6 ms·

The Point of the Banach Tarski Theorem (2015)

108 points by ColinWright 2y ago

ColinWright 2y ago
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. [0] https://news.ycombinator.com/item?id=40797598 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.
- g15jv2dp 2y ago
  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.
  - l33t7332273 2y ago
    I think GP was drawing a parallel between Euclid’s fifth axiom of and results like Banach-Tarski since the fifth axiom is independent of the other four, and Bnach-Tarski follows from the Axiom of Choice which is independent of the rest of set theory. The pun was intended, by the way.
  - ColinWright 2y ago
    > 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. Let me try to make the analogy more explicit. The interesting thing about the fifth postulate is that we show we can't prove it from the other axioms because there are models where the first four hold and the fifth doesn't. The interesting thing about the Banach-Tarski Theorem is that it shows we can't have all four desirable properties of a metric because there are constructions that show they they can't all hold at once. > ... you're completely glossing over the theory of measurable sets. I'm not glossing over it, I'm showing why it is necessary and important.
    - skhunted 2y ago
      If one doesn’t accept the Axiom of Choice but uses instead Dependent Choice the paradox no longer holds. Is it the case in this situation that the 4 desirable properties hold?
      - CaptainNegative 2y ago
        Assuming the usual consistency caveats, the paradox is no longer a theorem of ZF+DC, but its complement isn't either. So in that case the analogue to the fifth postulate is even stronger, as there are both models in which you get the counterintuitive results of unmeasurable sets and those in which you don't, and the axioms are not strong enough to distinguish the two.
        skhunted 2y ago
        In ZF+DC is it true that measures satisfy the desirable properties mentioned by Colin? I think the sticking point is isometry invariance. Are there measures in ZF+DC of R^3 that are finitely (countably?) additive and isometry invariant?
      - fmap 2y ago
        You don't get new theorems if you remove assumptions. Rather, you get the ability to add different assumptions. The Banach-Tarski paradox shows that classical set theory makes the wrong assumptions to intrinsically model measure theory and probability. There are other systems which don't suffer from this paradox and hence don't need all the machinery of sigma algebras and measurable sets. I wish there was a good accessible book/article/blog post about this, but as is you'd have to Google point-free topology or topos of probability (there are several).
- JHonaker 2y ago
  That was a great article. 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.
  - nyc111 2y ago
    “Once I realized they were often just defined…” I think definition is fundamental in nature because we need to define a unit in order to measure.
    - JHonaker 2y ago
      Yea, the more I learn about things, the more I realize that everything is defined relative to something else. Even measurements are defined relative to some standard (that beautifully manicured kilogram ball, the speed of light in a vacuum, your ruler's hand width or foot length, etc.) That's both very satisfying and extremely frustrating. Programming languages can't escape it either; see the tautologies at the top of your favorite programming language (metaclasses, metaobject protocols, etc).
  - bdjsiqoocwk 2y ago
    > Once I realized they were often just defined to be (...) Right, that's why I often flippantly say that maths is the study of all ideas which are interesting. We could also think about other things, but they haven't been considered interesting for one reason or another.
deleted 2y ago
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downvotetruth 2y ago
Dang or mod please remove the "?xf26hn" text from the end of the URL so it is able to used to find the past posts by scripts: 145 points 1 year ago 105 comments https://news.ycombinator.com/item?id=34482226 https://news.ycombinator.com/item?id=34482226 19 points 4 hours ago 5 comments https://news.ycombinator.com/item?id=40798224 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 2y ago
  I've taken it out in this case but we can't take them out in the general case because it sometimes changes what page gets displayed.
- g15jv2dp 2y ago
  How would you deal with <https://example.com/read_blog_post.php?id=1234 https://example.com/read_blog_post.php?id=1234>? The query param is absolutely necessary in that case.
  - downvotetruth 2y ago
    https://example.com https://example.com would be the submission & https://example.com/read_blog_post.php?id=1234 https://example.com/read_blog_post.php?id=1234 an added comment or preferably the text (first comment) field. Yes, this would almost surely cause the 1 URL per year HN submission rule to be violated, but the expense is being unable to at least attempt to force a unique constraint on the URL as the query params are not required to be in order like ?id=123&comments=1 and ?comments=t&id=123 both being valid leading to the combinatorial growth in URLs pointing to the same content and hindering exact URL matching and filtering. Given that, search engines are also unlikely to favor that site structure when indexing over a scheme such as https://example.com/read_blog_post/id/1234 https://example.com/read_blog_post/id/1234 There is still the possibility if a host wanted to get around such a restriction the server could create virtual directories to allow the same reordering as the query params, but that could be detected and the site flagged if necessary.
IshKebab 2y ago
> 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.
- ska 2y ago
  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.
