Theorems That Disappointed Mathematicians
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The world is full of fake mathematical models. Just ask the politicians.
The only theorem that came to my mind when I read the title was Gödel's incompleteness theorem. And now, after watching the video I also remember the problem of constructing a square and a circle with the same area
Won't the areas be the same if the side of the square will be the square root of pi times the radius of the circle?
@@roihemed5632 that's true, however there is no way to construct a side that length with just a straightedge (a ruler with no numbers written on it) and a compass.
@@roihemed5632 assuming you meant pi times the square of the radius: yes, the area would be the same but it is impossible to construct with the classic rules of geometry
@@dennismuller1141 Ok I got what you meant thanks. But I think you're wrong about the length. if x is the side of the square and x² = πr² then x = r√π
@@roihemed5632 Sorry for the confusion, I misread "square root of pi times the radius" as sqrt(pi * r) instead of sqrt(pi) * r, so I corrected it to sqrt(pi * r²)
I'd like to see someone draw the Weierstrass function without lifting their pen
right lool
real
Manufacturing a sufficiently thin/fine pen would be a challenge
@@SteveThePster if epsilon is line thickness, I think that for any epsilon greater than zero, the thickness is not sufficient
@sensorer My pen was manufactured in Surrealia
Godel's second incompleteness theorem states that in any collection of disappointing theorems, there will always be a disappointing theorem that is not included
You are EVIL.
Upvoted ...
This video has to be incomplete, otherwise it would be inconsistent.
Well that's very disappointing.
There is always a disappointment that you are not aware of. Disappointment never ends.
that's not what goedel states.
Was expecting the Gödel theorems on this list
Would be too easy. Also, it doesn't have to be all too bad.
It implies that mathematics has no end, since independent axioms are bound to just pop up.
Would be pretty boring if you could do all the math using one rote algorithm.
The real theorem that dissappointed me the most was the Halting Problem. Something about having undecideable problems in certain axiomatic systems, lacking computational knowledge, and not being able to compute the busy beavers (which, honestly without HP would be tremendous still) is just astounding to me
The halting problem, just like its cousin - Goedel incompleteness theorem, prohibits self-referential logic.
@@AlexagrigorieffAnd Tarski's undefinability theorem.
I still don't know how the halting problem manages to be so _practical._ All the proofs i've seen leave me with the feeling of "it's possible to construct some outlandish gigamess of a program that can't be analyzed to see if it stops." It seems like a completely separate and unexplained result that many such problems are simple to describe
@muskyoxes a good intro video to watch is about "the boundary of computation" and busy beaver numbers, it's the main video that got me interested in the halting problem
@@muskyoxesIt boils down to a simple contradiction like the set of all sets which do not include themselves. Or in practical terms: A sergeants is disappointed with the shaving attitude of his platoon. He thus commands one soldier to shave all soldiers in his platoon, who don't shave themselves, but only them.
3:44 I think Minkowskis question mark function denoted ?(x) Is weirder than this function.
It is continues and always increasing (strongly monotonic) yet its derivative is 0 almost everywhere!
You'd think such a function with derivative 0 a.e. would at least be constant a.e. but nope. This function is never constant and always increases!
Abel Ruffini is kind of a blessing though. Mathematics telling mathematicians, "Guys, guys. What are you doing? Stop solving single variable polynomial equations with roots. You have better things to be doing."
Continuity 👏 does 👏 not 👏 mean 👏 that 👏 a 👏 function 👏can 👏 be 👏drawn 👏withouth 👏lifting 👏 your 👏 pen
for 👏🏻 real-valued 👏🏻 functions 👏🏻 in 👏🏻 one 👏🏻 dimension 👏🏻 it 👏🏻 does
@@justtimo8638no
No, it does not! 1/x is a continuous function. The function indeed is simply not defined in 0. Simply it can't be extended for X=0 to a continuous function, but itself is continuous.
@lucacesarano3661 1/x is not continuous in x=0. Continuity in x=a means that f(a) = lim_x->a (x). A function is considered continuous on an interval I if the function is continuous for all x in I. Since f(0) doesn't exist for f(x) = 1/x, and the limit when x goes to 0 doesn't even exist, certainly it's not continuous in x=0. Hence, f(x) is not continuous over R.
