry (certain representable functors into the category of groups). What you mean during the classification of simple groups are just "groups"
So, for the 4 color theorem, it’s actually not true that 4 colors can color a map without same color elements sharing a border on maps of any complexity. There are some rules that the map has to follow. Say, for example, that there is a set of 4 countries that form a circular border, and each country occupies one quadrant of this circle. These four countries could not then be enclaves inside of a fifth country that encircles all of them, because no matter which of the four colors that the fifth country is colored in, it would match up with and border one of the quadrants of the same color.
I'm actually working on x⁵-x-1=0 right now. I have a hunch that while it cannot be solved algebraically (by radicals), it can be solved transcendentally (something containing e=2.718...). Even if I could do that, it would be short of a full explanation of higher-degree polynomials. It also might still be impossible to have a single formula for all quintics, but it's a step in the right direction.
That the solutions to equations like x^5-x-1 = 0 are transcendental is not a bad guess at first brush, but actually cannot be true by definition. Transcendental numbers are defined to be numbers that cannot be expressed as the solution to a polynomial with rational coefficients. So, for example, there is no polynomial with rational coefficients that gives e or pi as a solution. There is actually a general formula for the solution of quintic and higher degree finite polynomials, in terms of hypergeometric functions. The output of these functions is not radical (cannot be written as a rational power of a rational number), so there is no contradiction with Galios' result. However, these numbers are still not transcendental, since they are solutions to rational polynomial equations. In essence though, your intuition is correct: the general space of numbers that solves these equations is necessarily a larger group than just radicals. This set of numbers is actually called Algebraic Numbers, because they solve algebraic equations.
from educational pov i would add euclids parallel postulate before continuum hypothesis. or at least mention it as easy to understand analogy. im not sure if its ever stated as "an unsolved problem" however its solved as an axiom of choice.
That quintic equation solution impossibility is a cursed one! Galois died in his 20, also Niels Henrik Abel died at 25! Abel provided the first formal proof of that! Gauss of course beat them to it but he never published it formally, for him it was a near sure guess which people read in his notebook after his death!
Haken: pronounced HAH Ken. One son is a composer and another a computer scientist I think, associated like him with the University of Illinois at Urbana-Champaign (at least early on).
5:59 why does it say "can't color this with 4 colors"? You clearly can - just make the purple bit blue, and the little blue nubbin one green (or red or yellow)
@@tupoibaran3706 the main issue is that 4 color theorem is about contiguous planes, so case presented is invalid from the theorem perspective. Theorem is not about real appliances, when countries may have separate territories somewhere else.
@@em.1633 Nah, his sum of natural numbers= -1/12 is the biggest lie which people who want to learn about Maths believe it's true. I mean, saying that 1+2+3+4+...=-1/12 sounds pretty elegant once you see how he found that sum, but you need to dig further to understand that the sum of n from 1 to infinity diverges and that's on period, and since it diverges, there's a property which , by doing partial sums of the original sum, we can get different convergences (which proves, again, the big series diverges). But no one explains this to the newbies in math, they take the well-know value for the sum and get the wrong idea of Analysis.
Fermat's last theorem states that for all n>2 there are no integer solution to the equation aⁿ+bⁿ=cⁿ, what you presented in the video is just a specific case
ffs, so many statements are presented wrong. fermat last theorem said about any nth power bigger than 2, not just 3. 3rd power was prove impossible long before Wiles.
Insolvability of the quintic equation was actually first proved by Abel and Ruffini, Galois only later generalized the theorem and simplified the proof
You didn't state Fermat's last theorem correctly. The case of 3 as the exponent was proved shortly after Fermat's death. So was exponent 4. But the theorem said there was no equation for any integer exponent greater than 2.
I think you're underselling Grigori's contribution to the Poincaré conjecture in the way you bring up his use of Hamilton's work, he always admitted this and when he declined the prize he said it was because Hamilton's work had been equal to his own.
You need a pop stop for your mic badly, but other than that great vid. Also since I'm being a ballbreaker I might as well add this critique: you should speak more naturally, and less with the generic "youtuber giving lecture" monotonous tone
He butchers the last name too lol. Can't be helped since he's American but I wish he would just stick to an Anglican pronunciation so that it at least doesn't sound annoyingly pretentious.