- andrewla 2y ago
  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.
dang 2y ago
Related: The Point of the Banach-Tarski Theorem - https://news.ycombinator.com/item?id=34482226 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 https://news.ycombinator.com/item?id=9674286 - June 2015 (91 comments)
- SilasX 2y ago
  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: https://news.ycombinator.com/item?id=16586370 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=40797598 https://news.ycombinator.com/item?id=40798216 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".
  - dang 2y ago
    To compare two single datapoints across 6 years (!) is an extreme underestimate of the role played by randomness in HN moderation. I guess in that way it makes sense that the discrepancy felt personal to you. I promise it wasn't!
    - SilasX 2y ago
      But now you know of the violation! Don't you want to remind Colin of the rule now? Or maybe the rule isn't actually that important, for the reasons in my last comment, and you were just looking for a reason to make your criticism of my six-year-old/8-year-old comment more authoritative?
ajkjk 2y ago
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.
- moffkalast 2y ago
  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.
  - pvillano 2y ago
    The best/worst part is that there are useful branches of mathematics which assume the Axiom of Choice is true AND there are useful branches of mathematics which assume the Axiom of Choice is false. That's one reason I take the view that mathematics does not exist: if math did exist, there would only be one set of axioms consistent with nature.
    - 9991 2y ago
      You conclude the problem is with math. I conclude the problem is with reality.
      - alkyon 2y ago
        In physical reality there is no infinity of anything* so Axiom of Choice is irrelevant for it. * Universe could be itself inifinite but locally everything is finite.
        9991 2y ago
        > locally everything is finite. You don't know that.
mikhailfranco 2y ago
So what does it mean for physics? 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
- untilted 2y ago
  A physical sphere has a finite number of constituents, therefore the theorem does not apply
  - deleted 2y ago
    [deleted]
- mjburgess 2y ago
  None of these. 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.
- andrewla 2y ago
  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.
cvoss 2y ago
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?
tliltocatl 2y ago
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.
- 082349872349872 2y ago
  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?
  - tliltocatl 2y ago
    Second this, because I'm no mathematician myself and it doesn't look like there are much layman-available material out there. "Point-free" is also another name for tacit programming (i. e. without assigning names to variables, like Forth or bash pipelines), but I can't find any reference on if it is just named for superficial similarity or is there some deeper theoretical connection between point-free topology and point-free programming.
    - 082349872349872 2y ago
      In the tacit programming case, I'm pretty sure it's for (a) superficial resemblance, and (b) the excuse to call one's favoured technique "pointless programming". (there's probably something interesting to be gotten at by considering how much violence a particular function does to structure in the input space —thinking of compositions of homomorphisms being homomorphisms of compositions, or how mapping a monotonic function over keys allows one to avoid reindexing— but I'm pretty sure that's mostly orthogonal to whether one binds parameter names or not)
- deleted 2y ago
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andrewla 2y ago
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.
- gmadsen 2y ago
  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..
  - andrewla 2y ago
    The article says it better than me: > [Banach-Tarski] means there can be no measure satisfying the requirements even when weakened from countable additivity to finite additivity. Measure theory was an attempt to rescue us from the breakdown of the Riemann integral for poorly-behaved functions by adding a ton of abstraction of formalism. And it turns out that it isn't even successful at that. But it turns out that what "poorly behaved functions" means is basically "non-computable functions", and so you can back up even further and point to that idea as the problematic one in the classical foundations of set theory -- that constructibility is not only a nice-to-have but is in fact necessary for any sort of coherent theory. Put differently, Banach-Tarski shows that formalism-based systems that rely on the axiom of choice and the excluded middle too heavily result in behaviors which are clearly incorrect, but which are internally consistent.
    - rhelz 2y ago
      > behaviors which are clearly incorrect, but which are internally consistent. For mathematical objects, what other considerations are there than consistency? I mean, physics has given us all manner of counterintuitive things. At what point do you stop saying "so much the worse for the theory" and start saying "so much the worse for my intuitions?"
ysofunny 2y ago
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)
nayuki 2y ago
Vsauce: The Banach–Tarski Paradox (24m13s) [2015-08-01]: https://www.youtube.com/watch?v=s86-Z-CbaHA https://www.youtube.com/watch?v=s86-Z-CbaHA (explainer video) Matematikrevyen: Matematikrevyen 2011: Banach-Tarski (3m42s) [2011-12-30]: https://www.youtube.com/watch?v=uFvokQUHh08 https://www.youtube.com/watch?v=uFvokQUHh08 (music video)
chalcolithic 2y ago
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?
- ColinWright 2y ago
  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.

