Does maths have a fatal flaw? Yes, it makes my head hurt

@92RKID

11 күн бұрын

My brain hurts when it comes to math because I have Dyscalculia and math is a foreign language that I can't ever understand.

@savitatawade2403

11 күн бұрын

​@@92RKIDwth even is that?

@canyoupoop

11 күн бұрын

Maths make my headache go away I do it as a hobby sometimes

@arifhossain9751

11 күн бұрын

@@savitatawade2403 its like Dyslexia, but specific to numbers

@oldcowbb

10 күн бұрын

whose flaw is that

I watched the Vsauce video on the Banach-Tarski Paradox about 4 times before somewhat grasping the concept...

@jbassstrat

11 күн бұрын

Thats the one Vsauce video I simply cannot understand

@ILoveLuhaidan

11 күн бұрын

I watched it when I was binge watching all his videos for the first time when I was 11. I remember being genuinely dizzy after that.

@Simpson17866

11 күн бұрын

He went into a lot of the technical detail, but the most important base principle is ♾️ x 2 = ♾️ The practical-sounding description of “cutting up a ball and putting the pieces together into two balls” disguises the fact that at its core, this is like Hilbert’s Infinite Hotel

@vaibhavsharma2700

11 күн бұрын

@@Simpson17866 shouldn’t the set be of finite size? Or am I missing something?

@lyrichives7859

11 күн бұрын

Us

In fact, the Banach-Tarski paradox is an abbreviation. Full name is the Banach-Tarski Banach-Tarski paradox paradox

@abinashdas6040

11 күн бұрын

I see what you did there 🗿

@Martin-pb7ts

10 күн бұрын

Or the BTBTPP?

@lrwerewolf

9 күн бұрын

Wouldn't that make it the Banach-Tarski^n paradox^n, for arbitrarily high values of n, via iteration of the process?

@michaelwarren2391

9 күн бұрын

So it's a pair o' paradoxes?

my head hurts just thinking about the video 😭😭😭 but the animation is adorable omg

@jojorrentamaranatha9389

7 күн бұрын

Me too 😂

I first encountered the Banach-Tarski paradox in my subject for mathematical proof. When we discussed certain set-theoretical concepts, we naturally covered the Axiom of Choice. Our teacher introduced us to the Banach-Tarski paradox and promised we would eventually learn its proof as we attended higher mathematical classes. I needed to learn concepts from mathematical analysis and topology to actually understand the proof.

@Schmidtelpunkt

10 күн бұрын

What do I have to study to even just understand what this problem is about?

@JaybeePenaflor

10 күн бұрын

​@@Schmidtelpunkt If you read the proofs of the Banach-Tarski paradox, you'll need extensive knowledge on group theory, set theory, analysis, and linear algebra. The Axiom of Choice is usually introduced in foundational math courses (usually where principles of mathematical proof is introduced).

@JaybeePenaflor

10 күн бұрын

@@Schmidtelpunkt This proof is longer, but is more detailed. Plus, I love how the author manages to explain some of the concepts. www.diva-portal.org/smash/get/diva2:1672461/FULLTEXT01.pdf

@Jose_Hunters_EWF_Remixes

9 күн бұрын

@@Schmidtelpunkt The Axiom of Choice has such incredibly diverse and apparently formulations I once wrote a paper which merely listed these various formulation, with a discussion of how bizarrely unconnected these are I'll take a stab at it by calling it the mathematician's get out of jail free card for proof-writing - with the realization that it's an adequate conceptualization

@Jose_Hunters_EWF_Remixes

9 күн бұрын

​@@JaybeePenaflor No topology? I assumed BTP would require it. It _sounds_ like topology to me

You haven't explained why the axiom of choice makes the sphere construction possible

@julianbruns7459

11 күн бұрын

It makes it possible to deconstruct the (measurable) ball into non measurable sets that, if reassembled into a (measurable) set, happen to have a different volume. I don't think explaining that would fit into a Ted video aimed at a general audience.

@user-en6tz3iy1z

10 күн бұрын

if you have no mathematical knowledge besides highschool, it would probably take you around a year of studies that are necessary to understand the proof

@arleyantes9321

10 күн бұрын

Yes, because the tiny fraction of the audience that would understand it would be all made of people who already understood it. I guess one of the goals here is to inspire people, in particular young people, to seek that kind of knowledge. But the technical parts of it require years of intense study, of course. Anyway, understand the basics of mathematics, as in what is an axiom and what is a theorem, is much more accessible and was indeed covered here.

