Layman's Guide to the Banach-Tarski Paradox(kuro5hin.org)
kuro5hin.org
Layman's Guide to the Banach-Tarski Paradox
http://www.kuro5hin.org/story/2003/5/23/134430/275
21 comments
I think it was pretty good. Its not really for people who don't know math, but for people (like me) who know some math but have never looked into the paradox. The key here is to explain what pieces means and the whole cloud idea. With that, I sorta get it (I think so anyway, I'd need to understand the proof better to see if my intuitive feeling is correct).
That being said, I notice a problem immediately. Why do we need 4 pieces? From the explanation, its pretty much giving a sort of "algorithm" for creating the 4 (or 5) pieces. Essentially, start somewhere, and go around and around, and put each "atom" into one of 4 sets. I.e. the first "atom" goes into the first bucket, the second atom into the second bucket, etc... Then, take bucket 1 and 2, and translate them (really just to avoid intersection) and then rotate set 2 so that the atoms are "half way" between those of set 1, and do the same rotation for bucket 4.
So... why not just call buckets 1 and 3 "bucket A" and buckets 3 and 4 "bucket B" and just do the translation?
I know enough to know that my "algorithm" doesn't work, but its what the intuition here suggests. So, there is still some gap in my understanding.
I'm curious how far an intuitive explanation can go. I'll take a look at the link someone else gave below, but does anyone have an intuitive answer to my question?
That being said, I notice a problem immediately. Why do we need 4 pieces? From the explanation, its pretty much giving a sort of "algorithm" for creating the 4 (or 5) pieces. Essentially, start somewhere, and go around and around, and put each "atom" into one of 4 sets. I.e. the first "atom" goes into the first bucket, the second atom into the second bucket, etc... Then, take bucket 1 and 2, and translate them (really just to avoid intersection) and then rotate set 2 so that the atoms are "half way" between those of set 1, and do the same rotation for bucket 4.
So... why not just call buckets 1 and 3 "bucket A" and buckets 3 and 4 "bucket B" and just do the translation?
I know enough to know that my "algorithm" doesn't work, but its what the intuition here suggests. So, there is still some gap in my understanding.
I'm curious how far an intuitive explanation can go. I'll take a look at the link someone else gave below, but does anyone have an intuitive answer to my question?
The initial description deceived me (e.g. the layman :-)). It used the word "piece" in a very technical way. Piece means contiguous chunk in layman's terms, it typically does not mean a set of fragments of the object (that would be many pieces).
If someone told me you could, mathematically speaking, take 5 piles of fragments of a sphere and combine them into two identical spheres, it would still be slightly mysterious, but it would only take the words "infinite density" to make it clear, instead of three paragraphs explaining what "piece" means. :-)
Anyway, I still thought the following explanation was pretty clear. It was probably not the absolute correct mathematical intuition, but what summarization of extremely technical topics are completely correct?
If someone told me you could, mathematically speaking, take 5 piles of fragments of a sphere and combine them into two identical spheres, it would still be slightly mysterious, but it would only take the words "infinite density" to make it clear, instead of three paragraphs explaining what "piece" means. :-)
Anyway, I still thought the following explanation was pretty clear. It was probably not the absolute correct mathematical intuition, but what summarization of extremely technical topics are completely correct?
I love Maths, but this one got me. I admit publicly it is beyond me. When I was small my grand mother used to tell me this story:
One night as the priest was lying with his wife, he said, ‘Wife, if you love me, get up and light a candle, that I may write down a verse which has come into my head.’ His wife, getting up, lighted the candle, and brought him pen and inkstand. The priest wrote, and his wife said, ‘O Master of my soul, won’t you read to me what you have written?’ Whereupon he read, ‘Amongst the green leaves methinks I see a black hen go with a red bill’.
To this day I have not grasped what she meant with this story.
Please can you remove the Layman from the title?
One night as the priest was lying with his wife, he said, ‘Wife, if you love me, get up and light a candle, that I may write down a verse which has come into my head.’ His wife, getting up, lighted the candle, and brought him pen and inkstand. The priest wrote, and his wife said, ‘O Master of my soul, won’t you read to me what you have written?’ Whereupon he read, ‘Amongst the green leaves methinks I see a black hen go with a red bill’.
To this day I have not grasped what she meant with this story.
Please can you remove the Layman from the title?