108 points by ColinWright 2y ago

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. [0] https://news.ycombinator.com/item?id=40797598 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.

g15jv2dp 2y ago
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.
- l33t7332273 2y ago
  I think GP was drawing a parallel between Euclid’s fifth axiom of and results like Banach-Tarski since the fifth axiom is independent of the other four, and Bnach-Tarski follows from the Axiom of Choice which is independent of the rest of set theory. The pun was intended, by the way.
- ColinWright 2y ago
  > 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. Let me try to make the analogy more explicit. The interesting thing about the fifth postulate is that we show we can't prove it from the other axioms because there are models where the first four hold and the fifth doesn't. The interesting thing about the Banach-Tarski Theorem is that it shows we can't have all four desirable properties of a metric because there are constructions that show they they can't all hold at once. > ... you're completely glossing over the theory of measurable sets. I'm not glossing over it, I'm showing why it is necessary and important.
  - skhunted 2y ago
    If one doesn’t accept the Axiom of Choice but uses instead Dependent Choice the paradox no longer holds. Is it the case in this situation that the 4 desirable properties hold?
    - CaptainNegative 2y ago
      Assuming the usual consistency caveats, the paradox is no longer a theorem of ZF+DC, but its complement isn't either. So in that case the analogue to the fifth postulate is even stronger, as there are both models in which you get the counterintuitive results of unmeasurable sets and those in which you don't, and the axioms are not strong enough to distinguish the two.
      - skhunted 2y ago
        In ZF+DC is it true that measures satisfy the desirable properties mentioned by Colin? I think the sticking point is isometry invariance. Are there measures in ZF+DC of R^3 that are finitely (countably?) additive and isometry invariant?
    - fmap 2y ago
      You don't get new theorems if you remove assumptions. Rather, you get the ability to add different assumptions. The Banach-Tarski paradox shows that classical set theory makes the wrong assumptions to intrinsically model measure theory and probability. There are other systems which don't suffer from this paradox and hence don't need all the machinery of sigma algebras and measurable sets. I wish there was a good accessible book/article/blog post about this, but as is you'd have to Google point-free topology or topos of probability (there are several).
JHonaker 2y ago
That was a great article. 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.
- nyc111 2y ago
  “Once I realized they were often just defined…” I think definition is fundamental in nature because we need to define a unit in order to measure.
  - JHonaker 2y ago
    Yea, the more I learn about things, the more I realize that everything is defined relative to something else. Even measurements are defined relative to some standard (that beautifully manicured kilogram ball, the speed of light in a vacuum, your ruler's hand width or foot length, etc.) That's both very satisfying and extremely frustrating. Programming languages can't escape it either; see the tautologies at the top of your favorite programming language (metaclasses, metaobject protocols, etc).
- bdjsiqoocwk 2y ago
  > Once I realized they were often just defined to be (...) Right, that's why I often flippantly say that maths is the study of all ideas which are interesting. We could also think about other things, but they haven't been considered interesting for one reason or another.

deleted 2y ago

[deleted]

downvotetruth 2y ago

Dang or mod please remove the "?xf26hn" text from the end of the URL so it is able to used to find the past posts by scripts: 145 points 1 year ago 105 comments https://news.ycombinator.com/item?id=34482226 https://news.ycombinator.com/item?id=34482226 19 points 4 hours ago 5 comments https://news.ycombinator.com/item?id=40798224 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 2y ago
I've taken it out in this case but we can't take them out in the general case because it sometimes changes what page gets displayed.
g15jv2dp 2y ago
How would you deal with <https://example.com/read_blog_post.php?id=1234 https://example.com/read_blog_post.php?id=1234>? The query param is absolutely necessary in that case.
- downvotetruth 2y ago
  https://example.com https://example.com would be the submission & https://example.com/read_blog_post.php?id=1234 https://example.com/read_blog_post.php?id=1234 an added comment or preferably the text (first comment) field. Yes, this would almost surely cause the 1 URL per year HN submission rule to be violated, but the expense is being unable to at least attempt to force a unique constraint on the URL as the query params are not required to be in order like ?id=123&comments=1 and ?comments=t&id=123 both being valid leading to the combinatorial growth in URLs pointing to the same content and hindering exact URL matching and filtering. Given that, search engines are also unlikely to favor that site structure when indexing over a scheme such as https://example.com/read_blog_post/id/1234 https://example.com/read_blog_post/id/1234 There is still the possibility if a host wanted to get around such a restriction the server could create virtual directories to allow the same reordering as the query params, but that could be detected and the site flagged if necessary.