@@jorian_meeuseThere is a logical misunderstanding in the definition.
How would you evaluate the predicate "1=√-1"? True or false? The point is, it does make any sense to say it's true or false, since √-1 is even not defined.
You say correctly that in X=0 the function is not defined. Exactly because of it, it does not even make sense to speak about continuity in X=0.
That's because of the definition of continuity:
"A function f with variable x is continuous at the real number c, if the limit of 𝑓(𝑥) as x tends to c, is equal to 𝑓(𝑐)."
Well, if f(c) is not defined, simply the sentence "is equal to f(c)" can't be logically evaluated as true or false.
Could you logically evaluate the sentence "1 is a yellow number"? If you say it's false, then it means that the sentence "1 is not a yellow number" is true for you, and the property "not to be yellow" is properly defined. But it is not.
For this reason, one can't even say that 1/x is not continuous in x=0.
But it is absolutely a continuous function, since it is continuous on every point of its domain(!).
2:00 But Pythagoras would probably be glad to know that the square root of two has a periodic continued fraction *√2 = [1; 2, 2, 2, ...].*
I'd bet he would get a kick out of that.
Wait until he learns about transcendental numbers
@@JJean64 "Pythagoras, check these out!"
Hm, but did Pythagoras even know anything about the number π? 🤔
Maths is beautiful
*math ftfy
meth is more beautiful
Heh, although ironically this particular video is about instances where math refuses to be beautiful. 😄
Regarding Arrow’s theorem, there are non-ranked systems that satisfy these criteria. Approval Voting (pick as many or as few as you like with no upper limit) and Score Voting (Score every candidate, greatest sum of scores wins) both satisfy these conditions.
en.wikipedia.org/wiki/Duggan%E2%80%93Schwartz_theorem
What an interesting presentation. Loved it.
The weierstrass function ties into the idea of fractals (in fact, the function itself is an example of a fractal) and the idea that most things in nature are fractals, things that are not smooth or well-behaved.
"Most"?
"things that are not smooth or well-behaved."
So, like my wife? ;)
5:40 NO! I usually let this slide but this is a maths channel. The nonexistence of a universal algorithm does NOT mean that there are unsolvable instances of the problem. There could be a (different) algorithm for *every* instance of the problem, there is just not a (single, universal) for *all* of them. There may even be a way to describe strategies for all possible cases, while there is just no systematic way to select a strategy that works, so you might continue trying forever.
Just a minor correction. IIA in Arrow’s impossibility theorem isn’t about whether alternatives (added or taken out of the menu of alternatives) affect binary preference relations at the aggregate level. It’s about whether changing the preference profile outside the binary comparison yields a different aggregate preference between the two.
The Weierstrass function has a feature which is probably even more striking to non-mathematicians: it's a continuous function, which is not monotone (increasing, decreasing, or constant) on any interval, no matter how small. Seems impossible at first glance.
Also, I was anything but disappointed that quintic equations aren't solvable. People need to grow out of this "solving" mindset. To solve something only means to write it in some other way that you find simpler. Guess what, some things just can't be written in simple ways according to narrow minded definitions of simple. Conclusion: expand your horizon and move on.
There are actually some really interesting ways of solving quintic equations. One way is with power series, and the coefficients involve factorials related to multiples of 5. Klein also found a way to relate symmetries of the icosahedron to solving quintics.
Yeah but they don't always work
@@FishSticker What do you mean by that? The power series has a finite radius of convergence, but there are ways to get around that. As for the other method, I believe it always works, as long as the quintic is in the right form.
@@MathFromAlphaToOmega you won't get an exact answer
@@MathFromAlphaToOmega also you might not know if it exists or not
At 4:18, I recall that in analysis we usually define a smooth function as one that is infinitely continuously differentiable. Did you mean to say "a continuous curve should be differentiable almost everywhere"?
And then there are conjectures that are proven to be unprovable.
All conjectures can be proven to be unprovable modulo some axiomatic system.
No Gödel incompleteness theorems?
And wasn't there a theorem that showed that you couldn't prove that the infinity of the continuum was equal to Aleph-1?