@Jose_Hunters_EWF_Remixes

9 күн бұрын

​@@julianbruns7459 By measurable, you mean in the Lebesgue integration sense? Dear lord, that was my waterloo This from a person whose area of expertise is the pinnacle of an introductory graduate level course in abstract algebra Namely Galois Theory

@Jose_Hunters_EWF_Remixes

9 күн бұрын

​@@user-en6tz3iy1z I'll postulate that it would take anyone - under those condition - a heck of a lot longer Unless they were extremely gifted in math In which case, they'd have known that already and therefore have much more than a basic understanding

Nope. I don't understand. Have a nice day

@stephenj9470

10 күн бұрын

Yeah, this was a very poorly designed Ted Ed video. Usually more practical examples in other Ted Ed videos.

@aditisk99

4 күн бұрын

​@@stephenj9470 Or maybe we were not on a level of understanding 👀

@AshikurRahmanRifat

4 күн бұрын

This is to advanced

@sackeshi

2 күн бұрын

The Banach Tarski paradox is that its theoretically possible for there to be an infinite number of options and the same option to be picked every time. If you have a bag of M&Ms and put each M&M into a different box in a room with infinity number of boxes its theoretically possible for an all knowing being to choose those same M&Ms each time.

I’ve always loved that futurama references this with the professor’s duplicator machine in the episode where there are infinite benders.

Giving an alien you've just met an infinitely sharp knife might not be the smartest idea

@VVerVVurm

10 күн бұрын

I was thinking the same ..

What's an anagram of Banach-Tarksi? Banach-Tarski Banach-Tarski

@jonathan3372

8 күн бұрын

yes.

Couldnt understand

I studied math in college and you guys explained the axiom of choice so clearly that I learned something new

Let's not forget during the Banach-Tarski construction, the pieces the ball is cut into are in fact non-measurable, meaning there's no consistent way to assign a volume to each of them, making it even less realistic

@crazysasha1374

3 күн бұрын

I think that's kinda key, right? It's missing an axiom we use to model reality as we perveive it, right?

Not an infinite number (at 0:15), a finite number. From Wik: "Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. "

@Robertganca

11 күн бұрын

If it’s finite, then how are you able to recreate the ball twice with the same diameter and density? Or do those change?

@julianbruns7459

11 күн бұрын

The point is that those finite decompositions require an uncountable number of choices, so you first need to have an infinite amount of sets to chose the points from. The decompositions are still finite though.

@julianbruns7459

11 күн бұрын

​@@Robertgancadiameter and density don't change. (I don't think something like density exists in this context, its either solid or its not. In this case the starting ball and the ending balls are all solid)

@julianbruns7459

11 күн бұрын

​@@GuzMat-matematicas a single point has measure 0, you won't increase the measure of the ball by rotating and reassembling sets of measure 0. You need non measurable sets for this which the axiom of choice implies the existence of.

@e-dragon4691

11 күн бұрын

​@@Robertganca That's a good question, because this paradox questions our understanding of volume and density. The problem here is that the axiom of choice allows you to contruct sets, where assigning any value as "volume" to it would result in a contradiction. In measure theory we call these sets non-measurable sets and by allowing the axiom of choice, we have to accept them. The trick in Banach Tarski's paradox is to split the ball into unmeasurable sets in order to circumvent any volume restrictions.

The flaw is that I can’t do math

I loved this video! Animations are always on point. Learning about axioms in college was so complicated, but you guys made it so easy to digest.

One of my favourite videos to date, loved the animation as well as the analogies used!

the animation is ON POINT. beautifully done

Ted ed finally doing a video on this, nice🔥🔥

Michael had already done a great job in explaining this Paradox.

@canyoupoop

11 күн бұрын

This is like a trailer for that 22min abomination

@Nyramyss-jj8mj

6 күн бұрын

Well it turns out this video isn't really about the paradox, but makes a bigger point about its implications for the bases of mathematics

Love these vids, keep it up :)

TED-Ed math videos are always impeccable. There's a great ending quote on Spivak's Calculus from Jonathan Swift, when he lays the definitions of the reals: There was a most ingenious Architect who had contrived a new Method for building Houses, by beginning at the Roof, and working downwards to the Foundation.

As a lover of mathematics this video is really amazing great job

Fascinating! Thank you!

I wanted a video on this topic for a long time. You guys reading my minds! 😭 Maths hasn't reached a perfect stage *yet* .

the most beautiful thing about math to me is how well we've learned to talk about imaginary things with other people - this video boils this down and does it in an accessible and fun way too :) bravo to the makers!

This reminds me of Non-Euclidean video games like Antichamber and Superliminal.

