[HN Gopher] The Point of the Banach Tarski Theorem (2015)
___________________________________________________________________
The Point of the Banach Tarski Theorem (2015)
Author : ColinWright
Score : 96 points
Date : 2024-06-26 09:55 UTC (13 hours ago)
(HTM) web link (www.solipsys.co.uk)
(TXT) w3m dump (www.solipsys.co.uk)
| ColinWright wrote:
| 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
|
| [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 wrote:
| 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 wrote:
| 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 wrote:
| > 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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).
| skhunted wrote:
| I think the following is a valid question. At least I
| hope it isn't completely stupid.
|
| Is there a known set theory of the form ZF+(something)
| which relatively consistent with ZFC in which additive,
| isometry invariant measures exist?
|
| I guess what you are saying is that the only known,
| reasonable way around this is the topos notion you
| mentioned.
| lupire wrote:
| The difference is that it's very easy to construct a non-
| Euclidean geometry in the physical world, at home, on your
| table, but it's impossible to construct an object anywhere
| in the real Universe where the Uncountable Axiom of Choice
| applies. It's purely a mathematical game of pretend.
|
| They are as far apart from each other as possible, as
| similar as polar opposites.
|
| Banach-Tarski is resolved by deciding that "the real
| numbers" aren't real, and nothing is lost except for
| dubious overly-simple proofs.
| ColinWright wrote:
| Most paradoxes can be "resolved" by just saying "Don't do
| that."
|
| Problem is, there is a lot of mathematics that's widely
| used and which depends on the reals. Pretty much all of
| calculus, for example.
|
| Discarding the reals is pretty ambitious.
| gjm11 wrote:
| I think there's an important difference between the
| situations.
|
| With (non-)Euclidean geometry, you have a bunch of axioms
| and it turns out you _don 't have to_ accept the parallel
| postulate even if you accept all the others.
|
| With measure theory, you have a bunch of things you'd like
| to be true and it turns out you _can 't_ accept all of them
| at once.
|
| Those are quite different.
|
| On the other hand, the analogy between geometry and, say,
| _set theory_ is closer. There are a bunch of axioms for
| Euclidean geometry, the parallel postulate seems a bit
| dicey, and it turns out that you can accept the others and
| lose that one. There are a bunch of axioms for set theory,
| the axiom of choice seems a bit dicey, and it turns out
| that you can accept the others and lose that one.
|
| From this perspective, the role of the Banach-Tarski
| paradox is to help show _why_ the axiom of choice seems a
| bit dicey, maybe more so than the other commonly-adopted
| set-theoretic axioms.
|
| (In set theory, too, there are situations where we can
| write down a bunch of axioms we would like to be true but
| that actually can't all be true at once, just like with
| measure theory. Russell's paradox is the best-known
| example, and it led set theorists to abandon the otherwise
| very attractive axiom of "unrestricted comprehension".)
| JHonaker wrote:
| 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 wrote:
| "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 wrote:
| 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).
| downvotetruth wrote:
| 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
|
| 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 wrote:
| 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 wrote:
| How would you deal with
| <https://example.com/read_blog_post.php?id=1234>? The query
| param is absolutely necessary in that case.
| downvotetruth wrote:
| https://example.com would be the submission &
| 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 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.
| calfuris wrote:
| It seems like you're arguing on the wrong side of the is-
| ought divide. It doesn't matter how URLs _should_ be
| structured, HN has to decide how to deal with how they _are
| in fact_ structured by whatever site is being linked. HN
| exists to share articles, not to try to enforce a preferred
| URL structure, so making direct links impossible on sites
| that use a "non-preferred" structure is throwing the baby
| out with the bathwater. It would also be kind of
| hypocritical, considering that HN itself relies on query
| parameters to show everything but the front page.
| downvotetruth wrote:
| It does matter how URLs _should_ be structured as that
| defines and constrains the text that is accepted by the
| designated input element otherwise without such any plain
| text translatable to a resource would be acceptable thus
| the need for rules to determine _valid_ URLs.
|
| > HN exists to share articles
|
| Not to argue one way or another for that claim, but
| assuming it were true then enforcing a no query param
| constraint would allow greater visibility for the shared
| content as again it would prevent the specified case(s)
| of allowing it to get buried by submissions duplicating
| the link. Also, it seems pointless to go against the REST
| standard of using path params to identify a specific
| resource or resources while using query parameters to
| sort/filter those resources; disallowing direct linking
| to a filter does not make direct links to _articles_
| impossible. It is true that HN itself relies on query
| parameters to link resources that goes against the stated
| REST standard.