If you don't get it because you know nothing about set theory, unfortunately, the answer is to learn some about set theory or the notation will just go over your head.
If you don't get it because you are trying to visualize what the five pieces would look like, you can stop trying. It's not possible to visualize. Nobody knows what they "look" like, because there is no known algorithm to construct the five pieces. There is merely a proof that they exist, and that they are infinitely complicated little pieces. If you get everything before that, and you get everything after that, you actually get everything. The Axiom of Choice is basically the official mathematician version of "And magic happens here...". (I'm being glib, it's better characterized than that, but there's a large kernel of truth there.)
(I'm not "against" the Axiom of Choice, but I think it should always be clear when it is necessary, as I think the division between "proofs that might mean something in the real world" and "not" is a division worth keeping track of, even if mathematicians aren't supposed to care about such mundane matters.)
If you don't get it because you are trying to visualize what the five pieces would look like, you can stop trying. It's not possible to visualize. Nobody knows what they "look" like, because there is no known algorithm to construct the five pieces. There is merely a proof that they exist, and that they are infinitely complicated little pieces. If you get everything before that, and you get everything after that, you actually get everything. The Axiom of Choice is basically the official mathematician version of "And magic happens here...". (I'm being glib, it's better characterized than that, but there's a large kernel of truth there.)
(I'm not "against" the Axiom of Choice, but I think it should always be clear when it is necessary, as I think the division between "proofs that might mean something in the real world" and "not" is a division worth keeping track of, even if mathematicians aren't supposed to care about such mundane matters.)
> To this day I have not grasped what she meant with this story.
The story is from a 19th century Turkish book: "The Turkish Jester: Or The Pleasantries Of Cogia Nasr Eddin Effendi" (http://www.gutenberg.org/etext/16244).
The story is from a 19th century Turkish book: "The Turkish Jester: Or The Pleasantries Of Cogia Nasr Eddin Effendi" (http://www.gutenberg.org/etext/16244).
For many more (and rather better written) Nasredin stories (all of which are highly interesting, entertaining, illuminating, or all three at the same time), see my father's story-blog:
http://nasredin.blogspot.com/
Start from the first one:
http://nasredin.blogspot.com/2007/10/yes-there-is-nothing-ne...
http://nasredin.blogspot.com/
Start from the first one:
http://nasredin.blogspot.com/2007/10/yes-there-is-nothing-ne...
The entire thing hinges upon the fact that the set is defined to be infinitely dense, and that if you divide infinity by two (or, really, more), you will still have infinity - thus, if it is possible to subdivide a sphere into two sets with the same radius, and the original sphere was infinitely dense, the two resulting spheres will have the same radius, and be infinitely dense - and thus pretty much identicle to the original. The rest is just discussing various other aspects of it, so that their structure is also the same.
As for your grand-mother's story, I don't have a clue either - my best guess is that it's something about love not caring about how little sense something makes, but that doesn't entirely make sense.
As for your grand-mother's story, I don't have a clue either - my best guess is that it's something about love not caring about how little sense something makes, but that doesn't entirely make sense.
Simple version of B-T in 1D:
http://www.math.ucla.edu/~tao/resource/general/121.1.00s/tar...
http://www.math.ucla.edu/~tao/resource/general/121.1.00s/tar...
for those who want to have a go at a better/more mathematical exposition, http://www.math.hmc.edu/~su/papers.dir/banachtarski.pdf is an excellent rigorous exposition.
And Stan Wagon's book "The Banach-Tarski paradox" is outstandingly good. (But intended for mathematicians; look before you buy.)
I understood the paradox but did not buy the explanation at all.
It's not a great explanation b/c it doesn't go into what "measure" and "volume" mean.
Tough to do without going through a full course but a slightly-less-handwavy explanation would go like this:
There's a mathematical tool (called measure) that formalizes-and-generalizes the notion of "volume".
It generally behaves very much as you'd expect; your intuitions about how "volumes" combine and intersect will generally apply.
There's a catch, though: the way the tool is constructed leaves open the possibility of a non-measurable set, meaning a set for which the definition of the tool leaves you unable to assign that set a well-defined "volume"; you can't just assume that the measure of a set exists.
If you assume the Axiom of Choice then not only are such sets possible, but you can construct non-measurable sets.
