Given that this wonderful result has been posted again[0] I thought I would again provide a link to this explanation not of the result itself, but of why it's an important result in a wider context and not just a gimmick.
Here's the basic idea ...
In Classical Euclidean Geometry there are five axioms, and while the first four seem clear and obvious, the fifth seems a little contrived. So for centuries people tried to prove that the fifth was unnecessary and could be proven from the other four.
These attempts all failed, and we can show that they must fail, because there are systems that satisfy the first four, but do not satisfy the fifth. Hence the fifth cannot be a consequence of the first four. Such systems are (for obvious reasons) called Non-Euclidean Geometries.
So we can use explicit examples to demonstrate that certain proofs are impossible, and the Banach-Tarski Theorem is a result that proves that a "Measure"[1] cannot have all four obviously desirable characteristics.
That's the basic idea ... if you want more details, click through to the post. It's intended to be readable, but the topic is inherently complex, so it may need more than one read through. If you're interested.
[1] Technical term for a function that takes an object and returns a concept of its size. For lines it's length, for planar objects it's area, for 3D objects it's volume, and so on.
If I'd read this when I was first learning measure theory, I would have had a much easier time. In fact, it took me an embarrassingly long time to realize that sigma algebras were just the "nice sets and subsets" of things that we can extend measures from finite additivity to countable additivity.
I used to think mathematical objects were somehow "inherent". I was always amazed that people had discovered and proved so many interesting things about them. Once I realized they were often just defined to be the thing that has the property we want to prove something about, it got a lot less mysterious.
Note, I'm not saying we just stop there, or that this is somehow bad. The next obvious step taken by mathematicians is to start removing bits of the objects they study, and try to figure out what's still provable until we get to categories, logic, and start arguing about things like the axiom of choice.
Euclidean geometry and its fifth axiom are interesting, but unrelated to the Banach-Tarski paradox. I don't get the point you're trying to make. It's not a "wider context", it's a different thing altogether.
Also, as someone pointed out in the linked thread, you're completely glossing over the theory of measurable sets.
The Banach-Tarski paradox is really a deep critique of measure theory and specifically the axiom of choice. This is fun to study to get a feel for the places where formalism without connection to feasibility gets you.
In an intuitionist framework none of this applies.
It relies on the fact that not only does it not provide a constructive framework to produce such a division, but also that no such constructive framework is even theoretically possible! So this theorem tells us nothing about the nature of three dimensional objects or our ability to "measure" objects.
why is this a critique of measure theory? Measure theory is the answer to the paradox. The partition uses unmeasurable sets, so comparing the surface areas before and after doesn't make sense. You could do the partition a billion times and expand the volume as well..
Nevertheless it feels like the point of Banach-Tarski is that it proves math went wrong somewhere. Evidently the sets it's talking about are not objects which are interesting in reality.
Math is not reality. Physics isn't even reality, they're just theoretical frameworks that are easy to work with and somewhat align with what we've observed. "All models are wrong, but some are useful" as they say.
If you can prove something can or can't be done in math it doesn't mean shit, but it might end up being a useful guideline. Banach-Tarski assumes an infinite pointcloud (i.e. a mathematical sphere), which as you've realized, doesn't actually exist.
> makes you go looking for some kind of loophole. But there isn't one.
Doesn't it require fractal cuts? Seems like a loophole to me. It only seems paradoxical because you assume the resulting pieces are smooth at some scale, like real cuts are.
That’s opening a can of worms. Fundamentally, the real number system is larger and weirder than “we” mostly think it is (and hence its extensions to higher dimensions). If you start trying to remove objects it easily becomes a game of whack-a-mole; some of your nice, intuitive definitions elsewhere become muddy. Colin alludes to this in the article (e.g. should all sets be measureable, etc.). If there were easy and satisfying fixes for this, it would have been sorted out long ago!
This isn’t just aesthetic. Although probability is one of the oldest areas of mathematical thought, dating back millennia, it took measure theory to put it on a really solid basis, several decades ago. These approaches are powerful and useful, but some of the corners are certainly counterintuitive.
Fractal is understating it. It requires non-constructive cuts -- if you limit yourself to cuts that can be constructed through any finitely-expressed process then the theorem does not hold.
Also relevant: the paradox doesn't apply in point-free topology, because it allows for "locales" that doesn't contain any points but still have nonzero measure. So rather than giving up axiom of choice, we may instead accept that "a set of points" doesn't quite correspond to the intuitive notion of a shape.
Any suggestions for (as a CS person who has been finding quantales useful for my favourite applications, and is curious as to whether there may be connexions to other disciplines) getting into point-free topology?
Since you took notice of the discussion, I thought I'd point out that in the past, you'd given me flak for copying and reposting a previous comment of mine:
Since the rule isn't based on whether you like the content being copied, I thought you'd like to be aware so you can treat his comment the same way.
FWIW, I think it's a stupid rule: if you said something just right the first time around, why re-write? It makes perfect sense for Colin to make the same comment again, just as it did in my case! I think he did the right thing!