| calfuris wrote:
| You're talking about duplicate discussions, but if HN
| trims query parameters then how can it tell if a
| submission to a site that relies on query parameters is a
| duplicate? Every URL on such sites would become
| identical, making different pages appear to be
| duplicates. That would dramatically _hurt_ the visibility
| of such sites, unless HN turned off the automatic
| duplicate prevention for those sites, in which case the
| situation would be worse than the current situation.
|
| The second half of your comment is again on the wrong
| side of the is/ought divide. Websites that (a) have
| content worth sharing and (b) use query parameters to
| identify a specific resource do _in fact_ exist. The
| question is not whether those websites should be doing
| that, it is whether HN should make it impossible to link
| to specific resources on those websites or not.
| downvotetruth wrote:
| HN as is does not reliably detect duplicates even with
| query parameters untrimmed; see the point above about
| ?id=123&comments=1 and ?comments=t&id=123. Further, even
| if there was an exact match on the query params in a
| certain order, the REST standard makes no guarantee on
| what resource(s) are returned on subsequent identical
| calls such as when an addition is added with
| ?topic=Conjecture or ?topic=Theorem as they are by design
| meant to indicate the uses like filtering or sorting. So,
| it's not possible to rely on identical uniform _resource_
| locators with query params to detect duplicate resources.
| It would be an incentive for those sites that want to be
| able to have more frequent visibility of specific
| resources on aggregator sites with duplicate detection to
| conform to the standard. Note detection != blocking.
| Those non-conforming are not being forced to change and
| it could be argued that visibility would be _enhanced_ by
| preventing ambiguity of discussion on specific resources.
| If the title is set as required at the time of
| submission, then that field could be different and as
| discussed above any set of relevant URLs with query
| params could be added to the text field to specify
| resource(s) on a non-conforming site to bypass a
| duplicate check along with a [suggested not required]
| helpful description of what was seen at any query param
| URL at the time of submission. Obviously, the resources
| at non-query param URLs may change too depending on the
| whims of a site owner, but are suppose to be unigue
| resource IDs by REST.
| IshKebab wrote:
| > 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 wrote:
| 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 wrote:
| 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 wrote:
| Related:
|
| _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 wrote:
| 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
|
| 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".
| ajkjk wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| You conclude the problem is with math. I conclude the
| problem is with reality.
| alkyon wrote:
| 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.
| alkyon wrote:
| AC is an axiom (something assumed without proof) and not
| the result that would render mathematics itself
| inconsistent.
|
| Banach Tarski is not about AC, it is about existence of
| some pathological sets that have no measure.
|
| Edit: AC is generally assumed to be true because rejecting
| it leads to even stranger paradoxes than that of Banach-
| Tarski.
| ajkjk wrote:
| Yes, all models are wrong, some are useful, and the model
| math is using is evidently not very good. Probably there is a
| better one out there.
|
| Physics isn't reality but it does use "closeness to reality"
| as a metric for quality of a theory, and that's by the metric
| by which Banach-Tarski is irritating.
| lupire wrote:
| The better (more realistic) model is
| constructive/intuitionist mathematics - ZF without C, or at
| least without Uncountable Choice.
|
| Naturally, you should be suspicious of a theory of
| mathematics that assumes the existence of "uncountable"
| things.
| gjm11 wrote:
| You're conflating three separate things.
|
| 1. Accepting versus rejecting the axiom of choice.
|
| 2. Classical versus intuitionistic mathematics.
| Intuitionism goes much further than just rejecting the
| Axiom of Choice, and e.g. says that you haven't proved "
| _p_ or _q_ " until you have either proved _p_ or proved
| _q_. It denies not just the axiom of choice but the "law
| of the excluded middle" which says that for any
| proposition _p_ , either _p_ is true or _not p_ is true.
|
| 3. Accepting versus rejecting "large" sets -- in your
| case, you say we should be suspicious of anything that
| assumes the existence of uncountable things. (I think
| this is a bit unusual; there are _finitists_ who deny
| that there are any infinite sets, and _ultrafinitists_
| who go further and say there aren 't even arbitrarily big
| finite sets, but it's not so common to accept countable
| but not uncountable infinities.)