The core process in Banach-Tarksi looks like this:
(1) take the sphere (a 'nice' set, which we'll say has "volume" V)
(2) divide that sphere into some sub-sets that are non-measurable (are sets for which our tool cannot supply a measurement) (2.a) Effect on total volume: should have no impact, as the parts we have reassemble to an object of known volume
(3) move the subsets around by sliding-and-rotating them (3.a) Effect on total volume: should have no impact, as neither sliding nor rotating changes volume
(4) wind up with 2 spheres (both 'nice' sets, each with "volume" V)???
The paradox comes from getting double the volume through a sequence of operations that are apparently volume-conserving.
You can go with this a couple different directions.
I'm not convinced this should be an intuitive outcome.
An intuitive, hand-wavy explanation for what's "really going on" would be something like non-measurable sets carry around infinite amounts of finely-detailed structure (too finely-detailed to perceive using our measuring tool); depending on how you position some sets relative to each other their finely-detailed structure might either cancel out (adding no volume) or reinforce each other (adding lots of volume).
That said I think the real lesson here is that your intuition is trying to have its cake -- a non-measurable set -- and eat it too -- have the "volume" of a non-measurable set be preserved under volume-preserving operations.
Tough to do without going through a full course but a slightly-less-handwavy explanation would go like this:
There's a mathematical tool (called measure) that formalizes-and-generalizes the notion of "volume".
It generally behaves very much as you'd expect; your intuitions about how "volumes" combine and intersect will generally apply.
There's a catch, though: the way the tool is constructed leaves open the possibility of a non-measurable set, meaning a set for which the definition of the tool leaves you unable to assign that set a well-defined "volume"; you can't just assume that the measure of a set exists.
If you assume the Axiom of Choice then not only are such sets possible, but you can construct non-measurable sets.
The core process in Banach-Tarksi looks like this:
(1) take the sphere (a 'nice' set, which we'll say has "volume" V)
(2) divide that sphere into some sub-sets that are non-measurable (are sets for which our tool cannot supply a measurement) (2.a) Effect on total volume: should have no impact, as the parts we have reassemble to an object of known volume
(3) move the subsets around by sliding-and-rotating them (3.a) Effect on total volume: should have no impact, as neither sliding nor rotating changes volume
(4) wind up with 2 spheres (both 'nice' sets, each with "volume" V)???
The paradox comes from getting double the volume through a sequence of operations that are apparently volume-conserving.
You can go with this a couple different directions.
I'm not convinced this should be an intuitive outcome.
An intuitive, hand-wavy explanation for what's "really going on" would be something like non-measurable sets carry around infinite amounts of finely-detailed structure (too finely-detailed to perceive using our measuring tool); depending on how you position some sets relative to each other their finely-detailed structure might either cancel out (adding no volume) or reinforce each other (adding lots of volume).
That said I think the real lesson here is that your intuition is trying to have its cake -- a non-measurable set -- and eat it too -- have the "volume" of a non-measurable set be preserved under volume-preserving operations.
my first thought is: this happens all the time, if by tiny pieces you mean atoms. gently heat a balloon until it swells to twice its size. am i missing something?
A balloon is mostly empty space. The paradox deals with solid spheres rather than atomic matter.
true. what i meant was that my off-the-cuff physics radar did not automatically think this was impossible. i used a gas as an example, but solids can also be restructured into different volumes and shapes. for example, diamonds, charcoal and graphite. i'm no doubt missing the precise definition of the math problem...i was just responding from my practical layman perspective.
It occurred to me that a better way to approach explaining the B-T paradox to a layman might be through the story of the infinite hotel. If you aren't familiar with how an infinite hotel that is full can still make room for an infinite number of guests, a good introduction is here: http://diveintomark.org/archives/2003/12/04/infinite-hotel
Let's assume you know and understand the infinite hotel thing from one of Martin Gardner's books, the link above or any other source. Visualize a number line with the points 1, 2, 3, 4, ... on it marked as "rooms". Now when you make room for just five new people, by moving existing guests 1->6, 2->7 and so on, and freeing rooms 1-5, you can look at it as shifting all the "room points" five units to the right. When you need to make room for an infinity of new people, and you move guests 1->2, 2->4, 3->6 and so on, this isn't a simple shift, because points move non-uniformly: the farther away, the farther you move. But it turns out that that's just because you don't have much freedom of movement, so to speak, in one dimension.