Zeno's paradox tells us that if we take steps towards a door way such that each consecutive step is half that of the previous step, them we will never reach our destination. In reality we do due to convergence. In reality measurement is limited as shown by Heisenberg's uncertainty principle and paradoxes arise when we try to use "common" logic with infinities. They do not work because infinity is a concept, not a number and is therefore immeasurable AND it does not fulfil the axioms of real numbers.

Can't find anyone talking the animation here, I think it makes the video much easier to comprehend!

The Axiom of Choice(AC) can be substituted by the Axiom of Determinacy(AD).

@Jose_Hunters_EWF_Remixes

9 күн бұрын

Explain

One of my biggest math questions is about the order of operations. In my mother's time in school, all math was answered in the order it was written. For example; 1+2×3 would be 9. But when I was in school, we worked off BEDMAS. Using BEDMAS, the same question; 1+2×3 would end up with the answer being 7. How could the fundamental nature of math change in half a century and not throw the world into chaos?

@srizanchowdhury

10 күн бұрын

No, I think, the problem with the order of operations is caused by our teachers not being careful and fully informed. Besides, to make sure there is no ambiguity, we can use parenthesis and never skip a "x" when we imply multiplication. Take up a programming language and you will see how easy and clear the operations are.

@mattharper9602

9 күн бұрын

My interpretation is that underlying fundamental maths hasn’t changed, only the notation that’s used. As long as you have agreed the notation (e.g. order of operations, symbols, or base numbers) with the person you are communicating with, no problem is caused.

excellent explanation and presentation!

We learned this in my really analysis class about a month ago!

The graphics of this episode are excellent, particularly the math houses with differing foundations.

I only heard the name of this Banach-Tarski paradox and didn't really know what it was, but the animation definitely helped me understand what it was! 👍

Wait, I actually was able to follow through with this! Basically, math itself is pretty abstract but it becomes concrete once we apply it in a practical situation (a.k.a. reality). And there are alot of alternate "truths", I guess, that would lead to different realities. Cool stuff.

The catch with this and with Gödel is that a copy (or an infinite number of copies) is being made from the infinite

It makes me wonder if the axioms we choose can get closer to supporting all math, or is it actually just building different interesting homes. is one set of axioms "better" than the other or just simply different? I think it would be a great idea if we link back to ted's video on godel and "is math created or discovered?"

How much I love this channel

I love this so much! My brain feels expanded

Because math is a man-made tool rather than a natural science, it can contain such examples that likely have no parallel in the physical world. To my understanding this paradox stems down to being able to break down a set of values (i.e. ones that represent a sphere) into an infinite series of an infinitely-high resolution. The unobservable universe may very well be infinite, but applied physical situations within our observable reality don't seem to be infinite. If a physical sphere is made of a finite number of subatomic particles, and space as well may be of a finite resolution, we can't section a physical one an infinite number of times to make use of the mathematical phenomenon which is ∞=∞-1=∞-2... Point being, this paradox breaks our brain because it applies to a mathematical sphere values that don't exist in a physical sphere.

Beautifully done

Hey Ted -ed sugestion make a video about the 1992 riots of Los Angeles.

@blazer9547

11 күн бұрын

That goes against axiom of greatness of multiculturalism

@kirbya9545

11 күн бұрын

@@blazer9547the one axiom that still remains even where there is proof that says otherwise 😂

its usually is a rounding error

I was following along, then I got lost partway through. It’s a good thing I understand the math needed for everyday life! Not that I was terrible at math in school, it’s just not something I need day to day.

Flaws? Yes, it's not always fun to learn it.

Could we refine the AoC so that it’s usable in the sensical applications while not applicable in known non-sensical?

@gperm4941

11 күн бұрын

theres no such thing as "sensical applications". Axiom of choice naturally leads to these things.

@Otis151

11 күн бұрын

@@gperm4941 I defer to you, as I am not a math person. Still Euclid’s axiom was found not to be needed in certain situations. Perhaps AoC is similar? But I think I see what you’re saying. AoC is perfect and just because a paradoxical result is unintuitive doesn’t mean it’s wrong?

@gperm4941

10 күн бұрын

@@Otis151 My point is that you are stuck between having banac taramy and well ordering, and NOT having the fact that you can pick an item from a collection of nonempty sets. The finite axiom of choice is actually provable without axiom of choice, axiom of choice is really just for infinities. Thats why theres so many weird results from it

@Octa9on

10 күн бұрын

​​​​​@@Otis151I'm just an interested layman here but my understanding is that, without the axiom of choice, there are important (to us) mathematical questions that don't have an answer; so we can either just give up on those questions, or accept the sometimes bizarre answers that using the axiom gives us. there are weaker axioms that can be chosen in place of the axiom of choice, but as far as I'm aware no one's found a way to get the answers without the weirdness. the vast majority of mathematicians just accept the axiom of choice and its consequences and get on with their work

The spheres (circles) are constructed via an infinite process. Using that same infinite process one could have constructed two identical spheres (circles), instead. If I'm not mistaken, that is the idea behind the theorem. It is not saying that if you have two oranges, you can theoretically slice it up and combine the pieces into two identical oranges.