|
| I think intuitionism implies rejecting AC. I don't think
| any of the other possible implications between these
| three things holds; e.g., so far as I know most
| intuitionists have no particular problem with the
| existence of large infinite sets.
|
| (Actually, probably some theorems of the form "if there
| are no large sets, then the axiom of choice is true for
| boring reasons" are provable.)
| isotypic wrote:
| > and the model math is using is evidently not very good.
|
| There isn't really one "model of math". Plenty of people
| study alternative foundations of math or various axioms you
| can attach to ZF, etc. and its great that they do - choice,
| and lots of other set theoretic/foundational stuff, is very
| weird. But the reason mathematicians have largely settled
| on ZFC as a default is that it (and maybe nowadays + an
| inaccessible cardinal) also reduces some pathology and
| makes math more convenient.
|
| Let's list some things which are related to the axiom of
| choice: * The cartesian product of nonempty sets is
| nonempty (equivalent) * The reals can be partitioned into
| more parts than there are real numbers (consistent with the
| negation of choice) * All vector spaces have a basis
| (equivalent) * All commutative rings have a maximal ideal
| (equivalent) * All fields are contained in an algebraic
| closure (implied by choice)
|
| The first two, at least to me, are similar to Banach-Tarski
| in that they are things I would like to be true and false,
| respectively, and are not if we do not accept choice. The
| point here is that the weirdness of Banach-Tarski is as
| much related to the axiom of choice as it is to generally
| the fact that (uncountable) infinities are just very weird,
| and while introducing choice does introduce pathologies
| like Banach-Tarski, it also reduces some.
|
| The last three illustrate a more practical perspective.
| Suppose a universal decree, that math is no longer allowed
| to be done in ZFC, you could only use ZF, was imposed on
| mathematicians. This doesn't really change anything - all
| that mathematicians will do is, if they were perfectly fine
| with choice before, simply replace instances of "vector
| space" with "vector space with a basis", or "commutative
| ring" with "commutative ring with a maximal ideal", because
| the types of mathematical objects they care about are the
| ones with these desirable properties - they only use the
| axiom of choice because it ensures the general objects they
| work with also have those desirable properties. And
| mathematicians (often even the same ones as before, just
| wearing a different hat) who _do_ care about choice will
| continue studying those weird instances of the general
| object just as before.
|
| As a final point for the convenience of axiom of choice,
| there are lots of instances where a proof will use choice
| purely as a matter of convenience. Maybe with some
| technical set theory or a smarter argument its possible to
| completely eliminate the requirement of choice, or use a
| weaker, less-objectionable version of choice. Other times,
| while a general vector space having a basis requires
| choice, the vector spaces in your particular application
| have a basis regardless of your thoughts on the axiom of
| choice. But of course its much easier to simply assert
| "vector spaces have a basis" and introduce a (faux)
| dependence on the axiom of choice. For a concrete example
| of this last point, see https://mathoverflow.net/a/35772.
| While about inaccessible cardinals/"universes" rather than
| the axiom of choice, the principle I want to illustrate is
| the same. The proof of Fermat's last theorem, if you trace
| citations back, eventually depends on "universes". But the
| actual invocations of those theorems are applied to objects
| where that generality is not required.
|
| (Why not simply only have done the non-general scenario?
| Because "holds in general with additional axiom, holds in
| specificity without" is more knowledge than just "holds in
| specificity")
| ajkjk wrote:
| > The last three illustrate a more practical perspective.
| Suppose a universal decree, that math is no longer
| allowed to be done in ZFC, you could only use ZF, was
| imposed on mathematicians. This doesn't really change
| anything - all that mathematicians will do is, if they
| were perfectly fine with choice before, simply replace
| instances of "vector space" with "vector space with a
| basis", or "commutative ring" with "commutative ring with
| a maximal ideal", because the types of mathematical
| objects they care about are the ones with these desirable
| properties
|
| Lol, I would _love_ that. That 's the book I want to
| read. I am weird, maybe, but I find the full-generality
| of mathematics to be exhausting when I just want to
| understand how numbers and geometry work. I will never
| care about the details of infinite sets.