Unfortunately, there are quite a few mistakes in this. Just to name two: Niels Henrik ABEL proved that the quintic is generally insoluble, not Galois. Fermat's Last Theorem is for n > 2, not n = 3. EULER proved the n = 3 statement long before Wiles.
As a cuber, I am very confused how algebra is related to cubing. I mean, we use a completely different type of notation and there is no mathematical relation besides the R2s and stuff
There are 6 "basic" moves that can be performed on a Rubik's cube. These are the 90 degrees clockwise rotations of each of the 6 faces. (This is assuming we keep the cube in a fixed orientation, so the centre squares of each face do not move.) Each move is a rearrangement of the coloured squares on the cube. Moves can also be composed (i.e. performed in sequence) to further rearrange the squares. Moves can also be reversed, since each basic move can be undone by performing the corresponding counter-clockwise rotation. Each configuration of the squares on the cube can be described by a sequence of moves that takes the cube from the solved position to that particular position. (Although such a sequence of moves is not unique; for instance, RRLRR gives the same configuration as L.) In mathematics, a "group" is a collection of things that can be composed and reversed. The set of possible configurations of a Rubik's cube is a group. Group theory is a subfield of algebra. This is why Rubik's cubes can be studied using algebra. I presume you thought that "algebra" meant "equation solving" like one learns in high school. This is *part* of algebra, but in mathematics, algebra is a hundred times bigger than that. (And it is unfortunate that so few people know this.) Algebra involves the study of groups, rings, fields, modules, lattices, monoids, and possibly categories, depending on who you ask. These are all in the same vein as a group, in the sense that they are collections of things that can be "put together" somehow. (For instance, a monoid is like a group, but without the requirement that its elements be invertible.) I think the original meaning of the word "algebra" (or rather, the Arabic word which became "algebra" when borrowed into English) was actually something like "put together" or "broken apart". If you are wondering what it "looks like" to study the Rubik's cube group, Google "Rubik's cube group".
Imagine 5 (or 25 or as much as you want) countries meet at the pole. Then you can't use 4 colors, you have to use as many as there are those countries.
It also assumes that exclaves are treated as separate entities. Otherwise you can easily make 5 mutually bordering countries. I’m surprised that he even shows it in the graphic at 5:59 but doesn’t comment on it
the poincare conjecture isnt really about what the most general shape is. The way you formulated it in this video makes it seem that the circle is a more "general" shape than the square, which is kind of exactly what topology is not about. Im sure you know this, just wanted to point out that the formulation is super misleading for someone who doesnt know about topology.
Godel’s second incompleteness basically says: “completeness (all true statements are provable), consistency (only true statements are provable), and arithmetic-pick two”
Master
I choose consistency and arithmetic
@@alejrandom6592i choose consistency and completeness
one of the most succinct description of the theorem, vamos!
A correction. Fermat's last theorem was not just for third powers, that had been known for a long time and for quite high exponents. Wiles' achievement was to prove it for literally all positive integer exponents.
>2
Here after it has been solved.
❤🎉❤❤🎉🎉
Truly truly i say to you all Jesus is the only one who can save you from eternal death. If you just put all your trust in Him, you will find eternal life. But, you may be ashamed by the World as He was. But don't worry, because the Kingdom of Heaven is at hand, and it's up to you to choose this world or That / Heaven or Hell.
I say these things for it is written:
"Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, *teaching them* to observe all that I have commanded you; and behold, I am with you always, even to the end of seasonal". Amen."
-Jesus
-Matthew 28:19-20
🎉🎉🎉❤❤❤🎉❤
@@gamer__dud10 -- Stop spamming, *a-hole.*
The image you used for richard hamilton is not the mathematician but the artist. The mathematics Richard Hamilton is someone else
Fermat’s last theorem is more than just cubic numbers, it applied to all positive whole integer values of n where n is the power of x, y and z.
Just a small correction, AFAIK Wiles did not show that FLT follows from Taniyama-Shimura, that had been known for a long time and isn’t that hard.