IshKebab 2y ago

> 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.

ska 2y ago
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.
andrewla 2y ago
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.

dang 2y ago

Related: The Point of the Banach-Tarski Theorem - https://news.ycombinator.com/item?id=34482226 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 https://news.ycombinator.com/item?id=9674286 - June 2015 (91 comments)

SilasX 2y ago
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: https://news.ycombinator.com/item?id=16586370 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=40797598 https://news.ycombinator.com/item?id=40798216 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".
- dang 2y ago
  To compare two single datapoints across 6 years (!) is an extreme underestimate of the role played by randomness in HN moderation. I guess in that way it makes sense that the discrepancy felt personal to you. I promise it wasn't!
  - SilasX 2y ago
    But now you know of the violation! Don't you want to remind Colin of the rule now? Or maybe the rule isn't actually that important, for the reasons in my last comment, and you were just looking for a reason to make your criticism of my six-year-old/8-year-old comment more authoritative?

ajkjk 2y ago

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.

moffkalast 2y ago
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.
- pvillano 2y ago
  The best/worst part is that there are useful branches of mathematics which assume the Axiom of Choice is true AND there are useful branches of mathematics which assume the Axiom of Choice is false. That's one reason I take the view that mathematics does not exist: if math did exist, there would only be one set of axioms consistent with nature.
  - 9991 2y ago
    You conclude the problem is with math. I conclude the problem is with reality.
    - alkyon 2y ago
      In physical reality there is no infinity of anything* so Axiom of Choice is irrelevant for it. * Universe could be itself inifinite but locally everything is finite.
      - 9991 2y ago
        > locally everything is finite. You don't know that.

mikhailfranco 2y ago

So what does it mean for physics? 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

untilted 2y ago
A physical sphere has a finite number of constituents, therefore the theorem does not apply
- deleted 2y ago
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mjburgess 2y ago
None of these. 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.
andrewla 2y ago
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.

cvoss 2y ago

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?

tliltocatl 2y ago

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.

082349872349872 2y ago
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?
- tliltocatl 2y ago
  Second this, because I'm no mathematician myself and it doesn't look like there are much layman-available material out there. "Point-free" is also another name for tacit programming (i. e. without assigning names to variables, like Forth or bash pipelines), but I can't find any reference on if it is just named for superficial similarity or is there some deeper theoretical connection between point-free topology and point-free programming.
  - 082349872349872 2y ago
    In the tacit programming case, I'm pretty sure it's for (a) superficial resemblance, and (b) the excuse to call one's favoured technique "pointless programming". (there's probably something interesting to be gotten at by considering how much violence a particular function does to structure in the input space —thinking of compositions of homomorphisms being homomorphisms of compositions, or how mapping a monotonic function over keys allows one to avoid reindexing— but I'm pretty sure that's mostly orthogonal to whether one binds parameter names or not)
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andrewla 2y ago

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.

gmadsen 2y ago
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..
- andrewla 2y ago
  The article says it better than me: > [Banach-Tarski] means there can be no measure satisfying the requirements even when weakened from countable additivity to finite additivity. Measure theory was an attempt to rescue us from the breakdown of the Riemann integral for poorly-behaved functions by adding a ton of abstraction of formalism. And it turns out that it isn't even successful at that. But it turns out that what "poorly behaved functions" means is basically "non-computable functions", and so you can back up even further and point to that idea as the problematic one in the classical foundations of set theory -- that constructibility is not only a nice-to-have but is in fact necessary for any sort of coherent theory. Put differently, Banach-Tarski shows that formalism-based systems that rely on the axiom of choice and the excluded middle too heavily result in behaviors which are clearly incorrect, but which are internally consistent.
  - rhelz 2y ago
    > behaviors which are clearly incorrect, but which are internally consistent. For mathematical objects, what other considerations are there than consistency? I mean, physics has given us all manner of counterintuitive things. At what point do you stop saying "so much the worse for the theory" and start saying "so much the worse for my intuitions?"

ysofunny 2y ago

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)

nayuki 2y ago

Vsauce: The Banach–Tarski Paradox (24m13s) [2015-08-01]: https://www.youtube.com/watch?v=s86-Z-CbaHA https://www.youtube.com/watch?v=s86-Z-CbaHA (explainer video) Matematikrevyen: Matematikrevyen 2011: Banach-Tarski (3m42s) [2011-12-30]: https://www.youtube.com/watch?v=uFvokQUHh08 https://www.youtube.com/watch?v=uFvokQUHh08 (music video)

chalcolithic 2y ago

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?

ColinWright 2y ago
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.