|
| I am reminded by a quote from Jaynes' probability text:
| that in their opinion, infinite objects are only
| meaningful when explicitly provided via a limiting
| process from finite objects. It is not, as far as I can
| tell, a widespread stance, but it's the one I subscribe
| to.
|
| (Thanks for the lengthy reply, though. Just, it reminds
| me of the stuff I already find exhausting.)
| lupire wrote:
| Banach-Tarski assumes far more than the existence of a
| sphere. It assumes the existence of an uncountably infinitely
| complicated cut.
| ajuc wrote:
| Would you say that about imaginary numbers too?
| andoando wrote:
| Imaginary numbers are just 2D vectors. Its such a horrible
| name. The imaginary part isnt any more mysterious than the
| real part. Theyre just orthogonal properties.
| ajuc wrote:
| > Imaginary numbers are just 2D vectors.
|
| You meant complex numbers. I wrote "imaginary numbers" on
| purpose.
|
| > The imaginary part isnt any more mysterious than the real
| part.
|
| What's surprising is that imaginary numbers are required
| for QM. Not the 2d vector in one variable trick, actual
| imaginary numbers.
|
| See for example:
| https://www.nature.com/articles/s41586-021-04160-4
| lupire wrote:
| I think that paper is saying that you have to use N
| independent 2D numbers, not 2N independent 1D numbers.
|
| Complex numbers are literally the same as 2D real numbers
| with the rotational symmetry constraint.
|
| https://www.reddit.com/r/Physics/comments/11ujjjd/quantum
| _th...
| andoando wrote:
| What are imaginary numbers if you remove its relation to
| the real numbers? Do they not become regular real
| numbers?
|
| How I am seeing it, real and imaginary numbers are both
| equivalent constructions with the same
| properties/construction (you can
| add/substract/multiply/divide them the same way), and its
| only in the context of using both (in the context of the
| complex numbers), that they can be differentiated.
| fellowmartian wrote:
| They're not "just" 2D vectors, they have a bit more
| structure that makes them special, but sure there's
| obviously nothing "imaginary" about them.
| ajuc wrote:
| They are imaginary in the same way that Banach Tarski
| theorem is a paradox.
| lupire wrote:
| That's not quite right. Imaginary numbers are vectors with
| angle addition via multiplication. Real numbers all have
| zero angle, so their angle addition is trivial.
| andoando wrote:
| Is that a property of imaginary numbers, or a property of
| relating two orthogonal numbers? Legitimately asking
| because I can't find any special properties that
| imaginary numbers alone have in relation to themselves.
| fellowmartian wrote:
| Imaginary numbers don't allow you to create something of out
| nothing unlike the axiom of choice. Not all weird things are
| weird in the same way.
| ajkjk wrote:
| No, they're just plane geometry, albeit in a cludgy notation
| (which conflate vectors and rotation operators on vectors
| because they happen to be isomorphic in R^2). But plane
| geometry is a real thing.
| mikhailfranco wrote:
| 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 wrote:
| A physical sphere has a finite number of constituents,
| therefore the theorem does not apply
| mjburgess wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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)
| andrewla wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| > 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?"
| Davidzheng wrote:
| There are certainly considerations other than consistency
| in mathematical theories. But I don't think I'll be able
| to articulate them well here. Maybe someone else can
| help.
| Davidzheng wrote:
| I'd like to point out this is at odds with most of the
| mathematical community's interpretation. Caveat:
| mathematics itself does not have concerns about
| "interpretation" and your interpretation is completely
| valid of course.
|
| My interpretation is that powerful axioms often prevent
| theories which asks for "too much" because the power of the
| axioms then can form contradictions. For example asking for
| everything to be a set => Russel's paradox and asking for
| all sets to be measurable => Banach tarski type
| contradiction.
|
| Side note for mathematically trained peeps: Of course
| assuming all sets to be measurable and the measure to be
| translation and rotationally invariant leads to much easier
| contradictions with AOC. Banach-Tarski gives a finite
| decomposition contradiction (vs say countable). Also not a
| analyst myself, so take everything here with a grain of
| salt. But honestly analysis is basically completely not
| possible without excluded middle so I think analysts would
| have opinions more on classical side than myself.
| ysofunny wrote:
| 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 wrote:
| Vsauce: The Banach-Tarski Paradox (24m13s) [2015-08-01]:
| 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 (music
| video)
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