Also proving Taniyama-Shimura was an extremely important result for mathematics, so proving FLT was more of an icing on the cake.
I wonder if the AI will correct their mistake.
To correct this correction: Taniyama-Shimura WAS in fact proven by Wiles (and one other), so it wasn't "known for a long time", and neither is it not "that hard". It was regarded as a terribly difficult problem.
@@fysher3316The Taniyama-Shimura conjecture was not proven by Wiles. He proved a specific case of it (semistable elliptic curves) that included Fermat's last theorem (there is an amazing video by Aleph 0 on the topic). Using his work from 1995 on that proof a group of mathematicians finally proved the whole conjecture in 2001.
@@fysher3316proving that FLT follows from Taniyama-Shimura conjecture is not the same thing as proving the Taniyama-Shimura conjecture.
Hey, in 2:38 you used an image of a painter called Richard Hamilton from London. However, the actual mathematician is called Richard Streit Hamilton and lives in Ohio.
11:02 fire alarm chirp replace your batteries
there’s another one at 11:10
10:38 my understanding from taking discrete math years ago is that godels incompleteness theorm wasnt: “any math system has true statements that cannot be proven true and also cant prove that it isn’t inconsistent” but more so “any math system that doesn’t have true statements that can’t be proven true is inconsistent and any consistent math system has true statements that cant be proven true.” like it’s one or the other. A math system can only be useless (inconsistent and unprovable truths), have unprovable truths, or consistent.
Is that wrong?
One way to think of it:
1. A system is complete
2. A system is consistent
3. A system is recursively enumerable
4. A system can express basic arithmetic
You can only pick 3.
A system can be both complete and consistent, say Presburger arithmetic. It is strictly weaker (can’t even express multiplication) than Peano arithmetic, which is subject to Gödel incompleteness. Tarski even devised a complete axiomization of geometry, but it too fails to satisfy the hypothesis of Godel’s incompleteness theorem like Presburger arithmetic.
The hypothesis of Godel incompleteness is that it it can express arithmetic such as PA, once it reaches that threshold it can no longer be both complete and consistent.
Edit: #3 also makes it so this only applies to first order logic, as second order logic is not recursively enumerable.
Poincaré did not exist 800 years ago blud
He said a hundred
I still hear 800 even I saw 800 replies correcting it lol
One correction: "algebraic groups" are a concept from algebraic geometry (certain representable functors into the category of groups). What you mean during the classification of simple groups are just "groups"
Finally some solved problems. I’m always frustrated by the videos that describe a problem with no solution. 😂
crazy to see a video about advanced mathematics that seems to have been written by a high schooler. had to turn this off bc of all the errors.
So, for the 4 color theorem, it’s actually not true that 4 colors can color a map without same color elements sharing a border on maps of any complexity. There are some rules that the map has to follow. Say, for example, that there is a set of 4 countries that form a circular border, and each country occupies one quadrant of this circle. These four countries could not then be enclaves inside of a fifth country that encircles all of them, because no matter which of the four colors that the fifth country is colored in, it would match up with and border one of the quadrants of the same color.
I'm actually working on x⁵-x-1=0 right now. I have a hunch that while it cannot be solved algebraically (by radicals), it can be solved transcendentally (something containing e=2.718...). Even if I could do that, it would be short of a full explanation of higher-degree polynomials. It also might still be impossible to have a single formula for all quintics, but it's a step in the right direction.
That the solutions to equations like x^5-x-1 = 0 are transcendental is not a bad guess at first brush, but actually cannot be true by definition. Transcendental numbers are defined to be numbers that cannot be expressed as the solution to a polynomial with rational coefficients. So, for example, there is no polynomial with rational coefficients that gives e or pi as a solution.
There is actually a general formula for the solution of quintic and higher degree finite polynomials, in terms of hypergeometric functions. The output of these functions is not radical (cannot be written as a rational power of a rational number), so there is no contradiction with Galios' result. However, these numbers are still not transcendental, since they are solutions to rational polynomial equations.
In essence though, your intuition is correct: the general space of numbers that solves these equations is necessarily a larger group than just radicals. This set of numbers is actually called Algebraic Numbers, because they solve algebraic equations.