@@josepherhardt164what you're looking for is continuum hypothesis
@@antoine2571CH is the statement that the cardinality of the continuum equals aleph_1. The fact that CH is independent from ZFC should probably be called the Gödel-Cohen theorem, but that name doesn't seem to have caught on.
Why would one of the greatest theorems (and one of the greatest corollaries) in all of mathematics feature on a list of the most disappointing ones?
@SelvesteDovregubben I think it's because this theorem proves that there are theorems in mathematics that cannot be proven by other theorems. I'm a little disappointed, because there are truths that can't be proven, but it's better than mathematics being inconsistent. i think that theorems would be the real axioms, and all others mathematic theorem can be proven by this real axioms, but i don't know this is just a thought.
This Theorem is so boring that we're calling it a Lemma.
Arrow's Impossibility Theorem states only that you cannot achieve global independence of irrelevant alternatives in raked voting. But there are raked voting methods that are locally independent of irrelevant alternatives. In rated voting you can have global independence of irrelevant alternatives. But what is more important global of irrelevant alternatives is often criticized as something you may not want to have in your election system.
More interesting (and disappointing) is Gibbard-Satterthwaite theorem which states that there is no election system which isn't susceptible to tactical voting.
4:15 Smooth curve is differentiable everywhere by definition. Weierstrass function is continues, but not differentiable at any point. Weierstrass function is in C_{0} class of smoothness in all of this domain, which basically means that it is nowhere smooth. A smooth function is in C_{infinity} class of smoothness.
*which isn’t susceptible to tactical voting
@@drdca8263 Thank you, corrected ;)
The Abel-Ruffini theorem states that polynomials of degree greater than 4 cannot be solved *using radicals*. A radical is a solution to the equation x^n = a, or in lay terms an n-th root. E.g. sqrt(2) is a solution of x^2 = 2
If we go beyond plain radicals, and add to our toolbox the so-called Bring radicals, which are the unique real solutions to equations of the form x^5 + x = -a, then the general quintic becomes solvable!
So it isn't like quintic polynomials have this infinite complexity compared to lesser degree polynomials. Rather, quintics need a single additional function (the Bring radical function) to express their solutions.
Of course, Bring radicals cannot be computed exactly, only approximated, but neither can sqrt(2). Nothing special about it.
I want more videos like this.
This man needs more viewers
Me, an engineering student: *Laughs in Newton Raphson Iteration*
The solution proposed at 0:16 for the general cubic equation is wrong; it's, indeed, the solution of the depressed cubic y³ + py + q = 0, where y = x + b/(3a). Thus, the general solution you proposed needs a "-b/(3a)" in the second member to be correct.
Gödel: hold this beer, you're gonna need it!
Technically speaking Pitagorean Theorem said nothing about segment length, Pythagoras was only concerned with area of constructed squares
At least the upside of all these "there is no general solution/algorithm" theorems is that we can have secure cryptography
Solution: "Just invent new algebra!" 😅🤣 Why not!!!😉😅
Euler trying to prove i = sqrt(-1):
Proof: he made it up
@@parthpandey2030 Also known as "by definition"😅🤣
you should, just don’t “break” the old algebra 😉
@@bimrebeats Yes this is what I'm saying!!!😅🤣
Look up Bring radicals.
We're doing AI-generated thumbnails now too?
Can you tell us how to prepare for the IMO pls
I would also love to know hope he sees this comment
It's probably changed, but back in the day having AM-GM was always a good ticket. So is having intuition. That's probably not changed.
I think it's for the better that we don't know everything in mathematics and that not everything has a solution. A complete mathematical understanding would mean that we have no more reason to pursue the subject. How boring life would be!
This is why I rejoice whenever physics gets "broken." :)
"you can draw it without lifting your pen"...
For those of us who can draw nowhere differentiable graphs...
If you have Parkinson's, you have an advantage over the rest of us in drawing it.
I also think that the solution to Hilbert's 1. Problem is quite disappointing.
It is impossible to be absolutely sure about anything.
Technically, I would call these "conjectures".
You seriously made this video without bringing up Godel.
Interesting!
True
Arrow had it wrong anyway....that's a direct vote system and they all fail in some form, but if you were to change into an indirect vote, suddenly it solves actually quite easily.