It seems like, in practice, the actual rule is, "you can copy earlier comments, just don't own up to doing it or make it easier to find related discussion of the same point, like Silas did".
We are forced to relax one of the four named properties that we desire for the measure function. (And, well, Banach-Tarski rules out one of those options.) So we relax the requirement that mu be defined on all inputs.
But I wonder, isn't there an implicit 5th property that could be relaxed? That's the property that the codomain of mu is the reals. Is it viable to use, say, the hyperreals instead, or some other exotic extension that would allow us to name the (nonzero and nonreal!) number mu(V) such that a countable sum of mu(V) comes out to 1?
what if we interpret both spheres created out of the one as:
1. original "terrain" sphere
2. modeled version of the sphere. the virtual "map" sphere.
but because the abstractions are so thick (so to say, pardon the poetic language) — or the recursion so recursive, the "map" of the sphere accounts for it being a map by producing two duplicates virtual copies, one intended to reflect the terrain and the other the map (but both are virtual maps, but this is really hard to 'perceive'/'say' within the formalisms)
It's very neat but the gist is that one infinity equals two infinities - infinities are weird. While technically it's cool does it really add anything to our understanding?
The entire point of the post is exactly to show what the Banach-Tarski Theorem is adding to our understanding.
Summarising ...
Previously, people thought that it might be possible to define a measure on arbitrary sets and have the usual desirable properties of isometry invariance and finite additivity (we already know countable additivity won't work).
But the Banach Tarski Theorem shows us that's not possible. That is thereby adding to our understanding of what is and is not possible with measures.
So yes, it does add something to our understanding.
The disconnect between reality and Banach-Tarski is deep.
Important to the paradox is that it does not present a construction of the division. And more importantly, it relies on the fact that it is not possible to construct such a division.
If you accept some notion of the Universal Church-Turing Thesis as a matter of reality, then it is not possible even in principle for such a division to occur.
Note that this is not just a matter of "finding the right way to construct it" -- the theorem critically relies on the fact that given any way of constructing sets using finite representations (like computer programs or layout of molecules), the theorem does not hold.
The 'axiom of choice' isnt a property of the real world one way or another.
The issue is that physical measures (length, area, volume, etc.) require measure theory in mathematics to specify properly. The issue is the mathematics was too simple to model the relevant physical systems.
All of physics is specified in terms of a spacetime continuum, neither this, nor anything else, indicates that physics requires a revision.
[+] [-] ColinWright|1 year ago|reply
Here's the basic idea ...
In Classical Euclidean Geometry there are five axioms, and while the first four seem clear and obvious, the fifth seems a little contrived. So for centuries people tried to prove that the fifth was unnecessary and could be proven from the other four.
These attempts all failed, and we can show that they must fail, because there are systems that satisfy the first four, but do not satisfy the fifth. Hence the fifth cannot be a consequence of the first four. Such systems are (for obvious reasons) called Non-Euclidean Geometries.
So we can use explicit examples to demonstrate that certain proofs are impossible, and the Banach-Tarski Theorem is a result that proves that a "Measure"[1] cannot have all four obviously desirable characteristics.
That's the basic idea ... if you want more details, click through to the post. It's intended to be readable, but the topic is inherently complex, so it may need more than one read through. If you're interested.
[0] https://news.ycombinator.com/item?id=40797598
[1] Technical term for a function that takes an object and returns a concept of its size. For lines it's length, for planar objects it's area, for 3D objects it's volume, and so on.
[+] [-] JHonaker|1 year ago|reply
If I'd read this when I was first learning measure theory, I would have had a much easier time. In fact, it took me an embarrassingly long time to realize that sigma algebras were just the "nice sets and subsets" of things that we can extend measures from finite additivity to countable additivity.
I used to think mathematical objects were somehow "inherent". I was always amazed that people had discovered and proved so many interesting things about them. Once I realized they were often just defined to be the thing that has the property we want to prove something about, it got a lot less mysterious.
Note, I'm not saying we just stop there, or that this is somehow bad. The next obvious step taken by mathematicians is to start removing bits of the objects they study, and try to figure out what's still provable until we get to categories, logic, and start arguing about things like the axiom of choice.
[+] [-] g15jv2dp|1 year ago|reply
Also, as someone pointed out in the linked thread, you're completely glossing over the theory of measurable sets.
[+] [-] andrewla|1 year ago|reply
In an intuitionist framework none of this applies.
It relies on the fact that not only does it not provide a constructive framework to produce such a division, but also that no such constructive framework is even theoretically possible! So this theorem tells us nothing about the nature of three dimensional objects or our ability to "measure" objects.
[+] [-] gmadsen|1 year ago|reply
[+] [-] ajkjk|1 year ago|reply
[+] [-] moffkalast|1 year ago|reply
If you can prove something can or can't be done in math it doesn't mean shit, but it might end up being a useful guideline. Banach-Tarski assumes an infinite pointcloud (i.e. a mathematical sphere), which as you've realized, doesn't actually exist.