Fermat’s was proved by wiles through a remarkable application of elliptic curves and modular forms
from educational pov i would add euclids parallel postulate before continuum hypothesis. or at least mention it as easy to understand analogy.
im not sure if its ever stated as "an unsolved problem" however its solved as an axiom of choice.
I enjoyed the vid, great work and explanation, thumbs up!!
Not 800 years ago at 0:22
it was "a 100 years ago" 😄
It does sound like 800 tbf
That quintic equation solution impossibility is a cursed one! Galois died in his 20, also Niels Henrik Abel died at 25! Abel provided the first formal proof of that! Gauss of course beat them to it but he never published it formally, for him it was a near sure guess which people read in his notebook after his death!
I didn't read the rules but couldn't you solve Fermat's theorem by saying 0^3 + 0^3 = 0^3? Or 0 + 0 = 0?
Was that the correct picture of Richard Hamilton?
hey no, it's a wrong pic, Sorry about that. He is an artist Richard Hamilton :D
Hey ThoughtThrill, don't know if anyone told you, but I think you used the wrong image for Richard Hamilton... Otherwise, cool video!
arXiv is pronounced like "archive"
(yea, I know it's odd.. but it's also pretty cool)
So are there any infinite sizes between the natural numbers and real numbers?
The Continuum Hypothesis: yesn't
Can you make a video about every math problem that seems obviously possible but is proved impossible.
Examples would be: Euler's bridges problem, trisecting an angle, ...
Galois proved or helped prove two topics in this video and he was a teenager?!
Goddamn! I am an ape next to him.
How did Terrance Howard problem go unnoticed?
Haken: pronounced HAH Ken. One son is a composer and another a computer scientist I think, associated like him with the University of Illinois at Urbana-Champaign (at least early on).
Yo! FIX YOUR FIRE ALARM! Cool video. The incompletness thoerm makes me think consciousness is a solution outside our reality.
5:58 you can definitely colour that with four colours lol
I think poincaré's conjecture has been resolved in 2002
5:59 why does it say "can't color this with 4 colors"? You clearly can - just make the purple bit blue, and the little blue nubbin one green (or red or yellow)
The “little blue nubbin one” with an A has to be the exact color as the big blue square with an A. They are the same country so to speak.
@@tupoibaran3706 the main issue is that 4 color theorem is about contiguous planes, so case presented is invalid from the theorem perspective. Theorem is not about real appliances, when countries may have separate territories somewhere else.
you can't, but at the same time, this is not a 4 color theorem case (it doesn't satisfy contiguous condition).
Poincaré looks like if Jamie Hyneman had an alter ego. Not a bad thing
"Every unsolved problem math has solved"
Um, All of them?
Fun fact: trisecting an angle is trivial using origami method (folding)
no Hodge?😢
Henri Punkaré
The proof for the four colour theorem is kind of a let down. Either that or I'm misunderstanding the proof.
Fermat's theorem is proved by Wiles
Nice work on the video but it is too fast and not thorough enough to be insightful to any casual audience for the most part
Someone asked to me that: what's infinity.
Me: -1/12
Im pretty sure fermat's last theorem was solven not too long ago...
Well yeah, that’s why it’s in the video
Wow
❤
Change the battery in your smoke detector
😁
@@ThoughtThrill365 ily keep up the amazing content🫶
your video was very interesting but you have a horrible case of "youtuber voice"
sorry i didn't understand a single thing
Trisecting an angle has been done.
Hence why it’s here
Poincare didn't "study" topology. BRO INVENTED IT. Legend.
No ?
@@zeropol google "analysis situs" :p
i thought Euler invented topology with the Seven Bridges problem
@@atlassolid5946 isn't that graph theory?
@@zeropol he introduced the word topology
Galois was an absolute beast. His early death was probably one of the biggest setbacks math has ever had
Real
I'd argue Ramanujan was an even bigger loss
He was the ultimate simp.
@@em.1633 Nah, his sum of natural numbers= -1/12 is the biggest lie which people who want to learn about Maths believe it's true. I mean, saying that 1+2+3+4+...=-1/12 sounds pretty elegant once you see how he found that sum, but you need to dig further to understand that the sum of n from 1 to infinity diverges and that's on period, and since it diverges, there's a property which , by doing partial sums of the original sum, we can get different convergences (which proves, again, the big series diverges). But no one explains this to the newbies in math, they take the well-know value for the sum and get the wrong idea of Analysis.