As an example, instead of voting for which person is the best candidate for position XYZ, considering that the majority of the population doesn't even pay attention to politics and are voting on mostly false assumptions......we should be voting on the policies we want to see enacted or retracted. The candidate who scores the most in comparison to their own beliefs of what should be done against the voters that cover that area, should be the winner to the vote itself. To protect the population from tyrannical measures, a simple minimum requirement of 66% vote for those policy changes are the only ones that can actually be commanded for change....everything else is void until next term to be re-evaluated. This would work for all positions, and noting "policy" as different scenarios based on the field of position XYZ (so president will most detail command of military operations, while laws will deal with legislative branch, and punishment the judicial, etc.)
Hippasus, not Pythatorus discovered irrational numbers and for his discovery he was tossed into the sea to drown. A true "math martyr". The Pythagoreans did not like reality to intrude into their harmonious "rational bubble"
that AI person in the thumbnail looks like Hugh Dennis
I can't see the resemblance.
Not the AI thumbnail 😭
Standard equation for the perimeter of an ellipse? That C is not = pi * f(a, b) is seriously disappointing.
What is 0+0+0+0+0.... ∞ equal to?
Technically it should be 0.
But when you treat this series as a Geometric Progression and apply the formula for sum of infinite terms of G.P., the result comes to be 1/(1-(0/0)).
How is this possible?
If you treat this series as 0⁰ + 0¹ + 0² + 0³ + 0⁴ + 0⁵ + ... , when apply the geometric series formula you get 1/(1 - 0) = 1/1 = 1. Which makes sense because 0⁰ is usually defined to be equal to 1, so the series becomes 1 + 0 + 0 + 0 + 0 + 0 + ... = 1.
You can't use the infinite geometric series formula since it is only defined when the absolute value of the common ratio between terms is less than 1. In this case, one could make a decent argument that the common ratio is 1 since the number is not changing between terms, but of course since the first term is 0 you could call it any number you want, since 0*x=0 for all x. So, if you choose r where |r|>=1, use of the formula is just invalid, and if you use r where |r|
@@JJean64 why arbitrarily start at 0^0? Why not begin at 0^-1, or 0^57, or whatever? Usually the first term (a) is considered ar^0, and following terms are ar, ar^2, ar^3 etc. You seem to be thinking of 0 as the common ratio, and rolling with that, we should get 0*0^0 + 0*0^1 + 0*0^2 etc. to give an overall sum of 0.
@@figmentincubator7980
Why would 0 not be the common ratio?
@@JJean64 My point is that it could be anything (read the comment above the one you replied to). Starting at 0, you can multiply by anything to get 0. e.g. the sequence 0,0,0,0,0,... can be achieved by multiplying the first term be 0 every time, or by 1 every time, or by 276.5 every time, or literally any number you want since any number * 0 is still 0.
You should cite the Math Stack Exchange post "Theorems that Disappointed Mathematicians," assuming you based this on it
Wait, do you memorise all Maths Stack Exchange posts just in case?
@@xinpingdonohoe3978No, I was one of the main contributors to that post.
I don't think there's any legend about pythagoras discovering the irrationality of the square root of two, it's just that this is related, but there's other greek guy everyone says discovered this
tho the pythagorean theorem is related, not only we know for a fact it was not discovered by pythagoras (we don't even know if he existed) but you don't need this theorem to know about the square root of two, it's a very easy fact to discover that the diagonal of a square is the side of another square with double its area
I recall hearing a legend about some guy (with a name starting with “H”) proving that the square root of two being irrational, and subsequently being tied up with stone, put on a boat, and thrown into the sea.
Not making this up.
@@crix_h3eadshotgg992 ye and this was surely not pythagoras, and the legend says the pythagoreans threw him in the sea
I strongly believe this didn't happen tho
AI looking ahh thumbnail
Nonetheless, preferential voting is truer than first past the post.
That's a great thumbnail!
Yes; it took all of 10s to solve that first polynomial.
What's the solution, then?
I would add the Banach-Tarski Theorem
AI is art theft, you are telling every artist that sees this that you don't care about your work.
Correct.
Can we solve this problem (2+x)^14.33i where i is imaginary
But that's just a binomial with a complex exponent, isn't it?
Simply use Newton's binomial theorem.
