Does math have a major flaw? - Jacqueline Doan and Alex Kazachek
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- čas přidán 9. 05. 2024
- Practice more problem-solving at brilliant.org/teded
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A mathematician with a knife and ball begins slicing and distributing the ball into an infinite number of boxes. She then recombines the parts into five precise sections. Moving and rotating these sections around, she recombines them to form two identical, flawless, and complete copies of the original ball. How is this possible? Jacqueline Doan and Alex Kazachek explore the Banach-Tarski paradox.
Lesson by Jacqueline Doan and Alex Kazachek, directed by Mads Lundgård.
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Does maths have a fatal flaw?
Yes, it makes my head hurt
My brain hurts when it comes to math because I have Dyscalculia and math is a foreign language that I can't ever understand.
@@92RKIDwth even is that?
Maths make my headache go away I do it as a hobby sometimes
@@savitatawade2403
its like Dyslexia, but specific to numbers
whose flaw is that
I watched the Vsauce video on the Banach-Tarski Paradox about 4 times before somewhat grasping the concept...
Thats the one Vsauce video I simply cannot understand
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.
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
@@Simpson17866 shouldn’t the set be of finite size? Or am I missing something?
Us
my head hurts just thinking about the video 😭😭😭 but the animation is adorable omg
Me too 😂
In fact, the Banach-Tarski paradox is an abbreviation. Full name is the Banach-Tarski Banach-Tarski paradox paradox
I see what you did there 🗿
Or the BTBTPP?
Wouldn't that make it the Banach-Tarski^n paradox^n, for arbitrarily high values of n, via iteration of the process?
So it's a pair o' paradoxes?
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.
What do I have to study to even just understand what this problem is about?
@@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).
@@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
@@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
@@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
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.
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
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.
@@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
@@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
I’ve always loved that futurama references this with the professor’s duplicator machine in the episode where there are infinite benders.
What's an anagram of Banach-Tarksi?
Banach-Tarski Banach-Tarski
yes.
Giving an alien you've just met an infinitely sharp knife might not be the smartest idea
I was thinking the same ..
Nope. I don't understand. Have a nice day
Yeah, this was a very poorly designed Ted Ed video. Usually more practical examples in other Ted Ed videos.
@@stephenj9470 Or maybe we were not on a level of understanding 👀
This is to advanced
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.
One of my favourite videos to date, loved the animation as well as the analogies used!
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.
I studied math in college and you guys explained the axiom of choice so clearly that I learned something new
the animation is ON POINT. beautifully done
Ted ed finally doing a video on this, nice🔥🔥
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
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. "
If it’s finite, then how are you able to recreate the ball twice with the same diameter and density? Or do those change?
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.
@@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)
@@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.
@@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.
Michael had already done a great job in explaining this Paradox.
This is like a trailer for that 22min abomination
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
This reminds me of Non-Euclidean video games like Antichamber and Superliminal.
Love these vids, keep it up :)
Couldnt understand
As a lover of mathematics this video is really amazing great job
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.
Can't find anyone talking the animation here, I think it makes the video much easier to comprehend!
Fascinating! Thank you!
Could we refine the AoC so that it’s usable in the sensical applications while not applicable in known non-sensical?
theres no such thing as "sensical applications". Axiom of choice naturally leads to these things.
@@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?
@@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
@@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
We learned this in my really analysis class about a month ago!
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 Axiom of Choice(AC) can be substituted by the Axiom of Determinacy(AD).
Explain
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! 👍
The catch with this and with Gödel is that a copy (or an infinite number of copies) is being made from the infinite
excellent explanation and presentation!
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?"
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!
The graphics of this episode are excellent, particularly the math houses with differing foundations.
How about a video on the strength/weakenss of logic amd rules of inference.
How does the link work
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?
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.
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.
It's a similar concept to Hilbert paradox
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
@@user-dk1nr3tv8bThanks you explained it beautifully
its usually is a rounding error
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.
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.
@@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.
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.
I keep a Sly Eye on the Axiom of Choice ….. 👀
Hoping to see some action . Any action ?
So far … so good .
Beautifully done
So different axioms lead to different results. How do we define these axioms?