It's even more useful to look at the hotel process "in reverse": say you have all the rooms taken, now people in rooms 1, 3, 5, 7... all move out and renumber themselves, founding another hotel of the same kind, while people in rooms 2, 4, 6, 8... squeeze together, each moving to the room half their original number. So you start with one hotel and you get two identical ones, again with all the rooms taken. And again, the manner of movement here is non-uniform, but it's because one dimension is too crowded.
In two dimensions, there's a way to shift an infinity of points to become two identical infinities, but all the movement is simple shifting together or rotating together. You divide the points similarly to the even versus odd numbers division in the hotel example, but because you have a lot more space to move around in two dimensions, it turns out you can move all the "evens" and all the "odds" uniformly with respect to each other, as if you were shifting and rotating them together in the physical world. But the end result is the same: two infinities where one was, and the basic idea is just the one with the hotel rooms. The actual way you divide the points into two groups in two dimensions and shift/rotate them is the tricky technical part of the proof you'll have to take on faith here. It's not very complex math, but it does require some abstract higher math knowledge, at about the level of a math major college degree.
OK, so given all that, what do we do with a ball? In a ball, we first look at just its surface - the sphere - which is really two-dimensional. You can take an infinite mesh of points in two dimensions - the one we learned to "duplicate" with the hotel process above - and stretch them over the sphere, like a lattice. It's not too difficult to show that by wiggling around this lattice of points you can cover the whole sphere with its copies, and the tricky hotel-like rotation and shifting that you do with one lattice, you can do with all of them together in sync. So it looks like we are breaking the sphere down into two parts, and shifting/rotating them around to get two spheres next to each other. Each part is the composition of one half of the infinite lattice - one half of the "hotel rooms" in the two-dimensional hotel - collected over all the wiggled lattices together. Only it turns out to be more complex to unify them like that, so it requires four parts and not two.
Now, these four parts are unbelievably complex-looking. Just as with the original hotel puzzle there's a break with intuition where you get two of the same from one, here you do this infinitely many times at the same time, in two dimensions. Nothing like that could be done in the physical world. You're basically taking the sphere, breaking it down to individual points, and them juggling them very intricately hotel-like in an infinity of configurations together. The point is, any intuitive notion of "volume" or "space taken" by the sphere breaks down in this process, becomes irrelevant. With the infinite hotel, two hotels are also taking up twice more "space", but we don't perceive that as especially freaky on top of everything else, because they stretch to infinity anyway. But they don't have to; there's an infinity of points inside a fixed volume too. You could host the infinite hotel on a surface of a sphere if you were willing to make the rooms really tiny (one point each), and this is kinda what happens in the Banach-Tarski paradox. So the paradoxical sense of getting something from nothing is because in the physical world, we can never go to the scale of individual points, where the notion of "volume" loses relevance. But in math we can.
Well, back to the ball - if I convinced you, with lots of handwaving, that you can break down the sphere into four unbelievably complex-looking parts and reassemble them into two spheres, balls are now easy. Every time you do something to a point on the sphere, think of a ray from that point to the ball's center, and do exactly the same shifts and rotations to all the points on that ray. This way, you're sort of shifting and rotating many concentric spheres at the same time, all the way from a single point in the center to the surface of the ball. And each sphere gets reassembled into two of the same, so the entire ball gets reassembled into two of the same. You do need a bit of a special treatment for the very point in the center, and that's your fifth "part".
Let's assume you know and understand the infinite hotel thing from one of Martin Gardner's books, the link above or any other source. Visualize a number line with the points 1, 2, 3, 4, ... on it marked as "rooms". Now when you make room for just five new people, by moving existing guests 1->6, 2->7 and so on, and freeing rooms 1-5, you can look at it as shifting all the "room points" five units to the right. When you need to make room for an infinity of new people, and you move guests 1->2, 2->4, 3->6 and so on, this isn't a simple shift, because points move non-uniformly: the farther away, the farther you move. But it turns out that that's just because you don't have much freedom of movement, so to speak, in one dimension.
It's even more useful to look at the hotel process "in reverse": say you have all the rooms taken, now people in rooms 1, 3, 5, 7... all move out and renumber themselves, founding another hotel of the same kind, while people in rooms 2, 4, 6, 8... squeeze together, each moving to the room half their original number. So you start with one hotel and you get two identical ones, again with all the rooms taken. And again, the manner of movement here is non-uniform, but it's because one dimension is too crowded.