But the definition of complex exponentiation is even simpler:
*zʷ = exp(w·Ln(z)),*
where
*Ln(z) = ln(|z|) + 𝒊·arg(z) + 2𝒊πk, k ∈ ℤ*
is a _multivalued_ complex logarithm, so eventually you get _infinitely_ many values (and may choose one of them on a whim).
That's not a problem. That's just an expression. What are we trying to do with it? Equate it to 5? Expand it as a power series?
Continous everywhere means you can draw it without lifting your pen😂 mate get straight with your concepts
Fun fact: the degree of polynomials which always have solutions goes up to 4, just like the number of dimensions, coincidence?
This is true only in so far that polynomials up to degree four may have their roots solved by using a finite number of simple algebraic operations (re: summation, multiplication/division, and exponentiation/nth roots). Technically the roots of any polynomial in degree z (the "solutions") can be determined by solving a particular series of z nonlinear equations (hint: think about coefficients of algebraic polynomials as consisting as sums of combinations of those roots. Otherwise, evoke the complex domain). This is extremely difficult and so in practice we often rely on the use of approximations and perturbation methods instead
Fun fact: the above is wrong. All polynomials have solutions; you just ignore complex solutions. People using science as religion get facts wrong: coincidence?
@@user-gh4lv2ub2jx^2+1 😳
Yes, this is entirely a coincidence. The reason why all polynomials of degree at most 4 can be explicitly solved in terms of radicals is because every group of order at most 24 is solvable. This is not the case in general. For instance S_5 is not solvable. This is why the roots of the polynomial x^5 + x + 1 cannot be expressed in terms of radicals. The real reason is Galois theory, not some spurious coincidence that the number 4 appears in two places.
@@ethanbottomley-mason8447It seems very likely to be only a coincidence, but, I’m not sure that that description of why quintic not solvable, proves that it is. I suppose for that we might need a more precise definition of what it means for it to be a coincidence
If you are disappointed by any of those theorems, you are really not understanding them
please don’t use ai generated thumbnails
Why?
It got you to click, and it got me to click. The data is what is important here more than personal opinions. I think it’s a nice dramatic picture to generate a click.
@@JoelRosenfeld Any stance on this involves personal opinions to some degree. Here's some data for you: Of the comments currently present on this video 4% of them refer to the thumbnail as AI. Of those 6 comments, 1 is neutral and 5 are negative. I personally am a lot less likely to click on videos with thumbnails that are incoherent or particularly noticably AI-generated.
Here's some data we don't have: How many more or less views would this video get with a human-made thumbnail that took 5 minutes to make? 3 hours to make? If an artist was paid $30 to make a thumbnail, might it result in additional revenue that would cover those $30? Will the channel's community be affected by the use of AI in thumbnails? Will the relationship with the video's sponsor be negatively impacted?
I don't intend to cause you any offense, and I'm not here to try to make you feel one way or another about AI. I just wrote this comment because I know I've used naïve arguments to dismiss people, and I would rather be called out on it than not. Opinions matter, people's thoughts are worth consideration even (and especially) when they differ from your own.
Why not? He does what he wants
@@Nolys-bk4kd I asked 11 days ago mate. He's not gonna answer.
W
Can I get a heart pls been following for a long time gr8 vid btw
Please don't use AI-generated images.
IM THE 1000TH VIEWER IM FAMOUS!
25 seconds and no views bro fell off
12hours and 4k views
15 hours and almost 5k views bro fell off
19 hours and 6.4k views bro fell off
The npc's are coming out again
😐👎
Please like this comment so that u can solve millenium prize problem
Dislike, because WE NEED VIDEO ABOUT WHAT IS i^^i!
The ith tetration of i? Is that what you're asking?
Quite a quandary. Unfortunately, there isn't an agreed upon meaning for tetration outside of natural numbers yet. We have the property a↑↑(n+1)=a^(a↑↑n), and hence a↑↑n=log_a(a↑↑(n+1)), but that doesn't so easily apply to more general cases.
Powers are nice. (a^b)^c=a^bc is powerful. Find something like that for tetration, and we'll get somewhere.
Anyone does not known how I got this much likes
You ain't got any likes nigga 🤡
Very Cool