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 🎉🎉🎉❤❤❤🎉❤
Godel’s second incompleteness basically says: “completeness (all true statements are provable), consistency (only true statements are provable), and arithmetic-pick two”
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
@@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.
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
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.
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.
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.
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.
Abel was the first to prove you can't solve an equation of the 5th degree or higher. However, Galois generalized it further by proving the necessary and sufficient conditions on when a polynomial equation is soluble by radicals.
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"
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".
you said it yourself. 'notation'. math is all about generality, and abstraction generally tends to be the price of generality. And who would've guessed that 'abstract' algebra is what you use to study cubes eh. That's the power of the group. In some sense, solving certain types of equations is the same thing as solving a rubiks cube.
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.
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.
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
I bet the main reason none of the other problems have been solved is because a 1 million dollar novelty cheque would be an insult compared to the work and talent involved.
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.
Impossibility of proving quintic equation was first proved by Niels Henrik Abel. He was the first to demonstrate that in proving anything we should first chech whether it canbe proven or not. Galois reached the same path as Abel.
The classification of Finite Simple Groups was not completed in the 90s. There have been several stages where the classification was deemed complete, but it was only with Aschbacher and Smith's monumental work on the classification of quasithin groups, that the final piece was in place in 2004. I don't think any further gaps have come to light since then.
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.
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.
8:29 -- no, it's not *" **_WHOLE_** numbers"* ... The formulation is actually that: "There are NO *_NATURAL_** numbers **_GREATER than 2 (TWO)_* that satisfy this..." :P .
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!
11:55 Just so you know, for the French name "Jacques", the 'u', 'e' and 's' are silent. So it's pronounced more closely to "Jack", and not "Jakwi" :) This video was very interesting and well-made.
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.
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.
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
I would not describe the results about the Continuum Hypothesis (CH) as it being neither true nor false, Gödel and Cohen's negative results were about the impossibility of proving it or refuting it within certain formal systems, but the question of whether it is actually true or false remains subject of debate - in fact Gödel himself seemed to believe that CH has a definite truth value (true or false), we just don't know for sure which it is. That said there are also mathematicians, like Solomon Feferman, who believe that the truth value of CH is undefined, or even more, that it is not even a defined mathematical problem - Feferman has a paper about it using a semi-intuitionistic subsystem of Zermelo Fraenkel (ZF). There are also mathematicians that have had varying opinions on the subject, e.g. W. Hugh Woodin developed an argument against CH around the year 2000, however in 2010s he stated that he now believes CH to be true.
But couldn't we just say: "Let 'Φ' be a set such that any function 'f : ℕ-->Φ' and 'g : Φ-->ℝ" is non-exhaustive" (understanding that a set "B" is bigger than a set "A" if there's no exhaustive function "A-->B") and just check whether it creates some kind of contradiction or not? If the answer to whether there's no contradiction is "yes", then CH is true. If the answer is "no", then CH is false. If the answer is "it cannot be proven", then it means that we can't find any contradiction so technically it would be true.
@@GabriTell We already know that (if ZF is consistent) the existence of a set with cardinality strictly between that of N and R does not lead to contradiction (by Cohen's result on the impossibility to prove of CH in ZFC), however this does not says anything about whether it is true or false. First, statements are not true or false by themselves, you need a model and an interpretation of statements in that model. Gödel found a model, the universe L of constructible sets, in which CH is true. Cohen, using a technique called "forcing," found another model in which CH is false. When mathematicians claim CH to be "true" or "false" in an absolute way they presume that there is a model that fulfills exactly our intuition of "set," and such model cannot be Gödel's L - it is too small and seems to leave many sets outside it, L is in fact the minimal model in which the axioms of ZF are true. It cannot be Cohen's model either, since it is not even just a model, it is a collection of models (forcing provides a lot of flexibility in model construction), and they are clearly artificial. The tendency in ontology of set theory is to accept as sets as many collections of things as possible without causing contradiction - in other words, the universe V of "actual" sets should be maximal, not minimal. This has open the door to many axioms of large cardinals stretching the "height" of V as much as possible, but none of them tells us anything about its "width" (how many subsets a set has). The maximality principle would lead to the cardinality of R being waaaay larger than that of N, but we do not know very well how this combines with the maximality of the height of V. I remember a talk by John Conway (the author of the game of life) expressing his opinion that perhaps the cardinality of R is actually an inaccessible cardinal, but he was disappointed by Cohen's results which technically show CH cannot be proved but tell little to nothing about the nature of the "actual" universe of sets V. On the other hand, in a talk by W. Hugh Woodin's I learned about his work on the "Ultimate L", which seems like one of the most serious efforts to determine the characteristics of V - but we still need to see what other experts in the area have to say about it. My own position lean's towards Solomon Feferman's. I am not sure that our intuition of "set" is clear enough to determine an "actual" universe of sets with perfectly well defined properties - intuitions can be blurry and often even plain wrong, so I remain skeptical.