In two dimensions, there's a way to shift an infinity of points to become two identical infinities, but all the movement is simple shifting together or rotating together. You divide the points similarly to the even versus odd numbers division in the hotel example, but because you have a lot more space to move around in two dimensions, it turns out you can move all the "evens" and all the "odds" uniformly with respect to each other, as if you were shifting and rotating them together in the physical world. But the end result is the same: two infinities where one was, and the basic idea is just the one with the hotel rooms. The actual way you divide the points into two groups in two dimensions and shift/rotate them is the tricky technical part of the proof you'll have to take on faith here. It's not very complex math, but it does require some abstract higher math knowledge, at about the level of a math major college degree.
OK, so given all that, what do we do with a ball? In a ball, we first look at just its surface - the sphere - which is really two-dimensional. You can take an infinite mesh of points in two dimensions - the one we learned to "duplicate" with the hotel process above - and stretch them over the sphere, like a lattice. It's not too difficult to show that by wiggling around this lattice of points you can cover the whole sphere with its copies, and the tricky hotel-like rotation and shifting that you do with one lattice, you can do with all of them together in sync. So it looks like we are breaking the sphere down into two parts, and shifting/rotating them around to get two spheres next to each other. Each part is the composition of one half of the infinite lattice - one half of the "hotel rooms" in the two-dimensional hotel - collected over all the wiggled lattices together. Only it turns out to be more complex to unify them like that, so it requires four parts and not two.
Now, these four parts are unbelievably complex-looking. Just as with the original hotel puzzle there's a break with intuition where you get two of the same from one, here you do this infinitely many times at the same time, in two dimensions. Nothing like that could be done in the physical world. You're basically taking the sphere, breaking it down to individual points, and them juggling them very intricately hotel-like in an infinity of configurations together. The point is, any intuitive notion of "volume" or "space taken" by the sphere breaks down in this process, becomes irrelevant. With the infinite hotel, two hotels are also taking up twice more "space", but we don't perceive that as especially freaky on top of everything else, because they stretch to infinity anyway. But they don't have to; there's an infinity of points inside a fixed volume too. You could host the infinite hotel on a surface of a sphere if you were willing to make the rooms really tiny (one point each), and this is kinda what happens in the Banach-Tarski paradox. So the paradoxical sense of getting something from nothing is because in the physical world, we can never go to the scale of individual points, where the notion of "volume" loses relevance. But in math we can.
Well, back to the ball - if I convinced you, with lots of handwaving, that you can break down the sphere into four unbelievably complex-looking parts and reassemble them into two spheres, balls are now easy. Every time you do something to a point on the sphere, think of a ray from that point to the ball's center, and do exactly the same shifts and rotations to all the points on that ray. This way, you're sort of shifting and rotating many concentric spheres at the same time, all the way from a single point in the center to the surface of the ball. And each sphere gets reassembled into two of the same, so the entire ball gets reassembled into two of the same. You do need a bit of a special treatment for the very point in the center, and that's your fifth "part".
Your response is a reasonable explanation of Hilbert's hotel paradox, but that paradox is really different from the Banach-Tarski paradox. In particular, the Banach-Tarski paradox is false in two dimensions. (One technical reason for this is that you can embed a nonabelian free group into SO(3), the rotations on the sphere, but not into SO(2), the rotations on the circle.) This means that you should not think of the Banach-Tarski paradox as being a Hilbert-hotel-like statement about a two dimensional lattice.
There is always a gap between formalism and intuition and sometimes quite interesting things happen between these gaps. The Banach-Tarski paradox is labeled a paradox because it says something about how geometry and topology are formalized in set theory. There are other formalizations of geometry and topology where such things are not possible.
http://www.biblegateway.com/passage/?search=Matthew+14%3A13-...
Now, quick: what happened to the 2 little fishies?
Now, quick: what happened to the 2 little fishies?
http://searchyc.com/banach+tarski
I actually think the wikipedia article is pretty good:
http://en.wikipedia.org/wiki/Banach-Tarski_paradox
I think the article lunk to here is not actually a very good explanation. It mixes folksy and technical too freely, really being neither one nor the other, and even though I actually know the proof quite well (and its 1D and 2D relations) I still found it unhelpful and awkward.
YMMV - I'd be interested to see if anyone here finds it enlightening.