@julianbruns7459

9 күн бұрын

Thr idea behind the theorem is that you have non measurable sets, that, when rotated and translated, don't preserve volume. So if you decompose a ball into those sets, you can rearrange them into twice the original ball. The existence of those non measurable sets is only guaranteed because of the axiom of choice.

@sai_beo

9 күн бұрын

@@julianbruns7459 Sounds more like the idea behind the proof. A little challenging an intuition for the layman or even students beginning their course in measure theory.

Stopping people from saying first

I keep a Sly Eye on the Axiom of Choice ….. 👀 Hoping to see some action . Any action ? So far … so good .

How about a video on the strength/weakenss of logic amd rules of inference.

A "cameo" from Heptapods "Flapper" or "Raspberry" at the end would've been an amazingly apt reference.

How does the link work

It's a similar concept to Hilbert paradox

@user-dk1nr3tv8b

11 күн бұрын

Not really, the main point of this paradox is to show that there is no consistent way to assign a measure (i.e. volume) to every possible subset of three dimensional space, because otherwise you can transform a ball into two balls of the same size through seemingly volume-preserving transformations. Hilberts paradox is not really a paradox as it simply shows a few (unintuitive) differences between finite and infinite sets

@shubhamjat6926

10 күн бұрын

​@@user-dk1nr3tv8bThanks you explained it beautifully

You had me at 'Maths' Ted-ed. Love your videoes on maths.

Damn! YOU ARE PERFECT in making simple paradox more complicated 😒

@julianbruns7459

11 күн бұрын

If you find this explanation complicated, how can you find the actual paradox simple?

So different axioms lead to different results. How do we define these axioms?

Great video, thanks

Thanks Ted. This was actually quite beautiful

So basically the paradox in simple terms is ((1/infinite)x(infinite))=2?

I watched two different videos several times, I'll finally get it

When I saw the name Bancah-Tarski i immediately thought of the riddle ted ed did.The infinite gold riddle Where the name was in front of the little mans shirt.

Whilst I appreciate the message at the end about axioms being potentially non-universal, please could we not give our infinitely-sharp knife to an alien

AoC is necessary for proving that if two sets aren't the same size, one of them is bigger

@julianbruns7459

9 күн бұрын

Your language is confusing. What do you mean by "size"? Do you mean cardinality? In that case it is false, since both the integers and natural numbers have the same cardinality. Do you mean lebesgue measure? Then this is trivially not true. Are you referring to the generalized continuum hypothesis? Then the Axiom of Choice isn't necessary for proving it, in fact it is the exact opposite: ZF+ GCH implies AoC. Do you mean Cantors theorem, that the power set of a set has larger cardinality than that set? Then your communication would have been pretty poorly. Could you please elaborate what you mean?

@MichaelDarrow-tr1mn

9 күн бұрын

@@julianbruns7459 i mean cardinality in both the "bigger" part and the "size" part.

@julianbruns7459

8 күн бұрын

​@@MichaelDarrow-tr1mn then this seems more like a tautology that you don't need the axiom of choice for, just the definition of an order.

@MichaelDarrow-tr1mn

8 күн бұрын

@@julianbruns7459 pretty sure it's actually equivalent to the axiom of choice

@julianbruns7459

7 күн бұрын

@@MichaelDarrow-tr1mn would you be so kind and give me a source for that claim?

Very nice video

Sometimes you can follow all the rules and come to an unexpected result. This is like California declaring the bee a fish because it meets all the criteria. No it's not actually a fish but according to the rules we established it is legally a fish. The same goes for this. It's not technically possible in reality (as far as we know), however according to the rules we have established it's possible.

@armanahmadian4373

11 күн бұрын

Well in our universe it is not possible. In another one it may be. And no, this isn't sci-fi. 😂

We stan the axiom of choice

Axiom of Choice sounds like a powerful spirit!

Consider a hypothetical scenario, that in our universe we get a new law that if we take 2 and more 2 object and then add them then they will collapse and turn into 0. Now in this case will you say 2+2=0 just because we are seeing in the universe when 2 and 2 objects get add they collapse and turn into 0? Or you will say 2+2=4 because of logical consistency?