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.*
@@gamer__dud10???
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
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!
@@alejrandom6592 You get Peano theory.
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.
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
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.......
Insolvability of the quintic equation was actually first proved by Abel and Ruffini, Galois only later generalized the theorem and simplified the proof
The image you used for richard hamilton is not the mathematician but the artist. The mathematics Richard Hamilton is someone else
Change your smoke alarm battery 11:51
Again at 12:18
no
Dude, I thought I was the only one who heard it. Lol
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!
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.
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 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.
11:02 fire alarm chirp replace your batteries
there’s another one at 11:10
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.
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.
Perelman is such a good guy
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.
The way you pronounced both of these French gentlemen’s names 11:42 actually gave me cancer
the 2nd one was belgian, but i 2nd the sentiment
Abel was the first to prove you can't solve an equation of the 5th degree or higher. However, Galois generalized it further by proving the necessary and sufficient conditions on when a polynomial equation is soluble by radicals.
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.
"... began work to simplify his proof."
Oh neat!
"...to a mere 4000 pages!"
Oh...😅
One small thing to note: please change the batteries in your smoke detector
arxiv is pronounced like “archive” (ar[k]ive)
Replace the battery in your smoke alarm. 11:10.
Your smoke alarm needs a new battery
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...
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. 😂
I feel like Fermat was the old times equivalent of a legendary troll having made the theory around his death
Great video, although the picture of Richard Hamilton around minute 1:28 is that of the artist by that name, not the mathematician.
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".
you said it yourself. 'notation'. math is all about generality, and abstraction generally tends to be the price of generality. And who would've guessed that 'abstract' algebra is what you use to study cubes eh. That's the power of the group. In some sense, solving certain types of equations is the same thing as solving a rubiks cube.
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.
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.
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. 🥸
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
Euler proved that Fermat’s last theorem holds for cubes centuries before Wiles proved the whole of FLT 8:55
All my favorite theorems. Great video.
I bet the main reason none of the other problems have been solved is because a 1 million dollar novelty cheque would be an insult compared to the work and talent involved.
It doesn't matter if there's a prize or not
They've not been proven because they're like, super hard
Fermat’s was proved by wiles through a remarkable application of elliptic curves and modular forms
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.
I dont understand anything about math, but man i love some good math videos
Congratulations, this is well done, synthetic but informative
Impossibility of proving quintic equation was first proved by Niels Henrik Abel. He was the first to demonstrate that in proving anything we should first chech whether it canbe proven or not. Galois reached the same path as Abel.
1.Solve impossible math problem.
2. Refuse 1 Million dollars and an exclusive medal.
3. Refuses to elaborate further
Gigachad
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.
2:52 what an incredibly uncomfortable position to work in. that dude's back is going to be so messed up.
not me watching this video as if I could understand all these 💀
Poincaré did not exist 800 years ago blud
He said a hundred
I still hear 800 even I saw 800 replies correcting it lol
I love watching along like I understand the Richie Flow Method or anything else described
Love this channel. Just wonderful. Keep it up
The classification of Finite Simple Groups was not completed in the 90s. There have been several stages where the classification was deemed complete, but it was only with Aschbacher and Smith's monumental work on the classification of quasithin groups, that the final piece was in place in 2004. I don't think any further gaps have come to light since then.
I can hear your fire alarm beeping in the background (12:18) lol
I, as a speedcuber, can tell you that I don't use any algebra to solve a Rubik's cube. Just the solving part doesn't require any math really.
Not 800 years ago at 0:22
it was "a 100 years ago" 😄
It does sound like 800 tbf
Thanks for clearing that up 😂
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).
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.
did you ever change the battery in your smoke detector.
The Picture is from Richard Hamilton (artist), not Richard Streit Hamilton (mathematician)
8:29 -- no, it's not *" **_WHOLE_** numbers"* ... The formulation is actually that: "There are NO *_NATURAL_** numbers **_GREATER than 2 (TWO)_* that satisfy this..." :P
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11:01 change your smoke alarm batteries smh (great video)
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 think poincaré's conjecture has been resolved in 2002
Galois proved or helped prove two topics in this video and he was a teenager?!
Goddamn! I am an ape next to him.
trisection of the angle problem is actually about proving which angles are trisectable, every trisection is not impossible
5:58 you can definitely colour that with four colours lol
How?
Absolutly amazing! Keep up the great work!
11:55 Just so you know, for the French name "Jacques", the 'u', 'e' and 's' are silent. So it's pronounced more closely to "Jack", and not "Jakwi" :)
This video was very interesting and well-made.
So are there any infinite sizes between the natural numbers and real numbers?
The Continuum Hypothesis: yesn't
I enjoyed the vid, great work and explanation, thumbs up!!
The continuum hypothesis looks like two elephants except one is flipped
great video but please change the batteries in your smoke alarm
Fermat's theorem is proved by Wiles
Galois is a god
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.
