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.
> 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.
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).
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.
> 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.
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".)
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.
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?
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).
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.
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?
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 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..
> [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.
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.
Right, but those were self imposed requirements based on intuition. The axioms of sigma algebras are different than these requirements. If anything measure theory highlights the limits and conditions of when to consider something measureable. That is the point of the Paradox and the need for measure theory.
> 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?"
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.
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.
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.
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.)
> 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")
> 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.)
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.
>> Universe could be itself infinite but locally everything is finite.
> You don't know that.
I think they were saying that it could be that the Universe is infinite with everything still being locally finite. I do not think they were asserting that everything is locally finite, I think it was included in the "it could be" part.
But certainly you're right that we don't know that.
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.
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.
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.
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.
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.
> 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?
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.
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)
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".
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!
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?
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.
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.
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.
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.
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.
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.
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.
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