@user-bk7ip1do8g

10 күн бұрын

We wouldn't change the way + behaves, but we would make a new operation that is consistent with the new behavior of objects in our universe. So maybe 2#2=0

@MuffinsAPlenty

2 күн бұрын

@@user-bk7ip1do8g mod 4 arithmetic still uses a + sign.

I will not be handing an alien a sharp knife.

I learned basic math with apples and oranges, and of course I understand irrational numbers, but if the elements and variants have to be ordeal and not existing, then how can I know what you say is correct?

This flew over my head

I'm not smart enough to understand this but give the animation team a raise they did an amazing job

Did you get any of that?

Great video, but your explanation for the axiom of choice (2:20) was rather unclear - I’m already familiar with the aoc and still got lost in the metaphor. It might have been better to explain what the axiom actually is, before saying when a choice is valid and telling the story about the omniscient chooser.

No matter what axioms you choose, there will always be true statements that you will be unable to prove.

@stefanperko

11 күн бұрын

Almost! Its true if your axioms are strong enough for a portion of standard arithmetic. For weaker axiom systems it can be different.

@mujtabarehman5255

10 күн бұрын

@@stefanperko That sounds familiar, but makes no sense to me. I’ve been out of school for too long lol

Since this video is about Banach-Tarski Paradox but doesn't actually tell you what Banach-Tarski Paradox is. Basically it is how math becomes wonky with working with different types of infinite, where in most cases we just avoid the topic completely or the calculator spits out as error.

@julianbruns7459

9 күн бұрын

I don't see how this is related to calculators spitting out errors, or is this an analogy?

@mooing_cowmilk

9 күн бұрын

@@julianbruns7459 Kind of. Basically was saying that calculators avoid trying to work out math with infinite by just displaying a error instead

@julianbruns7459

8 күн бұрын

​@@mooing_cowmilk well wolframalpha can calculate me the infinite sum from k=1 to infinity of 1/k^2 to be (π^2)/6 so i guess that depends on what calculator you are talking about.

Ahhhh high school math class flashbacks. Didn't understand much and I zoned out.

So, at last, in this video what the heck is axiom of choice? Why they just mentioned but didn't explain about it?

@zmaj12321

10 күн бұрын

They did explain it, using the "marbles in boxes" metaphor.

Yea I most DEFINITELY understood that.

me before watching the video - what is Banach-Tarski paradox ?🤥 me after watching the video - what is Banach-Tarski paradox ?🤥

Math is everything ❤🎉 thank you

Banach-Tarski? Like the Banach-Tarski from the Infinite Gold riddle?

@thenovicenovelist

10 күн бұрын

That's what came to my mind as well.

@JoshRendall

6 күн бұрын

@@thenovicenovelist Really?

@Gordy-io8sb

6 күн бұрын

The Banach-Tarski paradox in set theory.

I swear mathematicians just enjoy making confusing problems that dont really make sense 😭

So that’s where the little man from the Infinite Coin Riddle got his name. Very interesting.

Good video.

Until we discover how our universe can support... - Infinitely sharp knives, - Infinitely divisible balls, and - Processes for being able to complete infinite numbers of actions in a finite amount of time... surely this is all moot? If we can accept the existence of, say, the square root of -1 (which we can manipulate mathematically but not manifest physically) then why should this be any different?

The mysterious omniscient chooser, as a mathematician, I can't stop laughing; this analogy was too good.

Babe wake up..Ted-Ed just dropped another banger

The "problem" seems to be the concept of infinity. Does "infinite" actually mean anything in the real world? (Genuine question)

Math is supposed to be abstract. One thing that strikes people about it is that it can never be completely understood. Even when you look at it in a different angle, there's still some areas that need analysis on. Equations are anything but perfect. People spend years just looking for the 'correct' answers when they probably aren't the best answers.

Hey Ted -ed sugestion make a video about the 1992 riots of Los Angeles

The animations are so good that they are distracting from the actual narration. had to watch twice.

"New! Shorts Replies!" Didn't I already have it?

Oh... NOW I understand the Strangle Little Man with the bag that makes infinite coins

That's obviously a contradiction. Something finite =/= something infinite.

You know a TedEd video is gonna be serious when it lists two writers in the title

That went over my head. :(

@tiziobelo

11 күн бұрын

Whatch the vsauce video 4 times

@alexandriacollins7119

11 күн бұрын

@@tiziobelo ?

@rgb2296

10 күн бұрын

@@alexandriacollins7119 There's a vsauce video about this

This video explains how I exactly feel abt math

Title: Does math have a major flaw? Intro: Picture a mathematician holding a knife