@@konstantinantonovmladenov5740 people not understanding his work doesn't detract from him being a possibly bigger loss
Fermat's last theorem states that for all n>2 there are no integer solution to the equation aⁿ+bⁿ=cⁿ, what you presented in the video is just a specific case
which ironically, Fermat gave a complete proof of.
I came up with a proof, but it's too big to fit in my mind.
Bro said "comp-ass"
Yeah I can get past a lot of the silly pronunciations but this is just hard to watch
Dude says mathematicians names like a native speaker yet we still get this lol
Bro i read ur comment and he says it litterally as i read comp ass, im dead😂
like the Burger King foot lettuce guy. I refuse to believe anyone talks like this in real life
@@stilts121He does not. His pronunciations of the non-Anglo names are pretty bad.
Just wondering if Evariste Galois had lived long enough he could have massive contribution in maths
Also Ramanujan too. I hope live longer until 60 years old but he died in 30's. 😢
Same deal with ramanujan. Lost way too soon
I dont know, maybe yes maybe no, he could be a Gauss or be a one hit wonder. Anyway we will never know it.
@@juaneliasmillasverawas gauss a one hit wonder?
@@FishStickerthe guy literally wrote “Gauss OR a one hit wonder”
ffs, so many statements are presented wrong. fermat last theorem said about any nth power bigger than 2, not just 3. 3rd power was prove impossible long before Wiles.
7:15 Forget Jigen People Dawg.......
for french, "poincaré" is like "point carré" which would means "square dot"
For me, he always will be not a point, not a square, but a closed manifold
@@atzuras Mmmm, doughnuts!
Change your smoke alarm battery 11:51
Again at 12:18
no
Dude, I thought I was the only one who heard it. Lol
The way you pronounced both of these French gentlemen’s names 11:42 actually gave me cancer
Insolvability of the quintic equation was actually first proved by Abel and Ruffini, Galois only later generalized the theorem and simplified the proof
Poincare didn't live "around 800 years ago," he lived from 1854 to 1912.
he said "a hundered years ago", not "8 hundered years ago".
He said "A hundred" not "800"
OK, I stand corrected! 🤣
It did sound to me like he said 800 years ago. 🥸
I like the video, but please look up the pronunciations of the names beforehand
Come piss. Not calm pass. For example.
He should probably look up the pronunciation of "compass" as well...
arxiv is pronounced like “archive” (ar[k]ive)
You didn't state Fermat's last theorem correctly. The case of 3 as the exponent was proved shortly after Fermat's death. So was exponent 4. But the theorem said there was no equation for any integer exponent greater than 2.
One small thing to note: please change the batteries in your smoke detector
Please for the love of god say compass normally
He's saying it normally you dummy
@@015FedeIn what dialect?
I think you're underselling Grigori's contribution to the Poincaré conjecture in the way you bring up his use of Hamilton's work, he always admitted this and when he declined the prize he said it was because Hamilton's work had been equal to his own.
Great video, although the picture of Richard Hamilton around minute 1:28 is that of the artist by that name, not the mathematician.
You need a pop stop for your mic badly, but other than that great vid.
Also since I'm being a ballbreaker I might as well add this critique: you should speak more naturally, and less with the generic "youtuber giving lecture" monotonous tone
did you ever change the battery in your smoke detector.
Perelman is such a good guy
"... began work to simplify his proof."
Oh neat!
"...to a mere 4000 pages!"
Oh...😅
FYI Henri Poincaré’s first name is pronounced “En-ree”, the H is silent in that French surname.
It's /ɑ̃.ʁi/. Don't try to transcribe French words into English spelling, it doesn't work.
It’s more like Awn-ree
He butchers the last name too lol. Can't be helped since he's American but I wish he would just stick to an Anglican pronunciation so that it at least doesn't sound annoyingly pretentious.
Your smoke alarm needs a new battery
Love this channel. Just wonderful. Keep it up
Unfortunately, there are quite a few mistakes in this.