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.
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
Not larger, but with more elements
"Unsolved math"
> look inside
> solved
Your pronunciation of long names is impeccable
Henry Ponkaray
He gets Godel very wrong
My man, Change your batteries in your smoke alarm please.
Fun fact: trisecting an angle is trivial using origami method (folding)
Yo! FIX YOUR FIRE ALARM! Cool video. The incompletness thoerm makes me think consciousness is a solution outside our reality.
Nobody's gonna talk about how crazy the fact that Evariste Galois literally created new maths in order to make his proofs is? 🚬
I would not describe the results about the Continuum Hypothesis (CH) as it being neither true nor false, Gödel and Cohen's negative results were about the impossibility of proving it or refuting it within certain formal systems, but the question of whether it is actually true or false remains subject of debate - in fact Gödel himself seemed to believe that CH has a definite truth value (true or false), we just don't know for sure which it is. That said there are also mathematicians, like Solomon Feferman, who believe that the truth value of CH is undefined, or even more, that it is not even a defined mathematical problem - Feferman has a paper about it using a semi-intuitionistic subsystem of Zermelo Fraenkel (ZF). There are also mathematicians that have had varying opinions on the subject, e.g. W. Hugh Woodin developed an argument against CH around the year 2000, however in 2010s he stated that he now believes CH to be true.
My limited understanding is the CH is independent of ZFC. It was not stated in this video this fact, which I think is crucial.
But couldn't we just say: "Let 'Φ' be a set such that any function 'f : ℕ-->Φ' and 'g : Φ-->ℝ" is non-exhaustive" (understanding that a set "B" is bigger than a set "A" if there's no exhaustive function "A-->B") and just check whether it creates some kind of contradiction or not?
If the answer to whether there's no contradiction is "yes", then CH is true.
If the answer is "no", then CH is false.
If the answer is "it cannot be proven", then it means that we can't find any contradiction so technically it would be true.
@@GabriTell We already know that (if ZF is consistent) the existence of a set with cardinality strictly between that of N and R does not lead to contradiction (by Cohen's result on the impossibility to prove of CH in ZFC), however this does not says anything about whether it is true or false. First, statements are not true or false by themselves, you need a model and an interpretation of statements in that model. Gödel found a model, the universe L of constructible sets, in which CH is true. Cohen, using a technique called "forcing," found another model in which CH is false. When mathematicians claim CH to be "true" or "false" in an absolute way they presume that there is a model that fulfills exactly our intuition of "set," and such model cannot be Gödel's L - it is too small and seems to leave many sets outside it, L is in fact the minimal model in which the axioms of ZF are true. It cannot be Cohen's model either, since it is not even just a model, it is a collection of models (forcing provides a lot of flexibility in model construction), and they are clearly artificial.
The tendency in ontology of set theory is to accept as sets as many collections of things as possible without causing contradiction - in other words, the universe V of "actual" sets should be maximal, not minimal. This has open the door to many axioms of large cardinals stretching the "height" of V as much as possible, but none of them tells us anything about its "width" (how many subsets a set has). The maximality principle would lead to the cardinality of R being waaaay larger than that of N, but we do not know very well how this combines with the maximality of the height of V. I remember a talk by John Conway (the author of the game of life) expressing his opinion that perhaps the cardinality of R is actually an inaccessible cardinal, but he was disappointed by Cohen's results which technically show CH cannot be proved but tell little to nothing about the nature of the "actual" universe of sets V. On the other hand, in a talk by W. Hugh Woodin's I learned about his work on the "Ultimate L", which seems like one of the most serious efforts to determine the characteristics of V - but we still need to see what other experts in the area have to say about it. My own position lean's towards Solomon Feferman's. I am not sure that our intuition of "set" is clear enough to determine an "actual" universe of sets with perfectly well defined properties - intuitions can be blurry and often even plain wrong, so I remain skeptical.
Poincaré looks like if Jamie Hyneman had an alter ego. Not a bad thing
7:20 continuum hypothesis is independent of ZFC
How did Terrance Howard problem go unnoticed?
It wasn't 4th dimension it was the 3rd dimension he solved
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
Sometimes math seems useless in the real Life but It Is really used everywhere. Buy i wonder if there are real world application for topology
Hey ThoughtThrill, don't know if anyone told you, but I think you used the wrong image for Richard Hamilton... Otherwise, cool video!
"Every unsolved problem math has solved"
Um, All of them?
perelman is legend.
Next video will be "Unsolved math problems solved by philologists"
😂 sounds interesting but no
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, ...
0:15 “around 1900”, 0:20 “lived around 800 years ago” oof nice math
i said "a hundred years ago" but it sounded like 800 years ago
arXiv is pronounced like "archive"
(yea, I know it's odd.. but it's also pretty cool)
wasnt fermats last theorem abt every integer bigger than, not only 3?