[+] [-] ajuc|1 year ago|reply
[+] [-] IshKebab|1 year ago|reply
Doesn't it require fractal cuts? Seems like a loophole to me. It only seems paradoxical because you assume the resulting pieces are smooth at some scale, like real cuts are.
[+] [-] ska|1 year ago|reply
This isn’t just aesthetic. Although probability is one of the oldest areas of mathematical thought, dating back millennia, it took measure theory to put it on a really solid basis, several decades ago. These approaches are powerful and useful, but some of the corners are certainly counterintuitive.
[+] [-] andrewla|1 year ago|reply
[+] [-] tliltocatl|1 year ago|reply
[+] [-] 082349872349872|1 year ago|reply
[+] [-] unknown|1 year ago|reply
[deleted]
[+] [-] nayuki|1 year ago|reply
Matematikrevyen: Matematikrevyen 2011: Banach-Tarski (3m42s) [2011-12-30]: https://www.youtube.com/watch?v=uFvokQUHh08 (music video)
[+] [-] dang|1 year ago|reply
The Point of the Banach-Tarski Theorem - https://news.ycombinator.com/item?id=34482226 - Jan 2023 (105 comments)
The Point of the Banach-Tarski Theorem – not just a curiosity - https://news.ycombinator.com/item?id=9674286 - June 2015 (91 comments)
[+] [-] SilasX|1 year ago|reply
https://news.ycombinator.com/item?id=16586370
Now, ColinWright is doing the same thing:
https://news.ycombinator.com/item?id=40797598
https://news.ycombinator.com/item?id=40798216
Since the rule isn't based on whether you like the content being copied, I thought you'd like to be aware so you can treat his comment the same way.
FWIW, I think it's a stupid rule: if you said something just right the first time around, why re-write? It makes perfect sense for Colin to make the same comment again, just as it did in my case! I think he did the right thing!
It seems like, in practice, the actual rule is, "you can copy earlier comments, just don't own up to doing it or make it easier to find related discussion of the same point, like Silas did".
[+] [-] cvoss|1 year ago|reply
But I wonder, isn't there an implicit 5th property that could be relaxed? That's the property that the codomain of mu is the reals. Is it viable to use, say, the hyperreals instead, or some other exotic extension that would allow us to name the (nonzero and nonreal!) number mu(V) such that a countable sum of mu(V) comes out to 1?
[+] [-] ysofunny|1 year ago|reply
1. original "terrain" sphere
2. modeled version of the sphere. the virtual "map" sphere.
but because the abstractions are so thick (so to say, pardon the poetic language) — or the recursion so recursive, the "map" of the sphere accounts for it being a map by producing two duplicates virtual copies, one intended to reflect the terrain and the other the map (but both are virtual maps, but this is really hard to 'perceive'/'say' within the formalisms)
[+] [-] chalcolithic|1 year ago|reply
[+] [-] ColinWright|1 year ago|reply
Summarising ...
Previously, people thought that it might be possible to define a measure on arbitrary sets and have the usual desirable properties of isometry invariance and finite additivity (we already know countable additivity won't work).
But the Banach Tarski Theorem shows us that's not possible. That is thereby adding to our understanding of what is and is not possible with measures.
So yes, it does add something to our understanding.
[+] [-] mikhailfranco|1 year ago|reply
Perhaps zero, one, more or all of the following:
- duplication of volume seems to duplicate mass and energy, which is impossible
- the Axiom of Choice is false in the real world
- the real world is not based on real numbers (ironic)
- mathematical measures are not related to real world measure(ment)s
- there are no realizable infinities or infinitesimals
- the real world is ultimately discrete at the lowest level
[+] [-] andrewla|1 year ago|reply
Important to the paradox is that it does not present a construction of the division. And more importantly, it relies on the fact that it is not possible to construct such a division.
If you accept some notion of the Universal Church-Turing Thesis as a matter of reality, then it is not possible even in principle for such a division to occur.
Note that this is not just a matter of "finding the right way to construct it" -- the theorem critically relies on the fact that given any way of constructing sets using finite representations (like computer programs or layout of molecules), the theorem does not hold.
[+] [-] mjburgess|1 year ago|reply
The 'axiom of choice' isnt a property of the real world one way or another.
The issue is that physical measures (length, area, volume, etc.) require measure theory in mathematics to specify properly. The issue is the mathematics was too simple to model the relevant physical systems.
All of physics is specified in terms of a spacetime continuum, neither this, nor anything else, indicates that physics requires a revision.
[+] [-] untilted|1 year ago|reply
[+] [-] unknown|1 year ago|reply
[deleted]
[+] [-] downvotetruth|1 year ago|reply
145 points 1 year ago 105 comments https://news.ycombinator.com/item?id=34482226
19 points 4 hours ago 5 comments https://news.ycombinator.com/item?id=40798224
Further suggest that HTML query params be disallowed in submissions; if URLs with params are relevant they can be added as comment(s).
[+] [-] dang|1 year ago|reply
[+] [-] g15jv2dp|1 year ago|reply