Theorems That Disappointed Mathematicians 

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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

I'd like to see someone draw the Weierstrass function without lifting their pen

Godel's second incompleteness theorem states that in any collection of disappointing theorems, there will always be a disappointing theorem that is not included

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

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.

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."

Was expecting the Gödel theorems on this list

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, ...].*

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!

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.

Maths is beautiful

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.

What an interesting presentation. Loved it.

This Theorem is so boring that we're calling it a Lemma.

No Gödel incompleteness theorems?

Continuity 👏 does 👏 not 👏 mean 👏 that 👏 a 👏 function 👏can 👏 be 👏drawn 👏withouth 👏lifting 👏 your 👏 pen

And then there are conjectures that are proven to be unprovable.

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.

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.

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.

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.

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.

We're doing AI-generated thumbnails now too?

You seriously made this video without bringing up Godel.

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

I want more videos like this.

Me, an engineering student: *Laughs in Newton Raphson Iteration*

"you can draw it without lifting your pen"...

This man needs more viewers

Gödel: hold this beer, you're gonna need it!

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.

I also think that the solution to Hilbert's 1. Problem is quite disappointing.

Isn't the last theorem a direct consequence of the quintic equation theorem?

Can you tell us how to prepare for the IMO pls

Paulo Ruffini gave a algebraic proof but the proof was flawed as it was found later. Niels Henrik Abel proved it using groups.

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!

It is impossible to be absolutely sure about anything.

Solution: "Just invent new algebra!" 😅🤣 Why not!!!😉😅

Interesting!

Technically, I would call these "conjectures".

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"?

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.)

that AI person in the thumbnail looks like Hugh Dennis

AI looking ahh thumbnail

Not the AI thumbnail 😭

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?

You should cite the Math Stack Exchange post "Theorems that Disappointed Mathematicians," assuming you based this on it

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"

Standard equation for the perimeter of an ellipse? That C is not = pi * f(a, b) is seriously disappointing.

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

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.

Can we solve this problem (2+x)^14.33i where i is imaginary

AI is art theft, you are telling every artist that sees this that you don't care about your work.

Funny that you choose a very solvable fifth-grade... Do your research.

I would add the Banach-Tarski Theorem

Continous everywhere means you can draw it without lifting your pen😂 mate get straight with your concepts

please don’t use ai generated thumbnails

Fun fact: the degree of polynomials which always have solutions goes up to 4, just like the number of dimensions, coincidence?

Can I get a heart pls been following for a long time gr8 vid btw

If you are disappointed by any of those theorems, you are really not understanding them

W

Please don't use AI-generated images.

IM THE 1000TH VIEWER IM FAMOUS!

25 seconds and no views bro fell off

Please like this comment so that u can solve millenium prize problem

Dislike, because WE NEED VIDEO ABOUT WHAT IS i^^i!

Anyone does not known how I got this much likes

Very Cool