Just to name two:
Niels Henrik ABEL proved that the quintic is generally insoluble, not Galois.
Fermat's Last Theorem is for n > 2, not n = 3.
EULER proved the n = 3 statement long before Wiles.
Your way of saying things is just too irritating
As a cuber, I am very confused how algebra is related to cubing. I mean, we use a completely different type of notation and there is no mathematical relation besides the R2s and stuff
There are 6 "basic" moves that can be performed on a Rubik's cube. These are the 90 degrees clockwise rotations of each of the 6 faces. (This is assuming we keep the cube in a fixed orientation, so the centre squares of each face do not move.) Each move is a rearrangement of the coloured squares on the cube. Moves can also be composed (i.e. performed in sequence) to further rearrange the squares. Moves can also be reversed, since each basic move can be undone by performing the corresponding counter-clockwise rotation. Each configuration of the squares on the cube can be described by a sequence of moves that takes the cube from the solved position to that particular position. (Although such a sequence of moves is not unique; for instance, RRLRR gives the same configuration as L.)
In mathematics, a "group" is a collection of things that can be composed and reversed. The set of possible configurations of a Rubik's cube is a group. Group theory is a subfield of algebra. This is why Rubik's cubes can be studied using algebra.
I presume you thought that "algebra" meant "equation solving" like one learns in high school. This is *part* of algebra, but in mathematics, algebra is a hundred times bigger than that. (And it is unfortunate that so few people know this.) Algebra involves the study of groups, rings, fields, modules, lattices, monoids, and possibly categories, depending on who you ask. These are all in the same vein as a group, in the sense that they are collections of things that can be "put together" somehow. (For instance, a monoid is like a group, but without the requirement that its elements be invertible.) I think the original meaning of the word "algebra" (or rather, the Arabic word which became "algebra" when borrowed into English) was actually something like "put together" or "broken apart".
If you are wondering what it "looks like" to study the Rubik's cube group, Google "Rubik's cube group".
I can hear your fire alarm beeping in the background (12:18) lol
good video but you used the wrong picture for richard s hamilton, you used the picture for richard hamilton who is an artist lol
Imagine 5 (or 25 or as much as you want) countries meet at the pole. Then you can't use 4 colors, you have to use as many as there are those countries.
I think this theorem implies that a border between two countries cannot be a single point and has to be an actual line.
It also assumes that exclaves are treated as separate entities. Otherwise you can easily make 5 mutually bordering countries.
I’m surprised that he even shows it in the graphic at 5:59 but doesn’t comment on it
not me watching this video as if I could understand all these 💀
Tom Clancy: What is the sum of all fears
Mathematicians: -1/12 fears
Lmao
Nah, integral from -inf to inf of x dx
Which is inf-inf, which means it doesn't converge.
I feel like Fermat was the old times equivalent of a legendary troll having made the theory around his death
Congratulations, this is well done, synthetic but informative
Too bad they are already solved, was trying to get some quick money because these problems are very easy
Bro that's not how you pronounce compass....
Yeah wth
It wasn't 4th dimension it was the 3rd dimension he solved
Why can't I just input thse problems in calculator and solve them?
Next video will be "Unsolved math problems solved by philologists"
😂 sounds interesting but no
the poincare conjecture isnt really about what the most general shape is. The way you formulated it in this video makes it seem that the circle is a more "general" shape than the square, which is kind of exactly what topology is not about. Im sure you know this, just wanted to point out that the formulation is super misleading for someone who doesnt know about topology.
I'm not smart enough to understand all this maths, but I'm pretty sure Poincarré did NOT live 800 years ago
Both the Poincare Conjecture and Fermat's Last Theorem were solved... so why?
Here's a problem that hasn't been solved: why do people use Comic Sans?
I hate the names of these french guys, so weird pronounciation...
Absolutly amazing! Keep up the great work!
All my favorite theorems. Great video.
Not larger, but with more elements
Why tf u pronounce compass like that lmao
Wrong Richard Hamilton photo.
Great video, but I would suggest looking up how to pronounce the names of famous mathematicians next time! (Gödel rhymes with turtle)
perelman is legend.