A "cameo" from Heptapods "Flapper" or "Raspberry" at the end would've been an amazingly apt reference.
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.
I love this so much! My brain feels expanded
You had me at 'Maths' Ted-ed. Love your videoes on maths.
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.
Seems like this could be used as a mathematical interpretation of particle/anti-particle creation and annihilation
How much I love this channel
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?
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
@@user-bk7ip1do8g mod 4 arithmetic still uses a + sign.
Damn!
YOU ARE PERFECT in making simple paradox more complicated 😒
If you find this explanation complicated, how can you find the actual paradox simple?
The flaw is that I can’t do math
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.
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?
Thanks Ted. This was actually quite beautiful
Axiom of Choice sounds like a powerful spirit!
So, at last, in this video what the heck is axiom of choice? Why they just mentioned but didn't explain about it?
They did explain it, using the "marbles in boxes" metaphor.
So basically the paradox in simple terms is ((1/infinite)x(infinite))=2?
Did you get any of that?
Very nice video
This flew over my head
Stopping people from saying first
Banach-Tarski? Like the Banach-Tarski from the Infinite Gold riddle?
That's what came to my mind as well.
@@thenovicenovelist Really?
The Banach-Tarski paradox in set theory.
Flaws? Yes, it's not always fun to learn it.
No matter what axioms you choose, there will always be true statements that you will be unable to prove.
Almost! Its true if your axioms are strong enough for a portion of standard arithmetic. For weaker axiom systems it can be different.
@@stefanperko That sounds familiar, but makes no sense to me. I’ve been out of school for too long lol
Great video, thanks
Yea I most DEFINITELY understood that.
That went over my head. :(
Whatch the vsauce video 4 times
@@tiziobelo ?
@@alexandriacollins7119 There's a vsauce video about this
AoC is necessary for proving that if two sets aren't the same size, one of them is bigger
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?
@@julianbruns7459 i mean cardinality
in both the "bigger" part and the "size" part.
@@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.
@@julianbruns7459 pretty sure it's actually equivalent to the axiom of choice
@@MichaelDarrow-tr1mn would you be so kind and give me a source for that claim?
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.
Hey Ted -ed sugestion make a video about the 1992 riots of Los Angeles.
That goes against axiom of greatness of multiculturalism
@@blazer9547the one axiom that still remains even where there is proof that says otherwise 😂
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.
Well in our universe it is not possible. In another one it may be. And no, this isn't sci-fi. 😂
Ahhhh high school math class flashbacks. Didn't understand much and I zoned out.
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.
I don't see how this is related to calculators spitting out errors, or is this an analogy?
@@julianbruns7459 Kind of. Basically was saying that calculators avoid trying to work out math with infinite by just displaying a error instead
@@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.
I watched two different videos several times, I'll finally get it
I'm not smart enough to understand this but give the animation team a raise they did an amazing job
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
I will not be handing an alien a sharp knife.
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.
This video explains how I exactly feel abt math
Babe wake up..Ted-Ed just dropped another banger
What was this video about exactly?
The animations are so good that they are distracting from the actual narration. had to watch twice.
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.
So that’s where the little man from the Infinite Coin Riddle got his name. Very interesting.
Hey Ted -ed sugestion make a video about the 1992 riots of Los Angeles
vsause did a video on it
We stan the axiom of choice
The mysterious omniscient chooser, as a mathematician, I can't stop laughing; this analogy was too good.
You know a TedEd video is gonna be serious when it lists two writers in the title
The "problem" seems to be the concept of infinity. Does "infinite" actually mean anything in the real world? (Genuine question)
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?
"New! Shorts Replies!"
Didn't I already have it?
That's obviously a contradiction. Something finite =/= something infinite.
me before watching the video - what is Banach-Tarski paradox ?🤥
me after watching the video - what is Banach-Tarski paradox ?🤥
Oh... NOW I understand the Strangle Little Man with the bag that makes infinite coins
It's hard to understand the paradox when you don't explain it. How does the cutting and slicing, then putting together again, produce two identical spheres? And why spheres? Does it work with any object, 2d, 3d or more?